Lingo 14 4c5v6e

[email protected]

C=. The interpretations of these operators are: Operator

Interpretation The constraint F(X) <= b is convex.

>C=

The constraint F(X) >= b is concave.

=C=

The function F(X) of the constraint F(X) = b is a convex function.

As an example, consider the nonlinear model: MODEL: MAX= 5 * W1 - 3 + + - W1 <= 1; W1 <= 1; X1 + Y1 >= 1; [NLROW1] W1 ^ 3

* X1 - 4 * Y1 5 * W2 - 3 * X2 - 4 * Y2 5 * W3 - 3 * X3 - 4 * Y3;

- X1 * Y1
- W2 <= 1; W2 <= 1; X2 + Y2 >= 1; [NLROW2] W2 ^ 3 - X2 * Y2 = 1;

[NLROW3] W3 ^ 3 - X3 * Y3
Model: CONVEX The three nonlinear rows in this model: NLROW1, NLROW2 and NLROW3, are convex, and we have marked them as such using the convexity inequality operator (
If you have a constraint for which F(x) is only quasi-convex or quasi-concave, you may still apply the convexity markers to the same advantage. A convex objective, Min = F(x), can be identified by rewriting it: Min = z; z>C = F(x);

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Maximum Problem Dimensions Some versions of LINGO limit one or more of the following model properties: total variables, integer variables, nonlinear variables, global variables, and constraints. The total variable limit is on the total number of optimizable variables in your model (i.e., variables LINGO was unable to determine as being fixed at a particular value). The integer variable limit applies to the total number of optimizable variables restricted to being integers with either the @BIN or @GIN functions. The nonlinear variable limit applies to the number of optimizable variables that appear nonlinearly in the model’s constraints. As an example, in the expression: X + Y, both X and Y appear linearly. However, in the expression: X ^ 2 + Y, X appears nonlinearly while Y appears linearly. Thus, X would count against the nonlinear variable limit. In some cases, nonlinear variables are allowed only if you have purchased the nonlinear option for your LINGO software. The global variable limit applies to the total number of nonlinear variables when using the global solver. The constraint limit refers to the number of formulas in the model that contain one or more optimizable variables. Keep in mind that a single @FOR function may generate many constraints. The maximum sized problem your LINGO can handle depends on the version you have. The current limits for the various versions are: Version Demo/Web Solver Suite Super Hyper Industrial Extended

Total Variables 300 500 2,000 8,000 32,000 Unlimited

Integer Variables 30 50 200 800 3,200 Unlimited

Nonlinear Variables 30 50 200 800 3,200 Unlimited

Global Variables 5 5 10 20 50 Unlimited

Constraints 150 250 1,000 4,000 16,000 Unlimited

You can also determine the limits of your version by selecting the About LINGO command from the Help menu in Windows, or by typing HELP at the command-line prompt on other platforms. If you determine you need a larger version of LINGO, upgrades are available from LINDO Systems. Please feel free to us for pricing and availability. Note 1: The limits of different LINGO versions are subject to change. Check our website, http://www.lindo.com, for the most current sizes. Note 2: In some versions of LINGO, the Nonlinear Variable limit will be 0 if you have not purchased the nonlinear option for your copy of LINGO. Similarly, the global variable limit will be o if the global solver option is not enabled.

44 CHAPTER 1 Note 3: LINGO has two other implicit limits not given by the table above—memory and time. Large models may require more memory to solve than is available on your system, or they may require more time to solve than one would normally be willing to wait. So, when building large models, be aware that just because your model falls within LINGO’s limits there is no guarantee it will be solvable in a reasonable amount of time on a particular machine.

How to LINDO Systems LINDO Systems can be reached at the following address and telephone numbers: LINDO Systems, Inc. 1415 North Dayton Street Chicago, IL 60642 Tel: Fax:

312-988-7422 312-988-9065

e-mail: [email protected] web: http://www.lindo.com For sales and product information, please us at: Tel: 1-800-441-2378 or 312-988-7422 e-mail: [email protected] For technical , we prefer you send your model and questions by electronic mail to [email protected]. You may also speak to our technical staff at 312-988-9421. Our technical staff can help you with questions regarding the installation and operation of LINGO. If you have simple modeling questions, we can generally help get you pointed in the right direction. If you have extensive modeling questions, we can recommend third party consultants well versed in the specifics of LINGO and mathematical modeling in general, who can assist you in your modeling efforts.

2 Using Sets As we mentioned in the previous chapter, whenever you are modeling situations in real life there will typically be one or more groups of related objects. Examples of such objects might include factories, customers, vehicles, or employees. LINGO allows you to group these related objects together into sets. Once the objects in your model are grouped into sets, you can make use of set-based functions to unleash the full power of the LINGO modeling language. Having given you a brief introduction into the use of sets in Chapter 1, Getting Started with LINGO, we will now go into greater depth as to how you construct sets and initialize set attributes with data. This will then give us the ability to begin constructing some interesting and useful examples. Once you’ve read this chapter, you should have a basic understanding of how to go about applying set-based modeling techniques to your own models.

Why Use Sets? Sets are the foundation of LINGO’s modeling language—the fundamental building block of the program’s most powerful capabilities. With an understanding of sets, you can write a series of similar constraints in a single statement and express long, complex formulas concisely. This allows you to express your largest models very quickly and easily. In larger models, you’ll encounter the need to express a group of several very similar calculations or constraints. Fortunately, LINGO’s ability to handle sets of information allows you to perform such operations efficiently. For example, preparing a warehouse-shipping model for 100 warehouses would be tedious if you had to write each constraint explicitly (e.g., “Warehouse 1 must ship no more than its present inventory, Warehouse 2 must ship no more than its present inventory, Warehouse 3 must ship no more than its present inventory…”, and so on). LINGO allows you to express formulas in the form easiest for you to read and understand (e.g., “Each warehouse must ship no more than its present inventory”).

What Are Sets? Sets are simply groups of related objects. A set might be a list of products, trucks, or employees. Each member in the set may have one or more characteristics associated with it. We call these characteristics attributes. Attribute values can be known in advance or unknowns that LINGO solves for. For example, each product in a set of products might have a price attribute; each truck in a set of trucks might have a hauling capacity attribute; and each employee in a set of employees might have a salary attribute, as well as a birth date attribute.

45

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Types of Sets LINGO recognizes two kinds of sets: primitive and derived. A primitive set is a set composed only of objects that can’t be further reduced. In the Wireless Widgets example (page 27), the WAREHOUSES set, which is composed of six warehouses, is a primitive set. Likewise, the set composed of eight vendors is a primitive set. A derived set is defined using one or more other sets. In other words, a derived set derives its from other preexisting sets. Again, using the Wireless Widgets example, the set composed of the links between the six warehouses and eight vendors (LINKS) is a derived set. It derives its from the unique pairs of of the WAREHOUSES and VENDORS sets. Although the LINKS set is derived solely from primitive sets, it is also possible to build derived sets from other derived sets as well. See the section below, Defining Derived Sets, for more information.

The Sets Section of a Model Sets are defined in an optional section of a LINGO model called the sets section. Before you use sets in a LINGO model, you have to define them in the sets section of the model. The sets section begins with the keyword SETS: (including the colon), and ends with the keyword ENDSETS. A model may have no sets section, a single sets section, or multiple sets sections. A sets section may appear anywhere in a model. The only restriction is you must define a set and its attributes before they are referenced in the model’s constraints.

Defining Primitive Sets To define a primitive set in a sets section, you specify:   

the name of the set, optionally, its (objects contained in the set), and optionally, any attributes the of the set may have.

A primitive set definition has the following syntax: setname [/ member_list /] [: attribute_list]; Note:

The use of square brackets indicates an item is optional. In this particular case, a primitive set’s attribute_list and member_list are both optional.

The setname is a name you choose to designate the set. It should be a descriptive name that is easy to . The set name must conform to standard LINGO naming conventions. In other words, the name must begin with an alphabetic character, which may be followed by up to 31 alphanumeric characters or the underscore (_). LINGO does not distinguish between upper and lowercase characters in names. A member_list is a list of the that constitute the set. If the set are included in the set definition, they may be listed either explicitly or implicitly. If set are not included in the set

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definition, then they may be defined subsequently in a data section of the model. For details on defining set in a data section, refer to Introduction to the Data Section. When listing explicitly, you enter a unique name for each member, optionally separated by commas. As with set names, member names must also conform to standard naming conventions. In the Wireless Widgets model, we could have used an explicit member list to define the set WAREHOUSES in the sets section as follows: WAREHOUSES / WH1 WH2 WH3 WH4 WH5 WH6/: CAPACITY;

When using implicit set member lists, you do not have to list a name for each set member. Use the following syntax when using an implicit set member list: setname / member1..memberN / [: attribute_list]; where member1 is the name of the first member in the set and memberN is the name of the last member. LINGO automatically generates all the intermediate member names between member1 and memberN. While this can be a very compact and convenient method for building a primitive set, there is one catch in that only certain formats of names are accepted for the initial and terminal member names. The following table details the available options: Implicit Member List Format

Example

Set

1..n

1..5

1, 2, 3, 4, 5

stringM..stringN

TRUCKS3..TRUCKS204

TRUCKS3, TRUCKS4, …, TRUCKS204

dayM..dayN

MON..FRI

MON, TUE, WED, THU, FRI

monthM..monthN

OCT..JAN

OCT, NOV, DEC, JAN

monthYearM..monthYearN

OCT2001..JAN2002

OCT2001, NOV2001, DEC2001, JAN2002

When using the 1..n format, n may be any positive integer value, and the initial member must always be a 1. The stringM..stringN format allows you to use any string to start both the initial and terminal member names as long as the string conforms to standard LINGO naming conventions. M and N must be nonnegative and integer, such that M ≤ N. The dayM..dayN format allows you to choose the initial and terminal member names for the names of the days of the week. All names are abbreviated to three characters. Thus, the available options are: Mon, Tue, Wed, Thu, Fri, Sat, and Sun. The monthM..monthN format allows you to select from the months of the year, where all names are abbreviated to three characters. The available options are: Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, and Dec. The monthYearM..monthYearN option allows you to specify a month and a four digit year.

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As further illustration, in the Wireless Widgets example, we could have also defined the WAREHOUSES set as: WAREHOUSES / 1..6/: CAPACITY;

As an alternative, when using this 1..n form of implicit definition, you may also place the length of the set in a data section, and then reference this length in a subsequent sets section as we do here: DATA: NUMBER_OF_WH = 6; ENDDATA SETS: WAREHOUSES / 1..NUMBER_OF_WH/: CAPACITY; ENDSETS

Set may have one or more attributes specified in the attribute_list of the set definition. An attribute is simply some property each member of the set displays. For instance, in the WAREHOUSES set above, there is a single attribute titled CAPACITY, which is used to represent the shipping capacity of the WAREHOUSES. Attribute names must follow standard naming conventions and be separated by commas. For illustration, suppose our warehouses had additional attributes related to their location and the number of loading docks. These additional attributes could be added to the attribute list of the set declaration as follows: WAREHOUSES / 1..6/: CAPACITY, LOCATION, DOCKS;

In addition to listing a primitive set's in a model's sets section, primitive set may also be listed in a model's data section. Some s may prefer this alternative approach in that a set's are actually input data for the model. Therefore, listing set in a model's data section, along with all other data, is a more natural approach that makes a model more readable. All the various techniques listed above for enumerating a primitive set's are also valid in a data section. Some examples of defining primitive set in a data section follow: SETS: WAREHOUSES: CAPACITY; ENDSETS DATA: WAREHOUSES = WH1 WH2 WH3 WH4 WH5 WH6; ENDDATA

Example 1: Listing a Primitive Set in a Data Section SETS: WAREHOUSES: CAPACITY; ENDSETS DATA: NUMBER_OF_WH = 6; WAREHOUSES = 1..NUMBER_OF_WH; ENDDATA

Example 2: Listing a Primitive Set in a Data Section

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Defining Derived Sets To define a derived set, you specify:    

the name of the set, its parent sets, optionally, its , and optionally, any attributes the set may have.

A derived set definition has the following syntax: setname(parent_set_list) [ / member_list /] [: attribute_list]; The setname is a standard LINGO name you choose to name the set. The parent_set_list is a list of previously defined sets, separated by commas. Without specifying a member_list element, LINGO constructs all combinations of from each parent set to create the of the new derived set. As an example, consider the following sets section: SETS: PRODUCT / A B/; MACHINE / M N/; WEEK / 1..2/; ALLOWED(PRODUCT, MACHINE, WEEK); ENDSETS

Sets PRODUCT, MACHINE, and WEEK are primitive sets, while ALLOWED is derived from parent sets, PRODUCT, MACHINE, and WEEK. Taking all the combinations of from the three parent sets, we come up with the following in the ALLOWED set: Index Member 1 (A,M,1) 2 (A,M,2) 3 (A,N,1) 4 (A,N,2) 5 (B,M,1) 6 (B,M,2) 7 (B,N,1) 8 (B,N,2) ALLOWED Set hip The member_list is optional, and is used when you want to limit the set to being some subset of the full set of combinations derived from the parent sets. The member_list may alternatively be specified in a model’s data section (for details on this see Introduction to the Data Section in Chapter 4, Data and Init Sections). If the member_list is omitted, the derived set will consist of all combinations of the from the parent sets. When a set does not have a member_list and, therefore, contains all possible combinations of , it is referred to as being a dense set. When a set includes a member_list that limits it to being a subset of its dense form, we say the set is sparse.

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A derived set’s member_list may be constructed using either:  

an explicit member list, or a hip filter.

When using the explicit member list method to specify a derived set’s member_list, you must explicitly list all the you want to include in the set. Each listed member must be a member of the dense set formed from all possible combinations of the parent sets. Returning to our small example above, if we had used an explicit member list in the definition of the derived set, ALLOWED, as follows: ALLOWED(PRODUCT, MACHINE, WEEK) / A M 1, A N 2, B N 1/;

then ALLOWED would not have had the full complement of eight . Instead, ALLOWED would have consisted of the three member sparse set: (A,M,1), (A,N,2), and (B,N,1). Note that the commas in the list of set are optional and were added only for readability purposes. If you have a large, sparse set, explicitly listing all can become cumbersome. Fortunately, in many sparse sets, the all satisfy some condition that differentiates them from the non. If you could just specify this condition, you could save yourself a lot of effort. This is exactly how the hip filter method works. Using the hip filter method of defining a derived set’s member_list involves specifying a logical condition that each potential set member must satisfy for inclusion in the final set. You can look at the logical condition as a filter to keep out potential that don’t satisfy some criteria. As an example of a hip filter, suppose you have already defined a set called TRUCKS, and each truck has an attribute called CAPACITY. You would like to derive a subset from TRUCKS that contains only those trucks capable of hauling big loads. You could use an explicit member list, and explicitly enter each truck that can carry heavy loads. However, why do all that work when you could use a hip filter as follows: HEAVY_DUTY(TRUCKS)|CAPACITY(&1) #GT# 50000:

We have named the set HEAVY_DUTY and have derived it from the parent set, TRUCKS. The vertical bar character (|) is used to mark the beginning of a hip filter. The hip filter allows only those trucks that have a hauling capacity (CAPACITY(&1)) greater than (#GT#) 50,000 into the HEAVY_DUTY set. The &1 symbol in the filter is known as a set index placeholder. When building a derived set that uses a hip filter, LINGO generates all the combinations of parent set . Each combination is then “plugged” into the hip condition to see if it es the test. The first primitive parent set’s member is plugged into &1, the second into &2, and so on. In this example, we have only one parent set (TRUCKS), so &2 would not have made sense. The symbol #GT# is a logical operator and means “greater than”.

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The logical operators recognized by LINGO are: #EQ# equal #NE# not equal #GE# greater-than-or-equal-to #GT# greater than #LT# less than #LE# less-than-or-equal-to In addition to listing a derived set's in a model's sets section, derived set may also be listed in a model's data section. Some s may prefer this alternative approach in that a set's are actually input data for the model. Therefore, listing set in a model's data section, along with all other data, is a more natural approach that makes a model more readable. All the various techniques listed above for enumerating a primitive set's are also valid in a data section, with the exception of the hip filter method. An example of defining derived set in a data section follow: SETS: PRODUCT; MACHINE; WEEK; ALLOWED( PRODUCT, MACHINE, WEEK); ENDSETS DATA: PRODUCT = P1..P3; MACHINE = M1..M3; WEEK = W1..W4; ALLOWED = P1 M1 W1 P2 M1 W3 P3 M1 W2 P3 M3 W4 ; ENDDATA

Inputting a Derived Set in a Data Section

Summary In summary, keep in mind that LINGO recognizes two types of sets—primitive and derived. Primitive sets are the fundamental objects in a model and can't be broken down into smaller components. Primitive sets can be defined using either an explicit or implicit list. When using an explicit list, you enter each member individually in the set member list. With an implicit list, you enter the initial and terminal set and LINGO generates all the intermediate . Derived sets, on the other hand, are created from other component sets. These component sets are referred to as the parents of the derived set, and may be either primitive or derived. A derived set can be either sparse or dense. Dense sets contain all combinations of the parent set (sometimes this is also referred to as the Cartesian product or cross of the parent sets). Sparse sets contain only a subset of the cross of the parent sets, and may be defined by two methodsexplicit listing or

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hip filter. The explicit listing method involves listing the of the sparse set. The hip filter method allows you to specify the sparse set compactly through the use of a logical condition that all must satisfy. The relationships amongst the various set types are illustrated in the graph below.Derived sets, on the other hand, are created from other component sets. These component sets are referred to as the parents of the derived set, and may be either primitive or derived. A derived set can be either sparse or dense. Dense sets contain all combinations of the parent set (sometimes this is also referred to as the Cartesian product or cross of the parent sets). Sparse sets contain only a subset of the cross of the parent sets and may be defined by two methodsexplicit listing or hip filter. The explicit listing method involves listing the of the sparse set. The hip filter method allows you to specify the sparse set compactly through the use of a logical condition all must satisfy. The relationships amongst the various set types are illustrated in the graph below:

LINGO Set Types At this point, you are probably thinking that set definition is, at best, somewhat complicated. We will be presenting you with plenty more examples in subsequent sections that should help to illustrate the concepts introduced in this section, and demonstrate that set definition is nowhere near as difficult as it may seem. For now, however, we will turn our attention to how data is input into a model. Then, we will examine a group of functions that are designed to operate on set . Once we have accomplished this, we will be able to bring together all we have learned in order to begin building some interesting and relevant examples of set-based modeling.

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The DATA Section Typically, you will want to initialize the of certain sets and assign values to certain set attributes. For this purpose, LINGO uses a second optional section called the data section. The data section allows you to isolate data from the equations of your model. This is a useful practice in that it leads to easier model maintenance and facilitates scaling up a model to larger dimensions. Similar to the sets section, the data section begins with the keyword DATA: (including the colon) and ends with the keyword ENDDATA. In the data section, you can have statements to initialize the sets and/or attributes you defined in a sets section. These expressions have the syntax: object_list = value_list; The object_list contains the names of a set and/or attributes you want to initialize, optionally separated by commas. If there is more than one attribute name on in the object list, then all attributes must be defined on the same set. Furthermore, if a set name appears in the object list, then it must be the parent set of any attributes also in the object list. The value_list contains the values to assign to the objects in the object list, optionally separated by commas. For example, consider the following model: MODEL: SETS: SET1: X, Y; ENDSETS DATA: SET1 = A B C; X = 1 2 3; Y = 4 5 6; ENDDATA END

We have two attributes, X and Y, defined on the set SET1. The three values of X are set to 1, 2, and 3, while Y is set to 4, 5, and 6. We could have also used the following compound data statement to the same end: MODEL: SETS: SET1: X, Y; ENDSETS DATA: SET1 X Y = A 1 4 B 2 5 C 3 6; ENDDATA END

An important fact to is that when LINGO reads a compound data statement's value list, it assigns the first n values in the list to the first position of each of the n objects in the object list, the second n values to the second position of each of the n objects, and so on. In other words, LINGO is expecting the input data in column format rather than row format, which mirrors the flat file approach used in relational databases.

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This section has served to give you a brief introduction into the use of the data section. In Data and Init Sections, you will learn more about the capabilities of the data section. You will learn data does not have to actually reside in the data section as shown in examples here. In fact, your data section can have OLE links to Excel, ODBC links to databases, and connections to text based data files.

Set Looping Functions We have mentioned the power of set-based modeling comes from the ability to apply an operation to all of a set using a single statement. The functions in LINGO that allow you to do this are called set looping functions. If your models don’t make use of one or more set looping function, then you are missing out on the power of set-based modeling and, even worse, you’re probably working too hard! Set looping functions allow you to iterate through all the of a set to perform some operation. There are currently four set looping functions in LINGO. The names of the functions and their uses are: Function @FOR @SUM @MIN @MAX @PROD

Use The most powerful of the set looping functions, @FOR is used to generate constraints over of a set. Probably the most frequently used set looping function, @SUM computes the sum of an expression over all of a set. Computes the minimum of an expression over all of a set. Computes the maximum of an expression over all of a set. Computes the product of an expression over all of a set.

The syntax for a set looping function is: @function(setname [ (set_index_list) [|conditional_qualifier]] : expression_list); where @function corresponds to one of the four set looping functions listed in the table above. setname is the name of the set you want to loop over. set_index_list is optional. It is used to create a list of indices. Each index corresponds to one of the parent, primitive sets that form the set specified by setname. As LINGO loops through the of the set setname, it will set the values of the indices in the set_index_list to correspond to the current member of the set setname. The conditional_qualifier is optional, and may be used to limit the scope of the set looping function. When LINGO is looping over each member of setname, it evaluates the conditional_qualifier. If the conditional_qualifier evaluates to true, then the @function is performed for the set member. Otherwise, it is skipped.

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The expression_list is a list of expressions that are to be applied to each member of the set setname. When using the @FOR function, the expression list may contain multiple expressions, separated by semicolons. These expressions will be added as constraints to the model. When using the remaining set looping functions (@SUM, @MAX, @MIN and @PROD), the expression list must contain one expression only. If the set_index_list is omitted, all attributes referenced in the expression_list must be defined on the set setname. The following examples should help to illustrate the use of set looping functions.

@SUM Set Looping Function In this example, we will construct several summation expressions using the @SUM function in order to illustrate the features of set looping functions in general, and the @SUM function in particular. Consider the model: MODEL: SETS: VENDORS: DEMAND; ENDSETS DATA: VENDORS, DEMAND = ENDDATA END

V1,5 V2,1 V3,3 V4,4 V5,6;

Each vendor of the VENDORS set has a corresponding DEMAND. We could sum up the values of the DEMAND attribute by adding the following expression after the ENDDATA statement: TOTAL_DEMAND = @SUM(VENDORS(J): DEMAND(J));

LINGO evaluates the @SUM function by first initializing an internal accumulator to zero. LINGO then begins looping over the in the VENDORS set. The set index variable, J, is set to the first member of VENDORS (i.e., V1) and DEMAND (V1) is then added to the accumulator. This process continues until all DEMAND values have been added to the accumulator. The value of the sum is then stored in the TOTAL_DEMAND variable. Since all the attributes in our expression list (in this case, only DEMAND appears in the expression list) are defined on the index set (VENDORS), we could have alternatively written our sum as: TOTAL_DEMAND = @SUM(VENDORS: DEMAND);

In this case, we have dropped the superfluous index set list and the index on DEMAND. When an expression uses this shorthand, we say the index list is implied. Implied index lists are not allowed when attributes in the expression list have different parent sets.

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Next, suppose we want to sum the first three elements of the attribute DEMAND. We can use a conditional qualifier on the set index to accomplish this as follows: DEMAND_3 = @SUM(VENDORS(J)|J #LE# 3: DEMAND(J));

The #LE# symbol is called a logical operator (see p. 381 for more details). This operator compares the operand on the left (J) with the one on the right (3), and returns true if the left operand is less-than-or-equal-to the one on the right. Otherwise, it returns false. Therefore, when LINGO computes the sum this time, it plugs the set index variable, J, into the conditional qualifier J #LE# 3. If the conditional qualifier evaluates to true, DEMAND(J) will be added to the sum. The end result is LINGO sums up the first three in DEMAND, omitting the fourth and fifth , for a total sum of 9. Note:

Before leaving this example, one subtle aspect to note in this last sum expression is the value that the set index J is returning. Note, we are comparing the set index variable to the quantity 3 in the conditional qualifier J #LE# 3. In order for this to be meaningful, J must represent a numeric value. Because a set index is used to loop over set , one might imagine a set index is merely a placeholder for the current set member. In a sense, this is true, but what set indices really return is the index of the current set member in its parent primitive set. The index returned is one-based. In other words, the value 1 is returned when indexing the first set member, 2 when indexing the second, and so on. Given that set indices return a numeric value, they may be used in arithmetic expressions along with other variables in your model.

@MIN and @MAX Set Looping Functions The @MIN and @MAX functions are used to find the minimum and maximum of an expression over of a set. Again, consider the model: MODEL: SETS: VENDORS: DEMAND; ENDSETS DATA: VENDORS, DEMAND = ENDDATA END

V1,5 V2,1 V3,3 V4,4 V5,6;

To find the minimum and maximum DEMAND, all one need do is add the two expressions: MIN_DEMAND = @MIN( VENDORS( J): DEMAND( J)); MAX_DEMAND = @MAX( VENDORS( J): DEMAND( J));

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The resulting model with the new statements in bold would then be as follows: MODEL: SETS: VENDORS: DEMAND; ENDSETS DATA: VENDORS, DEMAND = V1,5 V2,1 V3,3 V4,4 V5,6; ENDDATA MIN_DEMAND = @MIN( VENDORS( J): DEMAND( J)); MAX_DEMAND = @MAX( VENDORS( J): DEMAND( J)); END

As with the @SUM example, we can use an implied index list since the attributes are defined on the index set. Using implied indexing, we can recast our expressions as: MIN_DEMAND = @MIN( VENDORS: DEMAND); MAX_DEMAND = @MAX( VENDORS: DEMAND);

In either case, when we solve this model, LINGO returns the expected minimum and maximum DEMAND of: Variable MIN_DEMAND MAX_DEMAND

Value 1.000000 6.000000

For illustration purposes, suppose we had just wanted to compute the minimum and maximum values of the first three elements of DEMAND. As with the @SUM example, all we need do is add the conditional qualifier J #LE# 3. We then have: MIN_DEMAND3 = @MIN( VENDORS( J) | J #LE# 3: DEMAND( J)); MAX_DEMAND3 = @MAX( VENDORS( J) | J #LE# 3: DEMAND( J));

with solution: Variable MIN_DEMAND3 MAX_DEMAND3

Value 1.000000 5.000000

@FOR Set Looping Function The @FOR function is used to generate constraints across of a set. Whereas scalar based modeling languages require you to explicitly enter each constraint, the @FOR function allows you to enter a constraint just once, and LINGO does the work of generating an occurrence of the constraint for each set member. Thus, the @FOR statement provides the set-based modeler with a very powerful tool. To illustrate the use of @FOR, consider the following set definition: SETS: TRUCKS / MAC, PETERBILT, FORD, DODGE/: HAUL; ENDSETS

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Specifically, we have a primitive set of four trucks with a single HAUL attribute. If HAUL is used to denote the amount a truck hauls, then we can use the @FOR function to limit the amount hauled by each truck to 2,500 pounds with the following expression: @FOR(TRUCKS(T): HAUL(T) <= 2500);

In this case, it might be instructive to view the constraints LINGO generates from our expression. You can do this by using the LINGO|Generate|Display model command under Windows, or by using the GENERATE command on other platforms. Running this command, we find LINGO generates the following four constraints: HAUL(MAC) <= 2500 HAUL(PETERBILT) <= 2500 HAUL(FORD) <= 2500 HAUL(DODGE) <= 2500

In other words, as we anticipated, LINGO generated one constraint for each truck in the set limiting it to a load of 2,500 pounds. Here is a model that uses an @FOR statement (listed in bold) to compute the reciprocal of any five numbers placed into the VALUE attribute: MODEL: SETS: NUMBERS /1..5/: VALUE, RECIPROCAL; ENDSETS DATA: VALUE = 3 4 2 7 10; ENDDATA @FOR( NUMBERS( I): RECIPROCAL( I) = 1 / VALUE( I) ); END

Solving this model gives the following values for the reciprocals: Variable RECIPROCAL(1) RECIPROCAL(2) RECIPROCAL(3) RECIPROCAL(4) RECIPROCAL(5)

Value 0.3333333 0.2500000 0.5000000 0.1428571 0.1000000

Since the reciprocal of zero is not defined, we could put a conditional qualifier on our @FOR statement that causes us to skip the reciprocal computation whenever a zero is encountered. The following @FOR statement accomplishes this: @FOR(NUMBERS(I)| VALUE(I) #NE# 0: RECIPROCAL(I) = 1 / VALUE(I) );

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The conditional qualifier (listed in bold) tests to determine if the value is not equal (#NE#) to zero. If so, the computation proceeds. This was just a brief introduction to the use of the @FOR statement. There will be many additional examples in the sections to follow.

@PROD Set Looping Function The @PROD function is used to find the product of an expression across of a set. As an example, consider the model: MODEL: SETS: COMPONENTS: P; ENDSETS DATA: P = .95 .99 .98; ENDDATA P_FAIL = 1 - @PROD( COMPONENTS( I): P( I)); END

Here we have a system of three components arranged in a series. The probability that each component functions successfully (.95, .99, and .98) is loaded into attribute P in the model’s data section. We then compute the probability that the entire system will fail, P_FAIL, by taking the product of the component probabilities and subtracting it from 1: P_FAIL = 1 - @PROD( COMPONENTS( I): P( I));

As an aside, an interesting feature to note about this model is that we never initialized the COMPONENTS set. When LINGO sees that an attribute of an undefined primitive set being initialized to n values in a data section, it automatically initializes the parent primitive set to contain the : 1, 2, …, n. So, in this example, LINGO automatically assigned the member 1, 2 and 3 to the COMPONENTS set.

Nested Set Looping Functions The simple models shown in the last section use @FOR to loop over a single set. In larger models, you’ll encounter the need to loop over a set within another set looping function. When one set looping function is used within the scope of another, we call it nesting. An example of a nested set looping function can be found in the Wireless Widgets shipping model (p.27). If you , WW’s vendors had a demand for widgets that had to be met. The LINGO statement that enforces this condition is: ! The demand constraints; @FOR(VENDORS(J): @SUM(WAREHOUSES(I): VOLUME(I, J)) = DEMAND(J));

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Specifically, for each vendor, we sum up the shipments going from all the warehouses to that vendor and set the quantity equal to the vendor’s demand. In this case, we have nested an @SUM function within an @FOR function. @SUM, @MAX, and @MIN can be nested within any set looping function. @FOR functions, on the other hand, may only be nested within other @FOR functions.

Summary This section demonstrated that set looping functions can be very powerful and can simplify the modeler's task. If you aren't making use of sets and set looping functions, you will have a considerably more difficult time building your models. Furthermore, the difficulty will grow dramatically as the sizes of your models grow. We now know how to create sets, how to initialize sets and attributes using the data section, and how to work with sets using set looping functions. At this point, we now have the ability to start constructing some meaningful example models.

Set-Based Modeling Examples Recall from the earlier discussion in this chapter, there are four types of sets that can be created in LINGO. These set types are: 1. 2. 3. 4.

primitive, dense derived, sparse derived - explicit list, and sparse derived - hip filter.

If you would like to review the four set types, refer to the sections What are Sets? and The Sets Section of a Model at the beginning of this chapter. The remainder of this section will help develop your talents for set-based modeling by building and discussing four models, each introducing one of the set types listed above.

Primitive Set Example The following staff scheduling model illustrates the use of a primitive set. In a staff scheduling model, there is demand for staffing over a time horizon. The goal is to come up with a work schedule that meets staffing demands at minimal cost. The model used in this example may be found in the SAMPLES subdirectory off the main LINGO directory under the name STAFFDEM.

The Staff Scheduling Problem Suppose you run the popular Pluto Dogs hot dog stand that is open seven days a week. You hire employees to work a five-day workweek with two consecutive days off. Each employee receives the same weekly salary. Some days of the week are busier than others and, based on past experience, you know how many workers are required on a given day of the week. In particular, your forecast calls for

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these staffing requirements: Day Staff Req'd

Mon 20

Tue 16

Wed 13

Thu 16

Fri 19

Sat 14

Sun 12

You need to determine how many employees to start on each day of the week in order to minimize the total number of employees, while still meeting or exceeding staffing requirements each day of the week.

The Formulation The first question to consider when building a set-based model is, "What are the relevant sets and their attributes?". In this model, we have a single primitive set, the days of the week. If we call this set DAYS, we can begin by writing our sets section as: SETS: DAYS; ENDSETS

Next, we can add a data section to initialize the set of the DAYS set: SETS: DAYS; ENDSETS DATA: DAYS = MON TUE WED THU FRI SAT SUN; ENDDATA

Alternatively, we could use LINGO’s implicit set definition capability and express this equivalently as: SETS: DAYS; ENDSETS DATA: DAYS = MON..SUN; ENDDATA

We will be concerned with two attributes of the DAYS set. The first is the number of staff required on each day, and the second is the number of staff to start on each day. If we call these attributes REQUIRED and START, then we may add them to the sets section to get: SETS: DAYS: REQUIRED, START; ENDSETS

After defining the sets and attributes, it is useful to determine which of the attributes are data, and which are decision variables. In this model, the REQUIRED attribute is given to us and is, therefore, data. The START attribute is something we need to determine and constitutes the decision variables. Once you've identified the data in the model, you may go ahead and initialize it. We can do this by extending the data section as follows: DATA: DAYS = MON TUE WED THU FRI SAT SUN; REQUIRED = 20 16 13 16 19 14 12; ENDDATA

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We are now at the point where we can begin entering the model's mathematical relations (i.e., the objective and constraints). Let's begin by writing out the mathematical notation for the objective. Our objective is to minimize the total number of employees we start during the week. Using standard mathematical notation, this objective may be expressed as: Minimize: i STARTi The equivalent LINGO statement is very similar. Substitute "MIN=" for "Minimize:" and "@SUM( DAYS( I):" for i and we have: MIN = @SUM( DAYS( I): START( I));

Now, all that is left is to come up with our constraints. There is only one set of constraints in this model. Namely, we must have enough staff on duty each day to meet or exceed staffing requirements. In words, what we want is: Staff on duty today  Staff required today, for each day of the week The right-hand side of this expression, Staff required today, is easy to calculate. It is simply the quantity REQUIRED( I). The left-hand side, Staff on duty today, is a bit trickier to compute. Given that all employees are on a "five day on, two day off" schedule, the number of employees working today is: Number working today = Number starting today + Number starting 1 day ago + Number starting 2 days ago + Number starting 3 days ago + Number starting 4 days ago. In other words, to compute the number of employees working today, we sum up the number of people starting today plus those starting over the previous four days. The number of employees starting five and six days back don't count because they are on their days off. So, using mathematical notation, what one might consider doing is adding the constraint:

i = j-4, j STARTi  REQUIRED j , for j DAYS

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Translating into LINGO notation, we can write this as: @FOR( DAYS( J): @SUM( DAYS( I) | I #LE# 5: START( J - I + 1)) >= REQUIRED( J) );

In words, the LINGO statement says, for each day of the week, the sum of the employees starting over the five day period beginning four days ago and ending today must be greater-than-or-equal-to the required number of staff for the day. This sounds correct, but there is a slight problem. If we try to solve our model with this constraint we get the error message:

To see why we get this error message, consider what happens on Thursday. Thursday has an index of 4 in our set DAYS. As written, the staffing constraint for Thursday will be: START( 4 - 1 + 1) + START( 4 - 2 + 1) + START( 4 - 3 + 1) + START( 4 - 4 + 1) + START( 4 - 5 + 1) >= REQUIRED( 4);

Simplifying, we get: START( 4) + START( 3) + START( 2) + START( 1) + START( 0) >= REQUIRED( 4);

The START( 0) term is the root of our problem. START is defined for days 1 through 7. START( 0) does not exist. An index of 0 on START is considered "out of range.” We would like to have any indices less-than-or-equal-to 0 wrap around to the end of the week. Specifically, 0 would correspond to Sunday (7), -1 to Saturday (6), and so on. LINGO has a function that does just this: @WRAP. The @WRAP function takes two argumentscall them INDEX and LIMIT. Formally speaking, @WRAP returns J such that J = INDEX - K * LIMIT, where K is an integer such that J is in the interval [1, LIMIT]. Informally speaking, @WRAP will subtract or add LIMIT to INDEX until it falls in the range 1 to LIMIT. Therefore, this is just what we need to "wrap around" an index in multiperiod planning models.

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Incorporating the @WRAP function, we get the corrected, final version of our staffing constraint: @FOR( DAYS( J): @SUM( DAYS( I) | I #LE# 5: START( @WRAP( J - I + 1, 7))) >= REQUIRED( J) );

The Solution Below is our staffing model in its entirety: MODEL: SETS: DAYS: REQUIRED, START; ENDSETS DATA: DAYS = MON TUE WED THU FRI SAT SUN; REQUIRED = 20 16 13 16 19 14 12; ENDDATA MIN = @SUM( DAYS( I): START( I)); @FOR( DAYS( J): @SUM( DAYS( I) | I #LE# 5: START( @WRAP( J - I + 1, 7))) >= REQUIRED( J) ); END

Model: STAFFDEM

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Solving the model, we get the solution report: Global optimal solution found. Objective value: Infeasibilities: Total solver iterations: Variable REQUIRED( MON) REQUIRED( TUE) REQUIRED( WED) REQUIRED( THU) REQUIRED( FRI) REQUIRED( SAT) REQUIRED( SUN) START( MON) START( TUE) START( WED) START( THU) START( FRI) START( SAT) START( SUN)

22.00000 0.000000 5

Value 20.00000 16.00000 13.00000 16.00000 19.00000 14.00000 12.00000 8.000000 2.000000 0.000000 6.000000 3.000000 3.000000 0.000000

Reduced Cost 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.3333333

Row Slack or Surplus 1 22.00000 2 0.000000 3 0.000000 4 0.000000 5 0.000000 6 0.000000 7 0.000000 8 0.000000 Solution to STAFFDEM

Dual Price -1.000000 -0.3333333 0.000000 -0.3333333 0.000000 -0.3333333 -0.3333333 0.000000

The objective value of 22 means we need to hire 22 workers. We start our workers according to the schedule: Day Start

Mon 8

Tue 2

Wed 0

Thu 6

Fri 3

Sat 3

Sun 0

If we look at the surpluses on our staffing requirement rows (rows 2 - 7), we see that the slack values are 0 on all of the days. This means there are no more workers than required and we just meet staffing requirements on every day. Even though this is a small model, trying to come up with a solution this efficient "by hand" would be a difficult task.

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Dense Derived Set Example - Blending This following model illustrates, among other things, the use of a dense derived set in a blending model. In a blending model, one is blending raw materials into a finished product that must meet minimal quality requirements on one or more dimensions. The goal is to come up with a blend of the raw materials to satisfy the quality requirements at minimal cost. This model may be found in the SAMPLES subdirectory off the main LINGO directory under the name CHESS.

The Problem The Chess Snackfoods Co. markets four brands of mixed nuts. The four brands of nuts are called Pawn, Knight, Bishop, and King. Each brand contains a specified ratio of peanuts and cashews. The table below lists the number of ounces of the two nuts contained in each pound of each brand and the price the company receives per pound of each brand. Peanuts (oz.) Cashews (oz.) Selling Price ($)

Pawn 15 1 2

Knight 10 6 3

Bishop 6 10 4

King 2 14 5

Chess has contracts with suppliers to receive 750 pounds of peanuts/day and 250 pounds of cashews/day. Our problem is to determine the number of pounds of each brand to produce each day to maximize total revenue without exceeding the available supply of nuts.

The Formulation The primitive sets in this model are the nut types and the brands of mixed nuts. We can declare them in the sets section as follows: SETS: NUTS: SUPPLY; BRANDS: PRICE, PRODUCE; ENDSETS

The NUTS set has the single attribute SUPPLY, which we will use to store the daily supply of nuts in pounds. The BRANDS set has PRICE and PRODUCE attributes, where PRICE stores the selling price of the brands, and PRODUCE represents the decision variables of how many pounds of each brand to produce each day. We need one more set, however, which is the dense derived set we have been promising. In order to input the brand formulas, we will need a two dimensional table defined on the nut types and the brands. To do this, we will generate a derived set named FORMULA from the cross of the NUTS and BRANDS sets. Adding this derived set, we get the completed sets section: SETS: NUTS: SUPPLY; BRANDS: PRICE, PRODUCE; FORMULA( NUTS, BRANDS): OUNCES;

ENDSETS

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We have titled the derived set FORMULA. It has the single attribute OUNCES, which will be used to store the ounces of nuts used per pound of each brand. Since we have not specified the of this derived set, LINGO assumes we want the complete, dense set that includes all pairs of nuts and brands, for a total of eight (nut,brand) pairs. Now that our sets are declared, we can move on to building the data section. We initialize our three sets, NUTS, BRANDS and FORMULA, as well as the two data attributes SUPPLY and PRICE as follows: DATA: NUTS, PEANUTS CASHEWS

SUPPLY = 750 250;

BRANDS, PAWN KNIGHT BISHOPP KING

PRICE = 2 3 4 5;

FORMULA = 15 10 6 2 1 6 10 14;

ENDDATA

With the sets and data established, we can begin to enter our objective function and constraints. The objective function of maximizing total revenue is straightforward. We can express this as: MAX = @SUM( BRANDS( I): PRICE( I) * PRODUCE( I));

Our model has only one class of constraints: We can't use more nuts than we are supplied with on a daily basis. In words, we would like to ensure: For each nut i, the number of pounds of nut i used must be less-than-or-equal-to the supply of nut i. We can express this in LINGO as: @FOR( NUTS( I): @SUM( BRANDS( J): OUNCES( I, J) * PRODUCE( J) / 16) <= SUPPLY( I)

);

We divide the sum on the left-hand side by 16 to convert from ounces to pounds.

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The Solution Our completed blending model is: MODEL: SETS: NUTS: SUPPLY; BRANDS: PRICE, PRODUCE; NCROSSB( NUTS, BRANDS): FORMULA; ENDSETS DATA: NUTS, PEANUTS CASHEWS BRANDS, PAWN KNIGHT BISHOPP KING

SUPPLY = 750 250; PRICE = 2 3 4 5;

FORMULA = 15 10 6 2 1 6 10 14; ENDDATA MAX = @SUM( BRANDS( I): PRICE( I) * PRODUCE( I)); @FOR( NUTS( I): @SUM( BRANDS( J): FORMULA( I, J) * PRODUCE( J) / 16) <= SUPPLY( I) ); END

Model: CHESS An abbreviated solution report for the model follows: Global optimal solution found. Objective value: Infeasibilities: Total solver iterations: Variable PRODUCE( PAWN) PRODUCE( KNIGHT) PRODUCE( BISHOP) PRODUCE( KING) Row 1 2 3

Solution to CHESS

Value 769.2308 0.000000 0.000000 230.7692

Slack or Surplus 2692.308 0.000000 0.000000

2692.308 0.000000 2

Reduced Cost 0.0000000 0.1538461 0.7692297E-01 0.0000000 Dual Price 1.000000 1.769231 5.461538

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This solution tells us Chess should produce 769.2 pounds of the Pawn mix and 230.8 pounds of King for total revenue of $2692.30. Additional interesting information can also be found in the report. The dual prices on the rows indicate Chess should be willing to pay up to $1.77 for an extra pound of peanuts and $5.46 for an extra pound of cashews. If, for marketing reasons, Chess decides it must produce at least some of the Knight and Bishop mixes, then the reduced cost figures tell us revenue will decline by 15.4 cents with the first pound of Knight produced and 7.69 cents with the first pound of Bishop produced.

Sparse Derived Set Example - Explicit List In this example, we will introduce the use of a sparse derived set with an explicit listing. As you recall, when we use this technique to define a sparse set, we must explicitly list all belonging to the set. This will usually be some small subset of the dense set resulting from the full Cartesian product of the parent sets. For our example, we will set up a PERT (Project Evaluation and Review Technique) model to determine the critical path of tasks in a project involving the roll out of a new product. PERT is a simple, but powerful, technique developed in the 1950s to assist managers in tracking the progress of large projects. PERT is particularly useful in identifying the critical activities within a project, which, if delayed, will delay the project as a whole. These time critical activities are referred to as the critical path of a project. Having such insight into the dynamics of a project goes a long way in guaranteeing it won't get sidetracked and become delayed. In fact, PERT proved so successful, the Polaris project that it was first used on was completed 18 months ahead of schedule. PERT continues to be used successfully on a wide range of projects. For more information on PERT, and a related technique called M (Critical Path Method), please refer to Schrage (2005) or Winston (1995). The formulation for this model is included in the SAMPLES subdirectory off the main LINGO directory under the name PERT.

A Project Scheduling Problem Wireless Widgets is about to launch a new product: The Solar Widget. In order to guarantee the launch will occur on time, WW wants to perform a PERT analysis of the tasks leading up to the launch. Doing so will allow them to identify the critical path of tasks that must be completed on time in order to guarantee the Solar Widget's timely introduction. The tasks that must be accomplished before introduction and their anticipated times for completion are listed in the table below: Task Finalize Design Forecast Demand Survey Competition Set Prices Schedule Production Run Cost Out Train Salesmen

Weeks 10 14 3 3 7 4 10

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Certain of the tasks must be completed before others can commence. These precedence relations are shown in the following graph: Forecast Demand

Schedule Production Run

Cost Out

Survey Competition

Set Prices

Train Salesman

Finalize Design

Product Launch Precedence Relations For instance, the two arrows originating from the Forecast Demand node indicate the task must be completed before the Schedule Production Run and the Set Prices tasks may be started. Our goal is to construct a PERT model for the Solar Widget's introduction in order to identify the tasks on the critical path.

The Formulation We will need a primitive set to represent the tasks of the project. We can add such a set to the model using the set definition: SETS: TASKS: TIME, ES, LS, SLACK;

ENDSETS

We have associated four attributes with the TASKS set. The definitions of the attributes are: TIME

Time to complete the task

ES

Earliest possible start time for the task

LS

Latest possible start time for the task

SLACK

Difference between LS and ES for the task

The TIME attribute is given to us as data. We will compute the values of the remaining three attributes. If a task has a 0 slack time, it means the task must start on time or the whole project will be delayed. The collection of tasks with 0 slack time constitute the critical path for the project. In order to compute the start times for the tasks, we will need the precedence relations. The precedence relations can be viewed as a list of ordered pairs of tasks. For instance, the fact that the DESIGN task must be completed before the FORECAST task could be represented as the ordered pair (DESIGN, FORECAST). Creating a two-dimensional derived set on the TASKS set will allow us to input the list of precedence relations. Specifically, we add the derived set definition PRED: SETS: TASKS: TIME, ES, LS, SLACK; PRED( TASKS, TASKS);

ENDSETS

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Next, we can input the TASKS set and task times in the data section by including: DATA: TASKS, TIME = DESIGN 10 FORECAST 14 SURVEY 3 DUMMY 0 PRICE 3 SCHEDULE 7 COSTOUT 4 TRAIN 10 ;

ENDDATA

The set PRED is the sparse derived set with an explicit listing that we want to highlight in this example. The set is a subset derived from the cross of the TASKS set upon itself. The set is sparse because it contains only 8 out of a possible 49 found in the complete cross of TASKS on TASKS. The set is said to be an "explicit list" set, because we will explicitly list the we want included in the set. Explicitly listing the of a sparse set may not be convenient in cases where there are thousands of to select from, but it does make sense whenever set hip conditions are not well defined and the sparse set size is small relative to the dense alternative. Adding the initialization of PRED to the data set give us: DATA: TASKS, TIME = DESIGN 10 FORECAST 14 SURVEY 3 DUMMY 0 PRICE 3 SCHEDULE 7 COSTOUT 4 TRAIN 10 ; PRED = DESIGN, FORECAST, DESIGN, SURVEY, FORECAST, DUMMY FORECAST, SCHEDULE, SURVEY, PRICE, SCHEDULE, COSTOUT, PRICE, TRAIN, COSTOUT, TRAIN, DUMMY, PRICE ;

ENDDATA

Keep in mind that the first member of this set is the ordered pair (DESIGN, FORECAST)not just the single task DESIGN. Therefore, this set has a total of 8 that all correspond to a directed arc in the precedence relations diagram.

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Now, with our sets and data established, we can turn our attention to building the formulas of the model. We have three attributes to compute: earliest start (ES), latest start (LS), and slack time (SLACK). The trick is computing ES and LS. Once we have these times, SLACK is merely the difference of the two. Lets start by coming up with a formula to compute ES. A task cannot begin until all its predecessor tasks are completed. Thus, if we find the latest finishing time of all predecessors to a task, then we have also found its earliest start time. Therefore, in words, the earliest start time for task t is equal to the maximum over all predecessors of task t of the sum of the earliest start time of the predecessor plus its completion time. The corresponding LINGO notation is: @FOR( TASKS( J)| J #GT# 1: ES( J) = @MAX( PRED( I, J): ES( I) + TIME( I))

);

Note that we skip the computation for the first task by adding the conditional qualifier J #GT# 1. We do this because the first task has no predecessors. We will give the first task an arbitrary start time as shown below. Computing LS is slightly trickier, but very similar to ES. In words, the latest time for task t to start is the minimum over all successor tasks of the sum of the successor's earliest start minus the time to perform task t. If task t starts any later than this, it will prohibit at least one successor from starting at its earliest start time. Converting into LINGO syntax gives: @FOR( TASKS( I)| I #LT# LTASK: LS( I) = @MIN( PRED( I, J): ES( J) - TIME( I))

);

Here, we omit the computation for the last task since it has no successor tasks. Computing slack time is just the difference between LS and ES, and may be written as: @FOR( TASKS( I): SLACK( I) = LS( I) - ES( I));

We can set the start time of the first task to some arbitrary value. For our purposes, we will set it to 0 with the statement: ES( 1) = 0;

We have now input formulas for computing the values of all the variables with the exception of the latest start time for the last task. It turns out, if the last project were started any later than its earliest start time, the entire project would be delayed. So, by definition, the latest start time for the last project is equal to its earliest start time. We can express this in LINGO using the equation: LS( 7) = ES( 7);

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This would work, but it's probably not the best way to express the relation. Suppose you were to add some tasks to your model. You'd have to change the 7 in this equation to the new number of tasks was. The whole idea behind LINGO's set-based modeling language is the equations in the model should be independent of the data. Expressing the equation in this form violates data independence. Here's a better way to do it: LTASK = @SIZE( TASKS); LS( LTASK) = ES( LTASK);

The @SIZE function returns the size of a set. In this case, it will return the value 7, as desired. However, if we changed the number of tasks, @SIZE would also return the new, correct value. Thus, we preserve the data independence of our model's equations.

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The Solution The entire PERT formulation and portions of its solution appear below. MODEL: SETS: TASKS: TIME, ES, LS, SLACK; PRED( TASKS, TASKS); ENDSETS DATA: TASKS, TIME = DESIGN 10 FORECAST 14 SURVEY 3 DUMMY 0 PRICE 3 SCHEDULE 7 COSTOUT 4 TRAIN 10 ; PRED = DESIGN, FORECAST, DESIGN, SURVEY, FORECAST, DUMMY FORECAST, SCHEDULE, SURVEY, PRICE, SCHEDULE, COSTOUT, PRICE, TRAIN, COSTOUT, TRAIN, DUMMY, PRICE ; ENDDATA @FOR( TASKS( J)| J #GT# 1: ES( J) = @MAX( PRED( I, J): ES( I) + TIME( I)) ); @FOR( TASKS( I)| I #LT# LTASK: LS( I) = @MIN( PRED( I, J): LS( J) - TIME( I)); ); @FOR( TASKS( I): SLACK( I) = LS( I) - ES( I)); ES( 1) = 0; LTASK = @SIZE( TASKS); LS( LTASK) = ES( LTASK); END

Model: PERT

USING SETS Feasible solution found. Total solver iterations: Variable LTASK ES( DESIGN) ES( FORECAST) ES( SURVEY) ES( PRICE) ES( SCHEDULE) ES( COSTOUT) ES( TRAIN) LS( DESIGN) LS( FORECAST) LS( SURVEY) LS( PRICE) LS( SCHEDULE) LS( COSTOUT) LS( TRAIN) SLACK( DESIGN) SLACK( FORECAST) SLACK( SURVEY) SLACK( PRICE) SLACK( SCHEDULE) SLACK( COSTOUT) SLACK( TRAIN)

75

0 Value 7.000000 0.0000000 10.00000 10.00000 24.00000 24.00000 31.00000 35.00000 0.0000000 10.00000 21.00000 32.00000 24.00000 31.00000 35.00000 0.0000000 0.0000000 11.00000 8.000000 0.0000000 0.0000000 0.0000000

Solution to PERT The interesting values are the slacks for the tasks. Both SURVEY and PRICE have slack in their start times of 11 weeks and 8 weeks, respectively. Their start times may be delayed by as much as these slack values without compromising the completion time of the entire project. The tasks DESIGN, FORECAST, SCHEDULE, COSTOUT, and TRAIN, on the other hand, have 0 slack times. These tasks constitute the critical path for the project and, if any of their start times are delayed, the entire project will be delayed. Management will want to pay close attention to these critical path projects to be sure they start on time and are completed within the allotted amount of time. Finally, the ES( TRAIN) value of 35 tells us the estimated time to the start of the roll out of the new Solar Widget will be 45 weeks _35 weeks to get to the start of training, plus 10 weeks to complete training.

A Sparse Derived Set Using a hip Filter In this example, we introduce the use of a sparse derived set with a hip filter. Using a hip filter is the third method for defining a derived set. When you define a set using this method, you specify a logical condition each member of the set must satisfy. LINGO then filters out potential set that don't satisfy the hip condition. For our example, we will formulate a matching problem. In a matching problem, there are N objects we want to match into pairs at minimum cost. The pair (I,J) is indistinguishable from the pair (J,I). Therefore, we arbitrarily require I be less than J in the pair. Formally, we require I and J make a set of ordered pairs. In other words, we do not wish to generate redundant ordered pairs of I and J, but only those with I less than J. This requirement that I be less than J will form our hip filter.

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The file containing this model may be found in the SAMPLES subdirectory off the main LINGO directory under the name MATCHD.

A Matching Problem Suppose you manage your company’s strategic planning department. You have a total of eight analysts in the department. Furthermore, your department is about to move into a new suite of offices. There are a total of four offices in the new suite and you need to match up your analysts into 4 pairs, so each pair can be assigned to one of the new offices. Based on past observations, you know some of the analysts work better together than they do with others. In the interest of departmental peace, you would like to come up with a pairing of analysts that results in minimal potential conflicts. To this goal, you have come up with a rating system for pairing your analysts. The scale runs from 1 to 10, with a 1 rating of a pair meaning the two get along fantastically. Whereas, a rating of 10 means all sharp objects should be removed from the pair’s office in anticipation of mayhem. The ratings appear in the following table: Analysts 1 2 3 4 5 9 3 4 2 1 1 7 3 2 4 4 3 1 4 5 6 7 Analysts’ Incompatibility Ratings

6 1 5 2 5 8 -

7 5 2 9 5 7 2 -

8 6 1 2 2 6 3 4

Since the pairing of analyst I with analyst J is indistinguishable from the pairing of J with I, we have only included the above diagonal elements in the table. Our problem is to find the pairings of analysts that minimizes the sum of the incompatibility ratings of the paired analysts.

The Formulation The first set of interest in this problem is the set of analysts. This is a primitive set that can be declared simply as: SETS: ANALYSTS;

ENDSETS

The final set we want to construct is a set consisting of all the potential pairings. This will be a derived set that we will build by taking the cross of the ANALYST set on itself. As a first , we could build the dense derived set: PAIRS( ANALYSTS, ANALYST);

USING SETS

77

This set, however, would include both PAIRS( I, J) and PAIRS( J, I). Since only one of these pairs is required, the second is wasteful. Furthermore, this set will include "pairs" of the same analyst of the form PAIRS( I, I). As much as each of the analysts might like an office of their own, such a solution is not feasible. The solution is to put a hip filter on our derived set requiring each pair (I,J) in the final set to obey the condition J be greater than I. We do this with the set definition: PAIRS( ANALYSTS, ANALYSTS) | &2 #GT# &1;

The start of the hip filter is denoted with the vertical bar character (|). The &1 and &2 symbols in the filter are known as set index placeholders. Set index placeholders are valid only in hip filters. When LINGO constructs the PAIRS set, it generates all combinations in the cross of the ANALYSTS set on itself. Each combination is then "plugged" into the hip filter to see if it es the test. Specifically, for each pair (I,J) in the cross of the ANALYST set on itself, I is substituted into the placeholder &1 and J into &2 and the filter is evaluated. If the filter evaluates to true, (I,J) is added to the pairs set. Viewed in tabular form, this leaves us with just the diagonal elements of the (I,J) pairing table. We will also be concerned with two attributes of the PAIRS set. First, we will need an attribute that corresponds to the incompatibility rating of the pairings. Second, we will need an attribute to indicate if analyst I is paired with analyst J. We will call these attributes RATING and MATCH. We append them to the PAIRS set definition as follows: PAIRS( ANALYSTS, ANALYSTS) | &2 #GT# &1: RATING, MATCH;

The completed sets section containing both set declarations is then: SETS: ANALYSTS; PAIRS( ANALYSTS, ANALYSTS) | &2 #GT# &1: RATING, MATCH;

ENDSETS

Next, we initialize the ANALYSTS set and the RATING attribute to the incompatibility ratings in the data section: DATA: ANALYSTS = 1..8; RATING = 9 3 4 2 1 1 7 3 5 4 4 2 1 5 8

ENDDATA

5 2 9 5 7 2

6 1 2 2 6 3 4;

We will use the convention of letting MATCH( I, J) be 1 if we pair analyst I with analyst J, otherwise 0. Given this, the MATCH attribute contains the decision variables for the model.

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Our objective is to minimize the sum of the incompatibility ratings of all the final pairings. This is just the inner product on the RATING and MATCH attributes and is written as: MIN = @SUM( PAIRS( I, J): RATING( I, J) * MATCH( I, J));

There is just one class of constraints in the model. In words, it is: For each analyst, ensure that the analyst is paired with exactly one other analyst. Putting the constraint into LINGO syntax, we get: @FOR( ANALYSTS( I): @SUM( PAIRS( J, K) | J #EQ# I #OR# K #EQ# I: MATCH( J, K)) = 1

);

The feature of interest in this constraint is the conditional qualifier (J #EQ# I #OR# K #EQ# I) on the @SUM function. For each analyst I, we sum up all the MATCH variables that contain I and set them equal to 1. In so doing, we guarantee analyst I will be paired up with exactly one other analyst. The conditional qualifier guarantees we only sum up the MATCH variables that include I in its pairing. One other feature is required in this model. We are letting MATCH( I, J) be 1 if we are pairing I with J. Otherwise, it will be 0. Unless specified otherwise, LINGO variables can assume any value from 0 to infinity. Because we want MATCH to be restricted to being only 0 or 1, we need to apply the @BIN variable domain function to the MATCH attribute. Variable domain functions are used to restrict the values a variable can assume. Unlike constraints, variable domain functions do not add equations to a model. The @BIN function restricts a variable to being binary (i.e., 0 or 1). When you have a model that contains binary variables, it is said to be an integer programming (IP) model. IP models are much more difficult to solve than models that contain only continuous variables. Carelessly formulated large IPs (with several hundred integer variables or more) can literally take forever to solve! Thus, you should limit the use of binary variables whenever possible. To apply @BIN to all the variables in the MATCH attribute, add the @FOR expression: @FOR( PAIRS( I, J): @BIN( MATCH( I, J)));

The Solution The entire formulation for our matching example and parts of its solution appear below.

USING SETS MODEL: SETS: ANALYSTS; PAIRS( ANALYSTS, ANALYSTS) | &2 #GT# &1: RATING, MATCH; ENDSETS DATA: ANALYSTS = 1..8; RATING = 9 3 4 2 1 1 7 3 5 4 4 2 1 5 8

5 2 9 5 7 2

6 1 2 2 6 3 4;

ENDDATA MIN = @SUM( PAIRS( I, J): RATING( I, J) * MATCH( I, J)); @FOR( ANALYSTS( I): @SUM( PAIRS( J, K) | J #EQ# I #OR# K #EQ# I: MATCH( J, K)) = 1 ); @FOR( PAIRS( I, J): @BIN( MATCH( I, J))); END

Model: MATCHD

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CHAPTER 2 Global optimal solution found. Objective value: Extended solver steps: Total solver iterations: Variable MATCH( 1, 2) MATCH( 1, 3) MATCH( 1, 4) MATCH( 1, 5) MATCH( 1, 6) MATCH( 1, 7) MATCH( 1, 8) MATCH( 2, 3) MATCH( 2, 4) MATCH( 2, 5) MATCH( 2, 6) MATCH( 2, 7) MATCH( 2, 8) MATCH( 3, 4) MATCH( 3, 5) MATCH( 3, 6) MATCH( 3, 7) MATCH( 3, 8) MATCH( 4, 5) MATCH( 4, 6) MATCH( 4, 7) MATCH( 4, 8) MATCH( 5, 6) MATCH( 5, 7) MATCH( 5, 8) MATCH( 6, 7) MATCH( 6, 8) MATCH( 7, 8)

Value 0.0000000 0.0000000 0.0000000 0.0000000 1.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 1.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 1.000000 1.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

6.000000 0 0

Reduced Cost 9.000000 3.000000 4.000000 2.000000 1.000000 5.000000 6.000000 1.000000 7.000000 3.000000 5.000000 2.000000 1.000000 4.000000 4.000000 2.000000 9.000000 2.000000 1.000000 5.000000 5.000000 2.000000 8.000000 7.000000 6.000000 2.000000 3.000000 4.000000

Solution to MATCHD From the objective value, we know the total sum of the incompatibility ratings for the optimal pairings is 6. Scanning the Value column for 1s, we find the optimal pairings: (1,6), (2,7), (3,8), and (4,5).

Summary In this chapter, we’ve discussed the concept of sets, how to declare sets, and demonstrated the power and flexibility of set-based modeling. You should now have a foundation of knowledge in the definition and use of both primitive and derived sets. The next chapter will discuss the use of variable domain functions, which were briefly introduced in this chapter when we used @BIN in the previous matching model.

3 Using Variable Domain Functions Unless specified otherwise, variables in a LINGO model default to being non-negative and continuous. More specifically, variables can assume any real value from zero to positive infinity. In many cases, this default domain for a variable may be inappropriate. For instance, you may want a variable to assume negative values, or you might want a variable restricted to purely integer values. LINGO provides seven variable domain functions, which allow you to override the default domain of a variable. The names of these functions and a brief description of their usage are: @GIN @BIN @FREE @BND @SOS @CARD @SEMIC @PRIORITY

restricts a variable to being an integer value, makes a variable binary (i.e., 0 or 1), allows a variable to assume any real value, positive or negative, limits a variable to fall within a finite range, defines a set of binary variables and places restrictions on their collective values, defines a set of binary variables and places an upper limit on their sum, and restricts variables to being either zero or grater than a specified constant. used to assign branching priorities to variables.

In the remainder of this section, we'll investigate the mechanics of using these functions, and present a number of examples illustrating their usage.

Integer Variables LINGO gives the the ability to define two types of integer variablesgeneral and binary. A general integer variable is required to be a whole number. A binary integer variable is further required to be either zero or one. Any model containing one or more integer variables is referred to as an integer programming (IP) model. In many modeling projects, you will be faced with Yes/No types of decisions. Some examples would include Produce/Don’t Produce, Open Plant/Close Plant, Supply Customer I from Plant J/Don’t Supply Customer I from Plant J, and Incur a Fixed Cost/Don’t Incur a Fixed Cost. Binary variables are the standard method used for modeling these Yes/No decisions.

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General integer variables are useful where rounding of fractional solutions is problematic. For instance, suppose you have a model that dictates producing 5,121,787.5 blue crayons in your crayon factory. Whether you round the solution to 5,121,787 or 5,121,788 is inconsequential. On the other hand, suppose your planning model for NASA determines the optimal number of space stations to deploy is 1.5. Because building 0.5 space stations is impossible, you must very carefully consider how to round the results. When whole numbers are required and rounding can make a significant difference, general integer variables are appropriate. LINGO does not simply round or truncate values to come up with an integer answer. Rounding of a solution will typically lead to either infeasible or suboptimal solutions. To illustrate this point, consider the small model: MAX = X; X + Y = 25.5; X <= Y;

By examining this model, one can deduce the optimal solution is X=Y=12.75. Now, suppose we want an optimal solution with X being integer. Simply rounding X to 13 would make the model infeasible, because there would be no value for Y that would satisfy both the constraints. Clearly, the optimal solution is X=12 and Y=13.5. Unfortunately, “eyeballing” the optimal solution on larger models with many integer variables is virtually impossible. To solve these problems, LINGO performs a complex algorithm called branch-and-bound that implicitly enumerates all combinations of the integer variables to determine the best feasible answer to an IP model. Because of the extra computation time required by this algorithm, formulating your problem to avoid the use of integer variables is advised whenever possible. Even so, although computation times may grow dramatically when you add integer variables, it often makes sense to ask LINGO for integer solutions when fractional values are of little or no use.

General Integer Variables By default, LINGO assumes all variables in a model are continuous. In many applications, fractional values may be undesirable. You won’t be able to hire two-thirds of a person, or sell half an automobile. In these instances, you will want to make use of the general integer variable domain function, @GIN. The syntax of the @GIN function is: @GIN(variable_name);

VARIABLE DOMAIN FUNCTIONS

83

where variable_name is the name of the variable you wish to make general integer. The @GIN function may be used in a model anywhere you would normally enter a constraint. The @GIN function can be embedded in an @FOR statement to allow you to easily set all, or selected, variables of an attribute to be general integers. Some examples of @GIN are: Example 1: @GIN(X); makes the scalar variable X general integer, Example 2: @GIN(PRODUCE(5)); makes the variable PRODUCE(5) general integer, Example 3: @FOR(DAYS(I): @GIN(START(I))); makes all the variables of the START attribute general integer.

General Integer Example - CompuQuick Product-Mix To illustrate the use of @GIN in a full model, we will consider a variation on the CompuQuick Corporation model in Chapter 1, Getting Started with LINGO. CompuQuick has successfully rebalanced the Standard computer’s assembly line. In so doing, they are now able to build an additional 3 Standard computers on the line each day, for a daily total of 103 computers. As a result, the constraint on the Standard’s assembly line will now be: STANDARD <= 103;

Incorporating this constraint into the original CompuQuick model, we have: ! Here is the total profit objective function; MAX = 100 * STANDARD + 150 * TURBO; ! Constraints on the production line capacity; STANDARD <= 103; TURBO <= 120; ! Our labor supply is limited; STANDARD + 2 * TURBO <= 160;

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Solving this modified model, we get the solution: Global optimal solution found. Objective value: Infeasibilities: Total solver iterations:

14575.00 0.000000 0

Model Class:

LP

Total variables: Nonlinear variables: Integer variables:

2 0 0

Total constraints: Nonlinear constraints:

4 0

Total nonzeros: Nonlinear nonzeros:

6 0

Variable STANDARD TURBO Row 1 2 3 4

Value 103.0000 28.50000 Slack or Surplus 14575.00 0.000000 91.50000 0.000000

Reduced Cost 0.000000 0.000000 Dual Price 1.000000 25.00000 0.000000 75.00000

Note the new optimal number of Turbo computers, 28.5, is no longer an integer quantity. CompuQuick must produce whole numbers of computers each day. To guarantee this, we add @GIN statements to make both the STANDARD and TURBO variables general integer. The revised model follows: ! Here is the total profit objective function; MAX = 100 * STANDARD + 150 * TURBO; ! Constraints on the production line capacity; STANDARD <= 103; TURBO <= 120; ! Our labor supply is limited; STANDARD + 2 * TURBO <= 160; ! Integer values only; @GIN(STANDARD); @GIN(TURBO);


85

Solving the modified model results in the integer solution we were hoping for: Global optimal solution found. Objective value: Objective bound: Infeasibilities: Extended solver steps: Total solver iterations:

14550.00 14550.00 0.000000 0 0

Model Class:

PILP


2 0 2


4 0


6 0

Variable STANDARD TURBO Row 1 2 3 4


Reduced Cost -100.0000 -150.0000 Dual Price 1.000000 0.000000 0.000000 0.000000

Note that we now have a two new solution statistics: Extended solver steps and Objective bound. For models with integer variables, such as this one, the extended solver steps statistic is a tally of the number of times integer variables had to be forced to an integer value during the branch-and-bound solution procedure. In general, this value is not of much practical use to the normal , other than to give you a notion of how hard LINGO is working at finding an integer solution. If the number of steps gets quite large, LINGO is having a hard time finding good integer solutions to your model. In this case, given that the model is quite small, LINGO's preprocessor was able to find the optimal solution without having to resort to the branch-and-bound solver. The objective bound statistic gives us a bound on the best possible solution for the model. In other words, there is no feasible solution to the model with an objective value better than the objective bound statistic. Here we see that the bound and the objective value are both equal, which is further evidence that we have a globally optimal solution. On larger models that take a while to run, you may decide to interrupt LINGO before a global solution is found. In this case, the objective bound and the objective value will probably not agree. In this case, the bound will let you know how far you are from the true optimal solution. You may also find it useful to interrupt the solver once the bound and objective value get close to one another, with the idea being that any further potential gains in the best objective aren't worth the additional solve time.

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Also of interest is the Model Class, which has changed from LP (Linear Program) to PILP (Pure Integer Linear Program).

General Integer Example - Staff-Scheduling Recalling the staff-scheduling example in Chapter 2, Using Sets, for the Pluto hot dog stand, you will the solution told us how many employees to start on any given day of the week. You may also the optimal solution had us starting whole numbers of employees on every day even though we weren’t using integer variables. It turns out this was just a happy coincidence. Let’s return to the staffing model to demonstrate this. In the original staffing model, we required the following number of people on duty for the seven days of the week: 20, 16, 13, 16, 19, 14, and 12. Let’s change the second day requirement from 16 to 12 and the third day’s requirement from 13 to 18. Incorporating this change into the model, we have: MODEL: SETS: DAYS: REQUIRED, START; ENDSETS DATA: DAYS = MON TUE WED THU FRI SAT SUN; REQUIRED = 20 12 18 16 19 14 12; ENDDATA MIN = @SUM( DAYS( I): START( I)); @FOR( DAYS( J): @SUM( DAYS( I) | I #LE# 5: START( @WRAP( J - I + 1, 7))) >= REQUIRED( J) ); END

After making this modest change and re-solving, we no longer have a pure integer solution. In fact, all the START variables are now fractional as the following, abbreviated solution report shows: Global optimal solution found. Objective value: Total solver iterations: Variable START( MON) START( TUE) START( WED) START( THU) START( FRI) START( SAT) START( SUN)

Value 9.666667 2.000000 1.666667 5.666667 0.000000 4.666667 0.000000

23.66667 0 Reduced Cost 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.333333


87

In this particular model, we can always round the solution up and remain feasible. (In most models, we won’t tend to be as lucky. Rounding the continuous solution in one direction or the other can lead to an infeasible solution.) There may be some extra staff on some of the days, but, by rounding up, we will never have a day without enough staff. Rounding the continuous solution up gives an objective of 10+2+2+6+5=25 employees. Now, let’s apply integer programming to the revised staffing model. First, we will need to use the @GIN function to make the START variables general integers. We could do this by adding the following to our model: @GIN( @GIN( @GIN( @GIN( @GIN( @GIN( @GIN(

START( START( START( START( START( START( START(

MON)); TUE)); WED)); THU)); FRI)); SAT)); SUN));

However, an easier approach would be to embed the @GIN function in an @FOR function, so we can apply @GIN to each member of START using the single statement: @FOR( DAYS( I): @GIN( START( I)));

This new statement says, for each day of the week, make the variable corresponding to the number of people to start on that day a general integer variable. After inserting this @FOR statement at the end of our model and reoptimizing, we get the pure integer solution: Global optimal solution found. Objective value: Extended solver steps: Total solver iterations: Variable START( MON) START( TUE) START( WED) START( THU) START( FRI) START( SAT)

START( SUN)

Value 10.00000 2.000000 1.000000 6.000000 0.000000 5.000000

0.000000

24.00000 0 6 Reduced Cost 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

1.000000

Note that the objective of 24 beats the objective of 25 obtained by rounding. Thus, had we gone with the rounded solution, we would have hired one more employee than required.

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Binary Integer Variables A binary integer variable—also called a 0/1 variable—is a special case of an integer variable that is required to be either zero or one. It’s often used as a switch to model Yes/No decisions. The syntax of the @BIN function is: @BIN( variable_name); where variable_name is the name of the variable you wish to make binary. The @BIN function may be used in a model anywhere you would normally enter a constraint. The @BIN function can be embedded in an @FOR statement to allow you to easily set all, or selected, variables of an attribute to be binary integers. Some examples of @BIN are: Example 1: @BIN( X); makes the scalar variable, X, a binary integer, Example 2: @BIN( INCLUDE( 4)); makes the variable INCLUDE( 4) binary, Example 3: @FOR( ITEMS: @BIN( INCLUDE)); makes all variables in the INCLUDE attribute binary.

Binary Integer Example - The Knapsack Problem The knapsack model is a classic problem that uses binary variables. In this problem, you have a group of items you want to pack into your knapsack. Unfortunately, the capacity of the knapsack is limited such that it is impossible to include all items. Each item has a certain value, or utility, associated with including it in the knapsack. The problem is to find the subset of items to include in the knapsack that maximizes the total value of the load without exceeding the capacity of the knapsack. Of course, the knapsack euphemism shouldn’t lead one to underestimate the importance of this class of problem. The “knapsack” problem can be applied to many situations. Some examples are vehicle loading, capital budgeting, and strategic planning.


89

The Problem As an example, suppose you are planning a picnic. You’ve constructed a list of items you would like to carry with you on the picnic. Each item has a weight associated with it and your knapsack is limited to carrying no more than 15 pounds. You have also come up with a 1 to 10 rating for each item, which indicates how strongly you want to include the particular item in the knapsack for the picnic. This information is listed below: Item Ant Repellent Beer Blanket Bratwurst Brownies Frisbee Salad Watermelon

Weight 1 3 4 3 3 1 5 10

Rating 2 9 3 8 10 6 4 10

The Formulation We have only one set in this model—the set of items we are considering carrying in the knapsack. This is a primitive set, and we can define it in the sets section: SETS: ITEMS: INCLUDE, WEIGHT, RATING; ENDSETS

We have associated the three attributes INCLUDE, WEIGHT, and RATING with the set. INCLUDE will be the binary variables used to indicate if an item is to be included in the knapsack. WEIGHT is used to store the weight of each item, and RATING is used to store each item's rating. Next, we will need to construct a data section to input the set of set ITEMS and their associated weights and ratings. Here is a data section that accomplishes the task: DATA: ITEMS ANT_REPEL BEER BLANKET BRATWURST BROWNIES FRISBEE SALAD WATERMELON

WEIGHT RATING = 1 2 3 9 4 3 3 8 3 10 1 6 5 4 10 10;

KNAPSACK_CAPACITY = 15;

ENDDATA

Note that we have also included the knapsack’s capacity in the data section. This is a good practice in that it isolates data from the constraints of the model.

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Given that all the sets and data have been defined, we can turn to building our objective function. We want to maximize the sum of the ratings of the items included in our knapsack. Note that INCLUDE(I) will be 1 if item I is included. Otherwise, it will be 0. Therefore, if we take the inner product of INCLUDE with the RATING attribute, we will get the overall rating of a combination of included items. Putting this into LINGO syntax, we have: MAX = @SUM( ITEMS: RATING * INCLUDE);

Note that we did not specify a set index variable in the @SUM function. Since all the attributes in the function (RATING and INCLUDE) are defined on the index set (ITEMS), we can drop the set index variable and use implicit indexing. Our next step is to input our constraints. There is only one constraint in this model. Specifically, we must not exceed the capacity of the knapsack. In a similar manner as the objective, we compute the weight of a given combination of items by taking the inner product of the INCLUDE attribute with the WEIGHT attribute. This sum must be less-than-or-equal-to the capacity of the knapsack. In LINGO syntax, we express this as: @SUM(ITEMS: WEIGHT * INCLUDE) <= KNAPSACK_CAPACITY;

Finally, we must make the INCLUDE variable binary. We could do this by adding: @BIN( @BIN( @BIN( @BIN( @BIN( @BIN( @BIN( @BIN(

INCLUDE( INCLUDE( INCLUDE( INCLUDE( INCLUDE( INCLUDE( INCLUDE( INCLUDE(

@INDEX( @INDEX( @INDEX( @INDEX( @INDEX( @INDEX( @INDEX( @INDEX(

ANT_REPEL))); BEER))); BLANKET))); BRATWURST))); BROWNIES))); FRISBEE))); SALAD))); WATERMELON)));

(Note that the @INDEX function simply returns the index of a primitive set member in its set. ) However, a more efficient and data independent way of doing this would be to embed an @BIN function in an @FOR function as follows: @FOR( ITEMS: @BIN( INCLUDE));


91

The Solution The entire model for our knapsack example and excerpts from its solution are listed below. The model formulation file may be found in your SAMPLES subdirectory off the main LINGO directory under the name KNAPSACK: MODEL: SETS: ITEMS: INCLUDE, WEIGHT, RATING; ENDSETS DATA: ITEMS ANT_REPEL BEER BLANKET BRATWURST BROWNIES FRISBEE SALAD WATERMELON

WEIGHT RATING = 1 2 3 9 4 3 3 8 3 10 1 6 5 4 10 10;

KNAPSACK_CAPACITY = 15; ENDDATA MAX = @SUM( ITEMS: RATING * INCLUDE); @SUM( ITEMS: WEIGHT * INCLUDE) <= KNAPSACK_CAPACITY; @FOR( ITEMS: @BIN( INCLUDE));

END

Model: KNAPSACK

92

CHAPTER 3 Global optimal solution found. Objective value: Objective bound: Infeasibilities: Extended solver steps: Total solver iterations: Variable INCLUDE( ANT_REPEL) INCLUDE( BEER) INCLUDE( BLANKET) INCLUDE( BRATWURST) INCLUDE( BROWNIES) INCLUDE( FRISBEE) INCLUDE( SALAD) INCLUDE( WATERMELON)

38.00000 38.00000 0.000000 0 0

Value 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 0.000000 0.000000

Reduced Cost -2.000000 -9.000000 -3.000000 -8.000000 -10.00000 -6.000000 -4.000000 -10.00000

Solution to KNAPSACK Your knapsack is fully packed at 15 pounds, and we take along everything, but the salad and watermelon. Your lunch of beer, bratwurst and brownies may not be very healthy, but at least you will be happy!

An Extension - Modeling a Logical Or Condition Binary variables are very useful for modeling logical conditions. For instance, suppose your physician reviews your picnic plans and, fearing for your health, insists you must take either the salad or the watermelon along on your picnic. You could add this condition to your model by simply appending the constraint: INCLUDE( @INDEX( SALAD)) + INCLUDE( @INDEX( WATERMELON)) >= 1;

In order to satisfy this constraint, either the salad, the watermelon, or both must be included in the knapsack. Unfortunately, constraints of this form are not good practice in that they are not data independent. Suppose your list of picnic items changes. You may need to modify this new constraint to reflect those changes. A well formulated model should require no changes to the constraints as a result of changes to the data. The following model demonstrates a data independent way of incorporating your physician’s request (additions to the original model are listed in bold):


93

MODEL: SETS: ITEMS: INCLUDE, WEIGHT, RATING; MUST_EAT_ONE( ITEMS); ENDSETS DATA: ITEMS ANT_REPEL BEER BLANKET BRATWURST BROWNIES FRISBEE SALAD WATERMELON

WEIGHT RATING = 1 2 3 9 4 3 3 8 3 10 1 6 5 4 10 10;

MUST_EAT_ONE = SALAD WATERMELON; KNAPSACK_CAPACITY = 15; ENDDATA MAX = @SUM( ITEMS: RATING * INCLUDE); @SUM( ITEMS: WEIGHT * INCLUDE) <= KNAPSACK_CAPACITY; @FOR( ITEMS: @BIN( INCLUDE)); @SUM( MUST_EAT_ONE( I): INCLUDE( I)) >= 1;

END

We have derived a set called MUST_EAT_ONE from the original picnic items, and used an explicit list to include the items we must carry as . Then, at the end of the model, we added a constraint that forces at least one of the “must eat” items into the solution.

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For those interested, the solution to the modified model is: Global optimal solution found. Objective value: Objective bound: Infeasibilities: Extended solver steps: Total solver iterations: Variable INCLUDE( ANT_REPEL) INCLUDE( BEER) INCLUDE( BLANKET) INCLUDE( BRATWURST) INCLUDE( BROWNIES) INCLUDE( FRISBEE) INCLUDE( SALAD) INCLUDE( WATERMELON)

37.00000 37.00000 0.000000 0 0

Value 0.000000 1.000000 0.000000 1.000000 1.000000 1.000000 1.000000 0.000000

Reduced Cost -2.000000 -9.000000 -3.000000 -8.000000 -10.00000 -6.000000 -4.000000 -10.00000

In short, we drop the ant repellent and blanket, and replace them with the salad.

Binary Integer Example – Product-Mix with Fixed Costs In many situations, it is not unusual for a particular activity to incur a fixed cost. Examples where one might incur a fixed cost include opening a plant, producing a product, paying a commission on an order to buy stock, or retooling an assembly line. In this next example, we will put together a product-mix model much like the CompuQuick example from Chapter 1, Getting Started with LINGO. In this case, however, there is a fixed setup charge associated with the production of an item. In other words, whenever we produce any amount of a product, we incur a fixed charge independent of the output level of the product.

The Problem You’re the manager of an airplane plant and you want to determine the best product-mix of your six models to produce. The six models currently under production are the Rocket, Meteor, Streak, Comet, Jet, and Biplane. Each plane has a known profit contribution. There is also a fixed cost associated with the production of any plane in a period. The profit and fixed costs are given in the following table: Plane Rocket Meteor Streak Comet Jet Biplane

Profit 30 45 24 26 24 30

Setup 35 20 60 70 75 30


95

Each plane is produced using six raw materials—steel, copper, plastic, rubber, glass, and paint. The units of these raw materials required by the planes as well as the total availability of the raw materials are: Steel Copper Plastic Rubber Glass Paint

Rocket 1 4 0 2 2 1

Meteor 4 5 3 0 4 4

Streak 0 3 8 1 2 1

Comet 4 0 0 2 2 4

Jet 2 1 1 1 2 3

Biplane 1 0 0 5 4 4

Available 800 1160 1780 1050 1360 1240

The problem is to determine the final mix of products that maximizes net profit (gross profit - setup costs) without exceeding the availability of any raw material. Your brand new Meteor model has the highest profit per unit of anything you’ve ever manufactured and the lowest setup cost. Maybe you should build nothing but Meteors? Then again, maybe not.

The Formulation As you might guess, we will need two primitive sets in this model—one to represent the airplane models and one to represent the raw materials. We can construct these sets as follows: PLANES: PROFIT, SETUP, QUANTITY, BUILD; RESOURCES: AVAILABLE;

We added the following four attributes to the PLANES set:    

PROFIT SETUP QUANTITY BUILD

stores profit contribution for the plane, stores setup cost to begin producing the plane, a variable for quantity of planes to produce, and a binary variable, 1 if we produce the plane, else 0.

The AVAILABLE attribute on the RESOURCES set will be used to store the availability of each resource. We will also need to derive a dense set by taking the cross of the RESOURCES set with the PLANES set. We need this set in order to define a USAGE attribute to store the resource usage of each plane. We will call this derived set RXP, which, after inclusion into the sets section, gives us: SETS: PLANES: PROFIT, SETUP, QUANTITY, BUILD; RESOURCES: AVAILABLE; RXP( RESOURCES, PLANES): USAGE;

ENDSETS

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In our data section, we will initialize the set : PLANES and RESOURCES, along with the data attributes: PROFIT, SETUP, AVAILABLE, and USAGE. Here is the data section we will use: DATA: PLANES PROFIT SETUP = ROCKET 30 35 METEOR 45 20 STREAK 24 60 COMET 26 70 JET 24 75 BIPLANE 30 30; RESOURCES AVAILABLE = STEEL,800 COPPER,1160 PLASTIC,1780 RUBBER,1050 GLASS,1360 PAINT,1240; USAGE = 1 4 0 4 2 0 4 5 3 0 1 0 0 3 8 0 1 0 2 0 1 2 1 5 2 4 2 2 2 4 1 4 1 4 3 4;

ENDDATA

With the sets and data sections complete, we can now turn our attention to the objective function. For our objective, we want to maximize total net profit. Specifically, this is computed as the sum of profit times quantity produced of each plane, minus its setup cost multiplied by the BUILD binary variable. In LINGO syntax, we express the objective as: MAX = @SUM( PLANES: PROFIT * QUANTITY - SETUP * BUILD);

Since all attributes are defined on the index set, we can drop the set index variable and use implicit indexing. For our first set of constraints, we want to be sure raw material supplies are not exceeded. In words, what we want is: For each resource i, the sum over each plane j of the quantity of plane j built multiplied by the resource usage of resource i by plane j must be less-than-or-equal-to the availability of resource i.


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Given the vagaries of the English language, it's highly likely one would find the equivalent LINGO notation more concise and easier to understand: @FOR( RESOURCES( I): @SUM( PLANES( J): USAGE( I, J) * QUANTITY( J)) <= AVAILABLE( I) );

Our next set of constraints is not quite as intuitive. We are using the binary variable BUILD to represent if a plane is being built, so we can incorporate a fixed cost for the plane in the objective function. What we need is some constraint mechanism to force BUILD( I) to be 1 when we produce a nonzero quantity of plane I. The following constraint will do just that: @FOR( PLANES: QUANTITY <= 400 * BUILD; @BIN( BUILD) );

Given that BUILD is 0/1, as soon as QUANTITY goes nonzero the only feasible solution is for BUILD to go to 1. Constraints of this form used to force a binary variable to an appropriate value are sometimes referred to as forcing constraints. The coefficient of 400 in our forcing constraints was chosen because we know from scanning our data that no more than 400 of any plane can be built. Can you this? Coefficients used in this manner are sometimes called BigM coefficients. For solver efficiency reasons, it's best to try to keep BigM values as small as reasonably possible. Because the BigM coefficient of 400 is dependent upon the model's data, it is actually bad modeling practice to embed the coefficient in the model's constraints as we have done here. As we have discussed, it is best to try to keep the constraints of your model independent of the data to facilitate model maintenance. A more data independent formulation would actually involve calculations to come up with a good BigM value. Can you think of how you might add such a feature to this model? A reasonable question at this point would be: "We have the machinery to force BUILD to 1 when we build a plane. What forces BUILD to zero when we don't build a plane?" The fact that BUILD appears in the objective with a negative coefficient (we multiply it by SETUP and then subtract it from the objective) guarantees this. If a plane was not being built and the corresponding BUILD variable was 1, we could get a better solution by simply setting BUILD to 0. Since the goal is to maximize the objective, BUILD will always be driven to 0 when a plane is not built. One final feature of our forcing constraints to note is that we have piggybacked the @BIN function call onto the @FOR statement for the forcing constraints. As you recall from the discussion of set looping functions in Using Sets, an @FOR function may contain multiple expressions as long as they are separated by a semicolon. We have capitalized on this feature by including the @BIN expression as well. As a final feature, we can make the QUANTITY variables general integers with the expression: @FOR( PLANES: @GIN( QUANTITY));

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The Solution The formulation in its entirety and a selected portion of the solution appear below. The formulation file may be found in file PRODMIX. MODEL: SETS: PLANES: PROFIT, SETUP, QUANTITY, BUILD; RESOURCES: AVAILABLE; RXP( RESOURCES, PLANES): USAGE; ENDSETS DATA: PLANES PROFIT SETUP = ROCKET 30 35 METEOR 45 20 STREAK 24 60 COMET 26 70 JET 24 75 BIPLANE 30 30; RESOURCES AVAILABLE = STEEL,800 COPPER,1160 PLASTIC,1780 RUBBER,1050 GLASS,1360 PAINT,1240; USAGE = 1 4 0 4 2 0 4 5 3 0 1 0 0 3 8 0 1 0 2 0 1 2 1 5 2 4 2 2 2 4 1 4 1 4 3 4; ENDDATA MAX = @SUM( PLANES: PROFIT * QUANTITY - SETUP * BUILD); @FOR( RESOURCES( I): @SUM( PLANES( J): USAGE( I, J) * QUANTITY( J)) <= AVAILABLE( I); ); @FOR( PLANES: QUANTITY <= 400 * BUILD; @BIN( BUILD); ); @FOR( PLANES: @GIN( QUANTITY)); END

Model: PRODMIX

VARIABLE DOMAIN FUNCTIONS Global optimal solution found. Objective value: Objective bound: Infeasibilities: Extended solver steps: Total solver iterations: Variable QUANTITY( ROCKET) QUANTITY( METEOR) QUANTITY( STREAK) QUANTITY( COMET) QUANTITY( JET) QUANTITY( BIPLANE) BUILD( ROCKET) BUILD( METEOR) BUILD( STREAK) BUILD( COMET) BUILD( JET) BUILD( BIPLANE)

Value 96.00000 0.000000 195.0000 0.000000 191.0000 94.00000 1.000000 0.000000 1.000000 0.000000 1.000000 1.000000

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14764.00 14764.00 0.000000 7 296 Reduced Cost -30.00000 -45.00000 -24.00000 -26.00000 -24.00000 -30.00000 35.00000 20.00000 60.00000 70.00000 75.00000 30.00000

Solution to PRODMIX Surprisingly, we see from the solution that we build none of the "profitable" Meteors. Can you determine why this is so? On the other hand, the Rocket, Streak, Jet and Biplane are produced, and, as we anticipated, the BUILD variable for each of these planes has been correctly set to 1.

Dual Values and IP In Chapter 1, Getting Started with LINGO, we introduced the concept of dual values. The dual values of a solution are the reduced costs of the variables and dual prices on the constraints. We also discussed the useful information that can be obtained from dual values. Unfortunately, in IP models the interpretation of the dual values breaks down. Due to the discreet nature of IP models, the dual values in the solution to an IP model are of no practical use to the average . Given this, the dual values should be ignored when your model contains integer variables created through the use of @BIN or @GIN.

Variable Priorities The time required for branch-and-bound algorithm to converge is highly dependent on the order in which the solver branches on the integer variables. Some integer variables impact the solution more than others. An example would be a plant location model, where we are deciding a) which plants to open, and b) which customers to assign to which plants. Clearly, the decision of opening a plant influences the solution considerably more than the decision of which plant to assign a given customer to. In general, performance will improve if the solver branches on the more critical integer variables first, while branching on the less critical variables later. LINGO provides the @PRIORITY function for controlling the branching priority of the variables. The syntax of the @PRIORITY function is: @PRIORITY( variable_name, relative_priority);

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where variable_name is the name of the variable and relative_priority is a non-negative integer representing the relative priority of the variable. The @PRIORITY function may be used in a model anywhere you would normally enter a constraint. The @PRIORITY function can be embedded in an @FOR statement to allow you to easily set the priority of all, or selected, variables of an attribute. If a variable is not assigned a priority, it is assumed to have the lowest priority level of 0. Some examples of @PRIORITY are: Example 1: @PRIORITY( X, 100); assigns variable X a priority of 100, Example 2: @PRIORITY( PRODUCE( 5), 10); assigns variable PRODUCE( 5) a priority of 10, Example 3: @FOR( DAYS( I): @PRIORITY( START( I), 30)); assigns all the variables of the START a priority of 30.

Summary You should now be familiar with the use of the variable domain functions @BIN and @GIN, and how they are used to introduce integer variables into a model. This section has shown how integer variables bring a whole new dimension of power to the mathematical modeler. Given that we have only briefly delved into the topic of modeling with integer variables, the that would like to become more familiar with the many practical applications of integer programming and their formulations can refer to Schrage (2006), or Winston (1995).

Free Variables By default, a LINGO variable has a lower bound of zero and an upper bound of infinity. @FREE removes the lower bound of zero and lets a variable take negative values, rendering it unconstrained in sign, or free. The syntax is: @FREE( variable_name); where variable_name is the name of the variable you wish to make free. The @FREE function may be used in a model anywhere you would normally enter a constraint. The @FREE function can be embedded in an @FOR statement to allow you to easily make all, or selected, variables of an attribute to be free. Some examples of @FREE are: Example 1: @FREE( X); makes the scalar variable, X, free, Example 2: @FREE( QUANTITY( 4)); makes the variable QUANTITY( 4) free, Example 3: @FOR( ITEMS: @FREE( QUANTITY)); makes all variables in the QUANTITY attribute free.


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FREE Variable Example - Forecasting You are the inventory controller for the successful new Shack4Shades retail chain. Your business specializes exclusively in the retailing of sunglasses to the lover of the outdoors. You need to come up with a model to forecast sales of sunglasses in the coming quarter in order to build up inventory levels. You have created the following chart of your sales for the last eight quarters:

Looking at this chart, you theorize that sales are growing according to a linear trend line, but with rather sizable seasonal variations. Sales pick up in the summer months when people head to the beaches and again in winter when they head for the ski slopes. Given this, you have come up with the following theoretical function to forecast sales as a function of time: Predicted_Sales(t) = Seasonal_Factor(t) * (Base + Trend * t) where, Predicted_Sales(t) Seasonal_Factor(t) Base Trend

represents predicted sales for quarter t, is one of four multipliers (one for each quarter of the year) to for seasonal variations, is the y-intercept of the hypothesized linear function, and is the slope of the linear function.

You would like to come up with a LINGO model to estimate the six parameters of your function (i.e., the four seasonal factors, the trend line base, and the trend line slope). To do this, you will let LINGO choose values for the parameters that minimize the sum of the squared differences between predicted and observed sales for the historical data.

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The Formulation We will need two primitive sets in our model. The first set will have eight to represent the quarters that we have historical data for. The second set will have four corresponding to the four quarters of the year. This second set is used for defining the four seasonal factors. Here is our sets section that incorporates these two sets: SETS: PERIODS: OBSERVED, PREDICT, ERROR; QUARTERS: SEASFAC; ENDSETS

The three attributes on the PERIODS set—OBSERVED, PREDICT, and ERROR—correspond to the observed sales values, predicted sales values, and the prediction error. The prediction error is simply predicted sales minus observed sales. The SEASFAC attribute on the SEASONS set corresponds to the seasonal sales factors and will be computed by LINGO. We will also need to add a data section to initialize the set and the OBSERVED attribute with the historical sales data. We can do this with the following: DATA: PERIODS = P1..P8; QUARTERS = Q1..Q4; OBSERVED = 10 14 12 19 14 21 19 26; ENDDATA

Next, we must add a formula to compute the error . As mentioned, the error term in a period is the difference between the observed and predicted sales. We can express this in LINGO as: @FOR( PERIODS: ERROR = PREDICT - OBSERVED);

Our objective is to minimize the sum of the squared error , which may be written as: MIN = @SUM( PERIODS: ERROR ^ 2);

We choose to use squared error as a measure to minimize because we want to weight large errors relatively more heavily. Another option might be to minimize the sum of the absolute values of the errors, which would weight large and small errors proportionally the same. In order to compute the error , we will also need to compute predicted sales. Using our theoretical formula, we compute predicted sales as follows: @FOR( PERIODS( P): PREDICT( P) = SEASFAC( @WRAP( P, 4)) * ( BASE + P * TREND));

The @WRAP function is used here to allow us to apply the four seasonal factors over a time horizon exceeding four periods. Had we simply used the index P, instead of @WRAP(P, 4), we would have generated a subscript out of range error. For a more in depth explanation of the use of the @WRAP function, please see the staff-scheduling example on page 60. For esthetic reasons, we would like the seasonal factors to average out to a value of one. We can do this by adding the constraint: @SUM( QUARTERS: SEASFAC) = 4;


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Finally, it is possible for the error to be negative as well as positive. Given that variables in LINGO default to a lower bound of zero, we will need to use the @FREE function to allow the error to go negative. By embedding the @FREE function in an @FOR loop, we can apply @FREE to all the ERROR variables in the statement: @FOR( PERIODS: @FREE( ERROR));

The Solution The entire formulation and excerpts from the solution appear below. MODEL: SETS: PERIODS: OBSERVED, PREDICT, ERROR; QUARTERS: SEASFAC; ENDSETS DATA: PERIODS = P1..P8; QUARTERS = Q1..Q4; OBSERVED = 10 14 12 19 14 21 19 26; ENDDATA MIN = @SUM( PERIODS: ERROR ^ 2); @FOR( PERIODS: ERROR = PREDICT - OBSERVED); @FOR( PERIODS( P): PREDICT( P) = SEASFAC( @WRAP( P, 4)) * ( BASE + P * TREND)); @SUM( QUARTERS: SEASFAC) = 4; @FOR( PERIODS: @FREE( ERROR); @BND( -1000, ERROR, 1000)); END

Model: SHADES

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Objective value: Total solver iterations: Variable BASE TREND OBSERVED( P1) OBSERVED( P2) OBSERVED( P3) OBSERVED( P4) OBSERVED( P5) OBSERVED( P6) OBSERVED( P7) OBSERVED( P8) PREDICT( P1) PREDICT( P2) PREDICT( P3) PREDICT( P4) PREDICT( P5) PREDICT( P6) PREDICT( P7) PREDICT( P8) ERROR( P1) ERROR( P2) ERROR( P3) ERROR( P4) ERROR( P5) ERROR( P6) ERROR( P7) ERROR( P8) SEASFAC( Q1) SEASFAC( Q2) SEASFAC( Q3) SEASFAC( Q4)

1.822561 32

Value 9.718878 1.553017 10.00000 14.00000 12.00000 19.00000 14.00000 21.00000 19.00000 26.00000 9.311820 14.10136 12.85213 18.80620 14.44367 20.93171 18.40496 26.13943 -0.6881796 0.1013638 0.8521268 -0.1938024 0.4436688 -0.6828722E-01 -0.5950374 0.1394325 0.8261096 1.099529 0.8938789 1.180482

Solution to SHADES The solution is: TREND, 1.55; BASE, 9.72. The four seasonal factors are .826, 1.01, .894, and 1.18. The spring quarter seasonal factor is .826. In other words, spring sales are 82.6% of the average. The trend of 1.55 means, after the effects of season are taken into , sales are increasing at an average rate of 1,550 sunglasses per quarter. As one would expect, a good portion of the error are negative, so it was crucial to use the @FREE function to remove the default lower bound of zero on ERROR.


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Our computed function offers a very good fit to the historical data as the following graph illustrates:

Using this function, we can compute the forecast for sales for the quarter (quarter 1). Doing so gives: Predicted_Sales(9)

= Seasonal_Factor(1) * (Base + Trend * 9) = 0.826 * (9.72 + 1.55 * 9) = 19.55

Given this, inventory levels should be brought to a level sufficient to an anticipated sales level of around 19,550 pairs of sunglasses.

Bounded Variables Whereas @FREE sets the upper and lower bounds of the specified variable to plus and minus infinity (effectively removing any bounds on the variable), the @BND function lets you set specific upper and lower bounds on a variable. In other words, @BND limits a variable’s range within some specified interval. The syntax for @BND is: @BND(lower_bound, variable_name, upper_bound); where variable_name is the variable to be bounded below by the quantity lower_bound and bounded above by the quantity upper_bound. Both lower_bound and upper_bound must be either numeric values or variables whose values have been set in a data section or calc section. @BND may be used wherever you would normally enter a constraint in a model—including inside an @FOR looping function. In mathematical , LINGO interprets this @BND function as: lower_bound  variable_name  upper_bound

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It is certainly possible to add constraints in lieu of the @BND function, but, from the standpoint of the optimizer, @BND is an extremely efficient way of representing simple bounds on variables. Specifying variable bounds using @BND rather than explicitly adding constraints can noticeably speed up the solution times for larger models. Furthermore, @BND does not count against the limit on the total number of constraints LINGO imposes on some versions. So, in general, it is a good idea to use @BND in place of constraints whenever possible. Some examples of @BND are: Example 1: @BND( -1, X, 1); constrains the variable X to lie in the interval [-1,1], Example 2: @BND( 100, QUANTITY( 4), 200); constrains the variable QUANTITY( 4) to fall within 100 to 200, Example 3: @FOR( ITEMS: @BND( 10, Q, 20)); sets the bounds on all variables in the Q attribute to 10 and 20, Example 4: @FOR( ITEMS: @BND( QL, Q, QU)); sets the bounds on all variables in the Q attribute to QL and QU (QL and QU must have been previously set to some values in a data section).

SOS Variables LINGO s SOS (Special Ordered Sets) variables of Type 1, 2 and 3 via the @SOS1, @SOS2 and @SOS3 functions, respectively. The properties of the three SOS types are: SOS Type

Property

SOS1

At most, only one variable belonging to an SOS1 set will be greater than 0.

SOS2

At most, only two variables in an SOS2 set can be different from 0. If two variables are nonzero, then the variables will be adjacent to one another. SOS2 sets are particularly useful for implementing piecewise-linear functions in models.

SOS3

Exactly one variable from a given SOS3 set will be equal to 1. All remaining variables will be equal to 0.

Note:

Any variables added to an SOS set will count against the integer variable limit imposed in limited versions of LINGO.

The syntax for the @SOS declarations is as follows: @SOS{1|2|3}( 'set_name', variable_reference); The set_name argument is a unique label, or name, for the particular set of SOS variables. You add additional variables to an SOS set by making subsequent calls to the @SOS function using the same set name.


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Some examples of SOS sets are: Example 1: @SOS3( 'SUM_TO_1', X); @SOS3( 'SUM_TO_1', Y); @SOS3( 'SUM_TO_1', Z);

In this example, an SOS Type 3 set forces either X, Y or Z to be equal to 1. The remaining variables will be equal to 0. Example 2: @FOR( CUST( J): @FOR( PLANTS( I): @SOS1( 'SNGSRC_' + CUST( J), SHIP( I, J))));

Here, multiple SOS Type 1 sets force each customer to receive shipments from only one plant. There is one SOS1 set created for each customer, each bearing the name SNGSRC_customer_name. An example of using Type 2 sets follows in the next section.

Piecewise Linear Example - Type SOS2 Set As we mentioned above, SOS2 sets are particularly useful for implementing piecewise-linear functions. Many cost curves exhibit the piecewise-linear property. For example, suppose we want to model the following cost function, where cost is a piecewise-linear function of volume, X: 60

50

40

30

20

10

0 0

5

10

15

20

25

Piecewise-Linear Function Example The breakpoints of the curve lie at the following points: (0,22), (5,10), (12,41) and (20,49).

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The following sample model, SOSPIECE.LG4, uses a Type 2 SOS set to model this piecewise-linear function using what is referred to as the lambda method: MODEL: ! Demonstrates the lambda method for representing arbitrary, piecewise-linear curves using an SOS2 set; ! See "Optimization Modeling with Lingo", Section 11.2.7; SETS: ! 4 breakpoints in this example; B /1..4/: W, U, V; ENDSETS DATA: ! total cost at the breakpoints; V = 22 10 41 49; ! the breakpoints; U = 0 5 12 20; ENDDATA ! set x to any value in interval--the cost variable will automatically be set to the correct total cost; X = 8.5; ! calculate total cost; COST = @SUM( B( i): V( i) * W( i)); ! force the weights (w); X = @SUM( B( I): U( I) * W( i)); !weights must sum to 1; @SUM( B( I): W( I)) = 1; ! the weights are SOS2: at most two adjacent weights can be nonzero; @FOR( B( I): @SOS2( 'SOS2_SET', W( I))); END

Model: SOSPIECE We defined an attribute, W, whose act as weights, placing us on an particular segment of the curve. For example, if W(2)=W(3)=0.5, then we are exactly halfway between the second and third breakpoints : (5,10) and (12,41), i.e., at point (8.5,25.5). In the case where we lie exactly on a breakpoint, then only one of the W(i) will be nonzero and equal to 1.


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For this strategy to work correctly, only two, at most, of the W(i) may be nonzero, and they must be adjacent. As you recall, this is the definition of an SOS2 set, which we create at the end of the model with the expression: ! the weights are SOS2: at most two adjacent weights can be nonzero; @FOR( B( I): @SOS2( 'SOS2_SET', W( I)));

In particular, each weight W(i) is a member of the Type SOS2 set titled SOS2_SET. For this particular example, we have chosen to pick an x-value and then let LINGO compute the corresponding y-value, or cost. Running, the model, as predicted, we see that for an X value of 8.5, total cost is 25.5: Variable X COST W( 1) W( 2) W( 3) W( 4)

Value 8.500000 25.50000 0.000000 0.5000000 0.5000000 0.000000

Solution to SOSPIECE In addition to allowing the solver to work more efficiently, SOS sets also help to reduce the number of variables and constraints in your model. In this particular example, had we not had the SOS2 capability, we would have needed to add an additional 0/1 attribute, Z, and the following expressions to the model: ! Here's ! Can be w( 1) <= @for( b( w( i) ); @sum( b( @for( b(

Note:

what we eliminated by using @sos2: on only one line segment at a time; z( 1); w( @size( b)) <= z( @size( b)); i) | i #gt# 1 #and# i #lt# @size( b): <= z( i) + z( i + 1) i): z( i)) = 1; i): @bin( z( i)));

It may seem that piecewise linearity could be implemented in a more straightforward manner through the use of nested @IF functions. Certainly, the @IF approach would be more natural than the lambda method presented here. However, @IF functions would add discontinuous nonlinearities to this model. This is something to try and avoid, in that such functions are notoriously difficult to solve to global optimality. In the approach used above, we have maintained linearity, which allows LINGO to use its faster, linear solvers, and converge to a globally optimal solution.

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Cardinality Related to the SOS capability discussed above, LINGO also s cardinality sets of variables via the @CARD function. The cardinality feature allows you to specify a set of variables with a cardinality of N, meaning that, at most, N of the variables in the set will be allowed to be nonzero. As with SOS sets, cardinality sets help the integer solver branch more efficiently, and they reduce the number of variables and constraints in your models. Also, as with SOS sets, each variable added to a cardinality set will count against any integer variable limits imposed on your installation of LINGO. The syntax for the @CARD declarations is as follows: @CARD( 'set_name', variable_reference|set_cardinality);

The set_name argument is a unique label, or name, for the particular cardinality set. You add additional variables to an SOS set by making subsequent calls to the @CARD function using the same set name with a different variable_reference. In addition to calling @CARD once for each variable in a set, you will need to call @CARD once for each set ing an integer value as the second argument. This integer argument is the set_cardinality, and may be either an actual integer number or a variable set to an integer value in either a data or calc section. Some examples of @CARD sets are: Example 1:

@CARD( 'PICK2', 2); @CARD( 'PICK2', X); @CARD( 'PICK2', Y); @CARD( 'PICK2', Z);

In this example, at most, two out of the three variable X, Y, and Z will be nonzero. Example 2:

@FOR( PLANT( I): @CARD( 'OPENLIM', OPEN( I))); @CARD( 'OPENLIM', NCARD);

Here, we limit the maximum number of open plants to NCARD, where NCARD must be set beforehand to an integer value in either a data or calc section.

Semicontinuous Variables Many models require certain variables to either be 0 or lie within some nonnegative range, e.g., 10 to 20. Variables with this property are said to be semicontinuous. Modeling semicontinuity in LINGO in the past meant having to add an additional 0/1 variable and two additional constraints. LINGO now allows you to establish semicontinuous variables directly with the @SEMIC statement. The syntax for the @SEMIC declarations is as follows: @SEMIC( lower_bound, variable_reference, upper_bound);

This will restrict the variable, variable_reference, to be either 0 or to lie within the range [lower_bound, upper_bound].

VARIABLE DOMAIN FUNCTIONS Note:

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Each semi-continuous variable will be counted against any integer variable limit for your installation.

Some examples of @SEMIC usage are: Example 1: @SEMIC( 10, X, 20); In this example, X will be restricted to being either 0, or to lie within the range [10,20]. Example 2: @FOR( PLANT( I): @SEMIC( MIN_HOURS, HOURS( I), MAX_HOURS)); Here, we restrict the operating hours of each plant to be either 0, or to line in the range [MIN_HOURS,MAX_HOURS]. Note that MIN_HOURS and MAX_HOURS must have been set to explicit values beforehand in either a data or calc section. Below, we have taken our familiar transportation model and modified it, via the use of @SEMIC, to restrict shipments from each warehouse to each customer to be either 0, or between 3 and 10.

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CHAPTER 3 MODEL: ! A 3 Warehouse, 4 Customer Transportation Problem that uses the semi-continuous (@SEMIC) to restrict nonzero shipments to be between 3 and 10 units.; SETS: WAREHOUSE: CAPACITY; CUSTOMER: DEMAND; ROUTES( WAREHOUSE, CUSTOMER) : COST, VOLUME; ENDSETS DATA: WAREHOUSE,CAPACITY = WH1,30 WH2,25 WH3,21; CUSTOMER,DEMAND = C1,15 C2,17 C3,22 C4,12; COST = 6 2 6 7 4 9 5 3 8 8 1 5; ENDDATA ! The objective; [R_OBJ] MIN = @SUM( ROUTES: COST * VOLUME); ! The demand constraints; @FOR( CUSTOMER( J): [R_DEM] @SUM( WAREHOUSE( I): VOLUME( I, J)) >= DEMAND( J)); ! The supply constraints; @FOR( WAREHOUSE( I): [R_SUP] @SUM( CUSTOMER( J): VOLUME( I, J)) <= CAPACITY( I)); @FOR( ROUTES: @SEMIC( 3, VOLUME, 10)); END

Model: TRANSEMIC

VARIABLE DOMAIN FUNCTIONS Solving this model yields the following optimal values for the semicontinuous attribute, VOLUME: Global optimal solution found. Objective value: Objective bound: Infeasibilities: Extended solver steps: Total solver iterations: VOLUME( VOLUME( VOLUME( VOLUME( VOLUME( VOLUME( VOLUME( VOLUME( VOLUME( VOLUME( VOLUME( VOLUME(

Variable WH1, C1) WH1, C2) WH1, C3) WH1, C4) WH2, C1) WH2, C2) WH2, C3) WH2, C4) WH3, C1) WH3, C2) WH3, C3) WH3, C4)

Value 5.000000 10.00000 6.000000 0.000000 10.00000 0.000000 6.000000 9.000000 0.000000 7.000000 10.00000 3.000000

264.0000 264.0000 0.000000 1 32 Reduced Cost 0.000000 -6.000000 0.000000 2.000000 -1.000000 2.000000 0.000000 -1.000000 2.000000 0.000000 -5.000000 0.000000

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4 Data, Init and Calc Sections Typically, when dealing with a model’s data, you need to assign set to sets and give values to some set attributes before LINGO can solve your model. For this purpose, LINGO gives the three optional sections, the data section for inputting set and data values, the init section for setting the starting values for decision variables, and the calc section for performing computations on raw input data.

The DATA Section of a Model The DATA section allows you to isolate data from the rest of your model. This is a useful practice in that it facilitates model maintenance and scaling of a model’s dimensions.

Basic Syntax The data section begins with the keyword DATA: (including the colon) and ends with the keyword ENDDATA. In the data section, you can have statements to initialize set and/or the attributes of the sets you instantiated in a previous sets section. These expressions have the syntax: object_list = value_list; The object_list contains the names of the attributes and/or a set whose you want to initialize, optionally separated by commas. There can be no more than one set name in object_list, while there may be any number of attributes. If there is more than one attribute name in object_list, then the attributes must be defined on the same set. If there is a set name in object_list, then all attributes in object_list must be defined on this set. The value_list contains the values you want to assign to the of object_list, optionally separated by commas. As an example, consider the following model: SETS: SET1 /A, B, C/: X, Y; ENDSETS DATA: X = 1, 2, 3; Y = 4, 5, 6; ENDDATA

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We have two attributes, X and Y, defined on the SET1 set. The three values of X are set to 1, 2, and 3, while Y is set to 4, 5, and 6. We could have also used the following compound data statement to the same end: SETS: SET1 /A, B, C/: X, Y; ENDSETS DATA: X, Y = 1, 4, 2, 5, 3, 6; ENDDATA

Looking at this example, you might imagine X would be assigned the values 1, 4, and 2 because they are first in the values list, rather than the true values of 1, 2, and 3. When LINGO reads a data statement's value list, it assigns the first n values to the first position of each of the n attributes in the attribute list, the second n values to the second position of each of the n attributes, and so on. In other words, LINGO is expecting the input data in column form rather than row form. As mentioned, we can also initialize the set in the data section. Modifying our sample model to use this approach by moving the set from the sets section to the data section, we get: SETS: SET1: X, Y; ENDSETS DATA: SET1, X, Y = A 1 4 B 2 5 C 3 6; ENDDATA

This final method is, perhaps, the most elegant in that all model data—attribute values and set —are isolated within the data section.

Parameters You are not limited to putting attributes and sets on the left-hand side of data statements. You may also initialize scalar variables in the data section. When a scalar variable's value is fixed in a data section, we refer to it as a parameter. As an example, suppose your model uses an interest rate of 8.5% as a parameter. You could input the interest rate as a parameter in the data section as follows: DATA:

INTEREST_RATE = .085; ENDDATA

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As with set attributes, you can initialize multiple parameters in a single statement. Suppose you also add the inflation rate to your model. You could initialize both the interest rate and inflation rate in the same data statement as follows: DATA:

INTEREST_RATE, INFLATION_RATE = .085, .03; ENDDATA

What If Analysis In some cases, you may not be sure what values to input for the data in your model. For example, suppose your model uses the inflation rate as a parameter. You may be uncertain as to the exact rate of inflation in the future, but you know it will most likely fall within a range of 2 to 6 percent. What you would like to do is run your model for various values of the inflation rate within this range to see how sensitive the model's results are to inflation. We refer to this as what if analysis, and LINGO has a feature to facilitate this. To set up a parameter for what if analysis, input a question mark (?) as its value in place of a number as in the following example: DATA: INFLATION_RATE = ?; ENDDATA

LINGO will prompt you for a value for the INFLATION_RATE parameter each time you solve the model. Under Windows, you will receive a dialog box resembling:

Simply input the desired value for the inflation rate and then press the OK button. LINGO will then set INFLATION_RATE to the value you input and proceed with solving the model. On platforms other than Windows, LINGO will write a prompt to your screen asking you to input a value for INFLATION_RATE. Type in the value and then press the Enter key. In addition to parameters, you can perform what if analysis on individual of attributes by initializing them to question marks in the data section, as well. For an example of a model that uses what if analysis to compute the value of a home mortgage, see the Home Mortgage Calculation model in Appendix A, Additional Examples of LINGO Modeling.

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Initializing an Attribute to a Single Value Suppose you want to initialize all the elements of an attribute to a single value. You can do this by entering a single value on the right-hand side of the data statement. LINGO will initialize all the elements of the attribute to this value. To perform what if analysis on the attribute, initialize it to a single question mark and LINGO will prompt you for the values of all the of the attribute each time the model is solved. As an example, consider the following excerpt from a model: SETS: DAYS / MO, TU, WE, TH, FR, SA, SU/: NEEDS; ENDSETS DATA: NEEDS = 20; ENDDATA

LINGO will initialize all the of the NEEDS attribute to the value 20. If there are multiple attributes on the left-hand side of the data statement, you will need one value on the right-hand side for each attribute on the left. For instance, let's extend the previous example, so we have an additional attribute called COST: SETS: DAYS / MO, TU, WE, TH, FR, SA, SU/: NEEDS, COST; ENDSETS DATA: NEEDS, COST = 20, 100; ENDDATA

All seven of NEEDS will be initialized to 20 and all seven of COST to 100.

Omitting Values in a Data Section You can omit values in a data statement to indicate that you don’t want to fix the values of particular . For instance, suppose you have a manufacturing company and you need to do some capacity planning for the next 5 years. Furthermore, suppose it takes some time to boost capacity. As such, it would be impossible to increase capacity over the next two years. In such a case, you might do something like the following: SETS: YEARS /1..5/: CAPACITY; ENDSETS DATA: CAPACITY = 34, 34, , , ; ENDDATA


119

We have set CAPACITY for the first two years to 34, but have omitted values for the last three years. LINGO will assume, therefore, that it is free to determine the values for CAPACITY in the last three years. Note:

You must use commas when omitting values. If you do not use the commas, LINGO will think you did not enter the correct number of values for the attribute, which will trigger an error message.

The INIT Section of a Model The INIT section is another optional section offered by LINGO. In the init section, you enter initialization statements that look much like the data statements found in the data section. The values you input in the init section are used as starting points by LINGO’s solver. Unlike the variables that are initialized in the data section, the solver is free to alter the values of variables initialized in the init section. Note:

Starting points specified in an INIT section are only of use in nonlinear or integer models. Starting points currently offer no help in purely linear models. If you are not sure whether your model is linear or nonlinear, you can check the count of nonlinear constraints in the solver status window. If there are any nonlinear constraints, then your model is nonlinear. For more information on the nature of nonlinear models and how good starting points can be of assistance, please see Chapter 15, On Mathematical Modeling.

Note:

As an example, in a set defining a group of stocks, you may have a known price of each stock, but the amount to buy or sell of each stock is unknown. You would typically initialize the price attribute in the data section. If approximate values of the buy and sell attributes are known, you can tell LINGO this information by entering it in the init section. LINGO then uses the values specified as a starting point in its search for the optimal solution. If your starting point is relatively close to an optimal solution, you may save on the solution time required to run your model.

An init section begins with the keyword INIT: and ends with the keyword ENDINIT. The syntax rules for init statements in the init section are identical to the rules for data section statements. You can have multiple attributes on the left-hand side of a statement, you can initialize an entire attribute to a single value, you can omit values in an attribute, and you can use the question mark to have LINGO prompt you for an initialization value whenever you solve the model. As an example of how a good starting point may help to reduce solution times, consider the small model: Y <= @LOG(X);

X^2 + Y^2 <=1;

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The function @LOG(X) returns the natural logarithm of X. This model has only one feasible point of (X,Y) = (1,0). If we solve this model without an init section, we get the solution: Feasible solution found at step:

Variable Y X

12

Value 0.5721349E-03 1.000419

Note that it required 12 iterations to solve. Now, let’s add an init section to initialize X and Y to a point close to the solution, so we have: INIT:

X = .999; Y = .002; ENDINIT Y <= @LOG(X); X^2 + Y^2 <=1;

Solving this modified model, we get the solution: Feasible solution found at step:

Variable X Y

3

Value 0.9999995 0.0000000

Note that our solution required only 3 iterations compared to the 12 iterations required without the init section.

The CALC Section of a Model In many instances, your model’s raw input data will need additional massaging to get it into the proper form. As an example, suppose your raw data consists of daily observations of a number of securities’ closing prices. Furthermore, let’s suppose that your model ultimately requires the covariance matrix for the securities to be computed from the raw closing price data. You could certainly compute the covariance matrix as part of the constraint section in your model. However, entering simple computations as constraints will make your model artificially large. Another option, although inconvenient, would be to compute the covariance matrix outside of LINGO and it to LINGO as external data. Actually, what you would really like is a section in LINGO to perform data manipulation in such a way that it doesn’t increase the size of the final optimization model ed through to the solver engine. This is the function of the calc section. A CALC section begins with the keyword CALC: and ends with the keyword ENDCALC. You may input any expression in a calc section that you would in the constraint section of a model. However, each expression must be in the form of an assignment statement. In an assignment statement, a single variable appears on the left-hand side of an expression, followed by an equality sign, followed by an arbitrary mathematical expression on the right-hand side. Furthermore, the right-hand side expression may only contain references to variables that are set as part of the model’s input data (i.e., set in a previous data section or calc expression.)


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As an example, here’s a model with a calc section that computes the average of three variables: MODEL: DATA: X, Y, Z = 1, 2, 3; ENDDATA CALC: AVG = ( X + Y + Z) / 3; ENDCALC END

Example of a valid calc section Now, suppose we did not know the value of Y beforehand. The following model with Y dropped from the data section would trigger an error in LINGO. The error occurs because the value of Y is an unknown, which violates the requirement that all right-hand side variables in a calc expression must have already had their values established in a previous data or calc section: MODEL: DATA: X, Z = 1, 3; ENDDATA CALC: AVG = ( X + Y + Z) / 3; ENDCALC END

Example of an invalid calc section You may perform running calculations in a calc section, which means that you may break complex calc expressions down into a series of smaller expressions. Here we break the computation from above into two steps: MODEL: DATA: X, Y, Z = 1, 2, 3; ENDDATA CALC: AVG = X + Y + Z; AVG = AVG / 3; ENDCALC END

Example of a running calc expression

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There is no limit to the number of times that a variable may appear on the left-hand side of a calc expression. However, the final calc expression for the variable will determine its value in the final solution report. Calc expressions are computed sequentially in the order in which they appear in the model. So, if one calc expression feeds its value into a subsequent expression, then it must appear before its dependent expression. For example, the following calc section is valid: CALC: X = 1; Y = X + 1; ENDCALC

while this variation is not valid: CALC: Y = X + 1; X = 1; ENDCALC

In the second example, Y depends on X, but X is not defined until after Y. Of course, Set looping functions may also be used in calc expressions. For example, consider the following portfolio optimization model. In this model, we take the annual returns for three stocks and in a calc section compute the following three pieces of information for the stocks: average return, the covariance matrix, and the correlation matrix. This information is then used in a standard Markowitz model to determine an optimal portfolio that meets a desired level of return while minimizing overall risk. MODEL: SETS: STOCKS: AVG_RET, WEIGHT; DAYS; SXD( DAYS, STOCKS): RETURN; SXS( STOCKS, STOCKS): COVR, CORR; ENDSETS DATA: DAYS TARGET STOCKS RETURN

= 1..12; = .15; = ATT GMC = 0.300 0.225 0.103 0.290 0.216 0.216 -0.046 -0.272 -0.071 0.144 0.056 0.107 0.038 0.321 0.089 0.305 0.090 0.195 0.083 0.390 0.035 -0.072

USX; 0.149 0.260 0.419 -0.078 0.169 -0.035 0.133 0.732 0.021 0.131 0.006

DATA, INIT AND CALC SECTIONS 0.176

0.715

123

0.908;

ENDDATA CALC: !Average annual return for each stock; @FOR( STOCKS( S): AVG_RET( S) = ( @SUM( SXD( D, S): RETURN( D, S)) / @SIZE( DAYS)) ); !Covariance matrix; @FOR( SXS( S1, S2): COVR( S1, S2) = @SUM( DAYS( D):( RETURN( D, S1) - AVG_RET( S1)) * ( RETURN( D, S2) - AVG_RET( S2))) / @SIZE( DAYS) ); !Although not required, compute the correlation matrix; @FOR( SXS( S1, S2): CORR( S1, S2) = COVR( S1, S2) / ( COVR( S1, S1) * COVR( S2, S2))^.5; ); ENDCALC !Minimize the risk of the portfolio (i.e., its variance); [R_OBJ] MIN = @SUM( SXS( S1, S2): WEIGHT( S1) * WEIGHT( S2) * COVR( S1, S2)); !Must be fully invested; [R_BUDGET] @SUM( STOCKS: WEIGHT) = 1; !Must exceed target return; [R_TARGET] @SUM( STOCKS: AVG_RET * WEIGHT) >= TARGET; END

Model: MARKOW

Summary You should now be comfortable with adding basic data, init and calc sections to your models. Keep in mind that initialization performed in a data section fixes a variable's value. Initialization done in an init section is used only as a temporary starting point, which may be of benefit in finding solutions to nonlinear models. Initialization in a calc section holds until another calc expression redefining a variable’s value is encountered. The benefit of placing computations in a calc section as opposed to placing them in the general constraint section is that calc expressions are treated as side computations and aren’t fed into the main solver, thereby improving execution times. We have only touched on some of the basic features of data and init sections in this chapter. In subsequent sections, you will see how to add hooks in your data and init sections to external files, spreadsheets, and databases.

5 Windows Commands In this chapter, we will discuss the pull down menu commands available in the Windows version of LINGO. The following chapter, Command-line Commands, deals with the commands available through LINGO’s command-line interface. If you’re not using a Windows version of LINGO, then you will be primarily interested in the following chapter. If you are using a Windows version of LINGO, then you will be primarily interested in this chapter. Windows s will also be interested in the command-line interface if they plan to build command scripts to automate LINGO.

Accessing Windows Commands Under Windows, commands may be accessed by either selecting them from a pull down menu, pressing the command’s button in the toolbar, or, if applicable, entering the command’s keyboard equivalent (also referred to as its accelerator key).

Menus LINGO groups commands under the following five menus:     

File Edit LINGO Window Help

The File menu contains commands that primarily deal with handling input and output. The Edit menu contains commands for editing the document in the current window. The LINGO menu contains commands to solve a model and generate solution reports. The Window menu has commands that deal with the mechanics of handling multiple windows. The Help menu provides access to LINGO’s help facility.

The Toolbar By default, the toolbar runs along the top of the screen and is illustrated in the following picture:

LINGO’s toolbar “floats”. Thus, you can reposition it by dragging it to any part of the screen. You can also choose to suppress the toolbar by clearing the Toolbar button on the Interface tab of the LINGO|Options dialog box. Each button on the toolbar corresponds to a menu command. Not all menu commands have a toolbar button, but, in general, the most frequently used commands have an equivalent button.

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126 CHAPTER 5 LINGO displays “tool tips” for each button. When you position the mouse over a button, a short description of what the button does appears in a pop up window and in the status bar at the bottom of the screen. Here is a list of the buttons and their equivalent commands: File|New

Edit|Match Parenthesis

File|Open

LINGO|Solve

File|Save

LINGO|Solution

File|Print

LINGO|Options

Edit|Undo

LINGO|Picture

Edit|Redo

Window|Send To Back

Edit|Cut

Window|Close All

Edit|Copy

Window|Tile

Edit|Paste

Help|Topics

Edit|Find

Help|Pointer

Edit|Go To Line

Accelerator Keys Along with accessing commands via the menus and toolbar, most commands may also be accessed by a single, unique keystroke known as an accelerator. The equivalent accelerator key is listed alongside each command in the menus.

Windows Commands In Brief In this section, we give a brief listing of the commands available in the Windows version of LINGO. The commands are categorized into the five main menus:     

File Edit LINGO Window Help

The next section in this chapter contains an in-depth description of the commands.

WINDOWS COMMANDS

1. File Menu Commands: New Open Save Save As Close Print Print Setup Print Preview Log Output Take Commands Export File License Database Info Exit

Opens a new model window. Opens an existing model previously saved to disk. Saves the contents of the current window to disk. Saves the contents of the current window to a new name. Closes the current window. Prints the contents of the current window. Configures your printer. Displays the contents of the current window as it would appear if printed. Opens a log file for logging output to the command window. Runs a command script contained in a file. Exports a model in MPS or MPI file format. Prompts you for a new license to upgrade your system. Prompts you for a id and for database access via the @ODBC() function. Exits LINGO.

2. Edit Menu Commands: Undo Redo Cut Copy Paste Paste Special Select All Find Find Next Replace Go To Line Match Parenthesis Paste Function Select Font Insert New Object Links Object Properties

Undoes the last change. Redoes the last undo command. Cuts the current selection from the document. Copies the current selection to the clipboard. Pastes the contents of the clipboard into the document. Pastes the contents of the clipboard into the document, allowing choice as to how the object is pasted. Selects the entire contents of the current window. Searches the document for the occurrence of a specified text string. Repeats the find operation for the last string specified. Replaces a specified text string with a new string. Moves the cursor to a specified line number. Finds the parenthesis that closes a selected parenthesis. Pastes a template of a selected LINGO @function. Specifies a font for a selected block of text. Embeds an OLE (Object Linking and Embedding) object into the document. Controls the links to external objects in your document. Specifies the properties of a selected, embedded object.

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128 CHAPTER 5

3. LINGO Menu Commands: Solves the model in the current window. Generates a solution report window for the current model. Generates a range analysis report for the current window. Sets system options. Generates the algebraic representation for the current model. Displays a graphical picture of a model in matrix form. Tracks down formulation errors in infeasible and unbounded linear programs. Displays a brief report regarding the technical detail of a model. Generates a formulation report for the current window.

Solve Solution Range Options Generate Picture Debug Model Statistics Look

4. Window Menu Commands: Command Window Status Window Send to Back Close All Tile Cascade Arrange Icons

Opens a command window for command-line operation of LINGO. Opens the solver's status window. Sends the current window behind all other open windows. Closes all open windows. Arranges all open windows into a tile pattern. Arranges all open windows into a cascading pattern. Aligns all iconized windows at the bottom of the main frame window.

5. Help Menu Commands: Help Topics AutoUpdate About LINGO

Accesses LINGO's Help facility. s your version of LINGO online. Checks to see if an updated copy of LINGO is available for on the LINDO Systems Web site. Displays the version and size of your copy of LINGO, along with information on how to LINDO Systems.

WINDOWS COMMANDS

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Windows Commands In Depth In the remainder of this chapter, we will document all the commands specific to the Windows version of LINGO. The commands are categorized into the five main menus described in the Windows Commands In Brief section above.

1. File Menu LINGO's File menu is pictured below. This menu contains commands that generally pertain to the movement of files in and out of LINGO.

130 CHAPTER 5

File|New

F2

The New command opens a new, blank window. When you select the New command, you will be presented with the following dialog box:

You may then select the type of file you want to create. The file must be one of the four types: 1.

LINGO Model (*.lg4) The LG4 format was established with release 4.0 of LINGO. LG4 is the primary file format used by LINGO to store models under Windows. This format s multiple fonts, custom formatting, and OLE (Object Linking and Embedding). LG4 files are saved to disk using a proprietary binary format. Therefore, these files can’t be read directly into other applications or transported to platforms other than the PC. Use the LNG format (discussed next) to port a file to other applications or platforms.

2.

LINGO Model (Text Only) (*.lng) The LNG format is a portable format for storing your models. It was the standard file format used by LINGO in releases prior to 4.0 and remains in use on all platforms other than Windows. LNG files are saved to disk as ASCII text and may be read into any application or word processor that s text files. LNG files may also be ported to platforms besides the PC. LNG files do not multiple fonts, custom formatting, or OLE.

3.

LINGO Data (*.ldt) LDT files are data files typically imported into LINGO models using the @FILE function. @FILE can only read text files. Given this, all LDT files are stored as ASCII text. LDT files do not multiple fonts, custom formatting, or OLE.

4.

LINGO Command Script (*.ltf) LTF files are LINGO command scripts. These are ASCII text files containing a series of LINGO commands that can be executed with the File|Take Commands command. For more information on commands that can be used in a LINGO script, refer to the following chapter, Command-line Commands. LTF files do not multiple fonts, custom formatting, or OLE.

WINDOWS COMMANDS 5.

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LINDO Model (*.ltx) LTX files are model files that use the LINDO syntax. Longtime LINDO s may prefer LINDO syntax over LINGO syntax. LINDO syntax is convenient for quickly entering small to medium sized linear programs. As long as a file has an extension of .ltx, LINGO will assume that the model is written using LINDO syntax. Readers interested in the details of LINDO syntax may LINDO Systems to obtain a LINDO ’s manual.

When you simply press either the New toolbar button or the F2 key, LINGO assumes you want a model file. Thus, LINGO does not display the file type dialog box and immediately opens a model file of type LG4. If you have used the LINGO|Options command to change the default model file format from LG4 to LNG, LINGO will automatically open a model of type LNG when you press either the New button or the F2 key. You may begin entering text directly into a new model window or paste in text from other applications using the Windows clipboard and the Edit|Paste command in LINGO.

File|Open...

Ctrl+O

The Open command reads a saved file from disk and places it in a LINGO Window. The file can be a LINGO model file (*.LG4), or any other file. If the file is not in LG4 format, it must be in ASCII text format. After issuing the Open command, you will be presented with a dialog box resembling the following:

132 CHAPTER 5 You can enter a file name in the File name edit box, or select a file name from the list of existing files by double-clicking on a file. Press the Open button to open the file, the Cancel button to exit without opening a file, or the Help button for assistance. You may select a different file type from the Files of type list box causing LINGO to list only the files of that type. Once you have read in a LINGO model file (a LG4 or LNG file), you may use the LINGO|Solve command to solve the model. In addition to its native LG4 and LNG file formats, LINGO s the following three additional file formats :  MPS ¾ The MPS file format is an industry standard format developed by IBM, which is useful for ing linear and quadratic models from one solver or platform to another. 

MPI ¾ The MPI format was developed by LINDO Systems as a portable format for representing arbitrary math programming models.



LP ¾ The LP file format is another industry standard file format, as is MPS, for storing linear and quadratic model. However, with LP format equations are stored using standard algebraic format, making LP files much easier to read and interpret than MPS files. At present, LINGO only s linear models when reading LP format files.

If the file to be opened has an extension of .MPS, .MPI or .LP, then LINGO will invoke, respectively, its MPS, MPI or LP reader to parse the file. When importing a non-native file, LINGO reads the file from disk, converts it to an equivalent LINGO model, and places the model into a new model window. More details follow immediately below. LINGO can also write MPS and MPI format files (but not LP files); this is discussed in the File|Export File section below. Importing MPS, MPI or LP Files When LINGO reads an MPS, MPI or LP file, it converts the formulation to an equivalent LINGO model. As an example, consider the following, simple model: ObjRow) Maximize 20X + 30Y Subject To: Row1) X < 50 Row2) Y < 60 Row3) X + 2Y < 120

The MPS file for this model is: NAME OBJSENSE MAX ROWS N OBJROW L ROW1 L ROW2 L ROW3 COLUMNS X X X Y Y

SAMPLE

ROW3 OBJROW ROW1 OBJROW ROW2

1.0000000 20.0000000 1.0000000 30.0000000 1.0000000

WINDOWS COMMANDS Y

ROW3

2.0000000

ROW1 ROW2 ROW3

50.0000000 60.0000000 120.0000000

RHS RHS RHS RHS ENDATA

The MPI version of the model is: BEGINMODEL SAMPL ! Number of Objective Functions: ! Number of Constraints : ! Number of Variables : VARIABLES ! Name Lower Bound Upper Bound Type X 0 1e+030 C Y 0 1e+030 C OBJECTIVES OBJROW MAXIMIZE EP_USRCOD -101 EP_PUSH_NUM 20 EP_PUSH_VAR X EP_MULTIPLY EP_PUSH_NUM 30 EP_PUSH_VAR Y EP_MULTIPLY EP_PLUS CONSTRAINTS ROW1 L EP_USRCOD -101 EP_PUSH_VAR X EP_PUSH_NUM 50 EP_MINUS ROW2 L EP_USRCOD -101 EP_PUSH_VAR Y EP_PUSH_NUM 60 EP_MINUS ROW3 L EP_USRCOD -101 EP_PUSH_VAR X EP_PUSH_NUM 2 EP_PUSH_VAR Y EP_MULTIPLY EP_PLUS EP_PUSH_NUM 120 EP_MINUS ENDMODEL

And, the LP format version of the model is:

1 3 2 Initial Point 1.23457 1.23457

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134 CHAPTER 5 \ LP format example Maximize objrow: 20x + 30y Subject To row1: x <= 50 row2: y <= 60 row3: x + 2y <= 120 End

One thing to notice at this point is that MPS and MPI formats are not very compact methods for storing a model  they are designed for portability, as opposed to efficiency. Using the File|Open command to read either of these three versions of the model into LINGO, we are presented with the following window containing an equivalent LINGO model:

Note how the model is automatically converted from MPS, MPI or LP format to native LINGO format. Should you wish to save the file again using either MPS or MPI format rather than LINGO format, you must use the File|Export File|MPS Format… command. Note:

The MPS, MPI and LP file formats are intended primarily for exporting (importing) models to (from) other applications or platforms. These file formats are purely scalar in nature—all set-based information is lost upon saving a LINGO model in either MPS or MPI format (LINGO does not currently write LP format files). Thus, when saving copies of a model on your own machine, you should always use the File|Save command in order to save models in a native LINGO format (LG4 or LNG) in order to preserve your model in its entirety.

When it comes to acceptable constraint and variable names, the MPS, MPI and LP formats are less restrictive than LINGO. To compensate for this fact, LINGO attempts to patch names when reading a file, so that all the incoming names are compatible with its syntax. LINGO does this by substituting an underscore for any character in a name that is not issible. In most cases, this will work out OK. However, there is a chance for name collisions where two or more names get mapped into one. For instance, the variable names X.1 and X%1 would both get mapped into the single LINGO name X_1. Of course, situations such as this entirely alter the structure of the model, rendering it incorrect. However, you will be warned whenever LINGO has to patch a name with the following error message:

WINDOWS COMMANDS

135

This message displays the number of variable and row names that were patched to get them to conform to LINGO syntax. If name collisions are a problem, then LINGO has an option that will ensure that all names remain unique. This option involves using RC format for names encountered during MPS I/O. RC format involves renaming each row (constraint) in a model to be Rn, where n is the row’s index. Similarly, each column (variable) is renamed to Cn. In addition, LINGO renames the objective row to be ROBJ. To switch to RC format for MPS names, run the LINGO|Optionscommand, select the General Solver tab, then click the checkbox titled Use R/C format names for MPS I/O, as illustrated here:

As an example, we will once again import the same MPS format model as above. However, this time we will use RC naming conventions. Here is the model as it appears after importing it into LINGO:

Notice how the variable names now use RC format, guaranteeing that name collisions will not occur. Another potential conflict is that MPS and MPI allow variable names to be duplicated as constraint names and vice versa. LINGO does not allow for this. When you go to solve the model, you will receive either error message 28 (Invalid use of a row name), or error message 37 (Name already in use). Once again, you can switch to using RC name format to avoid this conflict. As a final note, LINGO only s free format MPS files, and does not fixed format MPS files. Therefore, variable and row names may not contain embedded blanks.

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File|Save

Ctrl+S

The Save command saves the contents of the active window to disk using the existing file name for the window. If the window has not been saved before, you will be prompted to provide a name for the file.

File|Save As...

F5

The Save As command allows you to save the contents of the active window under a new file name. When issuing the Save As command, you will be presented with a dialog box that resembles the following:

You can enter a new file name in the File name edit box, or select a file name from the list of existing files by double-clicking on it. If you do not specify a file extension, LINGO will append the extension of the default model format to the name. If you want to prevent LINGO from appending an extension to the name, place the file name in double quotes. Press the Save button to save the model, the Cancel button to exit without saving, or the Help button for assistance. You may select a different file type from the Save as type list box. If your model has special fonts or embedded objects, you must save it using the LG4 file format to preserve them. The LG4 format is a special binary format readable only by LINGO. If you wish to create a text copy of your model, then use the LNG file format. For further discussion of the available file formats under LINGO, refer to the New command above.

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File|Close

F6

Use the Close command to close the active (front most) window. If the window has been modified without being saved, you’ll be asked whether you want to save the changes.

File|Print...

F7

Use the Print command to send the contents of the active window to your printer. First, LINGO will display the Print dialog box:

Select the printer to route the output to from the Name list box. Modify the printer’s properties by pressing the Properties button. Select a range of pages to print in the Print range group box. If you need multiple copies, input the number desired in the Number of copies field and specify if you want the copies collated (assuming your printer is capable of collating). Finally, press the OK button to begin printing. Press the Cancel button to exit without printing.

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File|Print Setup...

F8

Use the Print Setup command to configure your printer. You should see a dialog box that resembles the following:

Select the target printer from the Name list box. Press the Properties button to set additional printer properties. Select the type of paper and tray from the Paper group box. In the Orientation group box, select whether you want portrait or landscape output. Press the Cancel button to exit without changing the printer configuration. Press the OK button to save changes and exit the Print Setup command.

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File|Print Preview

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Shift+F8

Use the Print Preview command to display each page of the active window as it will appear when printed. After issuing the Print Preview command, the contents of the active window will be placed in a Preview window as follows:

The Print button sends the file to the printer. The Next Page button brings the next page into the viewer. The Prev Page button brings the previous page into the viewer. The One Page button puts the viewer into single page mode, while the Two Page button puts the viewer into double page mode. The Zoom In button is used to have the viewer zoom in on a region of the document. The Zoom Out button undoes the effect of a Zoom In. Press the Close button to close the print viewer and return to the normal command mode of LINGO. If you would like to change some of the printer specifications, such as landscape output, use the Print Setup command (described above) before issuing the Print Preview command.

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File|Log Output...

F9

Normally, when you are using LINGO for Windows, it is operating in a menu driven mode, where you choose commands from the pull down menus and reports are displayed in individual windows. LINGO can also operate in command mode, where text commands or command script files drive the application and all output is routed to a window known as the command window. All input and output es through the command window when LINGO is in command mode. You can open a command window at anytime by issuing the Window|Command Window command. In general, you will only be interested in running LINGO in command mode if you are planning to embed LINGO in a larger application. If you do use LINGO in command mode, you will find that the command window can only hold a limited amount of output. Should you need to keep a disk-based copy of all that transpires in the command window, you will need to use the Log Output command. The Log Output command opens a standard Windows file dialog box from which you can name the log file. You can echo the output to the command window as well as the file by checking the Echo to screen checkbox. If you would like to append output to the end of an existing file, check the Append output checkbox. When you have selected a file for logging output, a check mark will appear in the File menu before the Log Output command. To turn off Log Output, select the command again and the check mark will disappear.

File|Take Commands...

F11

The Take Commands command is used to submit a LINGO command script file for processing. For more information on LINGO’s script language, refer to the following chapter, Command-line Commands. As an example, we will build a small script file that contains a small product-mix model and process it using Take Commands. To build a script file, issue the File|New command. LINGO will present you with the following dialog box:

Select item 4, LINGO Command Script, and press the OK button. LINGO will open a blank script file.

WINDOWS COMMANDS Now, enter the following into the script file:

This is a command script that inputs a small product-mix model, solves it, and puts the solution in a text file. Save the command script to a file titled MyScript.ltf using the File|Save As command. To run the script, issue the File|Take Commands command. You should see the following:

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142 CHAPTER 5 Double-click on the icon for MyScript.ltf to begin processing the command script. LINGO’s command window will now appear, and you should be able to watch LINGO’s progress at processing the script by watching commands and output as they are logged in the command window. When LINGO finishes the command script, the command window will resemble the following:

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Also of interest is the solution file, SOLU.LGR, created as part of our command script. If you open this file, you should find the following solution to the model: Variable X Y Row 1 2 3 4



The output that was routed to the command window can be routed to a file using the Log Output command described above.

File|Export File The File|Export File command allows you to either export MPS or MPI format files. The MPS file format is an industry standard format developed by IBM, and is useful for ing models from one solver or platform to another. MPI file format was developed by LINDO Systems as a way to store all math programs, from linear models to, in particular, nonlinear models

Exporting MPS Files The File|Export File|MPS format command generates the underlying algebraic formulation for the current model and then writes it to a selected disk file in MPS format. MPS format is a common format for representing linear programming models. MPS files can be ported to any solver that reads MPS files—this includes most commercial linear programming packages. As an example of exporting an MPS file, consider the model:

144 CHAPTER 5 After issuing the File|Export File|MPS format command and opening the file containing the MPS model, we will find: NAME NO_TITLE ***************************************************** * NOTICE: Generated by the MPS export utility for * a maximization type problem. * * The objective coefficients have flipped signs. * Interpret the objective value from the solution of * this model accordingly. ***************************************************** ROWS N 1 L 2 L 3 L 4 COLUMNS X 1 -20 X 2 1 X 4 1 Y 1 -30 Y 3 1 Y 4 2 RHS RHS1 2 50 RHS1 3 60 RHS1 4 120 BOUNDS ENDATA

Note 1: A model must be linear or quadratic to successfully export it in MPS format. Note 2: The MPS file format is intended primarily for exporting models to other applications or platforms. The MPS format is purely scalar in nature—all set-based information is lost upon converting a LINGO model to MPS format. Thus, when saving copies of a model on your own machine, you should always use the File|Save command in order to preserve your model in its entirety. Note 3: When exporting a stochastic program to MPS format, LINGO will write a total of three SMPS format files, consisting of the core model, its stochastic declarations and its time structure.

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Exporting MPI Files MPI file format was developed by LINDO Systems as a way to store all math programs, from linear models to, in particular, nonlinear models. As with MPS files, the MPI format is scalar-based. Thus, you will lose any sets in your model when saving it in this format. Most s will not have a need for MPI formatted files. However, LINDO API s can load these files directly and may find this feature useful. Note 4: When exporting a stochastic program to MPI format, LINGO will write a total of four SMPI format files, consisting of the core model, its stochastic declarations and its time structure.

File|License Some versions of LINGO require the to input a license key. Also, if you upgrade your copy of LINGO, then you will need to enter a new . The File|License command prompts you for a new license key. When you run the File|License command, you will be presented with the dialog box:

Carefully enter the key into the edit field, including hyphens, making sure that each character is correct. Click the OK button and, assuming the was entered correctly, LINGO will display the Help|About LINGO dialog box listing the features in the upgraded license. that these features correspond to the license you intended to install.

146 CHAPTER 5 Note:

If you were emailed your license key, then you have the option of cutting-and-pasting it into the dialog box. Cut the cut the key from the email that contains it. Then, press Ctrl+V to paste it into the LINGO File|License dialog box.

File|Database Info LINGO allows models to link directly with databases through use of the @ODBC() function. Many times, the database you link your model to will require a id and/or . To avoid having to enter your id and each time your model is run, you can input them once at the start of your session using this command. When you run the File|Database Info command, you will be presented with the following dialog box:

Enter any id and/or into the appropriate fields. For security reasons, LINGO does not store this information from one session to the next. So, you will need to run this command at the start of each session. If security is not a concern, and you would like to store your database information, then you can create an AUTOLG.DAT file containing a DBUID command and a DBPWD command. Commands in the AUTOLG.DAT file are executed automatically each time LINGO starts. Thus, DBUID and DBPWD commands contained in an AUTOLG.DAT file will restore your database information at the start of each LINGO run. For more information on the use of AUTOLG.DAT files, refer to LINGO Command Scripts section in Chapter 8, Interfacing with External Files.

File|Exit

F10

Use the Exit command to quit LINGO. If any unsaved files are open, you will be prompted to save them before LINGO shuts down.

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2. Edit Menu LINGO’s Edit menu, pictured at left, contains commands that generally pertain to editing and modifying the text within a window. Each command contained in the Edit menu is discussed below.

Edit|Undo

Ctrl+Z

Use the Undo Command to undo the last modification made to the contents of a Window. Undo can undo all operations except drag-and-drop. LINGO stores a limited amount of undo operations, so you won’t be able to depend on LINGO to undo extensive changes.

Edit|Redo

Ctrl+Y

This command will redo the last undo operation. LINGO stores a limited amount of redo operations, so you won’t be able to depend on LINGO to redo extensive changes.

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Edit|Cut

Ctrl+X

Use the Cut command to clear the selected block of text and place it on the clipboard for pasting. To select a block of text for cutting, place the cursor immediately before the block and press down on the left mouse button. Now, drag the mouse until the cursor appears immediately after the block of text. The text block should now be displayed in reverse video. Now, issue the Cut command to remove the selected text from the document, placing it in the Windows clipboard.

Edit|Copy

Ctrl+C

Use the Copy command to copy the selected text to the clipboard for pasting. To select a block of text for copying, place the cursor immediately before the block and press down on the left mouse button. Now, drag the mouse until the cursor appears immediately after the block of text. The text block should now be displayed in reverse video. Now, issue the Copy command to place a copy of the selected text in the Windows clipboard. The Copy command is a convenient way to transfer small amounts of data from LINGO to other applications.

Edit|Paste

Ctrl+V

Use the Paste command to replace the current selection in the active window with the contents of the Windows clipboard. The Paste command is a convenient way to import small amounts of data from other applications into your LINGO models.

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Edit|Paste Special... Use the Paste Special command to insert the contents from the Windows clipboard into the active window at the cursor insertion point. This command can do much more than insert just plain text as done by the standard Paste command. Paste Special can be used to insert other objects and links to other objects. This is particularly useful for adding links to ing data for your model. By inserting a link to your data sources, it is much easier to find and view them. As an example, suppose we have the following transportation model: ! A 3 Warehouse, 4 Customer Transportation Problem; SETS: WAREHOUSE / WH1, WH2, WH3/: CAPACITY; CUSTOMER / C1, C2, C3, C4/: DEMAND; ROUTES(WAREHOUSE, CUSTOMER): COST, VOLUME; ENDSETS ! The objective; MIN = @SUM(ROUTES: COST * VOLUME); ! The demand constraints; @FOR(CUSTOMER(J): @SUM(WAREHOUSE(I): VOLUME(I, J)) >= DEMAND(J)); ! The supply constraints; @FOR(WAREHOUSE(I): [SUP] @SUM(CUSTOMER(J): VOLUME(I, J)) <= CAPACITY(I)); ! Here are the parameters; DATA: CAPACITY = @OLE('D:\LNG\TRANLINKS.XLS'); DEMAND = @OLE('D:\LNG\TRANLINKS.XLS'); COST = @OLE('D:\LNG\TRANLINKS.XLS'); @OLE('D:\LNG\TRANLINKS.XLS') = VOLUME; ENDDATA

As we can see from the data section, we are importing data from the Excel file TRANLINKS.XLS and writing the solution back out to the same file.

150 CHAPTER 5 A nice feature would be to insert a link to the spreadsheet into our model file. This way, we could view the data and the solution without having to start Excel and load the spreadsheet. To do this, open Excel and load the spreadsheet as we have done here:

For complete information on importing data from Excel, see Chapter 9, Interfacing with Spreadsheets.

WINDOWS COMMANDS Now, select the range B2:F21 in the spreadsheet. Next, from Excel’s Edit menu, choose the Copy command. Now, click on LINGO, place the cursor right before the data section, and give the Edit|Paste Special command. Click on the Paste Link button in the dialog box, so you see the following:

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152 CHAPTER 5 Finally, click the OK button, and you should be able to see the spreadsheet contents in the LINGO model:

This link will be saved as part of your LINGO file. Therefore, whenever you open the model, the spreadsheet will be visible. Note that whenever you reopen the LINGO model, you may want to open the link, so the contents are updated automatically. You can do this by selecting the spreadsheet in the LINGO model, giving the Edit|Links command, and pressing the Open Links button in the dialog box. As a final note, LINGO's compiler ignores all embedded links and objects. Thus, you are free to insert links and objects wherever you choose in a model.

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Edit|Select All Use the Select All command to select the entire contents of the active window. This is useful when you want to copy the entire contents of the window elsewhere, or if you want to delete the contents of the window.

Edit|Find...

Ctrl+F

Use the Find command to search for a desired string of text in the active window. When you issue the Find command, you should see the following dialog box:

Enter the text you wish to search for in the Find what box. Check the Match whole word only box to have LINGO find only whole words of text (i.e., don’t search for occurrences of the text embedded in other words). Check the Match case box to have LINGO search only for instances of the text with the same capitalization. Click the Find Next button to find the next instance of the text.

Edit|Find Next

Ctrl+N

Use the Find Next command to find the next instance of the text most recently searched for using the Find command in the active window.

Edit|Replace

Ctrl+H

Use the Replace command to replace one string of text with another in the active window. When you issue the Replace command, you will see the following dialog box:

154 CHAPTER 5 Enter the name of the text you want to replace in the Find what box. Enter the text you want to replace the old text with in the Replace with box. Clicking the Find Next button will cause LINGO to find the next occurrence of the old text. Clicking the Replace button will cause the next occurrence of the old text to be replaced by the new text. The Replace All button will replace all occurrences of the old text with the new text throughout the entire document. Check the Match whole word only box to have LINGO replace only whole words of the text (i.e., don’t replace occurrences of the text embedded in other words). Check the Match case box to have LINGO replace only instances of the text with the same capitalization.

Edit|Go To Line...

Ctrl+T

Use the Go To Line command to jump to a selected line in the active window. When you issue the Go To Line command, you will see the following dialog box:

Enter a line number in the Go to line number box. Then, press the OK button and LINGO will jump to the desired line number. Press the Top button to go to the top of the document, or the Bottom button to go to the bottom.

Edit|Match Parenthesis

Ctrl+P

Select a parenthesis in a document. Then, use the Match Parenthesis command to find the closing parenthesis for the selected parenthesis. This command is useful when using nested statements such as: @FOR(FXA(I, J): JP(I, J) = MPF(I) * CAGF(I, J); JP(I, J) = MPA(J) * CFGA(I, J));

where it may be difficult to find the close of a given parenthesis. If no parenthesis is selected prior to issuing the Match Parenthesis command, LINGO will select the parenthesis nearest to the current cursor position.

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In addition to this command, there is one other way to find matching parentheses. LINGO will highlight matching parentheses in red when the Match Paren option is enabled under the LINGO|Options command. By placing the cursor immediately after one of the parentheses of interest, you will notice that the color of the parenthesis changes from black to red. LINGO will simultaneously display the matching parenthesis in red. These parentheses will remain displayed in red until you move the cursor to another position, at which point they will be returned to a black color.

Edit|Paste Function Use the Paste Function command to paste any of LINGO’s built-in functions at the current insertion point. Choose the category of the LINGO function you want to paste from the secondary menu, and then select the function from the cascading menu. In the following illustration, we have chosen the External Files category from the secondary menu:

On the right are all the functions that deal with external files. By selecting one of these functions, LINGO will paste a template for the selected function into your document, with a suggestive placeholder for each argument. You should then replace the argument placeholders with actual arguments that are relevant to your model.

Edit|Select Font Use the Select Font command to select a new font, size, style, color, or effect in which to display the selected text. You may find it easier to read models and solution reports if you select a mono-spaced font such as Courier. Custom fonts are preserved only when saving in the LG4 file format. (Refer to the File|New command above for a description of LINGO’s various file types.) Note:

You cannot change the display color of text if syntax coloring is enabled. If you need to use specific display colors in your document, you will need to disable syntax coloring.

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Edit|Insert New Object Use the Insert New Object command to insert an object or a link to an object into your model. As with the Edit|Paste Special command, this command is helpful in that it allows you to insert links to your model’s data sources. Unlike the Paste Special command, which links to portions of an external object, the Insert New Object command can add a link to an entire object. As an example, suppose you have the following staff-scheduling model:

From the model’s data section, we see that we are using the @ODBC function to retrieve the values for the NEED attribute from the STAFFING ODBC data source. We are also using the @ODBC function to send the optimal values for the START attribute back to the same data source. Because this data source is an integral part of our model, it would be nice to place a link to it in our model, so we can retrieve it easily each time we want to refer to it. We can do this with the Edit|Insert New Object command as follows: 1.

Position the cursor in the model where you would like the icon for the link to appear (Note, the LINGO parser ignores links to external objects, so you can insert the link anywhere you like).

WINDOWS COMMANDS 2.

Issue the Edit|Insert New Object command. You should see the following dialog box:

3.

Select the Create from File radio button.

4.

Type in the name of the database file containing your data.

5.

Click the Display As Icon button, so the box now resembles:

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158 CHAPTER 5 6.

Finally, click on the OK button, and an icon representing the linked database will appear in your LINGO model as pictured below:

Now, whenever you want to edit or view the ing database, all you need do is double-click on the icon. In this case, Microsoft Access will start and load the staffing database, so you will see the following on the screen:

As a final note, keep in mind linked objects are preserved only when a model is saved in LG4 format (see the File|New command above for details on the LG4 file format). For complete information on exchanging data and solution values with data sources, see Chapter 10, Interfacing with Databases.

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Edit|Links Use the Links command to modify the properties of the links to external objects in a LINGO document. The dialog box appears as follows:

Select the Automatic radio button to have LINGO automatically update the object when the source file is changed. The Manual radio button allows you to update the object only when you select the Update Now button. The Open Source button is used to open the connection to an automatic link. Once the link has been opened, any changes to the source document will be reflected in the view of the object in your LINGO model. The Change Source button is used to attach the link to a different source file. Finally, the Break Link button is used to break the connection to the external object.

Edit|Object Properties

Alt+Enter

Select a linked or embedded object in your model by single-clicking it, and then you can use the Object Properties command to modify the properties of the object. Properties you will be able to modify include: 1. 2. 3. 4. 5. 6.

display of the object, the object’s source, type of update (automatic or manual), opening a link to the object, updating the object, and breaking the link to the object.

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3. LINGO Menu The LINGO menu, pictured at left, contains commands that generally pertain to solving a model and generating reports. This menu also contains the Options command for customizing LINGO’s configuration.

LINGO|Solve

Ctrl+U

Use the Solve command to have LINGO solve the model in the active window. The Solve command is available only for model windows—report, script, and data windows cannot be solved. When you solve a model, LINGO first examines the model’s syntax to determine if it is valid. If LINGO finds a mistake in the syntax, you will be presented with a dialog box similar to the following:

In the Error Text box, LINGO prints the line number where the syntax error occurred, the text of the line, and points to where LINGO determines that the error occurred. In most cases, LINGO is good at pointing to where the error occurred. Sometimes, however, the error may not be located exactly where LINGO is pointing. Be sure to examine neighboring lines for possible flaws as well. In this particular example, the syntax error occurred in line 2, where we forgot to insert the multiplication signs (*) between the two coefficients and variable names.

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When you issue the Solve command (assuming your model has no further syntax errors), LINGO will post the solver status window. This window contains information about the composition of your model and keeps you posted as to the progress of the solver. The solver status window resembles the following:

For more information on the various fields in the solver status window, refer to Chapter 1, Getting Started with LINGO.

162 CHAPTER 5 Once the solver has completed processing your model, it will create a new window containing the Solution Report for your model. You can scroll through this window to examine its contents, save it to a text file, or queue it to your printer. The following is a sample solution report window:

LINGO|Solution

Ctrl+O

Use the Solution command to generate a solution report for the active window. After selecting the model window that you want to generate a solution for, issue the LINGO|Solution command and you will be presented with this dialog box:

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By clicking on the appropriate radio button on the Type of Output box, the solution report may be viewed in either text or chart format. If you select Text, LINGO will create a new window containing the solution in text format. If you select Chart, LINGO will create a new window containing the solution in one of several different graphical formats. Current ed chart formats are: bar, bubble, contour, curve, histogram, line, pie, radar, scatter and surface.

164 CHAPTER 5 Note:

LINGO maintains only one solution in memory. This is the solution to the last window you issued the LINGO|Solve command for. If you try to issue the Solution command for a window that LINGO does not currently have a solution, you will receive an error message. Thus, if you plan to work with two or more models that take a long time to solve, be sure to save copies of your solutions. This will allow you to refer to them later without having to re-solve your models.

Text Solution Reports In the Attribute(s) or Row Name(s) list box, select an attribute or row name for which you would like a report. If you do not select a name in this box, LINGO will generate a full solution report that includes all attributes and rows. You may also specify multiple objects in this box, in which case, each object will be included in the report. Both the multi-character (*) and single character (%) wildcards are ed in the Attribute(s) or Row Name(s) field. For example, inputting "X*" would cause all variables and rows beginning with the letter X to be displayed, while "X%1" will display all variables and rows with three-character names that begin with letter X and end with the digit 1. In the Header Text box, enter whatever text (e.g., "Values for X") you would like to appear at the head of the report. Check the Nonzeros Vars and Binding Rows Only box to see a report that contains only the variables with a nonzero value and constraints that are binding. When you click OK, LINGO creates a new solution window containing the solution report. You can use Cut and Paste commands to move the contents of the report to other applications. You may also use the File|Save command to save the report to a text file.

Charting Solutions If you choose to have the solution displayed as a chart, the box titled Chart Properties will be undimmed, which allows you to select options influencing the display of the chart. In the Chart Type box, you have the option of selecting a bar, bubble, contour, curve, histogram, line, netarc, netnode, pie, radar, scatter or surface chart. One easy way to familiarize yourself with the various chart types is to open and run the sample model CHARTS.LG4, which displays a sample of each of the chart types. If you've selected a histogram chart, then you will also have the option of specifying the number of bins in the histogram in the Histo Bins box. If the number of bins is set to 0, then LINGO will select a reasonable number of bins for the given data set. In the Values box, you can select to graph either primal or dual values.

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The Attribute(s) or Row Name(s) field works in the same way as mentioned in the previous section, with the additional feature that a colon may be inserted in between object names to indicate that you wish to display multiple series on the same graph. Each series of data will be displayed using a different color. The chart types that multiple series are: bar, bubble, curve, line, radar, scatter. Contour, histogram, pie and surface charts do not multiple data series. As an example, consider the following staff scheduling model, similar to the one discussed above in section A Staff Scheduling Problem: MODEL: SETS: DAYS: REQUIRED, START, ONDUTY; ENDSETS DATA: DAYS = MON TUE WED THU FRI SAT SUN; REQUIRED = 20 16 13 16 19 14 12; ENDDATA MIN = @SUM( DAYS( I): START( I)); @FOR( DAYS( J): ONDUTY( J) = @SUM( DAYS( I) | I #LE# 5: START( @WRAP( J - I + 1, 7))); ONDUTY( J) >= REQUIRED( J) ); END

Model: STAFFDEM2 The ONDUTY attribute tells us how many employees are working on each day of the week. To display a bar chart of this information, you would fill out the LINGO|Solution dialog box as below:

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After pressing OK, a new window will open with the desired chart:

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Along with charting the number on duty, it would also be useful to include the number of staffers required on each day. This would allow us in one glance to quickly see the days (if any) where we are over staffed. To do this, change the contents of Attribute(s) or Row Name(s) field from "ONDUTY" to "ONDUTY : REQUIRED". This tells LINGO we want to view both on duty and required staffing levels, with the colon inserted to indicate they should be displayed as separate series. Doing this yields the multi-bar chart:

Here we see that we just meet our staffing needs Monday through Saturday and are over staffed slightly on Sunday. Had we not inserted the colon separator between the attribute fields, then LINGO would have displayed the two attributes as a single series:

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The Bounds box gives you the option of placing bounds on the values to be included in the graph. If a number is entered in the Lower bound field, LINGO will only display points in the graph that are greater-than-or-equal-to the value. Conversely, if a value is placed in the Upper bound field, LINGO will only graph points less-than-or-equal-to the bound. For example, if we display a bar chart showing the number of employees starting each day of the week we'd see the following:

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Notice there are several "holes" in the chart due to no one starting on Tuesday, Wednesday and Sunday. Entering a lower bound of .1 in the Bounds box removes the holes in our chart:

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For charts with one-dimensional data (bar, histogram, line, pie and radar), the bounds will be applied to all data points in the first data series. If a point is eliminated in the first series, then the corresponding points in any additional series will also be eliminated, regardless of whether or not they lie within the specified bounds. Charts with two-dimensional data points (curve and scatter) and three-dimensional data (bubble, contour and surface) will have the bound applied only to the x-axis data. If the x-axis value lies outside the bounds, then the entire point will be eliminated from the chart. Of these higher dimension charts, only curve scatter and bubble allow for multiple series. In this case, the bounds will be applied to the x-axis value of each series, as opposed to just the first. Note:

Bounds are not currently applied to the two network chart types ¾ netarc and netnode.

Next in the Chart Properties box are options for controlling how the axes and legends are labeled. In both cases, the choices are:  Default - LINGO tries to make intelligent choices in choosing an appropriate labeling scheme.  None - No labels are displayed.  Set - Use one of the model's sets for labeling. Multiple sets may be specified, if needed, to supply sufficient labels.  Specified - Labels are entered explicitly into the Set(s) or Name(s) field, with individual labels being separated with colons.

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The final option in the Chart Properties box is the Use 3D and Shading checkbox. This option is on by default and will result in more modern looking charts that utilize shading and 3-dimensional effects. Disable this option to display simpler, 2-dimensional charts that may display better on certain printers.

Higher Dimension Charts In addition to the standard charts with one-dimensional data, LINGO offers 2-dimensional curve network and scatter charts, as well as 3-dimensional bubble, contour and surface charts. When a chart requires more than one dimension of data, you can supply the data in either of two ways. The first method is to supply one attribute for each dimension. So, for example, a 3-dimensional surface chart might be specified using the three attributes XVALS, YVALS and ZVALS. The first attribute always stores the x-axis data, the second stores the y-axis and, if needed, the third attribute stores the z-axis values. The second option is to provide a data series in an m x n table/attribute, where m is the number of data points and n is the dimension of the chart type. As an example, suppose we have the following model that generates points of the surface X * SIN( Y) + Y * SIN( X): MODEL: SETS: POINTS /1..21/; POINTS2( POINTS, POINTS): X, Y, Z; ENDSETS CALC: XS = @FLOOR( -(@SIZE( YS = XS; @FOR( POINTS2( I, J): X( I, J) = XS + I Y( I, J) = YS + J Z( I, J) = X( I, J) Y( I, J ) * @SIN( ); ENDCALC END

Model: CHARTSURF

POINTS) / 2) + .5); 1; 1; * @SIN( Y( I, J)) + X( I, J));

172 CHAPTER 5 A surface chart requires 3-dimensional data, which, in this case, is contained in the X, Y and Z attributes. To request a surface chart after solving the model, we fill out the LINGO|Solution dialog box as follows:

Note that we've listed each of the three attributes in the Attribute(s) or Row Name(s) field. Given that this is a 3-dimensional graph, the three attributes will be used to create a single chart. Clicking on OK then gives us the chart:

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174 CHAPTER 5 Below we list the dimensions of each of the various chart types and whether or not they allow for multiple data series: Chart Type

Dimension

s Multiple Series

Bar

1

Yes

Histogram

1

No

Line

1

Yes

Pie

1

No

Radar

1

Yes

Curve

2

Yes

Netarc

2

Yes

Netnode

2

Yes

Scatter

2

Yes

Bubble

3

Yes

Contour

3

No

Surface

3

No

Network Charts Network charts are specified slightly differently from other charts. Although network charts are technically 2-dimensional charts, they require 4 attributes of data. LINGO s two different formats for specifying network charts -- netarc and netnode. In netarc format, each arc is specified by two (X,Y) pairs, indicating the two end points of the arc. Whereas in netnode format, each node in the network is specified with one (X,Y) pair, and each arc between the nodes is specified by pair of arc indices from the arcs set. An example of both formats follows:

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MODEL: ! Illustrates the two formats for network charts: NETARC and NETNODE; SETS: ARCS /1..6/: X1, Y1, X2, Y2; NODES /1..4/: NODEX, NODEY; NODESINNET /1..6/: N1, N2; ENDSETS DATA: !Use NETARC format -- one arc for each (X1,Y1) <-> (X2,Y2) pair of coordinates; X1, Y1, X2, Y2 = 10, 10, 20, 10, 20, 10, 20, 40, 20, 40, 10, 40, 10, 40, 10, 10, 10, 40, 20, 10, 20, 40, 10, 10; !Use NETNODE format; !List locations of all the nodes; NODEX, NODEY = 10, 10, 20, 10, 20, 40, 10, 40; !And list the indices of the nodes that have arc between them; N1, N2 = 1, 2, 2, 3, 3, 4, 4, 1, 4, 2, 1, 3; ENDDATA END

Model: CHARTNET The CHARTNET model above uses both network chart formats for specifying a rectangular network of four nodes with each node pair being ed by an arc. After solving the model, you can display the network using netarc format by running the LINGO|Solution command and filling out the dialog box using the netarc attributes (X1, Y1, X2 and Y2) and clicking on the "Netarc" radio button as follows:

176 CHAPTER 5

To display the network using netnode format, you would fill out the dialog box as below, using the netnode attributes (NODEX, NODEY, N1 and N2) along with checking off the "Netnode" radio button:

In both cases, you should then generate a network chart resembling:

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LINGO|Range

177

Ctrl+R

Use the Range command to generate a range report for the model in the active window. A range report shows over what ranges you can: 1) change a coefficient in the objective without causing any of the optimal values of the decision variables to change, or 2) change a row's constant term (also referred to as the right-hand side coefficient) without causing any of the optimal values of the dual prices or reduced costs to change. Note:

The solver computes range values when you solve a model. Range computations must be enabled in order for the solver to compute range values. Range computations are not enabled by default. To enable range computations, run the LINGO|Options command, select the General Solver Tab, and, in the Dual Computations list box, choose the Prices and Ranges option. Range computations can take a fair amount of computation time, so, if speed is a concern, you don’t want to enable range computations unnecessarily.

178 CHAPTER 5 Note:

The example model below, when solved, yields the range report that follows. [OBJECTIVE] [ALIM] [CLIM] [T]

MAX = 20 * A + 30 * C; A <= 60; C <= 50; A + 2 * C <= 120;

Here is the range report: Ranges in which the basis is unchanged:

Variable A C Row ALIM CLIM T

Objective Coefficient Ranges Current Allowable Allowable Coefficient Increase Decrease 20.00000 INFINITY 5.000000 30.00000 10.00000 30.00000 Righthand Side Ranges Current Allowable Allowable RHS Increase Decrease 60.00000 60.00000 40.00000 50.00000 INFINITY 20.00000 120.0000 40.00000 60.00000

The first section of the report is titled Objective Coefficient Ranges. In the first column, titled Variable, all the optimizable variables are listed by name. The next column, titled Current Coefficient, lists the current coefficient of the variable in the objective row. The next column, Allowable Increase, tells us the amount that we could increase the objective coefficient without changing the optimal values for the variables. The final column, Allowable Decrease, lists the amount that the objective coefficient of the variable could decrease before the optimal values of the variables would change. Information on the allowable increases and decreases on objective coefficients can be useful when you need answers to questions like, "How much more (less) profitable must this activity be before we should be willing to do more (less) of it?". Referring to the Objective Coefficient Ranges report for our example, we can say, as long as the objective coefficient of A is greater-than-or-equal-to 15, the optimal values of the variables will not change. The same may be said for the objective coefficient of variable C, as long as it falls within the range of [0,40]. Note:

Ranges are valid only if you are planning to alter a single objective or right-hand side coefficient. The range information provided by LINGO cannot be applied in situations where one is simultaneously varying two or more coefficients. Furthermore, ranges are only lower bounds on the amount of change required in a coefficient to actually force a change in the optimal solution. You can change a coefficient by any amount up to the amount that is indicated in the range report without causing a change in the optimal solution. Whether the optimal solution will actually change if you exceed the allowable limit is not certain.

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The second section of the range report is titled Right-hand Side Ranges. The first column, Row, lists the names of all the optimizable rows, or constraints, in the model. The second column, Current RHS, gives the constant term, or right-hand side value, for the row. The next two columns, Allowable Increase and Allowable Decrease, tell us how far we can either increase or decrease the right-hand side coefficient of the row without causing a change in the optimal values of the dual prices or reduced costs. If you recall, the dual prices on rows are, effectively, shadow prices that tell us at what price we should be willing to buy (or sell) our resources for. The dual prices do not, however, tell us what quantity we should be willing to buy (or sell) at the dual price. This information is obtained from the allowable increases and decreases on the right-hand side coefficients for the row. So, for our example, the dual prices and reduced costs will remain constant as long as the right-hand side of row ALIM falls within the range [20,120], the right-hand side of CLIM is greater-than-or-equal-to 30, and the right-hand side of T is in [60,160]. Note:

We preceded all the constraints in our model with a name enclosed in square brackets. This is an important practice if you wish to generate range reports. If you do not name your constraints, LINGO assigns them a name that corresponds to the internal index of the constraint. This internal index will not always correspond to the order of the constraint in the text of the original model. So, to make the Right-hand Side Ranges section of range reports meaningful, be sure to name all your constraints. (See the section Constraint Names for details on asg constraint names.)

Note:

If a variable is nonlinear in the objective, its value in the Current Coefficient column will be displayed as NONLINEAR. Similarly, if a row is nonlinear, the value in the Current RHS column will be displayed as NONLINEAR.

Coefficients that can be increased or decreased indefinitely will display a range of INFINITY. Fixed variables are substituted out of a model and will not appear in a range report. Rows that contain only fixed variables are also substituted out of models and will also not appear in range reports. As an example, suppose we changed the following inequality in our sample model from: [ALIM]

A <= 60;

to the equality: [ALIM]

A = 60;

LINGO can now solve directly for the value of A. The variable A is considered fixed, as is the row ALIM (since it contains no optimizable variables.) Given this, the variable A will no longer appear in the Objective Coefficient Ranges section of the range report and the row ALIM will not appear in the Right-hand Side Ranges section. We can this by examining the updated range report: Ranges in which the basis is unchanged:

Variable C

Objective Coefficient Ranges Current Allowable Allowable Coefficient Increase Decrease 30.00000 INFINITY 30.00000

180 CHAPTER 5

Row CLIM T

Righthand Side Ranges Current Allowable Allowable RHS Increase Decrease 50.00000 INFINITY 20.00000 60.00000 40.00000 60.00000

Note:

LINGO maintains the range report for only one model in memory. This is the report for the window that you last issued the LINGO|Solve command for. If you try to issue the Range command for a window that LINGO does not currently have range information for, you will receive an error message. If you plan to work with two or more models that take a long time to solve, be sure to save copies of your range reports to disk, so you can refer to them later without having to re-solve your models.

Note:

The barrier crossover option must be enabled if you plan to do range analysis. Range computations cannot be performed if the final solution is not a basic solution.

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LINGO|Options

181

Ctrl+I

Use the LINGO|Options command to change a number of parameters that affect LINGO's interface, as well as the way LINGO solves your model. When issuing the Options command, you will be presented with the following dialog box:

Set these parameters to your personal preference and press the Apply button to set them for the extent of the current LINGO session. The currently selected settings are also applied when you click the OK button, with the one difference being that the OK button closes the dialog box. If you would like the current parameter settings to be maintained for use in subsequent LINGO sessions, click the Save button. The original default settings can be restored at any time by clicking the Default button.

182 CHAPTER 5 There are nine tabs in the Options dialog box:         

Interface General Solver Linear Solver Nonlinear Solver Integer Pre-Solver Integer Solver Global Solver Model Generator SP Solver

The first time you run the Options command during a session, the Interface tab will be selected. The Interface and General Solver tabs contain options of interest to most s. The remaining tabs (Linear Solver, Nonlinear Solver, Integer Pre-Solver, Integer Solver, Global Solver and SP Solver) contain advanced options that tend to be of interest primarily to the expert . Follow the links above for more details on the options available under each tab. Note:

LINGO uses the LINDO API as its solver engine. The LINDO API has a wealth of advanced parameter settings to control its various solvers. Most of the more relevant parameters may be set through the LINGO|Options command. However, some of the more advanced parameters must be set using the APISET command.

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Interface Tab The Interface tab on the Options dialog box (shown above) can be used to control the appearance of LINGO, LINGO’s output, and the default file format.

General Box The General box on the Interface tab:

allows you to set the following general options:       

Errors In Dialogs, Status Bar, Fill Out Ranges and Tables, Status Window, Splash Screen, Toolbar, and Output Level.

Errors In Dialogs If the Errors In Dialogs box is checked, LINGO will display error messages issued by the solver in a modal dialog box. This dialog box must be cleared before LINGO proceeds with any other operation. In some instances, you may have LINGO embedded in other applications, where it may not be desirable, or possible, to have s clearing error dialogs. By unchecking this option, LINGO will route the solver's error messages to the command window, where they will be displayed, and no intervention will be required to clear the messages. The default is for solver errors to be displayed in dialog boxes. Note:

This option allows you to route only those error messages generated by LINGO's solver to the report window. Error messages displayed by LINGO's interactive front-end will always be posted in dialog boxes.

184 CHAPTER 5

Status Bar If the Status Bar box is checked, LINGO displays a status bar along the bottom of the main frame window. Among other things, the status bar displays the time of day, location of the cursor, menu tips, and the current status of the program. To remove the status bar from the screen, clear the Status Bar checkbox. The default is for LINGO to display the status bar.

Fill Out ODBC Ranges LINGO can export a model’s solution to Excel and/or databases. When exporting to Excel, LINGO sends solutions to defined ranges in a workbook. Solutions exported to a database are sent to tables within the database. In either case, the target range or table may contain more space for values than you are actually exporting. In other words, there may be cells at the end of ranges or records at the end of tables that will not be receiving exported values from LINGO. The Fill Out Ranges option determines how these extra cells and records are treated. When the Fill Out Ranges option is enabled, LINGO overwrites the extra values with null entries. Conversely, when the option is not enabled, LINGO leaves any extra values untouched. The Fill Out Ranges option is disabled by default.

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Status Window If the Status Window box is checked, LINGO displays a solver status window whenever you issue the LINGO|Solve command. This window resembles the following:

The solver status window is useful for monitoring the progress of the solver and the dimensions of your model. It is updated every n seconds, where n is the value in the Update interval field in the lower right corner of the window. For a detailed description of the various fields in the solver status window, see the section Solver Status Window in Chapter 1, Getting Started with LINGO. The default is for the solver status window to be displayed.

Splash Screen If the Splash Screen box is checked, LINGO will display its splash screen each time it starts up. The splash screen lists the release number of LINGO and the software's copyright notice. If you disable this option, LINGO will not display the splash screen. The default is for the splash screen to be displayed.

186 CHAPTER 5

Output Level You can use the Output Level setting to control the amount of output LINGO generates. There are four settings available: 

Verbose—Causes LINGO to display the maximum amount of output, including full solution reports.



Terse—Less output than Verbose, with full solution reports suppressed. This is a good output level if you tend to solve large models. LINGO also suppresses Export Summary Reports generated when exporting data to spreadsheets or databases.



Errors Only—All output is suppressed, with the exception of error messages.



Nothing—LINGO suppresses all output. This level may be useful when taking advantage of the programming capabilities in LINGO, in which case, you will add statements to your model to generate all required output.

The default is for LINGO to be in verbose mode.

Toolbar If the Toolbar box is checked, LINGO displays its command toolbar containing buttons, which act as shortcuts to various commands contained in the LINGO menu. For definitions of the buttons on the toolbar, please see the section The Toolbar at the beginning of this chapter. If the Toolbar checkbox is unchecked, LINGO does not display its toolbar. The default is for LINGO to display its toolbar.

Solution Display Box The Solution Display box on the Interface tab:

allows you to set the following options:  

Show as 0, and, Precision.

Show as 0 On occasion, due to round-off error, some of the values returned by LINGO’s solver will be very small (less than 1e9.) In reality, the true values of these variables are either zero or so small as to be of no consequence. These tiny values can be distracting when interpreting a solution report. The Show as 0 parameter can be used to suppress small solution values. Any solution value less-than-or-equal-to Show as 0 will be reported as being zero. The default value for Show as 0 is 1e-9.

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Precision LINGO defaults to displaying seven significant digits for values in standard solution reports. In some cases, you may desire more or less than seven digits of precision, in which case, you will need to adjust the Precision parameter. For example, suppose we have a very simple model consisting of the single expression: X = 1/3;

The default solution report for this model using seven digits of precision would contain: Variable X Row 1

Value 0.3333333 Slack or Surplus 0.000000

Increasing the Precision parameter to 16 and resolving yields the following solution report with 16 significant digits: Variable X Row 1


The default value for Precision is seven significant digits.

File Format Box The File Format box on the Interface tab:

is used to select the default file format that LINGO uses to save models to disk. There are three different formats to choose from: LG4, LNG, or LTX. The LG4 format is a binary format readable only by LINGO. This format enables you to have custom formatting, fonts in your models, and to use LINGO as an OLE server and container. Files saved in the LG4 format are readable only by Windows versions of LINGO. The LNG format is a text-based format. Thus, models saved in the LNG format can be read into other applications. LNG format models are transferable to other platforms running LINGO. Models saved in LNG format cannot contain custom formatting or embedded objects. LTX files are model files that use the LINDO syntax. Longtime LINDO s may prefer LINDO syntax over LINGO syntax. LINDO syntax is convenient for quickly entering small to medium sized

188 CHAPTER 5 linear programs. As long as a file has an extension of .ltx, LINGO will assume that the model is written using LINDO syntax. Readers interested in the details of LINDO syntax may LINDO Systems to obtain a LINDO ’s manual. The default file format is LG4.

Syntax Coloring Box The Syntax Coloring box on the Interface tab:

is used to control the syntax coloring capability in LINGO’s editor. LINGO’s editor is “syntax aware.” In other words, when it encounters LINGO keywords, it displays them in blue. Comments are displayed in green, and all remaining text is displayed in black. Matching parentheses are also highlighted in red when you place the cursor immediately following a parenthesis. The controls available in this box are: Line Limit, Delay, and Paren Match.

Line Limit Syntax coloring can take a long time if you have very large files. The Line Limit field sets the maximum acceptable file size for syntax coloring. Files with line counts exceeding this parameter will not be syntax colored. Setting this parameter to 0 will disable the syntax coloring feature. The default line limit is 1000 lines.

Delay The Delay field sets the number of seconds LINGO waits after the last keystroke was typed before re-coloring modified text. s on slower machines may want to set this higher to avoid having syntax coloring interfere with typing. s on faster machines may want to decrease this value, so text is re-colored quickly. The default is 0 seconds.

Paren Match If the Paren Match box is checked, LINGO will highlight matching parentheses in red when you place the cursor immediately following a parenthesis. In other words, by placing the cursor immediately after one of the parentheses of interest, you will notice that the color of the parenthesis changes from black to red. LINGO will simultaneously display the matching parenthesis in red. These parentheses will remain displayed in red until you move the cursor to another position, at which point they will be returned to a black color. The default is for parenthesis matching to be enabled.

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Command Window Box The Command Window box on the Interface tab:

is used to customize the configuration of LINGO’s command window. LINGO’s command window can be opened by using the Window|Command Window command. This gives the a command-line interface to LINGO. This interface is identical to ones used by LINGO on platforms other than Windows. The command window is also useful for testing LINGO command scripts. For more information on the commands available under LINGO’s command-line interface, refer to Chapter 6, Command-line Commands.

Send Reports to Command Window If the Send Reports to Command Windowbox is checked, LINGO will send any reports it generates to the command window rather than to individual report windows. This is useful if you’d like to have two or more LINGO generated reports contained in a single window. The default is to not send reports to the command window.

Echo Input When you run a LINGO command script with File|Take Commands, the commands LINGO processes are normally not displayed. If the Echo Input box is checked, processed commands will be displayed in the command window. This can be a useful feature when you are trying to develop and debug a LINGO command script. The default is to not echo input.

Line Count Limits The Line Count Limits box on the Interface tab:

is used to control the total number of output lines that can be stored in the command window.

190 CHAPTER 5 When LINGO sends output to the command window, it places it at the bottom of the window. All previous output is scrolled up to make way for the new output. The Maximum field sets the maximum number of output lines allowed in the command window. When LINGO hits this limit, it starts removing lines from the top of the command window until there are n lines left, where n is the value of the Minimum field. In general, output to the command window will become slower as the maximum and minimum line counts are increased, or the difference between the maximum and minimum is decreased. If you have a long session you need to save, you can use the File|Log Output command to log all command window output to disk. The default value for Line Count Limits is 800 lines maximum and 400 lines minimum.

Page Size Limits The Page Size Limits box on the Interface tab:

is used to control the page length and width of the command window. If you would like LINGO to pause after a certain number of lines have been written to the command window, you can do so by setting the Length field in the Page Size Limits box. When LINGO hits this limit, it will display the following button on your screen:

LINGO will wait until you press the More button to display any subsequent output in the command window. The default is None, meaning no page length limit is imposed. When LINGO generates reports, it limits output lines to a certain width. In some reports, lines will be wrapped, so they fall within the line limit. In other reports, lines may be truncated. Because LINGO concatenates variable names in performing set operations, a variable name such as SHIPMENTS(WAREHOUSE1, CUSTOMER2) may result. This could be truncated in a solution report if too narrow an output width is used. You can control this line width limit through the Width field of the Page Size Limits box. You may set it anywhere between 64 and 200, with the default being 85.

WINDOWS COMMANDS

General Solver Tab The General Solver tab on the Options dialog box, shown here:

can be used to control several general parameters related to the functioning of LINGO’s solver.

191

192 CHAPTER 5

Multithreading Box The Multithreading box on the General Solver tab:

gives you control over LINGO's use of multithreading. With multithreading, LINGO's model generator and its solvers can utilize multiple processors to improve performance. The Threads option controls the maximum number of threads, or processors, to utilize at any given time. Interesting settings for the thread count are:   

Max Available — LINGO determines the number of processors on the system and sets the thread limit to this number. 1 — Only one processor will be used, causing LINGO to run in single-threaded mode. N — Here, N is some positive integer greater than 1 and less-than-or-equal-to 32, in which case, LINGO will set the maximum number of threads to N.

LINGO defaults to running in single-threaded mode with a thread limit of 1. The second option in the Multithreading box is Mode, which controls the multithreading mode that LINGO operates in. LINGO offers multicore extensions to its model generator and solvers. The multicore extensions are of two types: concurrent and parallel. Concurrent algorithms run two or more different serial algorithms on multiple copies of the same model, using a separate thread for each algorithm, terminating as soon as the winner thread finishes. These “different algorithms” may in fact be the same algorithm type, but using different strategies and/or parameters. Parallel algorithms, on the other hand, parallelize computationally intensive portions of the serial algorithm to distribute the workload across multiple threads.

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The following multicore extensions are currently available in LINGO: LINGO Component

Model Class

Concurrent Option

Linear Programs

Parallel Option Yes

Barrier Solver BNP Solver

Mixed Integer Programs

Yes

No

Global Solver

Nonlinear Programs

Yes

No

Integer Solver

Mixed Integer Programs

Yes

Yes

Linear Solver

Linear Programs

No

Yes

Model Generator

All

Yes

No

Multistart Solver

Nonlinear Programs

Yes

No

Stochastic Solver

Stochastic Programs

Yes

No

No

The available settings for the Mode, or multithreading mode, parameter are as follows:      

Solver Decides — The best available multithreading strategy, either parallel or concurrent, will be selected for each step of the solution process. Off in Solver — Multithreading will be disabled in the solvers, but allowed in the model generator if the number of threads is greater than 1. Prefer Parallel — If a parallel option is available for a particular solution step, then it will be selected, otherwise, a concurrent option will be selected when available. Parallel Only - If a parallel option is available for a particular solution step, then it will be selected, otherwise, the step will be executed in single-thread mode. Prefer Concurrent - If a concurrent option is available for a particular solution step, then it will be selected, otherwise, a parallel option will be selected when available. Concurrent Only - If a concurrent option is available for a particular solution step, then it will be selected, otherwise, the step will be executed in single-thread mode.

Note:

If the maximum thread count parameter, Threads, is set to 1, then the multithreading mode setting will be ignored, and LINGO will execute in single-threaded mode.

Note:

Setting the maximum thread count parameter, Threads, to N does not guarantee than N cores will be fully utilized throughout the entire solve. There may be serial sections inside parallel code that get executed from time to time, causing only one core to be utilized. Concurrent algorithms may also have fewer than N different serial algorithms to launch, meaning fewer than N cores will be occupied.

194 CHAPTER 5

Debugger Box The Debugger box on the General Solver tab:

gives you control over the output level and the solver used as part of the model debugging command, LINGO|Debug. The debugger is very useful in tracking down problems in models that are either infeasible or unbounded . The Output Level option controls how much output the model debugger generates. Possible output levels range from 1 (minimum output) to 15 (maximum output). In general, you will want to generate as much output as possible. The only reason to restrict the amount of output would be to speed debugging times on large models. The default setting for the debugger output level is 15. The Cold Start Solver and Warm Start Solver options control the solver used on linear models for cold starts (starting without an existing basis in memory) and warm starts (restarting from an existing basis) during the debugging process. In either case, the available options are    

Solver Decides — LINGO selects the solver it believes is the most appropriate, Primal — the primal simplex solver will be used, Dual — the dual simplex solver will be used, and Barrier — the barrier solver will be used (requires a barrier solver license).

With some models, you may find that choosing a particular solver improves overall performance of the debugger. LINGO defaults to Solver Decides for both the cold and warm debug solver.

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Runtime Limits Box The Runtime Limits box on the General Solver tab:

is used to control the length of time the solver spends on your model. The first field, Iterations, allows you to place an upper limit on the number of iterations the solver will perform. An iteration is the fundamental operation performed by the solver. At the risk of oversimplification, it is a process that involves forcing a variable, currently 0, to become nonzero until some other variable is driven to zero, improving the objective as we go. In general, larger models will take longer to perform an iteration and nonlinear models will take longer than linear models. The default iteration limit is None, meaning no limit is imposed on the iteration count. The second field in the Runtime Limits box, Time (sec), is a limit on the amount of elapsed time the solver is allowed when optimizing a model. The default time limit is None, meaning no limit is imposed on the length of time the solver can run. If the solver hits either of these limits, it returns to normal command mode. If the model contains integer variables, LINGO will restore the best solution found so far. You may need to be patient, however, because the solver may have to perform a fair amount of work to reinstall the current best solution after it hits a runtime limit. Note:

When the solver is interrupted, the only time it will return a valid solution is when an incumbent solution exists, as indicated by a Best Objective value in the Extended Solver Status box of LINGO's solver status window. Interrupting a model without and incumbent solution will result in an undefined solution. Models that can be interrupted include any model with integer variables, or nonlinear models solved with either the global or multistart solvers.

196 CHAPTER 5

Dual Computations Box The Dual Computations box on the General Solver tab:

is used to control the level of dual computations performed by the solver. The choices for this option are:    

None, Prices, Prices and Ranges, and Prices Opt Only.

When the None option is selected, LINGO does not compute any dual and range information. This option yields the fastest solution times, but is suitable only if you don’t require any dual information. In fact, the LINGO|Range command will not execute when dual computations are turned off. When the Prices option is selected, LINGO computes dual values, but not the ranges on the duals. When Prices & Ranges is selected, LINGO computes both dual prices and ranges When the Prices, Opt Only option is selected, LINGO computes dual values on the optimizable rows only—fixed rows simply receive a dual value of 0. Ranges are also not computed under this option. This can be a useful option if LINGO is spending a lot of time in the “Computing Duals…” phase of the solution process. This phase is devoted to the computation of dual values on the fixed rows. LINGO defaults to the Prices option (computing all the dual prices but not ranges.) Note:

If solution times are a concern, you should avoid unnecessarily enabling range computations.

@SOLVE Time Limit Box The @SOLVE Time Limits box on the General Solver tab:

is used to set time limits on the runtime of @SOLVE commands, which is a command available in calc sections for solving sub-models. The time limit, if specified, will be applied to each individual @SOLVE command encountered in calc sections.

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When the None option is selected for the Time field, LINGO does not impose a time limit. Any nonnegative value will be treated as a runtime limit, in seconds, for each @SOLVE command. If the time limit is hit, the @SOLVE command will be interrupted, and the best solution found, up to that point, will be returned. LINGO defaults to the no time limit on @SOLVE commands and will not kill scripting when interrupts occur

Scaling Warning Threshold The Scaling Warning Threshold parameter on the General Solver tab:

controls when LINGO displays its scaling warning message:

When LINGO generates a model, it keeps track of the largest and smallest coefficient values in the model. LINGO then computes the ratio of the largest to smallest coefficient, and if the absolute value of this ratio exceeds the setting for Scaling Warning Threshold, LINGO will display the above error message. The scaling warning is to reminder that, due to finite precision, round-off error in arithmetic calculations can result when the largest-to-smallest coefficient ratio becomes too large. Not all models will necessarily experience round-off problems when the scaling warning is issued, however, it is something you should be on the lookout for. The recommended technique for eliminating the scaling warning is to rescale the model so as to reduce the extreme coefficient ratio. Many times, just scaling units of measure (e.g., dollars to millions of

198 CHAPTER 5 dollars) is enough the bring a model's scaling into line. If this is not possible, or your model is performing well in light of the poor scaling, you may choose to increase the Scaling Warning Threshold in order to suppress the warning message. The default value for the Scaling Warning Threshold is 1.e12.

Variables Assumed Non-Negative When enabled, the Variables Assumed Non-Negative checkbox on the General Solver tab:

tells LINGO to place a default lower bound of 0 on all variables. In other words, unless otherwise specified, variables will not be allowed to go negative. Should you want a variable to take on a negative value, you may always override the default lower bound of 0 using the @BND() function. If this option is disabled, then LINGO’s default assumption is that variables are unconstrained and may take on any value, positive or negative. Unconstrained variables are also referred to as be being free

Use RC Format Names The Use R/C Format names for MPS I/O checkbox on the General Solver tab:

tells LINGO to convert all variable and row names to RC notation when performing MPS file format Input/Output. RC format involves renaming each row (constraint) in a model to be Rn, where n is the row’s index. Similarly, each column (variable) is renamed to Cn. In addition, LINGO renames the objective row to be ROBJ. Refer to the Importing MPS Files section under the File|Open command earlier in this chapter for a discussion of RC notation and why this option is useful. By default, LINGO disables the use of RC format names.

Favor Reproducibility Over Performance The Favor reproducibility over performance checkbox on the General Solver tab:

allows you to indicate if your preference is for reproducible solutions across multiple runs, or if your preference is for faster average runtimes.

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A number of solver steps have been found to perform better when using elapsed time as a measure of work performed. The problem with this approach is that the actual amount of work that gets performed over a fixed time interval will vary slightly across runs depending on the machine's load factor. Many models have alternate optimal solutions, with equal objectives but different variable values. LINGO is indifferent as to which solution is selected; it just wants to find a feasible solution with the best objective value, regardless of the variables values. The end result of this variability in work performed is that a different solution path may be selected from one run to the next, in which case, you may end up with a different alternate optimum from a previous run. On the other hand, enabling the Favor reproducibility over performance option causes LINGO to use fixed measures of work (such as iteration counts), allowing solutions to be reproducible across runs when using the same machine. By default, LINGO disables the Favor reproducibility over performance option. However, if reproducibility of results is important for your application, then you should enable this option.

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Model Generator Tab The Model Generator tab on the Options dialog box, shown here:

is used to control several parameters related to the generation of the model. The model generator takes the expressions in your LINGO model and converts them to a format understood by the solver engines that find the actual solutions to the model.

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Generator Memory Limit Box The Generator Memory Limit box on the Model Generator tab:

is used to control the amount of memory set aside to use as workspace for generating a model. Large models may run out of generator memory when you attempt to solve them. In this case, you will receive the error message "The model generator ran out of memory.” To avoid this error, increase the amount of memory in the Generator Memory Limit field. You will then need to click the Save button and restart LINGO. Since LINGO sets aside this memory when it starts, changes in LINGO's generator memory limit are not established until you restart the program. To determine exactly how much generator memory LINGO was able to successfully allocate, run the Help|About LINGO command. The About LINGO dialog box displays the amount of generator memory allocated at startup. The memory allocated to LINGO’s generator will not be available to the various solver engines contained in LINGO. Thus, you should not allocate overly excessive amounts of memory to the generator. If you set LINGO's generator memory limit to None, LINGO will allocate all available memory when it starts up. This is not a recommended practice. The default size for the workspace is 32Mb. Note:

By setting LINGO's generator memory limit abnormally high, both LINGO and Windows will resort to swapping virtual memory to and from the hard drive, which can slow down your machine dramatically and result in poor performance. In general, set the memory allocation to a level high enough to comfortably handle your largest models, but not too much higher than that. You can view the amount of memory used in the allotted workspace at any time by opening the solver status window and examining the Generator Memory Used field.

Unary Minus Priority The Unary Minus Priority box on the Model Generator tab:

is used to set the priority of the unary minus operator. The two available options are High and Low.

202 CHAPTER 5 There are two theories as to the priority that should be assigned to the unary minus (i.e., negation) operator in mathematical expressions. On the one hand, there is the Excel practice that the unary minus operator should have the highest priority, in which case, the expression -3^2 would evaluate to +9. On the other hand, there is the mathematicians’ preference for asg a lower priority to unary minus than is assigned to exponentiation, in which case, -3^2 evaluates to -9. Note that regardless which relative priority is used, one can force the desired result through the use of parenthesis. LINGO defaults to the Excel approach of setting a higher priority (High) on negation than on exponentiation.

Fixed Var Reduction The Fixed Var Reduction box on the Model Generator tab:

is used to control the degree to which fixed variables are substituted out of the ultimate math program ed to the solver engines. For example, consider the model: MAX= 20*X + 30*Y + 12*Z; X = 2*Y; X + Y + Z <= 110;

Y = 30;

If we run the LINGO|Generate command, we see that LINGO is able to reduce this model down to the equivalent, but smaller model: MAX= 12 * Z + 2100;

Z <= 20;

From the third constraint of the original model, it is obvious that Y is fixed at the value 30. Plugging this value for Y into the first constraint, we can conclude that X has a value of 60. Substituting these two fixed variables out of the original formulation yields the reduced formulation above. In most cases, substituting out fixed variables yields a smaller, more manageable model. In some cases, however, you may wish to avoid this substitution. An instance in which you might want to avoid substitution would be when equations have more than one root. When multiple roots are present, reduction may select a suboptimal root for a particular equation. On the other hand, the global and multistart solvers are adept at handling equations containing multiple roots. Thus, when using these solvers one may wish to forgo fixed variable reduction. The available options are:    

None, Always, Always, but linear only with global and multi, and Linear rows only.

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Selecting None disables all fixed variable reduction. Selecting Always enables reduction. When Always, but linear only with global and multi is selected, LINGO always enables reduction except when either the global or multistart solvers are selected, in which case it will only perform reduction on rows where the key variable appears linearly. The Linear rows only option always limits reduction to rows in which the key variable is linear. Note:

You should be careful when turning off fixed variable reduction. If the model generator is unable to substitute out fixed variables, you may end up turning a linear model into a more difficult nonlinear model.

LINGO defaults to selecting Always, but linear only with global and multi for fixed variable reduction.

Model Regeneration Box The Model Regeneration box on the Model Generator tab:

is used to control the frequency with which LINGO regenerates a model. Commands that will trigger the model generator are LINGO|Solve, LINGO|Generate, LINGO|Model Statistics, LINGO|Picture, LINGO|Debug, and File|Export File. The choices available under this option are:   

Only when text changes - LINGO regenerates a model only when a change has been made to the model’s text since the last generation took place. When text changes or with external references – LINGO regenerates whenever a change is made to the model text or when the model contains references to external data sources (e.g., text files, databases, or spreadsheets). Always - (default) LINGO always regenerates the model each time information regarding the generated model is needed.

LINGO defaults to always regenerating the model.

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Linearization The Linearization box on the Model Generator tab:

controls the linearization option in LINGO. Many nonlinear operations can be replaced by linear operations that are mathematically equivalent. The ultimate goal is to replace all the nonlinear operations in a model with equivalent linear ones, thereby allowing use of the faster and more robust linear solvers. We refer to this process as linearization. For more information on linearization, please refer to the section On Mathematical Modeling. Degree determines the extent to which LINGO will attempt to linearize models. The available options are:    

Solver Decides, None, Math only, Math and Logic.

Under the None option, no linearization occurs. With the Math Only option, LINGO linearizes the mathematical functions: @ABS(), @FLOOR(), @IF(), @MAX(), @MIN(), @SIGN(). @SMAX(), and @SMIN() along with any products of binary and continuous variables. The Math and Logic option is equivalent to the Math option plus LINGO will linearize all logical operators (#LT#, #LE#, #EQ#, #GT#, #GE#, and #NE#). Under the Solver Decides option, LINGO will do maximum linearization if the number of variables is less-than-or-equal-to 12, otherwise, LINGO will not perform any linearization. LINGO defaults to the Solver Decides setting. The Delta Coefficient is a tolerance indicating how closely you want the additional constraints added as part of linearization to be satisfied. Most models won’t require any changes to this parameter. However, some numerically challenging formulations may benefit from increasing Delta slightly. LINGO defaults to a Delta of 1.e-6. When LINGO linearizes a model, it will add forcing constraints to the mathematical program generated to optimize your model. These forcing constraints are of the form: f( x) = M • y where M is the BigM Coefficient and y is a 0/1 variable. The idea is that if some activity in the variables is occurring, then the forcing constraint will drive y to take on the value of 1. Given this, if we set the BigM value to be too small, we may end up with an infeasible model. Therefore, the astute reader might conclude that it would be smart to make BigM quite large, thereby minimizing the chance of an infeasible model. Unfortunately, setting BigM to a large number can lead to numerical stability problems in the solver resulting in infeasible or sub-optimal solutions. So, getting a good value for the BigM Coefficient may take some experimentation.

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As an example of linearization, consider the following model: MODEL: SETS: projects: baths, sqft, beds, cost, est; ENDSETS DATA: projects, p1 p2 p3 p4 p5 p6 p7 ; ENDDATA

beds, 5 2 1 4 4 3 3

baths, 4 1 1 3 2 1 2

sqft, 6200 820 710 4300 3800 2200 3400

cost = 559608 151826 125943 420801 374751 251674 332426

MIN = @MAX( projects: @abs( cost - est)); @FOR( projects: est = a0 + a1 * beds + a2 * baths + a3 * sqft ); END

Model: COSTING This model estimates the cost of home construction jobs based on historical data on the number of bedrooms, bathrooms, and square footage. The objective minimizes the maximum error over the sample project set. Both the @MAX() and @ABS() functions in the objective are non-smooth nonlinear functions, and, as a result, can present problems for LINGO’s default, local search NLP solver. Running the model under the default settings with linearization disabled, we get the following result: Local optimal solution found at step: Objective value: Variable A0 A1 A2 A3

Value 37441.55 27234.51 23416.53 47.77956

91 3997.347 Reduced Cost 0.000000 0.000000 0.000000 0.000000

Enabling linearization and re-optimizing yields the substantially better solution: Global optimal solution found at step: Objective value: Variable A0 A1 A2 A3

Value 46814.64 22824.18 16717.33 53.74674

186 1426.660

Reduced Cost 0.000000 0.000000 0.000000 0.000000

206 CHAPTER 5 Note that the maximum error has been reduced from 3,997 to 1,426! Linearization will substantially increase the size of your model. The sample model above, in unlinearized form, has a mere 8 rows and 11 continuous variables. On the other hand, the linearized version has 51 rows, 33 continuous variables, and 14 binary variables! Although linearization will cause your model to grow in size, you will tend to get much better solution results if the model can be converted entirely to an equivalent linear form. Note:

Linearization will be of most use when a nonlinear model can be 100% linearized. If LINGO can only linearize a portion of your model, then you may actually end up with a more difficult nonlinear model.

The linearization option is set to Solver Decides by default.

Allow Unrestricted Use of Primitive Set Member Names Check Box The Allow unrestricted use of primitive set member names checkbox on the Model Generator tab:

allows for backward compatibility with models created in earlier releases of LINGO. In many instances, you will need to get the index of a primitive set member within its set. Prior to release 4 of LINGO, you could do this by using the primitive set member’s name directly in the model’s equations. This can create problems when you are importing set from an external source. In this case, you will not necessarily know the names of the set beforehand. When one of the imported primitive set happens to have the same name as a variable in your model, unintended results can occur. More specifically, LINGO will not treat such a variable as optimizable. In fact, it would treat it as if it were a constant equal to the value of the index of the primitive set member! In short, different primitive set names can potentially lead to different results. Therefore, starting with release 4.0 of LINGO, models such as the following were no longer permitted: MODEL: SETS: DAYS /MO TU WE TH FR SA SU/; ENDSETS INDEX_OF_FRIDAY = FR; END

If you want the index of FR in the DAYS set, you should now use the @INDEX function: INDEX_OF_FRIDAY = @INDEX(DAYS, FR);

By default, LINGO disables the use of primitive set member names.

Assume Model Is Linear The Assume model is linear checkbox on the Model Generator tab:

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This option can be used for minimizing memory usage on models that are entirely linear. When this option is in effect, the model generator can take steps to dramatically reduce overall memory consumption without sacrificing performance. In fact, if all your models are linear, we recommend that you enable this option permanently as the default for your installation. The one restriction is that the model must prove to be entirely linear. If a single nonlinearity is detected, you will receive an error message stating that the model is nonlinear and model generation will cease. At which point, you should clear this option and attempt to solve the model again. By default, the Assume model is linear option is disabled.

Check for Duplicate Names The Check for duplicate names in data and model checkbox on the General Solver tab:

allows you to test your LINGO models from older releases for instances where primitive set appear in the model’s equations. The next time you run a model, LINGO will issue an error message if duplicate names appear as set and as variables in the model. Earlier releases of LINGO allowed you to use primitive set names in the equations of a model. Primitive set names in a model’s equations returned the index of the set member. Starting with release 4.0, LINGO required you to use the @INDEX function (see the Chapter 7, LINGO's Operators and Functions) to get the index of a primitive set member. By default, this option is disabled.

Enforce Bounds in CALC and DATA The Enforce Bounds In CALC and DATA checkbox on the General Solver tab:

causes LINGO to check the values of any variables input in calc and data sections against their bounds. Normally, variable bounds apply only to optimizable variables that appear in the model's constraints and not to variables fixed to constant values in calc and data sections. If you would also like LINGO to check variables fixed in calc and data sections against their bounds, then you will need to enable this option. By default, this option is disabled.

Loop Optimization The Loop optimization checkbox on the General Solver tab:

208 CHAPTER 5 is used to either enable or disable LINGO's loop optimization step. Loop optimization reformulates expressions containing set looping functions in order to make them more efficient, while maintaining mathematical equivalency. The end goal of loop optimization is to minimize the number of es through the inner loop of any nested loops in an expression. As an example, consider the following transportation model fragment, that just contains constraints for satisfying customer demand: MODEL: ! A transportation problem fragment; SETS: WAREHOUSE /1..50/ : CAPACITY; CUSTOMER /1..5000/ : DEMAND; ROUTES( WAREHOUSE, CUSTOMER) : COST, VOLUME; ENDSETS ! The demand constraints; @FOR( CUSTOMER( J): @SUM( ROUTES( I, J): VOLUME( I, J)) >= DEMAND( J) ); END

Transportation Model Fragment with Inefficient Constraints In the demand constraints expression there are two loops  an outer @FOR() over the CUSTOMER set and an inner @SUM() over the ROUTES set. As written, the inner loop must be executed 5000*50*5000=1.25 billion times. Note that a valid reformulation of these demand constraints would be: ! The demand constraints; @FOR( CUSTOMER( J): @SUM( WAREHOUSE( I): VOLUME( I, J)) >= DEMAND( J) );

With the expression rewritten in this manner, the inner loop will now only be executed 50*5000 times, reducing the total number of es by a factor of 5000. LINGO's loop optimizer seeks out such inefficient loops and, invisible to the , rewrites them before the model gets ed to the generator. Of course, the end result of such reformulations are faster generation times. In fact, the speedup from reformulation can be quite dramatic in many cases. In this particular example, the reformulated version ran over 400 times faster in the model generator. Note that in some cases, particularly with models that primarily contain sparse sets, loop optimization may actually increase runtimes. So, you should compare runtimes with, and without, the feature being enabled. By default, the Loop optimization feature is disabled.

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Minimize Memory Usage The Minimize memory usage checkbox on the General Solver tab:

may be used to guide LINGO’s memory usage. Enabling Minimize memory usage causes LINGO to opt for less memory usage when solving a model. The downside is that opting for less memory may result in longer runtimes. LINGO defaults to disabling Minimize memory usage.

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Linear Solver Tab The Linear Solver tab on the Options dialog box, pictured here:

can be used to control several options, discussed below, for tailoring the operation of LINGO’s linear solver. The linear solver is used on linear models and on mixed integer linear models as part of the branch-and-bound process.

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Method Box The Method box on the Linear Solver tab:

is used to control the algorithm LINGO’s linear solver employs. The current choices are:    

Solver Decides - LINGO selects the algorithm it determines is most appropriate. Primal Simplex - LINGO uses a primal simplex algorithm. Dual Simplex - LINGO uses a dual simplex algorithm. Barrier - LINGO uses a barrier algorithm (i.e., interior point).

The simplex algorithm moves around the exterior of the feasible region to the optimal solution, while the interior point algorithm, Barrier, moves through the interior of the feasible region. In general, it is difficult to say which algorithm will be fastest for a particular model. A rough guideline is Primal Simplex tends to do better on sparse models with fewer rows than columns. Dual Simplex does well on sparse models with fewer columns than rows. Barrier works best on densely structured models or very large models. The barrier solver is available only as an additional option to the LINGO package. Furthermore, if the model has any integer variables, the barrier solver will be used for solving the LP at the initial root node of the branch-and-bound tree, but may or may not be used on subsequent nodes. From a performance point-of-view, the barrier solver’s impact will be reduced on integer models. LINGO defaults to the Solver Decides option.

Model Reduction Box The Model Reduction box on the Linear Solver tab:

is used to control the amount of model reduction performed by LINGO’s linear solver. Your options are:   

Off - Disables reduction, On - Reduction is used on all models, and Solver Decides - LINGO decides whether or not to use reduction.

When this option is enabled, LINGO attempts to identify and remove extraneous variables and constraints from the formulation before solving. In certain cases, this can greatly reduce the size of the final model to be solved. Sometimes, however, reduction merely adds to solution times without trimming back much on the size of the model. LINGO defaults to the Solver Decides option.

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Feasibility Tolerance Boxes The Initial Linear Feasibility Tol. and the Final Linear Feasibility Tol. boxes on the Linear Solver tab:

are used to control the feasibility tolerances for the linear solver. These tolerances are related to how closely constraints must be satisfied in linear models. In general, if your models are well formulated, you should not have to modify these tolerances. However, access to these tolerances is provided for the expert . Due to the finite precision available for floating point operations on digital computers, LINGO can’t always satisfy each constraint exactly. Given this, LINGO uses these two tolerances as limits on the amount of violation allowed on a constraint while still considering it “satisfied”. These two tolerances are referred to as the Initial Linear Feasibility Tolerance (ILFT) and the Final Linear Feasibility Tolerance (FLFT). The default values for these tolerances are, respectively, .000003 and .0000001. The ILFT is used when the solver begins iterating. In the early stages of the solution process, having the solver less concerned with accuracy issues can boost performance. When LINGO thinks it has an optimal solution, it switches to the more restrictive FLFT. At this stage in the solution process, you want a relatively high degree of accuracy. Thus, the FLFT should be smaller than the ILFT. One instance where these tolerances can be of use is when LINGO returns a solution that is almost feasible. You can this by checking the values in the Slack or Surplus column in the model’s solution report. If there are only a few rows with small negative values in this column, then you have a solution that is close to being feasible. Loosening (i.e., increasing) the ILFT and FLFT may help you get a feasible solution. This is particularly true in a model where scaling is poor (i.e., very large and very small coefficients are used in the same model), and the units of measurement on some constraints are such that minor violations are insignificant. For instance, suppose you have a budget constraint measured in millions of dollars. In this case, a violation of a few pennies would be of no consequence. Short of the preferred method of rescaling your model, loosening the feasibility tolerances may be the most expedient way around a problem of this nature.

Pricing Strategies Box The Pricing Strategies box on the Linear Solver tab:

is used to control the pricing strategy used by LINGO’s simplex solvers. Pricing determines the relative attractiveness of the variables during the simplex algorithm.

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For the Primal Solver, you have the following choices:   

Solver Decides - LINGO selects the pricing method it believes is the most appropriate. Partial - LINGO prices out a small subset of variables at each iteration and intermittently prices out all the variables to determine a new subset of interesting variables. Devex - Devex prices out all columns at each iteration using a steepest-edge approximation (see below).

Partial pricing tends to yield faster iterations. Devex, while slower, results in fewer overall iteration and can be helpful when models are degenerate. Thus, it is difficult to determine what method is superior beforehand. For the Dual Solver, you have these options:     

Solver Decides - LINGO selects the pricing method it believes is the most appropriate. Dantzig - The dual simplex solver will tend to select variables that offer the highest absolute rate of improvement to the objective regardless of how far other variables may have to move per unit of movement in the newly introduced variable. Steepest Edge - The dual solver spends a little more time selecting variables by looking at the total improvement in the objective when adjusting a particular variable. Devex - Devex prices out all columns at each iteration using a steepest-edge approximation. Approximate Devex - An simplified implementation of true Devex pricing.

Dantzig pricing generally yields faster iterations, however, the other variables in the model may quickly hit a bound resulting in little gain to the objective. With the steepest-edge option, each iteration will tend to lead to larger gains in the objective resulting in fewer overall iterations, however, each iteration will tend to take more compute time due to increased time spent in pricing. The Devex options approximate true steepest-edge pricing.

Multi-Core Box The Multi-Core box on the Linear Solver tab:

may be used to perform parallel solves of linear programs on multiple cores. One of four different linear solvers is chosen for each core. LINGO will take the solution from the solver that finishes first and then interrupt the remaining solver threads. The idea behind this approach is that different linear solvers will have relatively better or worse performance on different classes of models. However, it may be difficult to predict beforehand the solver that is most likely to outperform. So, by enabling multi-core solves, we guarantee that we will

214 CHAPTER 5 always get top performance, even without knowledge beforehand of which solver is likely to run the fastest. Note:

The multi-core feature requires that your machine have at least one core free for each solver you wish to run. Using this feature with an inadequate number of cores will tend to decrease overall performance.

For the Cores to Use parameter, you have the following choices: Off, 2, 3, or 4. When the default Off option is selected, the multi-core feature is disabled, and LINGO will run only one solver on linear programs, namely the one specified as part of the Solver Method option detailed above. When either option 2, 3, or 4 is selected, LINGO will run linear solvers in the requested number of cores. When selecting two or more cores, you will have the option to specify which of the linear solvers to use in each of the running cores as part of the Core1 - Core4 list boxes. The available linear solvers are:    

Primal1 Dual Barrier Primal2

- Primal simplex algorithm 1 - Dual simplex algorithm - Barrier/Interior point solver (available as a option) - Primal simplex algorithm 2, installed as part of the Barrier option

As an example, the settings of the Multi-Core box below are requesting to run LP solvers in two cores, with core 1 running the dual simplex solver and core 2 running the barrier solver:

While LINGO is solving linear programs it normally displays solver statistics in the Solver Status Window. This will also be true with muti-core solves. However, LINGO reports the statistics from only one of the solvers, specifically, the solver selected to run in Core 1. Once optimization is complete, LINGO will populate the Solver Status Window with statistics from the solver that finished first. Finally, as part of the solution report, LINGO will display a line indicating the solver that finished first. In the solution report excerpt below, we see that the dual simplex solver was the first to completion: First returning solver: DUAL SIMPLEX Global optimal solution found. Objective value: Infeasibilities: Total solver iterations:

1272282. 0.9313226E-09 34862

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Linear Optimality Tolerance Box The Linear Optimality Tolerance box on the Linear Solver tab:

allows you to control the setting for the linear optimality tolerance. This tolerance is used to determine whether a reduced cost on a variable is significantly different from zero. You may wish to loosen this tolerance (make it larger) on poorly scaled and/or large models to improve performance. The default setting for the Linear Optimality Tolerance is 1.e-7.

Basis Refactor Frequency Box The Basis Refactor Frequency box on the Linear Solver tab:

allows you to control the how frequently the linear solver refactors the basis matrix. The options are either Solver Decides or some positive integer quantity. If an integer value, N, is selected, then the linear solver will refactor every N iterations. Numerically tough and/or poorly scaled models may benefit from more frequent refactoring. However, refactoring too frequently will cause the solver to slow down. The default setting for the Basis Refactor Frequency is Solver Decides, which will typically result in refactoring about once every 100 iterations.

Barrier Crossover The Barrier Crossover box on the Linear Solver tab:

is used to control whether or not the barrier solver performs a crossover operation. Unlike simplex algorithms, the barrier solver does not automatically find basic (cornerpoint) solutions. Very roughly speaking, basic solutions have the nice mathematical property that exactly m variables will have nonzero values, where m is the number of constraints. The crossover procedure takes the barrier’s non-basic solution, and, through the use of a simplex solver, converts the non-basic solution to a basic one. If the basic solution property is not important for your models, then you may wish to disable crossovers to improve performance when using the barrier solver. Note:

Barrier crossover must be enabled if you plan to do range analysis. Range computations cannot be performed if the final solution is not a basic solution.

216 CHAPTER 5 The default is to perform crossovers.

Matrix Decomposition The Matrix Decomposition box on the Linear Solver tab:

allows you to enable the matrix decomposition feature. Many large-scale linear and mixed integer problems have constraint matrices that are totally decomposable into a series of block structures. If total decomposition is possible, LINGO will solve the independent problems sequentially and report a solution for the original model, resulting in dramatic speed improvements. LINGO defaults to not using matrix decomposition.

Scale Model Checkbox The Scale Model box on the Linear Solver tab:

allows you to enable to matrix scaling option. This option rescales the coefficients in the model’s matrix, so the ratio of the largest to smallest coefficients is reduced. This reduces the chances of round-off error, which leads to greater numerical stability and accuracy in the linear solver. LINGO defaults to using scaling.

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Nonlinear Solver Tab The Nonlinear Solver tab on the Options dialog box, pictured here:

controls several options that affect the operation of LINGO’s solver on nonlinear models.

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Initial Nonlinear Feasibility Tolerance Final Nonlinear Feasibility Tolerance The Initial Nonl Feasibility Tol and the Final Nonl Feasibility Tol boxes on the Nonlinear Solver tab:

are used to control the feasibility tolerances for the nonlinear solver in the same manner that the Initial Linear and Final Linear Feasibility Tolerance are used by the linear solver. For information on how and why these tolerances are useful, refer to the Feasibility Tolerances section in the Linear Solver Tab section immediately above. Default values for these tolerances are, respectively, .001 and .000001.

Nonlinear Optimality Tolerance The Nonlinear Optimality Tol box on the Nonlinear Solver tab:

is used to control the adjustments to variables as described below. While solving a model, the nonlinear solver is constantly computing a gradient. The gradient gives the rate of improvement of the objective function for small changes in the variables. If the gradient’s rate of improvement computation for a given variable is less-than-or-equal-to the Nonlinear Optimality Tolerance, further adjustments to the variable’s value are not considered to be beneficial. Decreasing this tolerance towards a limit of 0 will tend to make the solver run longer and may lead to better solutions in poorly formulated or poorly scaled models. The default value for the Nonlinear Optimality Tolerance is .0000001.

Slow Progress Iteration Limit The Slow Progress Iteration Limit (SPIL) box on the Nonlinear Solver tab:

is used to terminate the solution process if little or no progress is being made in the objective value. Specifically, if the objective function’s value has not improved significantly in n iterations, where n is the value of SPIL, the nonlinear solver will terminate the solution process. Increasing this tolerance’s value will tend to force the solver to run longer and may be useful in models with relatively “flat” objective functions around the optimal solution. The default value for SPIL is 5 iterations.

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This option is available on the off chance that the older version of the nonlinear solver, Ver 1.0, should perform better on a particular model. LINGO defaults to Solver Decides for the nonlinear solver version.

Derivative Computation The Derivatives box on the Nonlinear Solver tab:

The First Order option determines how the nonlinear solver computes first order derivatives. There are two general methods available: numerical or analytical derivatives. Analytical derivatives are computed directly by symbolically analyzing the arithmetic operations in a constraint. Numerical derivatives are computed using finite differences. There are two types of numerical derivatives available using either central differences or forward differences. There are also two types of analytical derivatives available: backward analytical and forward analytical. Finally, a Solver Decides option is also available. LINGO defaults to the Solver Decides setting for the First Order option, which presently involves LINGO using backward analytical derivatives. However, one of the other choices may be more appropriate for certain classes on nonlinear models. We suggest you try the various derivative options to see which works best for your particular models. The Use Second Order option determines if the nonlinear solver will use second order derivatives. If used, second order derivatives will always be computed analytically. Computing second order derivatives will take more time, but the additional information they provide may lead to faster runtimes and/or more accurate solutions. LINGO defaults to not using second order derivatives.

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Strategies Box The Strategies box on the Nonlinear Solver tab:

allows you to set the following options:      

Crash Initial Solution, Quadratic Recognition, Selective Constraint Eval, SLP Directions, SLP Solver, and Steepest Edge.

Crash Initial Solution If the Crash Initial Solution box is checked, LINGO’s nonlinear solver will invoke a heuristic for generating a “good” starting point when you solve a model. If this initial point is relatively good, subsequent solver iterations should be reduced along with overall runtimes. LINGO defaults to not crashing an initial solution.

Quadratic Recognition If the Quadratic Recognition box is checked, LINGO will use algebraic preprocessing to determine if an arbitrary nonlinear model is actually a quadratic programming (QP) model. If a model is found to be a QP model and convex, then it can be ed to the faster quadratic solver. Note that the QP solver is not included with the standard, basic version of LINGO, but comes as part of the Barrier option. LINGO defaults to using quadratic recognition.

Selective Constraint Evaluation If the Selective Constraint Eval box is checked, LINGO’s nonlinear solver will only evaluate constraints on an as needed basis. Thus, not every constraint will be evaluated during each iteration. This generally leads to faster solution times, but can also lead to problems in models that have functions that are undefined in certain regions.

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LINGO may not evaluate a constraint for many iterations only to find that it has moved into a region where the constraint is no longer defined. In this case, there may not be a valid point for the solver to retreat to, and the solution process terminates with an error. Turning off selective constraint evaluation eliminates these errors. LINGO defaults to not using Selective Constraint Eval.

SLP Directions If the SLP Directions box is checked, LINGO’s nonlinear solver will use successive linear programming to compute new search directions. This technique uses a linear approximation in search computations in order to speed iteration times. In general, however, the number of total iterations will tend to rise when SLP Directions are used. LINGO defaults to not using SLP Directions.

SLP Solver If the SLP Solver box is checked, LINGO uses a successive linear programming (SLP) algorithm for its nonlinear solver. This technique uses a linear approximation of the true nonlinear model within successive, small regions. This then allows the use of the fast linear solver for optimizing each linear subregion. In general, the SLP solver will not be well suited for highly nonlinear models. However, it may offer better performance than the standard nonlinear solver when a model has few nonlinear variables, such that the model is "mostly linear". LINGO defaults to not enabling the SLP Solver option.

Steepest Edge If the Steepest Edge box is checked, LINGO’s nonlinear solver will use the steepest-edge strategy when selecting variables to iterate on. When LINGO is not in steepest-edge mode, the nonlinear solver will tend to select variables that offer the highest absolute rate of improvement to the objective, regardless of how far other variables may have to move per unit of movement in the newly introduced variable. The problem with this strategy is that other variables may quickly hit a bound, resulting in little gain to the objective. With the steepest-edge option, the nonlinear solver spends a little more time in selecting variables by looking at the rate that the objective will improve relative to movements in the other nonzero variables. Thus, on average, each iteration will lead to larger gains in the objective. In general, the steepest-edge option will result in fewer iterations. However, each iteration will take longer. LINGO defaults to not using the Steepest Edge option.

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Integer Pre-Solver Tab The Integer Pre-Solver tab on the Options dialog box, pictured here:

can be used to control several options for tailoring the operation of LINGO’s integer programming pre-solver. The integer pre-solver does a great deal of model reformulation, so that the final formulation ed to the branch-and-bound solver may be solved as fast as possible. The reformulated model is always mathematically equivalent to the original formulation, but it is structured in such a way that it is best suited for solution by the branch-and-bound integer programming algorithm.

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The integer pre-solver operates only with linear integer models (i.e., models that make use of the @BIN and @GIN functions to restrict one or more variables to integer values). Integer pre-solver option settings have no effect on nonlinear integer models.

Heuristics The Heuristics box on the Integer Pre-Solver tab:

controls the level of integer programming heuristics used by the integer solver. These heuristics use the continuous solution at each node in the branch-and-bound tree to attempt to quickly find a good integer solution. The Cutoff Criterion is used to control the criterion for terminating heuristics. Choices here are Solver Decides, Time, and Iterations. Under the Time setting, LINGO terminates heuristics after a certain amount of elapsed time. The Iterations option terminates heuristics after a certain number of iterations. In general, the Time setting results in the fastest performance. However, due to shifting computational loads on a machine, solution paths may change under the Time setting from one solve to the next, potentially resulting in non-reproducible solutions. If reproducibility of a runs is a concern, then the Iterations option should be selected. Under the Solver Decides setting, LINGO chooses the most appropriate strategy. The Min Seconds field specifies the minimum amount of time to spend on heuristics at each node. The default settings are 3 for Level, Solver Decides for Cutoff Criterion, and 0 for Min Seconds.

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Constraint Cuts Box The tolerances contained in the Constraint Cuts box on the Integer Pre-Solver tab:

can be used to control the solver’s cut generation phase on linear models. LINGO’s integer programming pre-solver performs extensive evaluation of your model in order to add constraint cuts. Constraint cuts are used to “cut” away sections of the feasible region of the continuous model (i.e., the model with integer restrictions dropped) that are not contained in the feasible region to the integer model. On most integer models, this will accomplish two things. First, solutions to the continuous problem will tend to be more naturally integer. Thus, the branch-and-bound solver will have to branch on fewer variables. Secondly, the bounds derived from intermediate solutions will tend to be tighter, allowing the solver to “fathom” (i.e., drop from consideration) branches higher in the branch-and-bound tree. These improvements should dramatically speed solution times on most integer models. Note:

Cuts are not applied to nonlinear models. Thus, modifying any of the tolerances in the Constraint Cuts box will have no bearing on nonlinear models.

Application In the Application drop-down box of the Constraint Cuts box:

you can control the nodes in the solution tree where the branch-and-bound solver adds cuts.

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If you pull down the selection list, you will find three options:   

Root Only, All Nodes, and Solver Decides.

Under the Root Only option, the solver appends cuts only at the first node, or root node, in the solution tree. With the All Nodes option, cuts are appended at each node of the tree. The Solver Decides option causes the solver to dynamically decide when it is best to append cuts at a node. The default is to let the solver decide when to append cuts. In general, this will offer superior performance. There may be instances, however, where one of the other two options prevails.

Relative Limit In the Relative Limit field of the Constraint Cuts box:

you can control the number of constraint cuts that are generated by the integer pre-solver. Most integer programming models benefit from the addition of some constraint cuts. However, at some point additional cuts take more time to generate than they save in solution time. For this reason, LINGO imposes a relative limit on the number of constraint cuts. The default limit is set to .75 times the number of true constraints in the original formulation. This relative limit may be overridden by changing it in the Relative Limit field.

Max es In the Max es box of the Constraint Cuts box:

you can control the number of iterative es the integer pre-solver makes through a model to determine appropriate constraint cuts to append to the formulation. In general, the benefits of each successive decline. At some point, additional es will only add to the total solution time. Thus, LINGO imposes a limit on the maximum number of es. The default limit is 100 es at the root node of the branch-and-bound tree, and 2 es at all subsequent nodes. You can override these limits by changing the values in the Root and Tree fields.

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Types The Types box of the Constraint Cuts box:

is used to enable or disable the different strategies LINGO uses for generating constraint cuts. LINGO uses twelve different strategies for generating constraint cuts. The default is for all cut generation strategies to be enabled with the exception of Objective cuts. It is beyond the scope of this manual to go into the details of the various strategies. Interested readers may refer to any good text on integer programming techniques. In particular, see Nemha and Wolsey (1988).

Probing Level The Probing Level option on the Integer Pre-Solver tab can be used on mixed integer linear programs to perform an operation known as probing. Probing involves taking a close look at the integer variables in a model and deducing tighter variable bounds and right-hand side values. In many cases, probing can tighten an integer model sufficiently to speed overall solution times. In other cases, however, probing may not be able to do much tightening, and the overall solution time will increase due to the extra time spent probing. Pulling down the selection list for the Probing Level field:

you will see that you can choose one of eight different probing levels. A probing level of 1 means probing is disabled, while levels 2 through 7 indicate successively higher degrees of probing. The default setting for this option, Solver Decides, leaves the decision up to LINGO to select the probing level.

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Integer Solver Tab The Integer Solver tab on the Options dialog box, pictured here:

can be used to control several tolerances for tailoring the operation of LINGO’s branch-and-bound solver used on integer models (i.e., models making use of the @BIN and @GIN functions to restrict one or more variables to integer values).

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Branching Box The Branching box on the Integer Solver tab:

contains the following two options for controlling the branching strategy used by LINGO’s branchand-bound solver:  Direction, and  Priority.

Direction LINGO uses a branch-and-bound solution procedure when solving integer programming models. One of the fundamental operations involved in the branch-and-bound algorithm is branching on variables. Branching involves forcing an integer variable that is currently fractional to either the next greatest or the next lowest integer value. As an example, suppose there is a general integer variable that currently has a value of 5.6. If LINGO were to branch on this variable, it would have to choose whether to set the variable first to 6 or 5. The Direction field controls how LINGO makes this branching decision. If you pull down the drop-down box for the Direction option, you’ll find the following:

The default, Both, involves LINGO making an intelligent guess as to whether it should branch up or down first on each individual variable. If Up is selected, LINGO will always branch up first. If Down is selected, LINGO will always branch down first. In most cases, the Both option will result in the best performance.

Priority When branching on variables, the branch-and-bound procedure can give priority to branching on the binary variables first, or it can make an intelligent guess as to the next best variable to branch on, regardless of whether it is binary or general. The Priority field controls how LINGO makes this branching decision. If you pull down the drop-down box for Priority, you’ll find the following:

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Select Binary to have LINGO give branching priority to the binary variables. Select LINGO Decides to have LINGO select the next integer variable for branching based on an intelligent guess, regardless of whether it is binary or general. The default is LINGO Decides, which should generally give the best results.

Integrality Box Due to the potential for round-off error on digital computers, it is not always possible for LINGO to find exact integer values for the integer variables. The Integrality box on the Integer Solver tab:

contains the following three options for controlling the amount of deviation from integrality that will be tolerated:  Absolute Integrality,  Relative Integrality, and  BigM Threshhold.

Absolute Integrality The Absolute Integrality tolerance is used by LINGO as a test for integrality in integer programming models. Due to round-off errors, the “integer” variables in a solution may not have values that are precisely integer. The absolute integrality tolerance specifies the absolute amount of violation from integrality that is acceptable. Specifically, if X is an "integer" variable and I is the closest integer to X, then X would be accepted as being integer valued if: | X – I | <= Absolute Integrality Tolerance. The default value for the absolute integrality tolerance is .000001. Although one might be tempted to set this tolerance to 0, this may result in feasible models being reported as infeasible.

Relative Integrality The Relative Integrality tolerance is used by LINGO as a test for integrality in integer programming models. Due to round-off errors, the “integer” variables in a solution may not have values that are precisely integer. The relative integrality tolerance specifies the relative amount of violation from integrality that is acceptable. Specifically, if I is the closest integer value to X, X will be considered an integer if: | X – I | <= Relative Integrality Tolerance. |X| The default value for the relative integrality tolerance is .000008. Although one might be tempted to set this tolerance to 0, this may result in feasible models being reported as infeasible.

BigM Threshold Many integer programming models have constraints of the form:

230 CHAPTER 5 f(x) ≤ M * z where f(x) is some function of the decision variables, M is a large constant term, and z is a binary variable. These types of constraints are called forcing constraints and are used to force the binary variable, z, to 1 when f(x) is nonzero. In many instances, the binary variable is multiplied by a fixed cost term in the objective; a fixed cost that is incurred when a particular activity, represented by f(x), occurs. The large constant term, M, Is frequently referred to as being a BigM coefficient. Setting BigM too small can lead to infeasible or suboptimal models. Therefore, the BigM value will typically have to be rather large in order to exceed the largest activity level of f(x). When BigM is large, the solver may discover that by setting z slightly positive (within normal integrality tolerances), it can increase f(x) to a significant level and thereby improve the objective. Although such solutions are technically feasible to tolerances, they are invalid in that the activity is occurring without incurring its associated fixed cost. The BigM threshold is designed to avoid this problem by allowing LINGO to identify the binary variables that are being set by forcing constraints. Any binary variable with a coefficient larger than the BigM threshold will be subject to a much tighter integrality tolerance. The default value for the BigM Threshold is 1.e8.

LP Solver Box In a mixed linear integer programming model, LINGO’s branch-and-bound solver solves a linear programming model at each node of the solution tree. LINGO has a choice of using the primal simplex, dual simplex, or barrier solver (assuming the barrier option was purchased with your license) for handling these linear programs. The LP Solver box on the Integer Solver tab:

contains the following two options for controlling this choice of linear program solver:  Warm Start, and  Cold Start

Warm Start The Warm Start option controls the linear solver that is used by the branch-and-bound solver at each node of the solution tree when a previous solution is present to use as a “warm start”. The cold start option, discussed below, determines the solver to use when a previous solution does not exist. If you pull down the drop-down box for Warm Start, you’ll find the following:

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The available options are:    

LINGO Decides – LINGO chooses the most appropriate solver. Barrier – LINGO uses the barrier method, assuming you have purchased a license for the barrier solver. Otherwise, the dual solver will be used. Primal – LINGO uses the primal solver exclusively. Dual – LINGO uses the dual solver exclusively.

In general, LINGO Decides will yield the best results. The barrier solver can’t make use of a pre-existing solution, so Barrier usually won’t give good results. In general, Dual will be faster than Primal for reoptimization in branch-and-bound.

Cold Start The Cold Start option controls the linear solver that is used by the branch-and-bound solver at each node of the solution tree when a previous solution is not present to use as a “warm start”. The Warm Start option, discussed above, determines the solver to use when a previous solution does exist. If you pull down the drop-down box for Cold Start, you’ll find the following:

The available options are:    

LINGO Decides – LINGO chooses the most appropriate solver at each node. Barrier – LINGO uses the barrier method, assuming you have purchased a license for the barrier solver. Otherwise, the dual solver will be used. Primal – LINGO uses the primal solver exclusively. Dual – LINGO uses the dual solver exclusively.

In general, LINGO Decides will yield the best results. However, experimentation with the other options may be fruitful.

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Optimality Box The Optimality Box on the Integer Solver tab:

is used to control three tolerances: Absolute, Relative, and Time to Relative. These tolerances control how close you want the solver to come to the optimal solution. Ideally, we’d always want the solver to find the best solution to a model. Unfortunately, integer programming problems are very complex, and the extra computation required to seek out the absolute best solution can be prohibitive. On large integer models, the alternative of getting a solution within a few percentage points of the true optimum after several minutes of runtime, as opposed to the true optimum after several days, makes the use of these tolerances quite attractive.

Absolute The Absolute Optimality tolerance is a positive value r, indicating to the branch-and-bound solver that it should only search for integer solutions with objective values at least r units better than the best integer solution found so far. In many integer programming models, there are huge numbers of branches with roughly equivalent potential. This tolerance helps keep the branch-and-bound solver from being distracted by branches that can’t offer a solution significantly better than the incumbent solution. In general, you shouldn’t have to set this tolerance. Occasionally, particularly on poorly formulated models, you might need to increase this tolerance slightly to improve performance. In most cases, you should experiment with the relative optimality tolerance, discussed below, rather than the absolute optimality tolerance in order to improve performance. The default value for the absolute optimality tolerance is 8e-8.

Relative The Relative Optimality tolerance is a value r, ranging from 0 to 1, indicating to the branch-and-bound solver that it should only search for integer solutions with objective values at least 100*r% better than the best integer solution found so far. The end results of modifying the search procedure in this way are twofold. First, on the positive side, solution times can be improved tremendously. Second, on the negative side, the final solution obtained by LINGO may not be the true optimal solution. You will, however, be guaranteed the solution is within 100*r% of the true optimum. Typical values for the relative optimality tolerance would be in the range .01 to .05. In other words, you would be happy to get a solution within 1% to 5% of the true optimal value. On large integer models, the alternative of getting a solution within a few percentage points of the true optimum after several minutes of runtime, as opposed to the true optimum after several days, makes the use of an optimality tolerance quite attractive.

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Generally speaking, the relative optimality tolerance is the tolerance that will most likely improve runtimes on integer models. You should be sure to set this tolerance whenever near optimal solutions are acceptable. Do keep in mind that when you set this option, LINGO may not return the true global optimum. However, you will be guaranteed the solution is within 100*r% of the true optimum.

The default for the relative optimality tolerance is 5e-8.

Time to Relative If an integer programming model is relatively easy to solve, then we would like to have the solver press on to the true optimal solution without immediately resorting to a relative optimality tolerance, discussed above. On the other hand, if, after running for a while, it becomes apparent that the optimal solution won’t be immediately forthcoming, then you might want the solver to switch to using a relative optimality tolerance. The Time to Relative tolerance can be used in this manner. This tolerance is the number of seconds before the branch-and-bound solver begins using the relative optimality tolerance. For the first n seconds, where n is the value of the time to relative tolerance, the branch-and-bound solver will not use the relative optimality tolerance and will attempt to find the true optimal solution to the model. Thereafter, the solver will use the relative optimality tolerance in its search. The default value for the time to relative tolerance is 100 seconds.

Tolerances Box The Tolerances box on the Integer Solver tab:

contains three miscellaneous tolerances for controlling the branching strategy used by the branch-and-bound solver on integer programming models. The three tolerances are Hurdle, Node Selection, and Strong Branch.

Hurdle If you know the objective value of a solution to a model, you can enter it as the Hurdle tolerance. This value is used in the branch-and-bound solver to narrow the search for the optimum. More specifically, LINGO will only search for integer solutions in which the objective is better than the hurdle value. This comes into play when LINGO is searching for an initial integer solution. LINGO can ignore branches in the search tree with objective values worse than the hurdle value, because a better solution exists (i.e., the solution whose objective value equals the hurdle tolerance) on some alternate branch. Depending on the problem, a good hurdle value can greatly reduce solution time. Once LINGO finds an initial integer solution, however, the hurdle tolerance no longer has an effect. At this point, the Relative Optimality tolerance comes into play.

234 CHAPTER 5 Note:

Be sure when entering a hurdle value that a solution exists that is at least as good or better than your hurdle. If such a solution does not exist, LINGO will not be able to find a feasible solution to the model.

The default hurdle value is None. In other words, the solver does not use a hurdle value.

Node Selection The branch-and-bound solver has a great deal of freedom in deciding how to span the branch-and-bound solution tree. The Node Selection option allows you to control the order in which the solver selects branch nodes in the tree. If you examine the pull down list for Node Selection, you will see the following:

The four choices function as follows:    

LINGO Decides – This is the default option. LINGO makes an educated guess as to the best node to branch on next. Depth First – LINGO spans the branch-and-bound tree using a depth first strategy. Worst Bound – LINGO picks the node with the worst bound. Best Bound – LINGO picks the node with the best bound.

In general, LINGO Decides will offer the best results. Experimentation with the other three choices may be beneficial with some classes of models.

Strong Branch The Strong Branch field uses a more intensive branching strategy during the first n levels of the branch-and-bound tree, where n is the value in Strong Branch. During these initial levels, LINGO picks a subset of the fractional variables as branching candidates. LINGO then performs a tentative branch on each variable in the subset, selecting as the final candidate the variable that offers the greatest improvement in the bound on the objective. Although strong branching is useful in tightening the bound quickly, it does take additional computation time. Therefore, you may want to try different settings to determine what works best for your model. The default strong branch setting is 10 levels.

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K-Best Solutions Box The K-Best Solutions box on the Integer Solver tab:

is used to set the number of solutions desired as part of the K-Best solutions feature of LINGO's mixed integer solver. Whenever this value is greater than 1, say K, LINGO will will return up to K unique solutions to the model. These solutions will have the property that they are the next best solutions available in of their objective values. Less than K solutions may be returned if a sufficient number of feasible solutions do not exist. An example of the K-Best feature follows.

K-Best Solutions Example In order to illustrate the K-Best feature, we will be using a variant of the knapsack model discussed above in the Binary Integer Variables section. You may want to refer back to the earlier discussion if you are not familiar with the knapsack model.

236 CHAPTER 5 Here's our model: MODEL: SETS: ITEMS: INCLUDE, WEIGHT, RATING; MYFAVORITES( ITEMS); ENDSETS DATA: KNAPSACK_CAPACITY = 15; ITEMS WEIGHT BRATS 3 BROWNIES 3 BEER 3 ANT_REPEL 7 BLANKET 4 FRISBEE 1 SALAD 5 WATERMELON 7

RATING = 1 1 1 1 6 6 10 9;

MYFAVORITES = BRATS BROWNIES BEER; ENDDATA MAX = @SUM( ITEMS: RATING * INCLUDE); @SUM( ITEMS: WEIGHT * INCLUDE) <= KNAPSACK_CAPACITY; @FOR( ITEMS: @BIN( INCLUDE)); NUMFAVE = @SUM( MYFAVORITES: INCLUDE); END

Model: KBEST In this example, we are packing a picnic basket for a picnic we will be taking with a friend. Our friend's ratings of the candidate picnic items is given in the data section above. It turns out that our friend is health conscious and does not care much for bratwurst, brownies nor beer. This is unfortunate, because these happen to be our favorite items, which we indicate with a new subset of ITEMS called MYFAVORITES.

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If we solve the model as is, thus solely maximizing our friend's preferences, we get the following solution: Global optimal solution found. Objective value: 25.00000 Variable NUMFAVE INCLUDE( BRATS) INCLUDE( BROWNIES) INCLUDE( BEER) INCLUDE( ANT_REPEL) INCLUDE( BLANKET) INCLUDE( FRISBEE) INCLUDE( SALAD) INCLUDE( WATERMELON)

Value 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000

As indicated by the NUMFAVE variable, none of our favorite items are included in the optimal basket. Now, we like our friend a lot, and we want him to be happy. However, we are wondering if there isn't another combination of items that our friend might like almost as much that includes at least one of our favorite items. To investigate this question, we set the Desired Number parameter of the K-Best Solutions box on the LINGO|Options Integer Solver tab to 4:

This means that we would like LINGO to generate the 4 best solutions to the model. We then click OK and then run the LINGO|Solve command. At which point, the integer solver sees that the K-Best feature is being requested, and it automatically generates the 4 best solutions to the model. At which point, we are presented with the following dialog box:

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In the Candidate Solutions window we see that the solver was able to find 4 feasible next-best solutions to the model. The solutions are ranked in order by their objective values. There is also a column labeled Tradeoff, which lists the value in each solution of a designated tradeoff variable. Any scalar variable in a model can be selected as the tradeoff variable. In this example, there is only one scalar variable, NUMFAVE, so it is automatically selected as the tradeoff variable. The idea behind the tradeoff variable is that it allows you to weigh the tradeoffs in a model's objective value with a secondary goal. In this case, our secondary goal is the number of our favorite items in the picnic basket. In particular, we see that there are three solutions with slightly worse objective values (23 vs. 25) that include one of our favorite items. For example, if we selected solution 2 and pressed the View button, we'd see the following solution containing one of our favorite items, bratwurst: Objective value: Variable KNAPSACK_CAPACITY NUMFAVE INCLUDE( BRATS) INCLUDE( BROWNIES) INCLUDE( BEER) INCLUDE( ANT_REPEL) INCLUDE( BLANKET) INCLUDE( FRISBEE) INCLUDE( SALAD) INCLUDE( WATERMELON)

23.00000 Value 15.00000 1.000000 1.000000 0.000000 0.000000 0.000000 1.000000 1.000000 1.000000 0.000000

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The following buttons are available along the bottom edge of the K-Best dialog box: Help Cancel View Select

Displays online help regarding the K-Best feature. Cancels out of K-Best mode, closing the dialog box. Displays any solutions selected in the Candidate Solutions box. Allow you to select one of the candidate solutions as the final solution to the model.

These buttons allow you to examine selected solutions returned by the K-Best solver. Once you find a solution you believe to be the best, you can select it as the final solution. Once a final solution is selected, all subsequent solution reports will be based on that particular solution.

BNP Solver The BNP Solver box on the Integer Solver tab:

contains two parameters - Blocks and Heuristic - for controlling the branch-and-price (BNP) solver. The BNP solver is a mixed integer programming solver for solving models with block structures like the following: Minimize:  c(k) * x(k) Subject To:

 A(k) * x(k) = d

x(k) in X(k), for all k

(linking constraints) (decomposition structure)

where d, c(k) and x(k) are vectors and A(k) is a matrix with appropriate dimensions. x(k) contains decision variables and X(k) denotes a linear feasible domain for x(k). The BNP solver is a hybrid of branch-and-bound, column generation, and Lagrangean relaxation methods. It can help to find either the optimal solution or a better lower bound (the Lagrangean bound) for a minimization problem. Based on the decomposition structure, the solver divides the original problem into several subproblems, or blocks, and solves them (almost) independently, exploiting parallel processing if multiple cores are available. BNP may perform better than the default MIP solver if: a) the number of linking constraints is small, b) the number of blocks is large and they are of approximately the same size, and c) the number of available processors (or cores) is large, e.g., 4 or more. Also, there may be some models for which BNP finds a good solution and good bound more quickly than the default MIP algorithm, although it may take longer to prove optimality.

240 CHAPTER 5 The Blocks option for the BNP solver controls the number of subproblems, or blocks, that the model will be partitioned into. Possible setting for the Blocks parameter are:  Row Names - Row names are constructed in such a way as to specify each row's block (an example is given below).  Off - This will disable the BNP solver, in which case, the standard MIP solver will be used to solve all mixed integer linear programs.  Specified - The explicitly specifies each row's block using the @BLOCKROW function.  N - A positive integer, greater-than-or-equal-to 2, indicating the number of independent blocks to try and partition the model into via one of the graph partitioning algorithms provided by LINGO. The actual heuristic used is chosen with the Heuristic parameter. The default setting for Blocks is Off, i.e., the BNP solver will not be used on integer programming models. The Block Heuristic parameter controls the heuristic used to partition the model into blocks. You may currently select from two graph partitioning algorithms named simply GP1 and GP2, with the default setting being GP1. As an example, consider the following model: MIN = x1 + x2 [c1] x1 + x2 constraint; [c2] x1 + x2 [c3] x2 [c4] [c5] @bin( x1); @bin( @bin( x4); @bin(

+ x3 + x4 + x5 + x6; + x3 + x4 + x5 + x6 >=3; + x3 x4 + x5 + x6 x4 + x6 x2); @bin( x3); x5); @bin( x6);

<=1; <=1; <=2; <=1;

!linking !block !block !block !block

1; 1; 2; 2;

The above model has six variables and five constraints. Constraint 1 will be the only linking constraint, with linking constraints referred to as being in block 0. Constraints 2 and 3 constitute the first independent subproblem, or block 1. Constraints 4 and 5 form block 2. Thus, for this particular model, you would want to set the Blocks parameter to be 2, corresponding to the two independent subproblem blocks. For the partitioning heuristic, you may choose either GP1 or GP2. When we solve this model, note that the beginning of the solution report returned by LINGO contains the following information: Global optimal solution found. Objective value: 3.000000 Objective bound: 3.000000 Infeasibilities: 0.000000 Extended solver steps: 0 Total solver iterations: 14 Number of branch-and-price blocks: 2

The interesting feature to note is the line: Number of branch-and-price blocks: 2

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indicating that LINGO partitioned the model in to two independent blocks and invoked the BNP solver. Note:

Note that in some cases the number of blocks listed in the solution report may be less than the number of blocks requested. This occurs when the partitioning heuristic is unable to find a partition with the full number of desired blocks.

Note:

The BNP solver can run the independent subproblems on separate threads to improve performance. So, if your machine has multiple cores, be sure to set the thread limit to at least the number of blocks. Refer to the Threads parameter on the General Solver Tab, discussed above. For this particular small example with its two independent blocks, you'd want to set the thread limit to at least 2.

The graph partitioning algorithms provided by LINGO can generally determine good partitioning schemes, however, for larger models, they may not be able to determine an optimal partitioning. In this case, you may prefer to explicitly specify a model's block structure. LINGO provides two ways to do this by either a) specifying the block structure as part of a model's row names, or b) specifying the block structure using the @BLOCKROW function. Examples of both follow. To specify a row's block using its row name, you should begin the row name with the string "BNP_N", where N is the row's block number. The block number should be some non-negative integer, with a value of 0 indicating that the row belongs to the set of linking constraints. For example, using the model above, we could have specified the block structure in the row names as follows: MIN = x1 + x2 + x3 + x4 + x5 + x6; !linking constraint [bnp_0_c1] x1 + x2 + x3 + x4 + x5 + x6 >=3; [bnp_1_c2] x1 + x2 <=1; !block [bnp_1_c3] x2 + x3 <=1; !block [bnp_2_c4] x4 + x5 + x6 <=2; !block [bnp_2_c5] x4 + x6 <=1; !block @bin( x1); @bin( x2); @bin( x3); @bin( x4); @bin( x5); @bin( x6);

1; 1; 2; 2;

You will also need to select the Row Names option for the Blocks parameter. When specifying block structure in row names, the Heuristic parameter is not relevant and will be grayed out. Alternatively, you could use the @BLOCKROW( BLOCK_NUMBER, ROW_NAME) function to specify block structure. Again, using our same sample model, we would enter:

242 CHAPTER 5 MIN = x1 + x2 + x3 + x4 + x5 + x6; [c1] x1 + x2 + x3 + x4 + x5 + x6 >=3; [c2] x1 + x2 <=1; [c3] x2 + x3 <=1; [c4] x4 + x5 + x6 <=2; [c5] x4 + x6 <=1; @bin( x1); @bin( x2); @bin( x3); @bin( x4); @bin( x5); @bin( x6); @blockrow( 0, c1); @blockrow( 1, c2); @blockrow( 1, c3); @blockrow( 2, c4); @blockrow( 2, c5);

!linking constraint; !block 1; !block 1; !block 2; !block 2;

In this case, you will need to select the Specified option for the Blocks parameter. When specifying block structure via @BLOCKROW, the Heuristic parameter is not relevant and will be grayed out.

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Global Solver Tab The Global Solver tab on the Options dialog box, pictured here:

can be used to control the operation of LINGO’s global solver capabilities. Please keep in mind that the global solver toolkit is an add-on option to LINGO. You must specifically purchase the global solver option as part of your LINGO license in order to make use of its capabilities.

243

244 CHAPTER 5 LINGO exploits the convex nature of linear models to find globally optimal solutions. However, we aren’t as fortunate with nonlinear models. LINGO’s default NLP solver uses a local search procedure. This can lead to LINGO stopping at locally optimal points when a model is non-convex and perhaps missing a global point lying elsewhere. You may refer to Chapter 15, On Mathematical Modeling, for more information on how and why this can happen. The global solver toolkit contains features designed to sift through the local points in search of the globally optimal point. The two primary features in LINGO’s global toolkit are a global solver and a multistart solver. The global solver uses range bounding and reduction techniques within a branch-and-bound framework to convert a non-convex model into a series of smaller, convex models. This divide-and-conquer strategy ultimately results in convergence to the guaranteed globally optimal point. The multistart solver, on the other hand, uses a heuristic approach of restarting the NLP solver several times from different initial points. It is not uncommon for a different starting point to lead to a different local solution point. Thus, if we restart from enough unique points, saving the best local solution as we go, we stand a much better chance of finding the true global solution.

Use Global Solver If the Use Global Solver box is checked, LINGO will invoke the global solver when you solve a nonlinear model. Many nonlinear models are non-convex and/or non-smooth (for more information see Chapter 15, On Mathematical Modeling.) Nonlinear solvers that rely on local search procedures (as does LINGO’s default nonlinear solver) will tend to do poorly on these types of models. Typically, they will converge to a local, sub-optimal point that may be quite distant from the true, global optimal point. Global solvers overcome this weakness through methods of range bounding (e.g., interval analysis and convex analysis) and range reduction techniques (e.g., linear programming and constraint propagation) within a branch-and-bound framework to find global solutions to non-convex models. The following example illustrates the usefulness of the global solver. Consider the simple, yet highly nonlinear, model: MODEL: MIN = X * @COS( 3.1416 * X); @BND( 0, X, 6);

END

The graph of the objective function is as follows:

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The objective function has three local minimal points over the feasible range. These points are summarized in the following table: Point 1

X 1.09

Objective -1.05

2

3.03

-3.02

3

5.02

-5.01

Clearly, the third local point is also the globally best point, and we would like the NLP solver to converge to this point. Below is the solution LINGO produces if the default nonlinear solver is invoked: Local optimal solution found at step: Objective value: Variable X Row 1

Value 1.090405 Slack or Surplus -1.046719

11 -1.046719 Reduced Cost 0.1181082E-07 Dual Price -1.000000

245

246 CHAPTER 5 Unfortunately, as you can see, we converged to the least preferable of the local minimums. However, after enabling the global solver by checking the Use Global Solver box, we do obtain the global solution: Global optimal solution found at step: Objective value: Variable X Row 1

Note:


35 -5.010083 Reduced Cost -0.7076917E-08 Dual Price -1.000000

There is one drawback to using the global solver; it runs considerably slower than the default nonlinear solver. Therefore, the preferred option is to always try and write smooth, convex nonlinear models. By doing this, the faster, default local solver can be successfully invoked.

Keep in mind that the global solver s most, but not all, of the functions available in the LINGO language. The following is a list of the nonlinear functions not currently ed by the global solver:    

All probability distributions — cumulative, inverse and pdf, with the exception of the normal distribution, which is fully ed @PFS() — Poisson finite source @PPL() — Poisson linear loss @( )— supplied function

Note:

The global solver will not operate on models containing one or more uned nonlinear operations that reference optimizable quantities; the default NLP solver will be called in this case.

Note:

The global solver can run on multiple cores to improve performance. So, if your machine has multiple cores, be sure to set the thread limit to something higher than 1, with the ideal setting being N, where N is the number of available cores on your machine. Refer to the Threads parameter on the General Solver Tab, discussed above, for more information on setting the thread limit.

The global solver is disabled by default.

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Variable Upper Bound Box The Variable Upper Bound box:

sets the default variable bounds while the global solver is running. If this parameter is set to d, then variables will not be permitted to assume values outside the range of [-d, d]. Setting this parameter as tightly as possible in the Value Field restricts the global solver from straying into uninteresting regions and will reduce run times. The default value for the Value Field is 1.e10. The Application list box has three options available: None, All and Selected. Selecting None removes the variable bounds entirely, and is not recommended. The All setting applies the bound to all variables. Finally, the Selected setting causes the global solver to apply the bound after an initial solver to find the first local solution. The bound will only be applied to a variable if it does not cutoff the initial local solution. LINGO defaults to the Selected setting.

Tolerances Box The Tolerances box:

contains two tolerances used by the global solver: Optimality and Delta. The Optimality tolerance specifies by how much a new solution must beat the objective value of the incumbent solution in order to become the new incumbent. The default value for Optimality 1.e-6. The Delta tolerance specifies how closely the additional constraints, added as part of the global solver’s convexification process, must be satisfied. The default value for Delta is 1.e-7.

Strategies Box The Strategies box:

allows you to control three strategies used by the global solver: Branching, Box Selection and Reformulation.

248 CHAPTER 5 The Branching strategy consists of six options to use when branching on a variable for the first time:      

Absolute Width, Local Width, Global Width, Global Distance, Absolute Violation, and Relative Violation.

The default setting for Branching is Relative Violation. The Box Selection option specifies the strategy to use for choosing between all active nodes in the global solver’s branch-and-bound tree. The choices are: Depth First and Worst Bound, with the default being Worst Bound. The Reformulation option sets the degree of algebraic reformulation performed by the global solver. Algebraic reformulation is critical for construction of tight, convex sub-regions to enclose the nonlinear and nonconvex functions. The available settings are None, Low, Medium and High, with High being the default.

Multistart Solver LINGO exploits the convex nature of linear models to find globally optimal solutions. However, we aren’t as fortunate with nonlinear models. With NLP models, LINGO’s default NLP solver uses a local search procedure. This can lead to LINGO stopping at locally optimal points, perhaps missing a global point lying elsewhere. You may refer to On Mathematical Modeling for more information on how and why this can happen. A strategy that has proven successful in overcoming this problem is to restart the NLP solver several times from different initial points. It is not uncommon for a different starting point to lead to a different local solution point. Thus, if we restart from enough unique points, saving the best local solution as we go, then we stand a much better chance of finding the true global solution. We refer to this solution strategy as multistart. The Multistart Solver Attempts box on the Global Solver tab:

is used to set the number of times the multistart solver restarts the standard NLP solver in its attempt to find successively better local solutions. Each new starting point is intelligently generated to maximize the chances of finding a new local point. The default option, Solver Decides, entails restarting 5 times on small NLPs and disabling multistart on larger models. Setting multistart to 1 causes the NLP solver to be invoked only once, effectively disabling multistart. Setting multistart to any value greater than 1 will cause the NLP solver to restart that number of times on all NLPs. In general, we have found that setting the number of multistarts to around 5 tends to be adequate for most models. Highly nonlinear models may require a larger setting.

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Keep in mind that multistart will dramatically increase runtimes, particularly if a large number of restarts is selected. Thus, one should avoid using multistart unnecessarily on convex models that will converge to a global point in a single without any additional prodding.

The following example illustrates the usefulness of multistart. Consider the simple, yet highly nonlinear, model: MODEL:

END

MIN = X * @COS( 3.1416 * X); @BND( 0, X, 6);

249

250 CHAPTER 5 The graph of the objective function is as follows:

The objective function has three local minimal points over the feasible range. These points are summarized in the following table: Point 1

X 1.09

Objective -1.05

2

3.03

-3.02

3

5.02

-5.01

Clearly, the third local point is also the globally best point, and we would like the NLP solver to converge to this point. Below is the solution you will get from LINGO if the multistart option is disabled: Local optimal solution found at step: Objective value: Variable X Row 1



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Unfortunately, as you can see, we converged to the least preferable of the local minimums. However, after setting the number of multistarts to five and re-solving, we do obtain the global solution: Local optimal solution found at step: Objective value: Variable X Row 1



Note:

Unlike the global solver, the multistart solver can only claim its solution to be locally optimal. This is because there may always be a better solution out there that the multistart solver may, or may not, be able to find with additional runs. The global solver, on the other hand, can claim global optimality by having partitioned the original model into a series of smaller, convex models.

Note:

The multistart solver can run on multiple cores to improve performance. So, if your machine has multiple cores, be sure to set the thread limit to something higher than 1, with the ideal setting being N, where N is the number of available cores on your machine. Refer to the Threads parameter on the General Solver Tab, discussed above, for more information on setting the thread limit.

252 CHAPTER 5

SP Solver Tab The SP Solver tab on the Options dialog box:

is used to control the operation of LINGO's SP (Stochastic Programming) solver. For more information on SP, refer to Chapter 14, Stochastic Programming. Please keep in mind that the SP solver is an add-on option to LINGO. You must specifically purchase the SP solver option as part of your LINGO license in order to make use of its capabilities.

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Default Sample Size/Stage The Default Sample Size/Stage parameter on the SP Solver tab:

is used to control the default sample size for random variables in stochastic programming (SP) whose outcomes are determined via sampling. In many SP models, LINGO will generate a set of sample values for the some or all of the random variables. This is particularly true when you have one or more continuous random variables, which have an infinite number of possible outcomes. In such a case, sampling is the only viable alternative. One way to specify the desired sample size for each stage is via the @SPSAMPSIZE function used directly in the model's text. If, on the other hand, all or most stages should have the same sample size, then you can use the Default Sample Size/Stage parameter to control the default sample size. Any stage which has not had its sample size specified with @SPSAMPSIZE will default to a sample size equal to the Default Sample Size/Stage parameter. Note:

In general, we prefer larger sample sizes to smaller ones in order to reduce sampling error. However, SP models can become quite large if sample sizes aren't kept reasonably small. This is particularly true for multiperiod models. For example, suppose we have a model with just one random variable and ten periods/stages. If the default sample size is set to 3, then there will be 3^10=59,049 possible scenarios. With this many scenarios, it would only take a handful of decision variables to end up with an underlying deterministic equivalent model approaching one million variables.

The Default Sample Size/Stage parameter defaults to a value of 2.

Random Generator Seed The Random Generator Seed parameter on the SP Solver tab:

is used to establish the seed for the random number generator used in generating samples for random variables in stochastic programming (SP) models. In many SP models, LINGO will generate a set of sample values for the random variables. This is particularly true when you have one or more continuous random variables, which leads to an infinite number of possible outcomes. In such a case, sampling is the only viable alternative. The Random Generator Seed parameter defaults to a value of 1031. Note:

The seed parameter has no effect when running demo versions of LINGO. Demo versions always use the default seed value, regardless of this parameter's setting.

254 CHAPTER 5

Sample Continuous Only The Sample Continuous Only option on the SP Solver tab:

is used to control whether LINGO samples continuous distributions only for the random variables in stochastic programs, or if it samples both continuous and discrete random variables. Obviously, continuous random variables must be subject to sampling, given the infinite number of possible outcomes. On the other hand, we have a choice when it comes to discretely distributed random variables. If this option is enabled, then LINGO will generate one outcome for each density point in the discrete distribution of the random variable. These outcomes will be weighted by their corresponding probabilities. Note:

If there are many density points in the distributions for the discrete random variables, the deterministic equivalent generated to solve the SP may become exceedingly large, in which case, disabling this option may be required. Once this option is disabled, discrete random variables will be sampled in the same way as the continuous random variables.

The Sample Continuous Only option defaults to being on.

SP Solver Method The SP Solver Method option:

on the SP Solver Tab is used to select the solution method for a stochastic programming (SP) model. Presently, the available choices are:   

Solver Decides  LINGO decides the most appropriate method for solving the SP model. Deterministic Equivalent  LINGO generates and directly solves the deterministic equivalent (DE) of the SP model. Nested Benders  The DE for realistic SP models can get to be quite large, in that the core model is replicated once for every scenario. Fortunately, the DE displays strong blockangular structure, which makes it adaptable to decomposition techniques, such as nested Bender decomposition (NBD). Note that the model must be entirely linear to use NBD.

The default setting for the SP Solver Method option is Solver Decides.

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Max Scenarios Limit The Max Scenarios Limit parameter on the SP Solver tab:

is used to establish a limit on the maximum number of scenarios in a stochastic programming (SP) model before forcing automatic sampling of the random variables. The Max Scenarios Limit defaults to a value of 40,000.

SP Big M Coefficient The Big M Coefficient parameter on the SP Solver tab:

is used by the SP solver in constructing forcing constraints that may be required in the deterministic models generated by the solver. Forcing constraints are generally added to force binary variables to 1 when some activity occurs. For example, suppose x is a continuous variable and z is a binary variable. A forcing constraint that would drive z to 1 whenever x is positive would be: xM*z where M is some large number. M in this case is the Big M coefficient. There are two things to note about M. First, if M is too small, a forcing constraint can become infeasible, making the entire model infeasible. In general, M should be at least as large as the largest possible value for x so as not to introduce an infeasibility. However, from an algorithmic point of view, M should not be too large. If M is unrealistically large, it will be tougher for the integer solver to converge, plus it also introduces the potential round off error. The default setting for M should be sufficient for most models. However, if your SP model is infeasible for no known reason, you may want to try increasing M. On the other hand, if your SP is running slow, you may want to try reducing M. The Big M Coefficient defaults to a value of 100,000,000.

256 CHAPTER 5

Calculate All Expected Value Statistics The Calculate All Expected Value Statistics option on the SP Solver tab:

controls whether LINGO displays information regarding the expected values for a number of statistics when solving stochastic programming (SP) models. To illustrate, when solving the SPGAS.LG4 model when this option is enabled, you will see the following expected values at the top of the solution report: Expected value of: Objective (EV): Wait-and-see model's objective (WS): Perfect information (EVPI = |EV - WS|): Policy based on mean outcome (EM): Modeling uncertainty (EVMU = |EM - EV|):

1400.000 1326.667 73.33333 1479.444 79.44444

These values are a guide as to how the stochastic nature of the model is impacting the objective value. The following is a brief description of these expected values: Expected Value of Objective (EV) - is the expected value for the model's objective over all the scenarios, and is the same as the reported objective value for the model. Expected Value of Wait-and-See Model's Objective (WS) - reports the expected value of the objective if we could wait and see the outcomes of all the random variables before making our decisions. Such a policy would allow us to always make the best decision regardless of the outcomes for the random variables, and, of course, is not possible in practice. For a minimization, it's true that WS <= EV, with the converse holding for a maximization. Technically speaking, WS is a relaxation of the true SP model, obtained by dropping the nonanticipativity constraints. Expected Value of Perfect Information (EVPI) - is the absolute value of the difference between EV and WS. This corresponds to the expected improvement to the objective were we to obtain perfect information about the random outcomes. As such, this is a expected measure of how much we should be willing to pay to obtain perfect information regarding the outcomes of the random variables. Expected Value of Policy Based On Mean Outcome (EM) - is the expected true objective value if we (mistakenly) assume that all random variables will always take on exactly their mean values. EM is computed using a two-step process. First, the values of all random variables are fixed at their means, and the resulting deterministic model is solved to yield the optimal values for the stage 0 decision variables. Next, a) the stage 0 variables are fixed at their optimal values from the previous step, b) the random variables are freed up, c) the

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nonanticipativity constraints are dropped, and d) this wait-and-see model is solved. EM is the objective value from this WS model. Expected Value of Modeling Uncertainty (EVMU) - is the absolute value of the difference EV - EM. It is a measure of what we can expect to gain by taking into uncertainty in our modeling analysis, as opposed to mistakenly assuming that random variables always take on their mean outcomes. Note:

The above approach for computing EM and EVMU makes unambiguous sense only for models with a stage 0 and a stage 1. If there are later random variables in stages 2, 3, etc., then there are complications. For example, for decisions in later stages, we have seen the outcomes from the random variables in earlier stages, so considering these random variables to take on their mean value does not make sense. For models with additional stages beyond 0 and 1, EVMU will merely be an approximation of the true expected value of modeling uncertainty.

Note:

Computing these expected value statistics can be very time consuming for large models. If speed is an issue, you may wish to disable this feature on the LINGO|Options|SP Solver tab.

The Calculate All Expected Value Statistics option is enabled by default.

LINGO|Generate...

Ctrl+G

Once you remove all the syntax errors from your LINGO model, there is still one very important step required: model verification. LINGO’s set-based modeling capabilities are very powerful, and they allow you to generate large, complex models quickly and easily. However, when you first develop a model you will need to that the model being generated matches up to the model you actually intended to generate. Many set-based models can be quite complex, and it is highly likely that logic errors may creep into one or more expressions, thereby causing your generated model to be flawed. The LINGO|Generate command is very useful for debugging such errors. It expands all of the model's compact set-based expressions and then writes out the full scalarbased equivalent of the LINGO model. The expanded model report explicitly lists all the generated constraints and variables in your model. You will find that the Generate report can be an invaluable tool in tracking down errors. When selecting the Generate command, you will be presented with a pop-up menu prompting you for one of the following options:      

Display model, Don’t display model, Display nonlinear rows, Dual model, Explicit Deteq, and Scenario.

258 CHAPTER 5

Display Model If you choose the Display model option of the LINGO|Generate command, LINGO will place a copy of the generated model in a new window, which you may scroll through to examine, print, or save to disk. As an example of the output from the Generate command, consider the transportation model developed in Chapter 1: MODEL: ! A 6 Warehouse 8 Vendor Transportation Problem; SETS: WAREHOUSES: CAPACITY; VENDORS: DEMAND; LINKS( WAREHOUSES, VENDORS): COST, VOLUME; ENDSETS DATA: !set ; WAREHOUSES = WH1 WH2 WH3 WH4 WH5 WH6; VENDORS = V1 V2 V3 V4 V5 V6 V7 V8; !attribute values; CAPACITY = 60 55 51 43 41 52; DEMAND = 35 37 22 32 41 32 43 38; COST = 6 2 6 7 4 2 5 9 4 9 5 3 8 5 8 2 5 2 1 9 7 4 3 3 7 6 7 3 9 2 7 1 2 3 9 5 7 2 6 5 5 5 2 2 8 1 4 3; ENDDATA ! The objective; [OBJECTIVE] MIN = @SUM( LINKS( I, J): COST( I, J) * VOLUME( I, J)); ! The demand constraints; @FOR( VENDORS( J): [DEMAND_ROW] @SUM( WAREHOUSES( I): VOLUME( I, J)) = DEMAND( J)); ! The capacity constraints; @FOR( WAREHOUSES( I): [CAPACITY_ROW] @SUM( VENDORS( J): VOLUME( I, J)) <= CAPACITY( I)); END

Model: WIDGETS

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The objective will generate one expression, there should be one demand constraint generated for each of the eight vendors and one supply constraint generated for each of the six warehouses, for a grand total of 15 rows in the expanded model. Running the generate command to this reveals the following report: MODEL: [OBJECTIVE] MIN= 6 * VOLUME_WH1_V1 + 2 * VOLUME_WH1_V2 + 6 * VOLUME_WH1_V3 + 7 * VOLUME_WH1_V4 + 4 * VOLUME_WH1_V5 + 2 * VOLUME_WH1_V6 + 5 * VOLUME_WH1_V7 + 9 * VOLUME_WH1_V8 + 4 * VOLUME_WH2_V1 + 9 * VOLUME_WH2_V2 + 5 * VOLUME_WH2_V3 + 3 * VOLUME_WH2_V4 + 8 * VOLUME_WH2_V5 + 5 * VOLUME_WH2_V6 + 8 * VOLUME_WH2_V7 + 2 * VOLUME_WH2_V8 + 5 * VOLUME_WH3_V1 + 2 * VOLUME_WH3_V2 + VOLUME_WH3_V3 + 9 * VOLUME_WH3_V4 + 7 * VOLUME_WH3_V5 + 4 * VOLUME_WH3_V6 + 3 * VOLUME_WH3_V7 + 3 * VOLUME_WH3_V8 + 7 * VOLUME_WH4_V1 + 6 * VOLUME_WH4_V2 + 7 * VOLUME_WH4_V3 + 3 * VOLUME_WH4_V4 + 9 * VOLUME_WH4_V5 + 2 * VOLUME_WH4_V6 + 7 * VOLUME_WH4_V7 + VOLUME_WH4_V8 + 2 * VOLUME_WH5_V1 + 3 * VOLUME_WH5_V2 + 9 * VOLUME_WH5_V3 + 5 * VOLUME_WH5_V4 + 7 * VOLUME_WH5_V5 + 2 * VOLUME_WH5_V6 + 6 * VOLUME_WH5_V7 + 5 * VOLUME_WH5_V8 + 5 * VOLUME_WH6_V1 + 5 * VOLUME_WH6_V2 + 2 * VOLUME_WH6_V3 + 2 * VOLUME_WH6_V4 + 8 * VOLUME_WH6_V5 + VOLUME_WH6_V6 + 4 * VOLUME_WH6_V7 + 3 * VOLUME_WH6_V8 ; [DEMAND_ROW_V1] VOLUME_WH1_V1 + VOLUME_WH2_V1 + VOLUME_WH3_V1 + VOLUME_WH4_V1 + VOLUME_WH5_V1 + VOLUME_WH6_V1 = 35 ; [DEMAND_ROW_V2] VOLUME_WH1_V2 + VOLUME_WH2_V2 + VOLUME_WH3_V2 + VOLUME_WH4_V2 + VOLUME_WH5_V2 + VOLUME_WH6_V2 = 37 ; [DEMAND_ROW_V3] VOLUME_WH1_V3 + VOLUME_WH2_V3 + VOLUME_WH3_V3 + VOLUME_WH4_V3 + VOLUME_WH5_V3 + VOLUME_WH6_V3 = 22 ; [DEMAND_ROW_V4] VOLUME_WH1_V4 + VOLUME_WH2_V4 + VOLUME_WH3_V4 + VOLUME_WH4_V4 + VOLUME_WH5_V4 + VOLUME_WH6_V4 = 32 ; [DEMAND_ROW_V5] VOLUME_WH1_V5 + VOLUME_WH2_V5 + VOLUME_WH3_V5 + VOLUME_WH4_V5 + VOLUME_WH5_V5 + VOLUME_WH6_V5 = 41 ; [DEMAND_ROW_V6] VOLUME_WH1_V6 + VOLUME_WH2_V6 + VOLUME_WH3_V6 + VOLUME_WH4_V6 + VOLUME_WH5_V6 + VOLUME_WH6_V6 = 32 ; [DEMAND_ROW_V7] VOLUME_WH1_V7 + VOLUME_WH2_V7 + VOLUME_WH3_V7 + VOLUME_WH4_V7 + VOLUME_WH5_V7 + VOLUME_WH6_V7 = 43 ; [DEMAND_ROW_V8] VOLUME_WH1_V8 + VOLUME_WH2_V8 + VOLUME_WH3_V8 + VOLUME_WH4_V8 + VOLUME_WH5_V8 + VOLUME_WH6_V8 = 38 ; [CAPACITY_ROW_WH1] VOLUME_WH1_V1 + VOLUME_WH1_V2 + VOLUME_WH1_V3 + VOLUME_WH1_V4 + VOLUME_WH1_V5 + VOLUME_WH1_V6 + VOLUME_WH1_V7 + VOLUME_WH1_V8 <= 60 ; [CAPACITY_ROW_WH2] VOLUME_WH2_V1 + VOLUME_WH2_V2 + VOLUME_WH2_V3 + VOLUME_WH2_V4 + VOLUME_WH2_V5 +

260 CHAPTER 5 VOLUME_WH2_V6 + VOLUME_WH2_V7 + VOLUME_WH2_V8 <= [CAPACITY_ROW_WH3] VOLUME_WH3_V1 + VOLUME_WH3_V2 VOLUME_WH3_V3 + VOLUME_WH3_V4 + VOLUME_WH3_V5 + VOLUME_WH3_V6 + VOLUME_WH3_V7 + VOLUME_WH3_V8 <= [CAPACITY_ROW_WH4] VOLUME_WH4_V1 + VOLUME_WH4_V2 VOLUME_WH4_V3 + VOLUME_WH4_V4 + VOLUME_WH4_V5 + VOLUME_WH4_V6 + VOLUME_WH4_V7 + VOLUME_WH4_V8 <= [CAPACITY_ROW_WH5] VOLUME_WH5_V1 + VOLUME_WH5_V2 VOLUME_WH5_V3 + VOLUME_WH5_V4 + VOLUME_WH5_V5 + VOLUME_WH5_V6 + VOLUME_WH5_V7 + VOLUME_WH5_V8 <= [CAPACITY_ROW_WH6] VOLUME_WH6_V1 + VOLUME_WH6_V2 VOLUME_WH6_V3 + VOLUME_WH6_V4 + VOLUME_WH6_V5 + VOLUME_WH6_V6 + VOLUME_WH6_V7 + VOLUME_WH6_V8 <= END

55 ; + 51 ; + 43 ; + 41 ; + 52 ;

Model: Generated Model Report for WIDGETS As expected, there are 15 rows in the generated model: [OBJECTIVE], [DEMAND_ROW_V1] through [DEMAND_ROW_V8], and [CAPACITY_ROW_WH1] through [CAPACITY_ROW_WH6]. As a side note, it’s interesting to compare the generated model to the original, set-based model. The generated model is the expanded version of the model, and has all sets and vector variables removed, resulting in a fully scalar model. The scalar-based, generated model is mathematically equivalent to the original vector-based model. However, we think most would agree that the set-based model is much easier to comprehend, thereby illustrating one of the primary benefits of modern algebraic languages over more traditional, scalar-based languages. In addition to ing that the correct number of rows is being generated, you should also examine each of the rows to determine that the correct variables are appearing in each row along with their correct coefficients. Note:

The reports generated by the LINGO|Generate command are valid LINGO models. You may load Generate reports into a model window and solve them as you would any other LINGO model.

One thing to keep in mind when examining generated model reports is that the LINGO model generator performs fixed variable reduction. This means that any variables that are fixed in value are substituted out of the generated model. For example, consider the simple model: MODEL: MAX = 200 * WS + 300 * NC; WS = 60; NC <= 40; WS + 2 * NC <= 120; END

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If we generate this model we get the following, reduced model: MODEL: MAX= 300 * NC + 12000 ; NC <= 40 ; 2 * NC <= 60 ; END

At first glance, it seems as if both the first constraint and the variable WS are missing from the generated model. Note that by the first constraint in the original model (WS = 60), WS is fixed at a value of 60. The LINGO model generator exploits this fact to reduce the size of the generated model by substituting WS out of the formulation. The final solution report will still contain the values for all the fixed variables, however, the fixed variables will not appear in the generated model report. If you would like to suppress fixed variable reduction so that all variables appear in your generated model, you may do so via the Fixed Var Reduction option.

Don't Display Model If you choose the Don’t display model option, LINGO will generate the model without displaying it, but will store the generated model for later use by the appropriate solver. This is a useful command for ing that your model contains no errors, while not actually having to it to the solver to be optimized.

Display Nonlinear Rows If you choose the Display Nonlinear Rows option of the Generate command, then LINGO will generate the model and then display only those rows that contain nonlinearities. Please refer to the Types of Constraints section for a discussion of linear vs. nonlinear expressions. In general, one would prefer to always have purely linear models. Linear models solve faster and will always converge to a global solution. Therefore, when developing a model, you will be interested in carefully evaluating nonlinearities to see if they can either be eliminated or rewritten in a linear fashion. The Display Nonlinear Rows is helpful in tracking down a model's nonlinearities.

Dual Model The third option of the LINGO|Generate command, Dual Model, displays the dual formulation of the current model. Every linear programming model has a corresponding, mirror-image formulation called the dual. If the original model has M constraints and N variables, then its dual will have N constraints and M variables. Some interesting properties of the dual are that any feasible solution to the dual model provides a bound on the objective to the original, primal model, while the optimal solution to the dual has the same objective value as the optimal solution to the primal problem. It's also true that the dual of the dual model is, once again, the original primal model. You may wish to refer to any good linear programming text for a further discussion of duality theory.

262 CHAPTER 5 If you run the LINGO|Generate|Dual Model command on the Widgets model shown above, you will receive the following formulation: MODEL:

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

MAX = 35 * DEMAND_ROW_V1 + 37 * DEMAND_ROW_V2 + 22 * DEMAND_ROW_V3 + 32 * DEMAND_ROW_V4 + 41 * DEMAND_ROW_V5 + 32 * DEMAND_ROW_V6 + 43 * DEMAND_ROW_V7 + 38 * DEMAND_ROW_V8 + 60 * CAPACITY_ROW_WH1 + 55 * CAPACITY_ROW_WH2 + 51 * CAPACITY_ROW_WH3 + 43 * CAPACITY_ROW_WH4 + 41 * CAPACITY_ROW_WH5 + 52 * CAPACITY_ROW_WH6; VOLUME_WH1_V1] DEMAND_ROW_V1 + CAPACITY_ROW_WH1 <= 6; VOLUME_WH1_V2] DEMAND_ROW_V2 + CAPACITY_ROW_WH1 <= 2; VOLUME_WH1_V3] DEMAND_ROW_V3 + CAPACITY_ROW_WH1 <= 6; VOLUME_WH1_V4] DEMAND_ROW_V4 + CAPACITY_ROW_WH1 <= 7; VOLUME_WH1_V5] DEMAND_ROW_V5 + CAPACITY_ROW_WH1 <= 4; VOLUME_WH1_V6] DEMAND_ROW_V6 + CAPACITY_ROW_WH1 <= 2; VOLUME_WH1_V7] DEMAND_ROW_V7 + CAPACITY_ROW_WH1 <= 5; VOLUME_WH1_V8] DEMAND_ROW_V8 + CAPACITY_ROW_WH1 <= 9; VOLUME_WH2_V1] DEMAND_ROW_V1 + CAPACITY_ROW_WH2 <= 4; VOLUME_WH2_V2] DEMAND_ROW_V2 + CAPACITY_ROW_WH2 <= 9; VOLUME_WH2_V3] DEMAND_ROW_V3 + CAPACITY_ROW_WH2 <= 5; VOLUME_WH2_V4] DEMAND_ROW_V4 + CAPACITY_ROW_WH2 <= 3; VOLUME_WH2_V5] DEMAND_ROW_V5 + CAPACITY_ROW_WH2 <= 8; VOLUME_WH2_V6] DEMAND_ROW_V6 + CAPACITY_ROW_WH2 <= 5; VOLUME_WH2_V7] DEMAND_ROW_V7 + CAPACITY_ROW_WH2 <= 8; VOLUME_WH2_V8] DEMAND_ROW_V8 + CAPACITY_ROW_WH2 <= 2; VOLUME_WH3_V1] DEMAND_ROW_V1 + CAPACITY_ROW_WH3 <= 5; VOLUME_WH3_V2] DEMAND_ROW_V2 + CAPACITY_ROW_WH3 <= 2; VOLUME_WH3_V3] DEMAND_ROW_V3 + CAPACITY_ROW_WH3 <= 1; VOLUME_WH3_V4] DEMAND_ROW_V4 + CAPACITY_ROW_WH3 <= 9; VOLUME_WH3_V5] DEMAND_ROW_V5 + CAPACITY_ROW_WH3 <= 7; VOLUME_WH3_V6] DEMAND_ROW_V6 + CAPACITY_ROW_WH3 <= 4; VOLUME_WH3_V7] DEMAND_ROW_V7 + CAPACITY_ROW_WH3 <= 3; VOLUME_WH3_V8] DEMAND_ROW_V8 + CAPACITY_ROW_WH3 <= 3; VOLUME_WH4_V1] DEMAND_ROW_V1 + CAPACITY_ROW_WH4 <= 7; VOLUME_WH4_V2] DEMAND_ROW_V2 + CAPACITY_ROW_WH4 <= 6; VOLUME_WH4_V3] DEMAND_ROW_V3 + CAPACITY_ROW_WH4 <= 7; VOLUME_WH4_V4] DEMAND_ROW_V4 + CAPACITY_ROW_WH4 <= 3; VOLUME_WH4_V5] DEMAND_ROW_V5 + CAPACITY_ROW_WH4 <= 9; VOLUME_WH4_V6] DEMAND_ROW_V6 + CAPACITY_ROW_WH4 <= 2; VOLUME_WH4_V7] DEMAND_ROW_V7 + CAPACITY_ROW_WH4 <= 7; VOLUME_WH4_V8] DEMAND_ROW_V8 + CAPACITY_ROW_WH4 <= 1; VOLUME_WH5_V1] DEMAND_ROW_V1 + CAPACITY_ROW_WH5 <= 2; VOLUME_WH5_V2] DEMAND_ROW_V2 + CAPACITY_ROW_WH5 <= 3; VOLUME_WH5_V3] DEMAND_ROW_V3 + CAPACITY_ROW_WH5 <= 9; VOLUME_WH5_V4] DEMAND_ROW_V4 + CAPACITY_ROW_WH5 <= 5; VOLUME_WH5_V5] DEMAND_ROW_V5 + CAPACITY_ROW_WH5 <= 7; VOLUME_WH5_V6] DEMAND_ROW_V6 + CAPACITY_ROW_WH5 <= 2; VOLUME_WH5_V7] DEMAND_ROW_V7 + CAPACITY_ROW_WH5 <= 6; VOLUME_WH5_V8] DEMAND_ROW_V8 + CAPACITY_ROW_WH5 <= 5; VOLUME_WH6_V1] DEMAND_ROW_V1 + CAPACITY_ROW_WH6 <= 5; VOLUME_WH6_V2] DEMAND_ROW_V2 + CAPACITY_ROW_WH6 <= 5;

WINDOWS COMMANDS [ VOLUME_WH6_V3] DEMAND_ROW_V3 + CAPACITY_ROW_WH6 [ VOLUME_WH6_V4] DEMAND_ROW_V4 + CAPACITY_ROW_WH6 [ VOLUME_WH6_V5] DEMAND_ROW_V5 + CAPACITY_ROW_WH6 [ VOLUME_WH6_V6] DEMAND_ROW_V6 + CAPACITY_ROW_WH6 [ VOLUME_WH6_V7] DEMAND_ROW_V7 + CAPACITY_ROW_WH6 [ VOLUME_WH6_V8] DEMAND_ROW_V8 + CAPACITY_ROW_WH6 @FREE( DEMAND_ROW_V1); @FREE( DEMAND_ROW_V2); @FREE( DEMAND_ROW_V3); @FREE( DEMAND_ROW_V4); @FREE( DEMAND_ROW_V5); @FREE( DEMAND_ROW_V6); @FREE( DEMAND_ROW_V7); @FREE( DEMAND_ROW_V8); @BND( -0.1E+31, CAPACITY_ROW_WH1, 0); @BND( -0.1E+31, CAPACITY_ROW_WH2, 0); @BND( -0.1E+31, CAPACITY_ROW_WH3, 0); @BND( -0.1E+31, CAPACITY_ROW_WH4, 0); @BND( -0.1E+31, CAPACITY_ROW_WH5, 0); @BND( -0.1E+31, CAPACITY_ROW_WH6, 0); END

<= <= <= <= <= <=

263

2; 2; 8; 1; 4; 3;

Dual Formulation: WIDGETS You will notice that in the dual formulation the variables from the primal model become the rows of the dual. Similarly, the rows in the primal become the variables in the dual. Note:

The row names from the primal problem will become the variable names in the dual formulation. For this reason, it is strongly recommended that you name all the rows in the primal model. If a row is unnamed, then a default name will be generated for the corresponding dual variable. The default name will consist of an underscore followed by the row's internal index. These default names will not be very meaningful, and will make the dual formulation difficult to interpret.

Explicit Deteq The fourth option of the LINGO|Generate command, Explicit Deteq, displays the explicit deterministic equivalent (DE) for the current model, assuming the current model is a stochastic program (SP). Viewing the DE can be very helpful in determining if you have properly set up the SP components of your model. You can refer to Chapter 14, Stochastic Programming, for more information on SP. When LINGO solves an SP model, it is really solving the deterministic equivalent of the original SP. The DE contains one instance of the original core model for each scenario, where the random variables in each instance are replaced by their sampled values for the particular scenario. These core instances are tied together by a probabilistically weighted objective of all the objectives form the individual scenarios, plus a set of nonanticipativity constraints. The nonanticipativity constraints enforce the condition that all scenarios with the same history must implement the same decisions, otherwise, the DE would have the ability to anticipate future events.

264 CHAPTER 5 As an example, if you were to load the SP gas buying sample model, SPGAS1.LG4, into LINGO and run the LINGO|Generate|Explicit Deteq command, you will receive the following report, containing the model's DE: MODEL: TITLE _DETEQ; [ _1] MIN = 0.3333333333333333 * PURCHASE_COST_SC1 + 0.3333333333333333 * HOLD_COST_SC1 + 0.3333333333333333 * PURCHASE_COST_SC2 + 0.3333333333333333 * HOLD_COST_SC2 + 0.3333333333333333 * PURCHASE_COST_SC3 + 0.3333333333333333 * HOLD_COST_SC3; [ _2_SC01] PURCHASE_COST_SC1 - 5 * PURCHASE_1_SC1 - 5 * PURCHASE_2_SC1 = 0; [ _3_SC01] HOLD_COST_SC1 - INVENTORY_1_SC1 INVENTORY_2_SC1 = 0; [ _4_SC01] INVENTORY_1_SC1 - PURCHASE_1_SC1 = - 100; [ _5_SC01] - INVENTORY_1_SC1 + INVENTORY_2_SC1 PURCHASE_2_SC1 = - 100; [ _2_SC02] PURCHASE_COST_SC2 - 5 * PURCHASE_1_SC2 - 6 * PURCHASE_2_SC2 = 0; [ _3_SC02] HOLD_COST_SC2 - INVENTORY_1_SC2 INVENTORY_2_SC2 = 0; [ _4_SC02] INVENTORY_1_SC2 - PURCHASE_1_SC2 = - 100; [ _5_SC02] - INVENTORY_1_SC2 + INVENTORY_2_SC2 PURCHASE_2_SC2 = - 150; [ _2_SC03] PURCHASE_COST_SC3 - 5 * PURCHASE_1_SC3 - 7.5 * PURCHASE_2_SC3 = 0; [ _3_SC03] HOLD_COST_SC3 - INVENTORY_1_SC3 INVENTORY_2_SC3 = 0; [ _4_SC03] INVENTORY_1_SC3 - PURCHASE_1_SC3 = - 100; [ _5_SC03] - INVENTORY_1_SC3 + INVENTORY_2_SC3 PURCHASE_2_SC3 = - 180; [ _NAC01] INVENTORY_1_SC1 - INVENTORY_1_SC2 = 0; [ _NAC02] PURCHASE_1_SC1 - PURCHASE_1_SC2 = 0; [ _NAC03] INVENTORY_1_SC1 - INVENTORY_1_SC3 = 0; [ _NAC04] PURCHASE_1_SC1 - PURCHASE_1_SC3 = 0; END

Deterministic Equivalent: SPGAS1 The first row of the DE is the probabilistically weighted objective over the three scenarios (Normal, Cold and Very Cold). Note that in the original core model, we had a variable called PURCHASE_COST, which was used to compute the cost of the gas purchases. In the DE we now have three PURCHASE_COST variables, or one for each of the three scenarios. LINGO appends the strings _SC1, _SC2 or _SC3 to PUCHASE_COST so as to indicate the appropriate scenario. Each of the constraints from the core model are also replicated across each of the scenarios with the appropriate scenario tag appended to each row's name. Finally, at the bottom of the model, there are four constraints beginning with the string: _NAC, which are the nonanticipativity constraints. It's worthwhile pointing out that this report on the DE for SPGAS1 is a valid LINGO model. In fact, you can copy this report into a new model window and solve it directly, in which case, you will get the optimal solution to the original SP model.

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Scenario As mentioned in the previous section, the ability to generate the deterministic equivalent (DE) of a stochastic programming (SP) model can be very useful when initially formulating and debugging an SP model. The only problem is that the DE can become quite unwieldy for all but the simplest SPs. For example, if the core model has 500 variables and there are 1,000 scenarios, then the DE will have 500,000 variables. Tracking down problems in such a large model can be difficult. Fortunately, in most cases when an SP is either infeasible or unbounded, LINGO will report the index of the first scenario that is causing the problem. With such information, we can focus our attention on the isolated scenario that's causing the problem. This is where the Scenario command comes in  it allows us to generate the formulation for an individual scenario, as opposed to the entire DE, which should substantially cut down on the burden of our debugging task. As an example, we will work again with the SPGAS.LG4 gas buying model. After loading the model and issuing the LINGO|Generate|Scenario command, we are prompted for the index of the scenario we wish to view:

For our purposes, suppose we are interested in viewing scenario 2, in which case, we enter a 2 in the edit box and then press the OK button. LINGO the displays the following report: MODEL: TITLE Scenario 2; [ _1] MIN = PURCHASE_COST + HOLD_COST; [ _2] PURCHASE_COST - 5 * PURCHASE_1 - 6 * PURCHASE_2 = 0; [ _3] HOLD_COST - INVENTORY_1 - INVENTORY_2 = 0; [ _4] INVENTORY_1 - PURCHASE_1 = - 100; [ _5] - INVENTORY_1 + INVENTORY_2 - PURCHASE_2 = - 150; END

You will note that this is the core model, with the period 2 random variable DEMAND replaced by its sampled value of 150 in row 5. Nonanticipativity constraints are not generated as part of individual scenario models.

266 CHAPTER 5

LINGO|Picture

Ctrl+K

The LINGO|Picture command displays a model in matrix form. Viewing the model in matrix form can be helpful in a couple of instances. First and perhaps most importantly, is the use of nonzero pictures in debugging formulations. Most models have strong repetitive structure. Incorrectly entered sections of the model will stand out in a model’s matrix picture. Secondly, a nonzero picture can be helpful when you are attempting to identify special structure in your model. As an example, if your model displays strong block angular structure, then algorithms that decompose the model into smaller fragments might prove fruitful. As an example, we loaded the DNRISK.LG4 model from LINGO’s sample model set. Issuing the Picture command, we see the following:

Positive coefficients are represented with blue tiles, negatives with red, and variables that appear in a row nonlinearly show up as black tiles. You can zoom in on a selected range in the matrix for closer viewing. To do this, place the cursor on the upper left corner of the range you wish to view, press and hold down the left mouse button. Next, drag the mouse to the lower right-hand corner of the desired range. Now, release the left mouse button and LINGO will zoom in on the selected range. As an example, here is a view of the matrix after zooming in on a 4x4 range:

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Note, we have zoomed in far enough to be able see the actual coefficient values, row names, and variable names. Scroll bars have also appeared to allow scrolling through the matrix. The matrix picture window s a number of additional interactive features. To access these features, place the cursor over the matrix picture, and press and hold the right mouse button. This will bring up the following menu:

268 CHAPTER 5 A brief description of these features follows:              

Unpermuted - Displays the matrix in its original form Lower Triangular - Displays the matrix picture to lower triangular, or almost lower triangular form Block Triangular - GP1 - The matrix is displayed in block triangular form using LINGO'S GP1 graph partitioning heuristic Block Triangular - GP2 - The matrix is displayed in block triangular form using LINGO'S GP1 graph partitioning heuristic Block Triangular - Row Names - The matrix is displayed using a block structure specified by the as part of the row names Block Triangular - Specified - The matrix is displayed using a block structure specified by the via the @BLKROW function Zoom In - Zooms the view in centered around the current cursor position Zoom Out - Zooms the view out centered around the current cursor position View All - Zooms all the way out to give a full view of the matrix Row Names - Toggles the display of row names on and off Var Names - Toggles the display of variable names on and off Scroll Bars - Toggles scroll bars on and off Print - Prints the matrix picture Close - Closes the matrix picture window

The Lower Triangular option is an interesting feature. The rows and columns are automatically permuted to place the matrix into mostly lower-triangular form. If a matrix is mostly lower triangular, then, in general, the model should prove relatively easier to solve. This is opposed to a model that contains a high degree of simultaneity that can't be permuted into lower triangular form. Refer to the BNP Solver section above for more information on the various Block Triangular options (GP1, GP2, Row Names and Specified).

LINGO|Debug In the ideal world, all models would return an optimal solution. Unfortunately, this is not the case. Sooner or later, you are bound to run across either an infeasible or unbounded model. This is particularly true in the development phase of a project when the model will tend to suffer from typographical errors. Tracking down an error in a large model can prove to be a daunting task. The Debug command is useful in narrowing the search for problems in both infeasible and unbounded linear programs. A small portion of the original model is isolated as the source of the problem. This allows you to focus your attention on a subsection of the model in search of formulation or data entry errors. The Debug command identifies two types of sets: sufficient and necessary. Removing any sufficient set object from the model is sufficient to fix the entire model. Not all models will have a sufficient set. In which case, they will have a necessary set with the property that removing any object from this set fixes the remaining objects within that set.

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As an example, suppose you have an infeasible model. If the complete model would be feasible except for a bug in a single row, that row will be listed as part of the sufficient set. If the model has a necessary set, then, as long as all of them are present, the model will remain infeasible. The following example illustrates. The coefficient .55 in row 4 should have been 5.5:

When we attempt to solve this formulation, we get the following error:

270 CHAPTER 5 Next, if we run the LINGO|Debug command, we are presented with the following report:

The Debug command has correctly identified that the erroneous ROW4, when eliminated, is sufficient to make the entire model feasible. Debug operates in a similar manner for unbounded models. In the following example, we introduced an error by placing a minus sign instead of a plus sign in front of variable Z3 in ROW3. A look at ROW3 reveals that Z3 can be increased indefinitely, leading to an unbounded objective.

The resulting model is unbounded and, when issuing the LINGO|Solve command, we receive the unbounded error message:

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Issuing the Debug command, we receive the following breakdown:

The Debug command has successfully determined that bounding Z3 is sufficient to bound the entire model. Typically, the Debug command helps to substantially reduce the search effort. The first version of this feature was implemented in response to a who had an infeasible model. The had spent a day searching for a bug in a model with 400 constraints. The debug feature quickly found a necessary set with 55 constraints, as well as one sufficient set constraint. The immediately noticed that the right-hand side of the sufficient set constraint was incorrect. Note:

Prior to release 10.0 of LINGO, the debugger was only capable of processing linear models. Starting with release 10.0, all classes of models (LP, QP, IP and NLP) may now be debugged.

LINGO|Model Statistics The Model Statistics command lists summary statistics for your model. The statistics vary slightly depending on whether the model you’re working with is linear or nonlinear. In the following example, we open the linear transportation model, TRAN.LG4, issue the Model Statistics command, and then discuss some of the details of the report. Here is the output generated by Model Statistics for TRAN.LG4:

The statistics report consists of five lines. In line one, the number of rows (constraints), variables (columns), and integer variables are shown. The report also specifies when the model is linear by stating that all variables are linear.

272 CHAPTER 5 Line two of the report gives a count of the number of nonzero coefficients appearing in the model. The first count is the number of nonzero coefficients in the entire model. The Constraint nonz count is the number of coefficients on the left-hand sides of all the constraints, excluding the nonzero objective and right-hand side coefficients. Next, is a count of the number of constraint coefficients that are plus or minus one. In general, a linear programming model is easier to solve when the number of percentage of +/- 1 coefficient increases. Finally, LINGO reports a Density figure, which is defined as: (total nonzeros) / [(number of rows) * (number of columns + 1)]. For large models, densities under .01 are common. High densities can mean that a problem will take longer to solve. Line three lists the smallest and largest coefficients in the model in absolute value. For stability reasons, the ratio of the largest coefficient to the smallest should, ideally, be close to 1. Also, in absolute , it is best to keep coefficient values in the range of 0.0001 to 100,000. Values outside this range can cause numerical difficulties for the solver. Line four lists the number of constraints by type (<, =, and >), the sense of the objective, and an upper bound on the number of Generalized Upper Bound (GUB) constraints. A GUB constraint is a constraint that does not intersect with the remainder of the model. Given this, the GUB statistic is a measure of model simplicity. If all the constraints were nonintersecting, the problem could be solved by inspection by considering each constraint as a separate problem. Line five lists the number of variables that appear in only one row. Such a variable is effectively a slack. If you did not explicitly add slack variables to your model and the single column count is greater than zero, then it suggests a misspelled variable name. The following report was generated by the Model Statistics command for the nonlinear model, DNRISK.LG4:

The statistics report for nonlinear models drops information about the range of coefficient values, the number of +/- 1 coefficient, and the GUB upper bound. A count of the number of nonlinear variables and rows is added in line two. The nonlinear rows count includes the objective, while the nonlinear constraint count does not.

WINDOWS COMMANDS

LINGO|Look...

273

Ctrl+L

Use the Look command to generate a report containing your model’s formulation. The Look command’s dialog box, pictured below, lets you choose All or Selected rows for viewing from the Rows to View:

When you choose Selected rows, the Beginning Row and Ending Row text boxes are available for entry in the Selected Rows box. You must enter the indices of the range of rows you wish displayed. LINGO will display the requested lines with line numbers in a new window.

4. Window Menu The Window menu, pictured at left, contains commands that generally pertain to managing open windows.

274 CHAPTER 5

Window|Command Window

Ctrl+1

In addition to the pull down menu commands, LINGO’s features can also be accessed through a command language. For more details on the command language, refer to the following chapter, Command-line Commands. A script file that contains LINGO commands may be run using the File|Take Commands command. Alternatively, you can interactively enter script commands into LINGO’s command window. The Window|Command Window command opens LINGO’s command window. The following window should appear on your screen:

WINDOWS COMMANDS

275

You may enter any valid LINGO script commands to the colon prompt in the upper left corner of the window. In the following example, we enter a small model with the MODEL command, display the formulation with the LOOK ALL command, and then solve it using the GO command ( input is shown in bold type):

In general, you will probably prefer to use the pull down menus and toolbar when using LINGO interactively. The command window interface is primarily provided for s wishing to interactively test command scripts.

276 CHAPTER 5

Window|Status Window

Ctrl+2

When you invoke LINGO’s Solve command, a status window is displayed on your screen that resembles the following:

This window allows you to monitor the progress of the solver. You can close the status window at any time. If you close the status window, it may be reopened with the Window|Status Window command. If you would like to prevent LINGO from opening a status window, see the LINGO|Options command above. For more information on the interpretation and use of the status window, see page 7.

Window|Send To Back

Ctrl+B

The Window|Send To Back command sends the active window behind all others on the screen. This command is useful when switching between a model and a solution window.

Window|Close All

Ctrl+3

The Window|Close All command closes all open windows. If you made a change to a model window without saving it, you will be prompted to save the model before it is closed.

WINDOWS COMMANDS

Window|Tile

277

Ctrl+4

The Window|Tile command arranges all the open windows in a tiled pattern. Each window is resized, so all windows appear on the screen and are of roughly the same size. When you issue the Window|Tile command, you will see the dialog box:

You have the choice of tiling the windows horizontally or vertically. If you tile Horizontally (or Vertically), LINGO will maximize the horizontal (or vertical) dimension of each window. If there are more than three open windows, LINGO will tile the windows, but the choice of horizontal or vertical will no longer make a difference.

Window|Cascade

Ctrl+5

The Window|Cascade command arranges all open windows in a cascade pattern starting in the upper left corner of the mainframe window. The currently active window remains on top.

Window|Arrange Icons

Ctrl+6

If you have minimized any open windows, so they appear as icons on the screen, you can issue the Window|Arrange Icons command to line all the icons up in the lower left-hand corner of the frame window.

5. Help Menu The Help menu, pictured at left, contains commands that generally pertain to LINGO’s Help system, copyright notice, and version specific information.

278 CHAPTER 5

Help|Help Topics A portion of the dialog box displayed by the Help Topics command is displayed below:

Select the Contents tab to display a table of contents for the Help system. You can select any of the topics that are of interest by double clicking on them. Select the Index tab to display an index of topics for the Help system. Select an item for viewing by double clicking on it. Go to the Find tab to search the Help system for a particular item.

WINDOWS COMMANDS

279

Help| Use the Help| command to your version of LINGO online. You will need a connection to the Internet open for this command to work. When you issue the command, you will be presented with the following dialog box:

280 CHAPTER 5 Enter your personal information and select the button. Your information will be sent directly to LINDO Systems via the Internet. Once your registration is complete, the following dialog box will appear on your screen:

Select the OK button to be returned to the main LINGO environment. LINDO Systems is constantly working to make our products faster and easier to use. ing your software with LINDO ensures that you will be kept up-to-date on the latest enhancements and other product news. You can also through the mail or by fax using the registration card included with your software package.

Help|AutoUpdate Turn the Help|AutoUpdate command on to have LINGO automatically check every time you start the LINGO software whether there is a more recent version of LINGO available for on the LINDO Systems website. You will need a connection to the internet open for this command to work. When you issue the AutoUpdate command or start a version of LINGO with AutoUpdate enabled, LINGO will search the Internet to see if an updated version of the LINGO software is available for . If you currently have the most recent version, then you will be returned to the main LINGO environment. If you have an outdated version of the software, you will be presented with the following dialog box:

at which point, you may wish to go to the LINDO Systems Web site, www.lindo.com, to the latest build of the software. If you want to disable the AutoUpdate feature, then select the Disable AutoUpdate button from the AutoUpdate dialog box. The AutoUpdate feature is disabled by default.

WINDOWS COMMANDS

281

Help|About LINGO When you issue the About LINGO command, you will be presented with a dialog box resembling the following:

The first box lists size and release information of your copy of LINGO.

282 CHAPTER 5 The second box tells you where you can get in touch with LINDO Systems. The third box, titled Limits for this Installation, lists various capacity limits of your version and the current number of bytes allocated to LINGO's model generator. The maximum sized problem your LINGO software can handle depends on the version you have. The current limits for the various versions are: Version Demo/Web

Total Variables 300

Integer Nonlinear Global Variables Variables Variables 30 30 5

Constraints 150

500

50

50

5

250

Super

2,000

200

200

10

1,000

Hyper

8,000

800

800

20

4,000

Industrial

32,000

3,200

3,200

50

16,000

Extended

Unlimited

Unlimited

Unlimited

Unlimited

Unlimited

Solver Suite

For more information on the definitions of these limits see section Maximum Problem Dimensions. In addition to the maximum problem limits, this box also lists the amount of memory allocated to LINGO’s model generator. You can adjust the size of generator memory allocation on the General Solver tab of the LINGO|Options dialog box. The fourth box titled License Expiration lists the date at which your license expires. If your license does not have an expiration date, this field will display Perpetual. The box labeled License Usage lists whether your license is for commercial or educational use. Educational licenses are restricted to use by faculty, staff, and students of educational institutions for instructional or research purposes. Commercial licenses are not restricted to any particular use. The box titled Licenses lists the number of s licensed to use your copy of LINGO. The API Version box lists the version number of the LINDO API in use by your copy of LINGO. The LINDO API is the library of solver tools used by LINGO to optimize your models. The License Location box displays the location of the license file in effect for the current LINGO session, while the Config Location box gives the location of LINGO's configuration file. The configuration file is where LINGO stores non-default option settings, which are controlled by the LINGO|Options command. The final box, Additional License Information, contains information relevant to your particular license. In most cases, your LINGO serial number can be found in this field. Scrolling through this field, you will also find information as to the optional features included with your license (e.g., the barrier, conic, nonlinear, global and stochastic solvers.)

Help|Pointer Press this button to switch the cursor into Help mode. Once the cursor is in Help mode, you can select a menu command or a toolbar button and LINGO will display help information on the selected item.

6 Command-Line Commands This chapter discusses all of the command-line commands available to the LINGO . On platforms other than Windows based PC’s, the interfaces with LINGO entirely through text commands issued to LINGO’s command-line colon prompt. If you are using a Windows version of LINGO, you will primarily be interested in the previous chapter, Windows Commands, which details the commands available in the pull down menus of LINGO’s Windows version. However, in Windows versions, LINGO command-line commands may be entered using the command window (see the Window|Command Window section in Chapter 5, Windows Commands) and may also be used to build command scripts. Command scripts may be run automatically at startup or whenever the desires. Command scripts are useful to both the Windows and s on other platforms. Thus, the Windows may find this chapter of interest, too. We will begin by briefly listing all the command-line commands according to their general function. This will be followed up by an in-depth explanation of the commands.

The Commands In Brief 1. Information CAT COM HELP MEM

lists categories of available commands lists available commands by category provides brief help on commands provides statistics about model generator memory usage

2. Input FRMPS MODEL RLPF RMPI RMPS TAKE

retrieves a model in free MPS format begins input of a new model retrieves a model in LP format retrieves a model in MPI format retrieves a model in fixed MPS format runs a command script from an external file

283

284 CHAPTER 6

3. Display DUAL GEN HIDE LOOK PICTURE SCENE SHOWNL STATS XDETEQ

generates and displays the dual formulation for the model generates the algebraic formulation for the model protects the current model displays the current model displays a picture of the model’s nonzero structure generates the algebraic formulation for a specified scenario of a larger stochastic program generates the entire model, but only displays rows containing nonlinearities gives summary statistics about the properties of a generated model generates the algebraic formulation for a explicit deterministic equivalent of a stochastic program

4. File Output DIVERT RVRT SAVE SMPI SMPS

opens a file for receiving output closes a file previously opened with DIVERT saves the current model to disk exports a model in MPI format sends a copy of the current model to a file in MPS format

5. Solution DEBUG GO NONZ RANGE SOLU

tracks down formulation errors in infeasible and unbounded models solves the current model generates a nonzeros only solution report generates a range analysis report generates a solution report

6. Problem Editing ALTER DELETE EXTEND

edits the contents of the model deletes a selected row from the model adds rows to the end of the current model

7. Conversational Parameters PAGE PAUSE TERSE VERBOSE WIDTH

sets the page/screen length pauses for keyboard input output level switches to verbose output mode sets terminal display and input width

COMMAND-LINE COMMANDS

285

8. Tolerances APISET DBPWD DBUID FREEZE SET

allows access to advanced parameters in the LINDO API, which is the solver library used by LINGO sets the for database access via @ODBC sets your id for database access via @ODBC saves current tolerance settings to disk overrides a number of LINGO defaults and tolerances

9. Miscellaneous ! QUIT TIME

inserts a comment exits LINGO displays current elapsed time since start of session

The Commands In Depth Each LINGO command-line command is discussed in detail in this section. Commands are grouped by category based upon their general function. Note:

input in the examples below is indicated through the use of bold typeface.

1. Information The Information category contains commands related to on-line information.

CAT The CAT command displays the nine categories of commands available in LINGO. You will be prompted to input a number corresponding to one of the categories. If you input a number, LINGO will display the commands available under the corresponding category. To exit out of the command, input a blank line.

COM The COM command lists all the command-line commands available in LINGO by category.

HELP The HELP command combined with another LINGO command gives you information on the command specified. The information is usually quite brief, but is often all that is needed. The HELP command without an argument will give you general information about your version of LINGO, along with the maximum number of constraints and variables that your version of LINGO can handle.

286 CHAPTER 6

MEM The MEM command displays statistics about the model generator's memory usage. The following is some sample output from the MEM command: : MEM Total generator memory Peak generator memory usage

Current generator memory usage Total handles Peak handle usage

Current handle usage Total bytes moved Total blocks moved

Total heap compacts Fragmentation ratio

5242880 12048

1312 96 9

5

1552 6

0

0.002

:

The Total generator memory figure is the amount of memory LINGO has allocated for a working memory heap for model generation. You can control the size of the heap using the SET command. Peak generator memory usage refers to the maximum amount of memory the model generator used during the current session. Current memory usage lists the amount of working memory currently in use by the model generator. Total handles is the maximum number of memory blocks LINGO can allocate. Peak handle usage lists the maximum number of memory blocks LINGO allocated at any one time during this session. Current handle usage represents the number of memory blocks currently in use by the model generator. Total bytes moved lists the number of memory bytes the generator has had to move so far in order to reallocate memory. Total blocks moved lists the number of memory blocks moved due to reallocation. Total heap compacts lists the number of times the generator has had to compact the heap to make room for growing memory needs. If the number of heap compacts is abnormally large, you should allocate more working memory using the SET command. The Fragmentation ratio is a statistic measuring how fragmented the memory heap is. A value of 1 would indicate high fragmentation, whereas a value of 0 indicates no fragmentation

2. Input The Input category contains commands that initiate input into LINGO

FRMPS / RMPS The FRMPS and RMPS commands are used to read MPS formatted models. The MPS file format is an industry standard format developed by IBM, and is useful for ing models from one solver or platform to another. FRMPS reads an MPS file in free format, while RMPS reads fixed format MPS files.


287

When LINGO reads an MPS file, it converts the formulation to an equivalent LINGO model. As an example, consider the following, simple model: ObjRow) Maximize 20X + 30Y Subject To: Row1) X < 50 Row2) Y < 60 Row3) X + 2Y < 120 An equivalent MPS file for this model is: NAME OBJSENSE MAX ROWS N OBJROW L ROW1 L ROW2 L ROW3 COLUMNS X X X Y Y Y RHS RHS RHS RHS ENDATA

SAMPLE

ROW3 OBJROW ROW1 OBJROW ROW2 ROW3 ROW1 ROW2 ROW3

1.0000000 20.0000000 1.0000000 30.0000000 1.0000000 2.0000000 50.0000000 60.0000000 120.0000000

As an aside, one thing to notice about the MPS representation is that it is not a very compact method for storing a model. In the following session, we read this MPS file into LINGO and then display the model with the LOOK command. Note how the model is automatically converted from MPS format to LINGO format: : rmps c:\sample.mps : look all 1] 2] 3] 4] 5]

TITLE SAMPLE; [ OBJROW] MAX = 20 * X + 30 * Y; [ ROW1] X <= 50; [ ROW2] Y <= 60; [ ROW3] X + 2 * Y <= 120;

: Should you wish to save the file again using MPS format rather than LINGO format, you may use the SMPS command. When it comes to acceptable constraint and variable names, MPS format is less restrictive than LINGO. MPS allows for embedded blanks and other additional characters in names. To compensate for this fact, LINGO attempts to patch names when reading an MPS file so that all the incoming names

288 CHAPTER 6 are compatible with its syntax. LINGO does this by substituting an underscore for any character in a name that is not issible. In most cases, this will work out OK. However, there is a chance for name collisions where two or more names get mapped into one. For instance, the variable names X.1 and X%1 would both get mapped into the single LINGO name X_1. Of course, situations such as this entirely alter the structure of the model rendering it incorrect. You will be warned whenever LINGO has to patch a name with the following error message: [Error Code:

179]

The model translator had to patch names to make them compatible: var names patched: 1 row names patched: 0 Name collisions may have occurred.

This message displays the number of variable and row names that were patched to get them to conform to LINGO syntax. If name collisions are a problem, then LINGO has an option that will ensure that all names remain unique. This option involves using RC format for names encountered during MPS I/O. RC format involves renaming each row (constraint) in a model to be Rn, where n is the row’s index. Similarly, each column (variable) is renamed to Cn. In addition, LINGO renames the objective row to be ROBJ. To switch to RC format for MPS names, you will need to use the SET command as follows: : SET RCMPSN 1

This will cause LINGO to use RC naming conventions for all MPS reads and saves. To cancel the use of RC names, type: : SET RCMPSN 0

As an example, we will once again read the same MPS format model we read above, but this time we will switch to RC naming conventions. : set rcmpsn 1 Parameter RCMPSN

Old Value 0

New Value 1

: rmps c:\sample.mps : look all 1] 2] 3] 4] 5]

TITLE SAMPLE; [ ROBJ] MAX = 20 * C1 + 30 * C2; [ R1] C1 <= 50; [ R2] C2 <= 60; [ R3] C1 + 2 * C2 <= 120;

Notice how the variable names now use RC format, guaranteeing that name collisions will not occur. Another potential conflict is that MPS allows variable names to be duplicated as constraint names, and vice versa. LINGO does not allow for this. When you go to solve the model, you will either receive error message 28 (Invalid use of a row name), or error message 37 (Name already in use). However, once again, you can switch to using RC format for names to avoid this conflict.


289

MODEL Use the MODEL command to begin inputting a new model into LINGO. LINGO prompts for each new line of the model with a question mark. When you are through entering the model, enter END on a single line by itself. LINGO will then return to normal command mode (indicated by the colon prompt). In the following example, we enter a small model with the MODEL command, display it with the LOOK command, and then solve it with the GO command: : MODEL ? ? ? ? ?

!How many years does it take to double an investment growing 10% per year?; 1.1 ^ YEARS = 2; END

: LOOK ALL

1]!How many years does it take 2]to double an investment growing 3]10% per year?; 4]1.1 ^ YEARS = 2; : GO Feasible solution found at step: Variable YEARS Row 1

0


:

RMPI The RMPI command is used to read MPI (Math Programmming Interface) formatted models. The MPI file format was developed by LINDO Systems as a portable format for storing arbitrary math programming models. When LINGO reads an MPI file, it converts the formulation to an equivalent LINGO model. As an example, consider the following, simple model: ObjRow) Maximize 20X + 30Y Subject To: Row1) X < 50 Row2) Y < 60 Row3) X + 2Y < 120

290 CHAPTER 6 The equivalent MPI file for this model is: BEGINMODEL SAMPLE ! Number of Objective Functions: ! Number of Constraints : ! Number of Variables : VARIABLES ! Name Lower Bound Upper Bound Type X 0 1e+030 C Y 0 1e+030 C OBJECTIVES OBJROW MAXIMIZE EP_USRCOD -101 EP_PUSH_NUM 20 EP_PUSH_VAR X EP_MULTIPLY EP_PUSH_NUM 30 EP_PUSH_VAR Y EP_MULTIPLY EP_PLUS CONSTRAINTS ROW1 L EP_USRCOD -101 EP_PUSH_VAR X EP_PUSH_NUM 50 EP_MINUS ROW2 L EP_USRCOD -101 EP_PUSH_VAR Y EP_PUSH_NUM 60 EP_MINUS ROW3 L EP_USRCOD -101 EP_PUSH_VAR X EP_PUSH_NUM 2 EP_PUSH_VAR Y EP_MULTIPLY EP_PLUS EP_PUSH_NUM 120 EP_MINUS ENDMODEL

1 3 2 Initial Point 1.23457 1.23457

As an aside, one thing to notice about the MPI representation is that it is not a very compact method for storing a modelMPI is designed for portability, as opposed to efficiency.


291

In the following session, we read this MPI file into LINGO and then display the model with the LOOK command. Note how the model is automatically converted from MPI format to native LINGO format : : rmpi c:\sample.mpi : look all 1] 2] 3] 4] 5]

TITLE SAMPLE; [OBJROW] MAX = 20 * X + 30 * Y; [ROW1] X <= 50; [ROW2] Y <= 60; [ROW3] X + 2 * Y <= 120;

: Should you wish to save the file again using MPI format rather than LINGO format, you may use the SMPI command. Note:

The MPI file format is intended primarily for exporting models to other applications or platforms. The MPI format is purely scalar in nature—all set-based information is lost upon converting a LINGO model to MPI format. Thus, when saving copies of a model on your own machine, you should always use the SAVE command in order to save models in native LINGO format in order to preserve your model in its entirety.

When it comes to acceptable constraint and variable names, MPI format is less restrictive than LINGO. MPI allows for embedded blanks and other additional characters in names. To compensate for this fact, LINGO attempts to patch names when reading an MPI file so that all the incoming names are compatible with its syntax. LINGO does this by substituting an underscore for any character in a name that is not issible. In most cases, this will work out OK. However, there is a chance for name collisions where two or more names get mapped into one. For instance, the variable names X.1 and X%1 would both get mapped into the single LINGO name X_1. Of course, situations such as this entirely alter the structure of the model rendering it incorrect. You will be warned whenever LINGO has to patch a name with the following error message: [Error Code:

179]


This message displays the number of variable and row names that were patched to get them to conform to LINGO syntax. If name collisions are a problem, then LINGO has an option that will ensure that all names remain unique. This option involves using RC format for names when translating non-native file formats. RC format involves renaming each row (constraint) in a model to be Rn, where n is the row’s index. Similarly, each column (variable) is renamed to Cn. In addition, LINGO renames the objective row to be ROBJ. To switch to RC format for MPS names mode, you will need to use the SET command as follows: : SET RCMPSN 1

292 CHAPTER 6 This will cause LINGO to use RC naming conventions for all MPI reads and saves. To cancel the use of RC names, type: : SET RCMPSN 0

As an example, we will once again read the same MPI format model we read above, but this time we will switch to RC naming conventions. : set rcmpsn 1 Parameter RCMPSN

Old Value 0

New Value 1

: rmpi c:\sample.mpi : look all 1] 2] 3] 4] 5]

TITLE SAMPLE; [ROBJ] MAX = 20 * C1 + 30 * C2; [R1] C1 <= 50; [R2] C2 <= 60; [R3] C1 + 2 * C2 <= 120;

Notice how the variable names now use RC format, guaranteeing that name collisions will not occur. Another potential conflict is that MPI allows variable names to be duplicated as constraint names, and vice versa. LINGO does not allow for this. When you go to solve the model, you will either receive error message 28 (Invalid use of a row name), or error message 37 (Name already in use). However, once again, you can switch to using RC format for names to avoid this conflict.

RLPF Command The RLPF command is used to read LP formatted models. The LP file format is an industry standard, and is useful for ing models from one solver or platform to another. When LINGO reads an LP format file, it converts the formulation to an equivalent LINGO model. As an example, consider the following, simple model: ObjRow) Maximize 20X + 30Y Subject To: Row1) X < 50 Row2) Y < 60 Row3) X + 2Y < 120 An equivalent LP format file for this model is: \ LP format example Maximize objrow: 20x + 30y Subject To row1: x <= 50 row2: y <= 60 row3: x + 2y <= 120 End

In the following session, we read this LP file into LINGO and then display the model with the LOOK command. Note how the model is automatically converted from LP format to LINGO format:


293

: rlpf c:\sample.lp : look all MODEL: MAX= 20 * X + 30 * Y; [ROW1] X <= 50; [ROW2] Y <= 60; [ROW3] X + 2 * Y <= 120; END :

Note:

Unlike with MPS and MPI formats, LINGO does not currently exporting/saving files in LP format.

When it comes to acceptable constraint and variable names, LP format is less restrictive than LINGO. To compensate for this fact, LINGO attempts to patch names when reading an LP file so that all the incoming names are compatible with its syntax. LINGO does this by substituting an underscore for any character in a name that is not issible. In most cases, this will work out OK. However, there is a chance for name collisions where two or more names get mapped into one. For instance, the variable names X.1 and X%1 would both get mapped into the single LINGO name X_1. Of course, situations such as this entirely alter the structure of the model rendering it incorrect. You will be warned whenever LINGO has to patch a name with the following error message: [Error Code:

179]


This message displays the number of variable and row names that were patched to get them to conform to LINGO syntax. If name collisions are a problem, then LINGO has an option that will ensure that all names remain unique. This option involves using RC format for names when translating non-native file formats. RC format involves renaming each row (constraint) in a model to be Rn, where n is the row’s index. Similarly, each column (variable) is renamed to Cn. In addition, LINGO renames the objective row to be ROBJ. To switch to the RC format for MPS names mode, you will need to use the SET command as follows: : SET RCMPSN 1

This will cause LINGO to use RC naming conventions for all names in an LP format file. To cancel the use of RC names, type: : SET RCMPSN 0

As an example, we will once again read the same LP format format model we read above, but this time we will switch to RC naming conventions. : set rcmpsn 1 Parameter

Old Value

New Value

294 CHAPTER 6 RCMPSN

0

1

: rmps c:\sample.lp : look all 1] 2] 3] 4] 5]

TITLE SAMPLE; [ ROBJ] MAX = 20 * C1 + 30 * C2; [ R1] C1 <= 50; [ R2] C2 <= 60; [ R3] C1 + 2 * C2 <= 120;

Notice how the variable names now use RC format, guaranteeing that name collisions will not occur. Another potential conflict is that LP format allows variable names to be duplicated as constraint names, and vice versa. LINGO does not allow for this. When you go to solve the model, you will either receive error message 28 (Invalid use of a row name), or error message 37 (Name already in use). However, once again, you can switch to using RC format for names to avoid this conflict.

TAKE The TAKE command is used to 1) read models saved to disk using the SAVE command, and 2) execute command scripts contained in external files. The syntax for the TAKE command is: TAKE [filename] If you omit a filename, LINGO will prompt you for one. As an example, suppose you used the SAVE command to save a model to the file C:\LINGOMOD\MYMODEL.LNG. You can read it back into LINGO by giving the command: : TAKE C:\LINGOMOD\MYMODEL.LNG


295

As a second example, we will use the TAKE command to execute a LINGO command script. A command script is simply a text file that contains a series of LINGO commands. Suppose we have built the following command script in an editor and have saved it in the text file D:\LNG\MYSCRIPT.LTF: MODEL: !For a given probability P, this model returns the value X such that the probability that a unit normal random variable is less than or equal to X is P; ! Here is the probability; P = .95; ! Solve for X; P = @PSN(X); END !Terse output mode; TERSE !Solve the model; GO !Report X; SOLU X

We can use the TAKE command to run the script as follows: : TAKE D:\LNG\MYSCRIPT.LTF Feasible solution found at step: Variable X

0

Value 1.644854

:

3. Display This category contains commands that display information.

DUAL The DUAL command displays the dual formulation of the current model. Every linear programming model has a corresponding, mirror-image formulation called the dual. If the original model has M constraints and N variables, then its dual will have N constraints and M variables. Some interesting properties of the dual are that any feasible solution to the dual model provides a bound on the objective to the original, primal model, while the optimal solution to the dual has the same objective value as the optimal solution to the primal problem. It's also true that the dual of the dual model is, once again, the original primal model. You may wish to refer to any good linear programming text for a further discussion of duality theory.

296 CHAPTER 6 As an example, consider the following small transportation model: MODEL: ! A 3 Warehouse, 4 Customer Transportation Problem; SETS: WAREHOUSE / WH1, WH2, WH3/ : CAPACITY; CUSTOMER / C1, C2, C3, C4/ : DEMAND; ROUTES( WAREHOUSE, CUSTOMER) : COST, VOLUME; ENDSETS ! The objective; [OBJ] MIN = @SUM( ROUTES: COST * VOLUME); ! The demand constraints; @FOR( CUSTOMER( J): [DEM] @SUM( WAREHOUSE( I): VOLUME( I, J)) >= DEMAND( J)); ! The supply constraints; @FOR( WAREHOUSE( I): [SUP] @SUM( CUSTOMER( J): VOLUME( I, J)) <= CAPACITY( I)); ! Here are the parameters; DATA: CAPACITY = 30, 25, 21 ; DEMAND = 15, 17, 22, 12; COST = 6, 2, 6, 7, 4, 9, 5, 3, 8, 8, 1, 5; ENDDATA END

Model: TRAN.LNG


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If the sample session below, we load the sample model TRAN.LNG and use the DUAL command to generate its dual formulation: : take \lingo\samples\tran.lng : dual MODEL: MAX = 15 * DEM_C1 + 17 * DEM_C2 + 22 * DEM_C3 + 12 * DEM_C4 + 30 * SUP_WH1 + 25 * SUP_WH2 + 21 * SUP_WH3; [ VOLUME_WH1_C1] DEM_C1 + SUP_WH1 <= 6; [ VOLUME_WH1_C2] DEM_C2 + SUP_WH1 <= 2; [ VOLUME_WH1_C3] DEM_C3 + SUP_WH1 <= 6; [ VOLUME_WH1_C4] DEM_C4 + SUP_WH1 <= 7; [ VOLUME_WH2_C1] DEM_C1 + SUP_WH2 <= 4; [ VOLUME_WH2_C2] DEM_C2 + SUP_WH2 <= 9; [ VOLUME_WH2_C3] DEM_C3 + SUP_WH2 <= 5; [ VOLUME_WH2_C4] DEM_C4 + SUP_WH2 <= 3; [ VOLUME_WH3_C1] DEM_C1 + SUP_WH3 <= 8; [ VOLUME_WH3_C2] DEM_C2 + SUP_WH3 <= 8; [ VOLUME_WH3_C3] DEM_C3 + SUP_WH3 <= 1; [ VOLUME_WH3_C4] DEM_C4 + SUP_WH3 <= 5; @BND( -0.1E+31, SUP_WH1, 0); @BND( -0.1E+31, SUP_WH2, 0); @BND( -0.1E+31, SUP_WH3, 0); END :

You will notice that in the dual formulation the variables from the primal model become the rows of the dual. Similarly, the rows in the primal become the variables in the dual. Note:

The row names from the primal problem will become the variable names in the dual formulation. For this reason, it is strongly recommended that you name all the rows in the primal model. If a row is unnamed, then a default name will be generated for the corresponding dual variable. The default name will consist of an underscore followed by the row's internal index. These default names will not be very meaningful, and will make the dual formulation difficult to interpret.

GEN Once you remove all the syntax errors from your LINGO model, there is still one very important step required: model verification. LINGO’s set-based modeling capabilities are very powerful, and they allow you to generate large, complex models quickly and easily. However, when you first develop a model you will need to that the model being generated matches up to the model you actually intended to generate. Many set-based models can be quite complex, and it is highly likely that logic errors may creep into one or more expressions, thereby causing your generated model to be flawed. The GEN (short for generate) command is very useful for debugging such errors. It expands all of the model's compact set-based expressions and then writes out the full scalar-based equivalent of the LINGO model. The expanded model report explicitly lists all the generated constraints and variables in your model. You will find that the Generate report can be an invaluable tool in tracking down errors.

298 CHAPTER 6 As an example of the output from the generate command, consider the transportation model developed in Chapter 1: MODEL: ! A 6 Warehouse 8 Vendor Transportation Problem; SETS: WAREHOUSES: CAPACITY; VENDORS: DEMAND; LINKS( WAREHOUSES, VENDORS): COST, VOLUME; ENDSETS DATA: !set ; WAREHOUSES = WH1 WH2 WH3 WH4 WH5 WH6; VENDORS = V1 V2 V3 V4 V5 V6 V7 V8; !attribute values; CAPACITY = 60 55 51 43 41 52; DEMAND = 35 37 22 32 41 32 43 38; COST = 6 2 6 7 4 2 5 9 4 9 5 3 8 5 8 2 5 2 1 9 7 4 3 3 7 6 7 3 9 2 7 1 2 3 9 5 7 2 6 5 5 5 2 2 8 1 4 3; ENDDATA ! The objective; [OBJECTIVE] MIN = @SUM( LINKS( I, J): COST( I, J) * VOLUME( I, J)); ! The demand constraints; @FOR( VENDORS( J): [DEMAND_ROW] @SUM( WAREHOUSES( I): VOLUME( I, J)) = DEMAND( J)); ! The capacity constraints; @FOR( WAREHOUSES( I): [CAPACITY_ROW] @SUM( VENDORS( J): VOLUME( I, J)) <= CAPACITY( I)); END

Model: WIDGETS


299

The objective will generate one expression; there should be one demand constraint generated for each of the eight vendors and one supply constraint generated for each of the six warehouses, for a grand total of 15 rows in the expanded model. Running the generate command to this reveals the following report: MODEL: [OBJECTIVE] MIN= 6 * VOLUME_WH1_V1 + 2 * VOLUME_WH1_V2 + 6 * VOLUME_WH1_V3 + 7 * VOLUME_WH1_V4 + 4 * VOLUME_WH1_V5 + 2 * VOLUME_WH1_V6 + 5 * VOLUME_WH1_V7 + 9 * VOLUME_WH1_V8 + 4 * VOLUME_WH2_V1 + 9 * VOLUME_WH2_V2 + 5 * VOLUME_WH2_V3 + 3 * VOLUME_WH2_V4 + 8 * VOLUME_WH2_V5 + 5 * VOLUME_WH2_V6 + 8 * VOLUME_WH2_V7 + 2 * VOLUME_WH2_V8 + 5 * VOLUME_WH3_V1 + 2 * VOLUME_WH3_V2 + VOLUME_WH3_V3 + 9 * VOLUME_WH3_V4 + 7 * VOLUME_WH3_V5 + 4 * VOLUME_WH3_V6 + 3 * VOLUME_WH3_V7 + 3 * VOLUME_WH3_V8 + 7 * VOLUME_WH4_V1 + 6 * VOLUME_WH4_V2 + 7 * VOLUME_WH4_V3 + 3 * VOLUME_WH4_V4 + 9 * VOLUME_WH4_V5 + 2 * VOLUME_WH4_V6 + 7 * VOLUME_WH4_V7 + VOLUME_WH4_V8 + 2 * VOLUME_WH5_V1 + 3 * VOLUME_WH5_V2 + 9 * VOLUME_WH5_V3 + 5 * VOLUME_WH5_V4 + 7 * VOLUME_WH5_V5 + 2 * VOLUME_WH5_V6 + 6 * VOLUME_WH5_V7 + 5 * VOLUME_WH5_V8 + 5 * VOLUME_WH6_V1 + 5 * VOLUME_WH6_V2 + 2 * VOLUME_WH6_V3 + 2 * VOLUME_WH6_V4 + 8 * VOLUME_WH6_V5 + VOLUME_WH6_V6 + 4 * VOLUME_WH6_V7 + 3 * VOLUME_WH6_V8 ; [DEMAND_ROW_V1] VOLUME_WH1_V1 + VOLUME_WH2_V1 + VOLUME_WH3_V1 + VOLUME_WH4_V1 + VOLUME_WH5_V1 + VOLUME_WH6_V1 = 35 ; [DEMAND_ROW_V2] VOLUME_WH1_V2 + VOLUME_WH2_V2 + VOLUME_WH3_V2 + VOLUME_WH4_V2 + VOLUME_WH5_V2 + VOLUME_WH6_V2 = 37 ; [DEMAND_ROW_V3] VOLUME_WH1_V3 + VOLUME_WH2_V3 + VOLUME_WH3_V3 + VOLUME_WH4_V3 + VOLUME_WH5_V3 + VOLUME_WH6_V3 = 22 ; [DEMAND_ROW_V4] VOLUME_WH1_V4 + VOLUME_WH2_V4 + VOLUME_WH3_V4 + VOLUME_WH4_V4 + VOLUME_WH5_V4 + VOLUME_WH6_V4 = 32 ; [DEMAND_ROW_V5] VOLUME_WH1_V5 + VOLUME_WH2_V5 + VOLUME_WH3_V5 + VOLUME_WH4_V5 + VOLUME_WH5_V5 + VOLUME_WH6_V5 = 41 ; [DEMAND_ROW_V6] VOLUME_WH1_V6 + VOLUME_WH2_V6 + VOLUME_WH3_V6 + VOLUME_WH4_V6 + VOLUME_WH5_V6 + VOLUME_WH6_V6 = 32 ; [DEMAND_ROW_V7] VOLUME_WH1_V7 + VOLUME_WH2_V7 + VOLUME_WH3_V7 + VOLUME_WH4_V7 + VOLUME_WH5_V7 + VOLUME_WH6_V7 = 43 ; [DEMAND_ROW_V8] VOLUME_WH1_V8 + VOLUME_WH2_V8 + VOLUME_WH3_V8 + VOLUME_WH4_V8 + VOLUME_WH5_V8 + VOLUME_WH6_V8 = 38 ; [CAPACITY_ROW_WH1] VOLUME_WH1_V1 + VOLUME_WH1_V2 + VOLUME_WH1_V3 + VOLUME_WH1_V4 + VOLUME_WH1_V5 + VOLUME_WH1_V6 + VOLUME_WH1_V7 + VOLUME_WH1_V8 <= 60 ; [CAPACITY_ROW_WH2] VOLUME_WH2_V1 + VOLUME_WH2_V2 + VOLUME_WH2_V3 + VOLUME_WH2_V4 + VOLUME_WH2_V5 +

300 CHAPTER 6 VOLUME_WH2_V6 + VOLUME_WH2_V7 + VOLUME_WH2_V8 <= [CAPACITY_ROW_WH3] VOLUME_WH3_V1 + VOLUME_WH3_V2 VOLUME_WH3_V3 + VOLUME_WH3_V4 + VOLUME_WH3_V5 + VOLUME_WH3_V6 + VOLUME_WH3_V7 + VOLUME_WH3_V8 <= [CAPACITY_ROW_WH4] VOLUME_WH4_V1 + VOLUME_WH4_V2 VOLUME_WH4_V3 + VOLUME_WH4_V4 + VOLUME_WH4_V5 + VOLUME_WH4_V6 + VOLUME_WH4_V7 + VOLUME_WH4_V8 <= [CAPACITY_ROW_WH5] VOLUME_WH5_V1 + VOLUME_WH5_V2 VOLUME_WH5_V3 + VOLUME_WH5_V4 + VOLUME_WH5_V5 + VOLUME_WH5_V6 + VOLUME_WH5_V7 + VOLUME_WH5_V8 <= [CAPACITY_ROW_WH6] VOLUME_WH6_V1 + VOLUME_WH6_V2 VOLUME_WH6_V3 + VOLUME_WH6_V4 + VOLUME_WH6_V5 + VOLUME_WH6_V6 + VOLUME_WH6_V7 + VOLUME_WH6_V8 <= END

55 ; + 51 ; + 43 ; + 41 ; + 52 ;

Model: WIDGETS As expected, there are 15 rows in the generated model: [OBJECTIVE], [DEMAND_ROW_V1] through [DEMAND_ROW_V8], and [CAPACITY_ROW_WH1] through [CAPACITY_ROW_WH6]. As a side note, it’s interesting to compare the generated model to the original, set-based model. We think most would agree that the set-based model is much easier to comprehend, thereby illustrating one of the primary benefits of modern algebraic languages over more traditional, scalar-based languages. In addition to ing that the correct number of rows is being generated, you should also examine each of the rows to determine that the correct variables are appearing in each row along with their correct coefficients. Note:

The reports generated by the GEN command are valid LINGO models. You may load Generate reports into LINGO and solve them as you would any other model.

One thing to keep in mind when examining generated model reports is that the LINGO model generator performs fixed variable reduction. This means that any variables that are fixed in value are substituted out of the generated model. For example, consider the simple model: MODEL: MAX = 200 * WS + 300 * NC; WS = 60; NC <= 40; WS + 2 * NC <= 120; END

If we generate this model we get the following, reduced model: MODEL: MAX= 300 * NC + 12000 ; NC <= 40 ; 2 * NC <= 60 ; END

At first glance, it seems as if both the first constraint and the variable WS are missing from the generated model. Note that by the first constraint in the original model ( WS = 60), WS is fixed at a


301

value of 60. The LINGO model generator exploits this fact to reduce the size of the generated model by substituting WS out of the formulation. The final solution report will still contain the values for all the fixed variables; however, the fixed variables will not appear in the generated model report. If you would like to suppress fixed variable reduction so that all variables appear in your generated model, you may do so via the Fixed Var Reduction option. Note:

To capture the results of the GEN command in a file, use the DIVERT command to open an output file before issuing the GEN command.

HIDE The HIDE command hides the text of a model from viewing by the . This may be useful if you are trying to protect proprietary ideas contained in your model. When you enter the HIDE command, you’ll be prompted for a . You may enter any with up to eight characters. LINGO will prompt you for this once more for verification. LINGO is sensitive to the case of the alphabetic characters in the . Once a model is hidden, commands allowing the to view the model text (GEN, GENL, LOOK, SMPS) are disabled. All other commands, however, will function as normal with the exception of ALTER. If a model is hidden, ALTER will perform modifications, but they will not be echoed to the screen. When a hidden model is saved to disk, its text will be encrypted. This prevents the from viewing the model from outside of LINGO as well. You will want to distribute the encrypted version of the model to those using your application. However, you should always keep an unhidden version of the model at your site for safekeeping in the event you forget the . A hidden model may be returned to the normal unhidden state by once again issuing the HIDE command with the correct .

302 CHAPTER 6 A sample session illustrating the use of the HIDE command follows: : TAKE TRAN.LNG !Read in a model : LOOK 4 6 !Display some rows 4] SUPPLY / WH1, WH2, WH3/ : CAP; 5] DEST / C1, C2, C3, C4/ : DEM; 6] LINKS(SUPPLY, DEST) : COST, VOL; : HIDE !Now hide the model ? TIGER Please reenter to : TIGER Model is now hidden. : ! Model is hidden so LOOK will fail : LOOK ALL [Error Code: 111] Command not available when model is hidden. : ! We can still solve it though : TERSE : GO Global optimal solution found at step: Objective value:

6 161.0000

: !And get a solution report : NONZ VOL Variable VOL(WH1, C1) VOL(WH1, C2) VOL(WH1, C3) VOL(WH2, C1) VOL(WH2, C4) VOL(WH3, C3)

Value 2.000000 17.00000 1.000000 13.00000 12.00000 21.00000

Reduced Cost 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

: !Now, unhide the model : HIDE ? TIGER Model is no longer hidden. : !Once again, we can view the model : LOOK 4 6 4] SUPPLY / WH1, WH2, WH3/ : CAP; 5] DEST / C1, C2, C3, C4/ : DEM; 6] LINKS(SUPPLY, DEST) : COST, VOL; :


303

LOOK The LOOK command displays all or part of the current model. The syntax of the LOOK command is: LOOK row_index|beg_row_index end_row_index|ALL Thus, you can specify the index of a single row to view, a range of rows, or ALL to view the entire model. In this next example, we use several forms of the LOOK command to view the current model: : LOOK ALL 1]!For a given probability P, this 2] model returns the value X such 3] that the probability that a unit 4] normal random variable is less 5] than or equal to X is P; 6] 7]! Here is the probability; 8] P = .95; 9] 10]! Solve for X; 11]P = @PSN(X); 12] : LOOK 8 8] P = .95; : LOOK 10 11 10]! Solve for X; 11]P = @PSN(X); :

304 CHAPTER 6

PICTURE The PICTURE command displays the model in matrix form. For small to medium sized models, the PICTURE command is a useful way to obtain a visual impression of the model and to hunt for formulation errors. The following letter codes are used to represent the linear coefficients in the PICTURE output: Letter Code Z Y X W V U T A B C D E F G

Coefficient Range (.000000, .000001) (.000001, .00001) (.00001, .0001) (.0001, .001) (.001, .01) (.01, .1) (.1, 1 ) (1, 10) (10, 100) (100, 1000) (1000, 10000) (10000, 100000) (100000, 1000000) > 1000000

Single digit integers are shown explicitly rather than being displayed as a code. This is especially handy, because many models have a large number of coefficients of positive or negative 1, which can affect the solution procedure. If a variable appears nonlinearly in a row, then the PICTURE command will represent its coefficient with a question mark.


305

In this example, we read in a copy of the small transportation model supplied with LINGO and use the PICTURE command to view the logical structure of the model: : take \lingo\samples\tran.lng : pic V O L U M E ( W H 1 , C 1 ) OBJ: DEM(C1): DEM(C2): DEM(C3): DEM(C4): SUP(WH1): SUP(WH2): SUP(WH3):

V O L U M E ( W H 1 , C 2 )

V O L U M E ( W H 1 , C 3 )

V O L U M E ( W H 1 , C 4 )

V O L U M E ( W H 2 , C 1 )

V O L U M E ( W H 2 , C 2 )

6 2 6 7 4 9 1 ' 1 ' 1' ' '1 ' 1 ' ' 1 1 1'1 1 ' ' ' 1 1 ' '

V O L U M E ( W H 2 , C 3 )

V O L U M E ( W H 2 , C 4 )

V O L U M E ( W H 3 , C 1 )

V O L U M E ( W H 3 , C 2 )

V O L U M E ( W H 3 , C 3 )

V O L U M E ( W H 3 , C 4 )

5 3 8 8 1 5 ' 1 ' ' ' 1 ' 1 ' 1 ' 1 ' 1 ' ' ' ' 1 1 ' ' 1 1 1 1

MIN > B > B > B > B < B < B < B

In this model, all the right-hand side values are in the range [12, 30]. Thus, they are all represented using the letter B. Row names are displayed running down the left-hand side of the matrix, while variable names are displayed along the top. The sense of the objective row and each of the constraints are shown. Spaces stand in for zero coefficients, and single quote marks are inserted to give a grid-like background. Note:

The PICTURE command is best used on small models. The amount of output generated for large models can be cumbersome. For larger models, the LINGO|Picture command in Windows versions of LINGO can compress the matrix picture of large models into a single screen for easier viewing.

SCENE The ability to generate the deterministic equivalent (DE) of a stochastic programming (SP) model via the XDETEQ command can be very useful when initially formulating and debugging an SP model. The only problem is that the DE can become quite unwieldy for all but the simplest SPs. For example, if the core model has 500 variables and there are 1,000 scenarios, then the DE will have 500,000 variables. Tracking down problems in such a large model can be difficult. Fortunately, in most cases when an SP is either infeasible or unbounded, LINGO will report the index of the first scenario that is causing the problem. With such information, we can focus our attention on the model for the isolated

306 CHAPTER 6 scenario that's causing the problem. This is where the Scenario command comes in  it allows us to generate the formulation for an individual scenario, as opposed to the entire DE, which should substantially cut down on the burden of our debugging task. As an example, we will work again with the SPGAS gas buying model. In the following example, we load the SP model and then use the SCENE command to generate the algebraic formulation of the second scenario: : take \lingo\samples\spgas.lng : scene 2 MODEL: TITLE Scenario 2; [ _1] MIN = PURCHASE_COST + HOLD_COST; [ _2] PURCHASE_COST - 5 * PURCHASE_1 - 6 * PURCHASE_2 = 0; [ _3] HOLD_COST - INVENTORY_1 - INVENTORY_2 = 0; [ _4] INVENTORY_1 - PURCHASE_1 = - 100; [ _5] - INVENTORY_1 + INVENTORY_2 - PURCHASE_2 = - 150; END :

You will note that this is the core model, with the period 2 random variable DEMAND replaced by its sampled value of 150 in row 5. Nonanticipativity constraints are not generated as part of individual scenario models.

SHOWNL The SHOWNL command will cause LINGO to generate the model and then display only those rows that contain nonlinearities. Please refer to the Types of Constraints section for a discussion of linear vs. nonlinear expressions. In general, one would prefer to always have purely linear models. Linear models solve faster and will always converge to a global solution. Therefore, when developing a model, you will be interested in carefully evaluating nonlinearities to see if they can either be eliminated or rewritten in a linear fashion. The SHOWNL command is helpful in tracking down a model's nonlinearities.


307

STATS The STATS command lists summary statistics for your model. The statistics vary slightly depending on whether the model you’re working with is linear or nonlinear. In this next example, we will read in a linear transportation model, run the STATS command, and explain some of the details of the report. : take \lingo\samples\tran.lng : stats Rows= 8 Vars= 12 No. integer vars= 0 (all are linear) Nonzeros= 43 Constraint nonz= 24( 24 are +- 1) Density=0.413 Smallest and largest elements in abs value= 1.00000 30.0000 No. < : 3 No. =: 0 No. > : 4, Obj=MIN, GUBs <= 4 Single cols= 0

The STATS report for linear models consists of five lines. In line one, we see the number of rows (constraints), variables (columns), and integer variables. The STATS command lets us know the model is linear by stating that all the variables are linear. Line two of the report gives a count of the number of nonzero coefficients appearing in the model. The first count is the number of nonzero coefficients in the entire model. The Constraint nonz count is the number of coefficients on the left-hand sides of all the constraints, excluding the nonzero objective and right-hand side coefficients. Next, STATS gives a count of the number of constraint coefficients that are plus or minus one. In general, a linear programming model is easier to solve when the number of unity coefficients increases. Finally, STATS reports a Density figure, defined as: (total nonzeros) / [(number of rows) * (number of columns + 1)]. For large models, densities under .01 are common. High densities can mean that a problem will take longer to solve. Line three lists the smallest and largest coefficients in the model in absolute value. For stability reasons, the ratio of the largest coefficient to the smallest should, ideally, be close to 1. Also, in absolute , it is best to keep coefficient values in the range of 0.0001 to 100,000. Values outside this range can cause numerical difficulties for the linear solver. Line four lists the number of constraints by type (<, =, and >), the sense of the objective, and an upper bound on the number of Generalized Upper Bound (GUB) constraints. A GUB constraint is a constraint that does not intersect with the remainder of the model. Given this, the GUB statistic is a measure of model simplicity. If all the constraints were nonintersecting, the problem could be solved by inspection by considering each constraint as a separate problem. Line five lists the number of variables that appear in only one row. Such a variable is effectively a slack. If you did not explicitly add slack variables to your model and the single column count is greater than zero, then it suggests a misspelled variable name.

308 CHAPTER 6 In the next example, we read a nonlinear model, DNRISK.LG4, into LINGO and review it’s model statistics. : take c:\lingo\samples\dnrisk.lng : stats Rows= 17 Vars= 17 No. integer vars= 0 Nonlinear rows= 1 Nonlinear vars= 7 Nonlinear constraints= Nonzeros= 68 Constraint nonz= 52 Density=0.222 No. < : 0 No. =: 8 No. > : 8, Obj=MIN Single cols=

0 0

The nonlinear STATS report drops information about the range of coefficient values, the number of +/-1 coefficients, and the GUB upper bound. A count of the number of nonlinear rows and variables is added in line two. The nonlinear rows count includes the objective, while the nonlinear constraint count does not.

XDETEQ The XDETEQ command displays the explicit deterministic equivalent (DE) for the current model, assuming the current model is a stochastic program (SP). Viewing the DE can be very helpful in determining if you have properly set up the SP components of your model. You can refer to Chapter 14, Stochastic Programming, for more information on SP. When LINGO solves an SP model, it is really solving the deterministic equivalent of the original SP. The DE contains one instance of the original core model for each scenario, where the random variables in each instance are replaced by their sampled values for the particular scenario. These core instances are tied together by a probabilistically weighted objective of all the objectives form the individual scenarios, plus a set of nonanticipativity constraints. The nonanticipativity constraints enforce the condition that all scenarios with the same history must implement the same decisions, otherwise, the DE would have the ability to anticipate future events. As an example below, we load the SP gas buying sample model, SPGAS, and run the XDETEQ command:


309

: take \lingo\samples\spgas.lng : xdeteq MODEL: TITLE _DETEQ; [ _1] MIN = 0.3333333333333333 * PURCHASE_COST_SC1 + 0.3333333333333333 * HOLD_COST_SC1 + 0.3333333333333333 * PURCHASE_COST_SC2 + 0.3333333333333333 * HOLD_COST_SC2 + 0.3333333333333333 * PURCHASE_COST_SC3 + 0.3333333333333333 * HOLD_COST_SC3; [ _2_SC01] PURCHASE_COST_SC1 - 5 * PURCHASE_1_SC1 - 5 * PURCHASE_2_SC1 = 0; [ _3_SC01] HOLD_COST_SC1 - INVENTORY_1_SC1 INVENTORY_2_SC1 = 0; [ _4_SC01] INVENTORY_1_SC1 - PURCHASE_1_SC1 = - 100; [ _5_SC01] - INVENTORY_1_SC1 + INVENTORY_2_SC1 PURCHASE_2_SC1 = - 100; [ _2_SC02] PURCHASE_COST_SC2 - 5 * PURCHASE_1_SC2 - 6 * PURCHASE_2_SC2 = 0; [ _3_SC02] HOLD_COST_SC2 - INVENTORY_1_SC2 INVENTORY_2_SC2 = 0; [ _4_SC02] INVENTORY_1_SC2 - PURCHASE_1_SC2 = - 100; [ _5_SC02] - INVENTORY_1_SC2 + INVENTORY_2_SC2 PURCHASE_2_SC2 = - 150; [ _2_SC03] PURCHASE_COST_SC3 - 5 * PURCHASE_1_SC3 - 7.5 * PURCHASE_2_SC3 = 0; [ _3_SC03] HOLD_COST_SC3 - INVENTORY_1_SC3 INVENTORY_2_SC3 = 0; [ _4_SC03] INVENTORY_1_SC3 - PURCHASE_1_SC3 = - 100; [ _5_SC03] - INVENTORY_1_SC3 + INVENTORY_2_SC3 PURCHASE_2_SC3 = - 180; [ _NAC01] INVENTORY_1_SC1 - INVENTORY_1_SC2 = 0; [ _NAC02] PURCHASE_1_SC1 - PURCHASE_1_SC2 = 0; [ _NAC03] INVENTORY_1_SC1 - INVENTORY_1_SC3 = 0; [ _NAC04] PURCHASE_1_SC1 - PURCHASE_1_SC3 = 0; END :

The first row of the DE is the probabilistically weighted objective over the three scenarios (Normal, Cold and Very Cold). Note that in the original core model, we had a variable called PURCHASE_COST, which was used to compute the cost of the gas purchases. In the DE we now have three PURCHASE_COST variables, or one for each of the three scenarios. LINGO appends the strings _SC1, _SC2 or _SC3 to PUCHASE_COST so as to indicate the appropriate scenario. Each of the constraints from the core model are also replicated across each of the scenarios with the appropriate scenario tag appended to each row's name. Finally, at the bottom of the model, there are four constraints beginning with the string: _NAC, which are the nonanticipativity constraints.

310 CHAPTER 6 It's worthwhile pointing out that this report on DE for SPGAS is a valid LINGO model, too. In fact, you can copy this report from into a new model file and solve it directly, in which case, you will get the optimal solution to the original SP model.

4. File Output The File Output category contains commands that output model and session information to a file.

DIVERT The DIVERT command opens a file and causes LINGO to route all subsequent reports (e.g., SOLUTION, RANGE, and LOOK commands) from the screen to the file. This command captures the reports in text format in the file you specify. Since the files created by the DIVERT command are in text format, they may be read into other programs, such as word processors and spreadsheets, or they may be queued to your printer. The syntax for the DIVERT command is: DIVERT filename where filename is the name of the file you wish to create. The RVRT command reverses a DIVERT command by closing the DIVERT file and then rerouting output back to the screen. In the following example, we create a small model with the MODEL command, solve it with the GO command, and then use the DIVERT command to create a file containing the formulation and solution: : : ? ? ? ? ? : : :

!Enter a small model MODEL MAX = 20*X + 30*Y; X <= 50; Y <= 60; X + 2*Y <= 120; END !Solve the model TERSE GO

Global optimal solution found at step: Objective value: : : : : : : :

1 2050.000

!Create a DIVERT file with !the formulation & solution DIVERT MYFILE.TXT !Opens the file LOOK ALL !Sends model to file SOLU !Sends solution to file RVRT !Closes DIVERT file


311

Opening the DIVERT file created in this example, we find the following file with the formulation and solution: 1]MAX = 20*X + 30*Y; 2]X <= 50; 3]Y <= 60; 4]X + 2*Y <= 120; Variable X Y Row 1 2 3 4



Note 1: Keep in mind that, when a DIVERT command is in effect, you will see little or no output on your screen. This is because the majority of output is being routed to the DIVERT file rather than to the screen. Note 2: Also, be sure you choose a DIVERT filename different from your model filename. If not, you will overwrite your model file and will be unable to retrieve it!

RVRT The RVRT command closes an output file opened with the DIVERT command. For an example of its use, see the DIVERT command immediately above.

SAVE The SAVE command saves the current model to a file. The syntax is: SAVE filename where filename is the name of the file to save your model in. LINGO saves the model in text format. You can read the model back into LINGO with the TAKE command. We recommend you use an extension of .LNG on your model files, so you can readily identify them. You may want to use your own text editor to modify your model. If you do, be sure to save the LINGO model in text (ASCII) format. Use the TAKE command to reopen the model in LINGO when you are through editing it.

312 CHAPTER 6 In the following example, we input a small model and save it in the file titled MYMODEL.LNG: : : ? ? ? ? ? : : :

!Enter a small model MODEL MAX = 20*X + 30*Y; X <= 50; Y <= 60; X + 2*Y <= 120; END !Save model to a file SAVE MYMODEL.LNG

If you open the model file, MYMODEL.LNG, in a text editor, you should see the following: MODEL: 1]MAX = 20*X + 30*Y; 2]X <= 50; 3]Y <= 60; 4]X + 2*Y <= 120; END

SMPI The SMPI command saves your model in a special format called Mathematical Programming Interface (MPI). MPI is a special format developed by LINDO Systems for representing all classes of mathematical programs  linear, integer, and nonlinear. This format is not intended for permanent storage of your models. LINDO API s may be interested in this format for exporting models to the LINDO API. Note 1: At present, LINGO does not read MPI format files. Thus, it is important that you do not use this format for permanent storage. Use the SAVE command, discussed above, to permanently save your files for later retrieval. Note 2: When exporting a stochastic program to MPI format, LINGO will write a total of four SMPI format files, consisting of the core model, its stochastic declarations and its time structure.

SMPS The SMPS command generates the underlying algebraic formulation for the current model and then writes it to a disk file in MPS format. MPS format is a common format for representing linear programming models. MPS files can be ported to any solver that reads MPS files—this includes most commercial linear programming packages. The syntax for the SMPS command is: SMPS filename where filename is the name of the file you wish to save the MPS representation of the model under.


313

In the following example, we input a small model and then save it in an MPS file: : : ? ? ? ? ? : : :

!Enter a small model MODEL MAX = 20*X + 30*Y; X <= 50; Y <= 60; X + 2*Y <= 120; END !Save model to an MPS file SMPS MYMODEL.MPS

If you open the MPS file created in a text editor, you should find: NAME LINGO GENERATED MPS FILE(MAX) ROWS N 1 L 2 L 3 L 4 COLUMNS Y 1 30.0000000 Y 3 1.0000000 Y 4 2.0000000 X 1 20.0000000 X 2 1.0000000 X 4 1.0000000 RHS RHS 2 50.0000000 RHS 3 60.0000000 RHS 4 120.0000000 ENDATA

Note 1: Your model must be entirely linear to be able to successfully export it using SMPS. If a model is nonlinear, the MPS file will contain question marks in place of numbers for coefficients of nonlinear variables. Note 2: SMPS truncates all variable names to 8 characters. For instance, the two distinct LINGO names SHIP(WH1, C1) and SHIP(WH1, C2) would both be truncated to the single 8 character name SHIPWH1C under SMPS . Either choose names to avoid collisions of truncated names or enable the RCMPSN option for converting names to RC format when doing MPS I/O. LINGO will print an error message if potential collisions exist. Note 3: The MPS file format is intended primarily for exporting models to other applications or platforms. The MPS format is purely scalar in nature—all set-based information is lost upon converting a LINGO model to MPS format. Thus, when saving copies of a model on your own machine, you should always use the SAVE command instead of the SMPS command.

314 CHAPTER 6 Note 4: When exporting a stochastic program to MPS format, LINGO will write a total of three SMPS format files, consisting of the core model, its stochastic declarations and its time structure.

5. Solution The Solution category contains commands for viewing a model’s solution.

DEBUG In the ideal world, all models would return an optimal solution. Unfortunately, this is not the case. Sooner or later, you are bound to run across either an infeasible or unbounded model. This is particularly true in the development phase of a project when the model will tend to suffer from typographical errors. Tracking down an error in a large model can prove to be a daunting task. The DEBUG command is useful in narrowing the search for problems in both infeasible and unbounded linear programs. A small portion of the original model is isolated as the source of the problem. This allows you to focus your attention on a subsection of the model in search of formulation or data entry errors. The DEBUG command identifies two types of sets: sufficient and necessary. Removing any sufficient set object from the model is sufficient to fix the entire model. Not all models will have a sufficient set. In which case, they will have a necessary set with the property that removing any object from this set fixes the remaining objects within that set. As an example, suppose you have an infeasible model. If the complete model would be feasible except for a bug in a single row, that row will be listed as part of the sufficient set. If the model has a necessary set, then, as long as all of them are present, the model will remain infeasible.


315

The following example illustrates. The coefficient .55 in ROW4 should have been 5.5: : look all MODEL: 1][ROW1] 2][ROW2] 3][ROW3] 4][ROW4] END

Max = 3*X + 7*Y; X + 2*Y <= 3; 2*X + Y <= 2; 0.55*X + Y >=4;

When we attempt to solve this formulation, we get the following error: : go [Error Code:

81]

No feasible solution found. Variable X Y Row ROW1 ROW2 ROW3 ROW4

Value 50.00000 -23.50000 Slack or Surplus 0.000000 0.000000 -74.50000 0.000000

Reduced Cost 0.000000 0.000000 Dual Price -1.000000 8.500000 0.000000 -10.00000

Next, if we run the DEBUG command, we are presented with the following report: : debug Sufficient Rows: ROW4] .55 X + Y >= Necessary Rows: ROW2] X + 2 Y <=

4 3

Necessary Variable Bounds: Y >= 0

The DEBUG command has correctly identified that the erroneous ROW4, when eliminated, is sufficient to make the entire model feasible. The debug feature operates in a similar manner for unbounded models. In the following example, we introduced an error by placing a minus sign instead of a plus sign in front of variable Z3 in ROW3. A look at ROW3 reveals that Z3 can be increased indefinitely, leading to an unbounded objective.

316 CHAPTER 6 : look all MODEL: 1][ROW1] 2] 3][ROW2] 4][ROW3] 5][ROW4] END

Max = 12*X1 + 13*X2 + 22*Y1 + 23*Z1 + 28*Z2 + X3 + Y3 + Z3; X1 + X2 + X3 <= 400; Y1 + Y2 + Y3 - Z3 <= 500; Z1 + Z2 <= 500;

The resulting model is unbounded and, when issuing the LINGO|Solve command, we receive the unbounded error message: : go [Error Code:

82]

Unbounded solution.

Issuing the DEBUG command, we receive the following breakdown: : debug Sufficient Variables: Z3 Necessary Variables: Y1

The DEBUG command has successfully determined that bounding Z3 is sufficient to bound the entire model. Typically, the DEBUG command helps to substantially reduce the search effort. The first version of this feature was implemented in response to a who had an infeasible model. The had spent a day searching for a bug in a model with 400 constraints. The debug feature quickly found a necessary set with 55 constraints, as well as one sufficient set constraint. The immediately noticed that the right-hand side of the sufficient set constraint was incorrect.

GO The GO command compiles and then solves the current model. When LINGO compiles the model, it produces an internally executable version of the model and then runs it to produce the solution. When LINGO finishes solving the model, it displays a full solution report on your screen. To suppress the full solution report, issue the TERSE command before the GO command.


317

To capture the solution report generated by the GO command in a file, use the DIVERT command before the GO command. To set various parameters pertaining to the operation of LINGO’s solver, see the SET command later in this chapter.

NONZ The NONZ, or NONZEROS, command displays an abbreviated version of the solution for the current model. NONZ is identical to the SOLUTION command with the exception that NONZ displays information only about nonzero variables and binding rows (i.e., the slack or surplus is 0). The syntax of the NONZ command is: NONZ ['header_text'] [var_or_row_names] For a standard NONZ solution report, omit the two optional arguments and enter the NONZ command by itself. LINGO will print primal and dual values for all nonzero variables and binding rows. LINGO will label all the columns in the report. The first optional field, header_text, will be displayed as a title header in the solution report. If the header_text argument is included, LINGO prints primal values only, omitting all labels in the report. The second optional field, var_or_row_names, is a variable and/or row name list that, if included, will limit the report to the given variables or rows. The standard wild card characters (* and %) are ed in the variable and row names. As an example, in the following session, we load the Chess Snackfoods example from the Using Sets section and then generate several solution reports using NONZ:

318 CHAPTER 6 : TAKE CHESS.LNG : TERSE : GO Global optimal solution found at step: Objective value:

0 2692.308

: !Generate a standard NONZ report : NONZ Variable SUPPLY( PEANUTS) SUPPLY( CASHEWS) PRICE( PAWN) PRICE( KNIGHT) PRICE( BISHOP) PRICE( KING) PRODUCE( PAWN) PRODUCE( KING) FORMULA( PEANUTS, PAWN) FORMULA( PEANUTS, KNIGHT) FORMULA( PEANUTS, BISHOP) FORMULA( PEANUTS, KING) FORMULA( CASHEWS, PAWN) FORMULA( CASHEWS, KNIGHT) FORMULA( CASHEWS, BISHOP) FORMULA( CASHEWS, KING) Row 1 2 3

Value 750.0000 250.0000 2.000000 3.000000 4.000000 5.000000 769.2308 230.7692 15.00000 10.00000 6.000000 2.000000 1.000000 6.000000 10.00000 14.00000

Reduced Cost 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Slack or Surplus 2692.308 0.000000 0.000000

Dual Price 1.000000 1.769231 5.461538

: !Generate a NONZ report for PRODUCE : NONZ PRODUCE Variable PRODUCE( PAWN) PRODUCE( KING)

Value 769.2308 230.7692

Reduced Cost 0.000000 0.000000

: !Now add a header and use a wildcard in the name : NONZ 'NONZERO PRODUCTION VALUES:' PROD* NONZERO PRODUCTION VALUES: 769.2308 230.7692

If you would like to capture the solution report in a file, use the DIVERT command before the NONZ command. For more information on the interpretation of the various fields in the NONZEROS report, see Getting Started with LINGO.

COMMAND-LINE COMMANDS Note:

319

If the solution report is scrolling off the screen, you can use the PAGEcommand to set the page length to n lines, so LINGO will pause every time n lines are printed and wait until you are ready to proceed with the next page.

RANGE Use the RANGE command to generate a range report for the model in the active window. A range report shows over what ranges you can: 1) change a coefficient in the objective without causing any of the optimal values of the decision variables to change, or 2) change a row’s constant term (also referred to as the right-hand side coefficient) without causing any of the optimal values of the dual prices or reduced costs to change. Note:

The solver computes range values when you solve a model. Range computations must be enabled in order for the solver to compute range values. Range computations are not enabled by default, so you will need to switch them on with the command: SET DUALCO 2

Range computations can take a fair amount of computation time. If speed is a concern, you don’t want to enable range computations unnecessarily. The example model below, when solved, yields the range report that follows: [OBJECTIVE] [ALIM] [CLIM] [T]

MAX = 20 * A + 30 * C; A <= 60; C <= 50; A + 2 * C <= 120;

Here is the range report: Ranges in which the basis is unchanged:

Variable A C Row ALIM CLIM T

Objective Coefficient Ranges Current Allowable Allowable Coefficient Increase Decrease 20.00000 INFINITY 5.000000 30.00000 10.00000 30.00000 Right-hand side Ranges Current Allowable Allowable RHS Increase Decrease 60.00000 60.00000 40.00000 50.00000 INFINITY 20.00000 120.0000 40.00000 60.00000

320 CHAPTER 6 The first section of the report is titled Objective Coefficient Ranges. In the first column, Variable, all the optimizable variables are listed by name. The next column, Current Coefficient, lists the current coefficient of the variable in the objective row. The third column, Allowable Increase, tells us the amount that we could increase the objective coefficient without changing the optimal values for the variables. The final column, Allowable Decrease, lists the amount that the objective coefficient of the variable could decrease before the optimal values of the variables would change. Information on the allowable increases and decreases on objective coefficients can be useful when you need answers to questions like, “How much more (less) profitable must this activity be before we should be willing to do more (less) of it?” Referring to the Objective Coefficient Ranges report for our example, we can say, as long as the objective coefficient of A is greater-than-or-equal-to 15, the optimal values of the variables will not change. The same may be said for the objective coefficient of variable C, as long as it falls within the range of [0-40]. Note:

Ranges are valid only if you are planning to alter a single objective or right-hand side coefficient. The range information provided by LINGO cannot be applied in situations where one is simultaneously varying two or more coefficients. Furthermore, ranges are only lower bounds on the amount of change required in a coefficient to actually force a change in the optimal solution. You can change a coefficient by any amount up to the amount that is indicated in the range report without causing a change in the optimal solution. Whether the optimal solution will actually change if you exceed the allowable limit is not certain.

The second section of the range report is titled Right-hand side Ranges. The first column, Row, lists the names of all the optimizable rows, or constraints, in the model. The second column, Current RHS, gives the constant term, or right-hand side value, for the row. The next two columns, Allowable Increase and Allowable Decrease, tell us how far we can either increase or decrease the right-hand side coefficient of the row without causing a change in the optimal values of the dual prices or reduced costs. If you recall, the dual prices on rows are, effectively, shadow prices, which tell us at what price we should be willing to buy (or sell) our resources for. The dual prices do not, however, tell us what quantity we should be willing to buy (or sell) at the dual price. This information is obtained from the allowable increases and decreases on the right-hand side coefficients for the row. So, for our example, the dual prices and reduced costs will remain constant as long as the right-hand side of row ALIM falls within the range [20-120], the right-hand side of CLIM is greater-than-or-equal-to 30, and the right-hand side of T is in [60-160]. Note:

We preceded all the rows in our model with a name enclosed in square brackets. This is an important practice if you wish to generate range reports. If you do not name your rows, LINGO assigns them a name that corresponds to the internal index of the row. This internal index will not always correspond to the order of the row in the text of the original model. To make the Right-hand side Ranges section of range reports meaningful, be sure to name all your rows. For details on asg names to rows, see page 38.

If a variable is nonlinear in the objective, its value in the Current Coefficient column will be displayed as NONLINEAR. Similarly, if a row is nonlinear, the value in the Current RHS column will be displayed as NONLINEAR. Coefficients that can be increased or decreased indefinitely will display a range of INFINITY.


321

Fixed variables are substituted out of a model and will not appear in a range report. Rows that contain only fixed variables are also substituted out of models and will not appear in range reports. As an example, suppose we changed the following inequality in our sample model from: [ALIM]

A <= 60;

to the equality: [ALIM]

A = 60;

LINGO can now solve directly for the value of A. The variable A is considered fixed; as is the row ALIM (since it contains no optimizable variables). Given this, the variable A will no longer appear in the Objective Coefficient Ranges section of the range report, and the row ALIM will not appear in the Right-hand Side Ranges section. We can this by examining the updated range report: Ranges in which the basis is unchanged:

Variable C Row CLIM T

Objective Coefficient Ranges Current Allowable Allowable Coefficient Increase Decrease 30.00000 INFINITY 30.00000 Right-hand Side Ranges Current Allowable Allowable RHS Increase Decrease 50.00000 INFINITY 20.00000 60.00000 40.00000 60.00000

As a final note, if the range report is scrolling off the screen, you can use the PAGE n command to set the page length to n lines, so LINGO will pause every time n lines are printed and wait until you are ready to proceed with the next page. In addition, if you would like to capture the solution report in a file, use the DIVERT command before the SOLU command.

SOLU The SOLU, or SOLUTION, command displays a solution report for the current model. The syntax of the SOLU command is: SOLU ['header_text'] [ var_or_row_names] For a standard solution report, omit the two optional arguments and enter the SOLU command by itself. LINGO will print primal and dual values for all the variables and rows in the model. LINGO will label all the columns in the report. The first optional field, header_text, will be displayed as a title header in the solution report. If the header_text argument is included, LINGO prints primal values only, omitting all labels in the report. The second optional field, var_or_row_names, is a variable and/or row name list that, if included, will limit the report to the given variable or row name. The standard wild card characters (* and %) are ed in the variable and row names. As an example, in the following session, we load the Chess Snackfoods example from the Using Sets section and then generate several solution reports using SOLU:

322 CHAPTER 6 : TAKE CHESS.LNG : TERSE : GO Global optimal solution found at step: 0 Objective value: 2692.308 : !Generate a standard SOLU report : SOLU Variable SUPPLY( PEANUTS) SUPPLY( CASHEWS) PRICE( PAWN) PRICE( KNIGHT) PRICE( BISHOP) PRICE( KING) PRODUCE( PAWN) PRODUCE( KNIGHT) PRODUCE( BISHOP) PRODUCE( KING) FORMULA( PEANUTS, PAWN) FORMULA( PEANUTS, KNIGHT) FORMULA( PEANUTS, BISHOP) FORMULA( PEANUTS, KING) FORMULA( CASHEWS, PAWN) FORMULA( CASHEWS, KNIGHT) FORMULA( CASHEWS, BISHOP) FORMULA( CASHEWS, KING) Row 1 2 3

Value 750.0000 250.0000 2.000000 3.000000 4.000000 5.000000 769.2308 0.000000 0.000000 230.7692 15.00000 10.00000 6.000000 2.000000 1.000000 6.000000 10.00000 14.00000

Slack or Surplus 2692.308 0.000000 0.000000

Reduced Cost 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.1538461 0.7692297E-01 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 Dual Price 1.000000 1.769231 5.461538

: !Generate a SOLU report for PRODUCE : SOLU PRODUCE Variable PRODUCE( PAWN) PRODUCE( KNIGHT) PRODUCE( BISHOP) PRODUCE( KING)

Value 769.2308 0.000000 0.000000 230.7692

: !Now add a header and use a wildcard : SOLU 'PRODUCTION QUANTITIES' PROD* PRODUCTION QUANTITIES 769.2308 0.000000 0.000000 230.7692

Reduced Cost 0.0000000 0.1538461 0.7692297E-01 0.0000000


323

If you would like to capture the solution report in a file, use the DIVERT command before the SOLU command. For more information on the interpretation of the various fields in the solution report, see Getting Started with LINGO. If the solution report is scrolling off the screen, you can use the PAGE command to set the page length to n lines, so LINGO will pause every time n lines are printed and wait until you are ready to proceed with the next page.

6. Problem Editing The Problem Editing category contains commands used in editing and modifying models.

ALTER The ALTER command is used to edit the current model. The syntax of ALTER is: ALTER [line_number|line_range|ALL] 'old_string'new_string' where, line_number line_range ALL old_string new_string

is the index of a single line to edit, is a range of lines to edit, means edit all the lines in the model, is the old string to search for and replace, and is the string to replace all occurrences of old_string with in the specified line range.

324 CHAPTER 6 In the following sample session, we read in a small knapsack model and perform two ALTER commands to modify the model: : TAKE ALTER.LNG : LOOK ALL 1]SETS: 2] THINGS /1..4/: VALUE, WEIGHT, X; 3]ENDSETS 4]DATA: 5] VALUE = 8 6 4 3; 6] WEIGHT = 66 44 35 24; 7]ENDDATA 8] MAX = @SUM(THINGS: VALUE * X); 9] @SUM(THINGS: WEIGHT * X) >= 100; 10] @FOR(THINGS: @BIN(X)); : !Change the direction of the constraint : ALTER 9 '>='<=' 9] @SUM(THINGS: WEIGHT * X) <= 100; : !Change 'THINGS' to 'ITEMS' in ALL rows : ALTER ALL 'THINGS'ITEMS' 2] ITEMS /1..4/: VALUE, WEIGHT, X; 8] MAX = @SUM(ITEMS: VALUE * X); 9] @SUM(ITEMS: WEIGHT * X) <= 100; 10] @FOR(ITEMS: @BIN(X)); : LOOK ALL 1]SETS: 2] ITEMS /1..4/: VALUE, WEIGHT, X; 3]ENDSETS 4]DATA: 5] VALUE = 8 6 4 3; 6] WEIGHT = 66 44 35 24; 7]ENDDATA 8] MAX = @SUM(ITEMS: VALUE * X); 9] @SUM(ITEMS: WEIGHT * X) <= 100; 10] @FOR(ITEMS: @BIN(X)); :

Note:

In addition to the single quote character ('), LINGO also allows the use of the double quote character (") for delimiting the text fields of the ALTER command.


325

DELETE The DELETE command is used to delete one or more lines of text from the current model. The syntax of DELETE is: DELETE [line_number|line_range|ALL] where, line_number line_range ALL

is the index of a single line to delete, is a range of lines to delete, and means delete the entire model.

Some examples of the DELETE command follow: Example 1: DELETE 3 deletes line 3 of the model, Example 2: DEL 2 10 deletes lines 2 through 10 of the model, and Example 3: DEL ALL deletes the entire model.

EXTEND The EXTEND command allows you to append lines to the current model. It puts LINGO in model input mode just after the last line of the current model. When you use the EXTEND command, you’ll see LINGO’s question mark prompt. Start entering your new lines of model text. When you’re done, enter END at the prompt. In the following sample session, we use the EXTEND command to append an additional constraint to a small model: : LOOK ALL 1]MAX 20*X + 30*Y; 2]X <= 50; 3]Y <= 60; 4]X + 2*Y <=120; : ! Use EXTEND to add another line : EXTEND ? X >= 30; ? END : LOOK ALL 1]MAX 20*X + 30*Y; 2]X <= 50; 3]Y <= 60; 4]X + 2*Y <=120; 5]X >= 30; :

326 CHAPTER 6

7. Conversational Parameters The Conversational Parameters category contains commands that control how information is displayed.

PAGE The PAGE command sets the length of the page or screen size in lines. The syntax for PAGE is: PAGE n where n is the desired number of lines per page of output. For instance, PAGE 25 will cause the display to pause after 25 lines and await a carriage return before displaying the next 25 lines. The PAGE command is convenient when you wish to page through long reports and not have them scroll off the top of the screen. When 0 is entered as the argument to PAGE, paging is turned off entirely. LINGO will no longer stop output to wait for a carriage return. Entering PAGE 0 at the top of any command script is helpful in that you generally want command scripts to run uninterrupted. The PAGE command is equivalent to the SET LENPAG command and is maintained for backward compatibility

PAUSE The PAUSE command causes screen display to pause until a carriage return is typed. If you enter text on the same line as the PAUSE command, the text will be displayed. The PAUSE command is useful in command scripts for conveying information to the .

TERSE The TERSE command causes LINGO to suppress the automatic display of a solution report after a model is solved with the GO command. When TERSE is enabled, you will need to use the NONZ or SOLU commands to view the solution. When LINGO is in terse output mode, export summary reports are also suppressed. Export summary reports are normally generated each time you export solutions to spreadsheets or databases. Once you enter the TERSE command, LINGO stays in terse output mode until you enter the VERBOSE command (see below). The TERSE command is equivalent to the SET TERSEO 1 command and is maintained for backward compatibility.

VERBOSE The VERBOSE command undoes the effects of the TERSE command, and places LINGO in verbose output mode. Verbose output mode is the default mode. It results in the automatic display of solution reports after solving a model. Verbose output mode also results in the automatic display of export summary reports whenever export operations are performed to spreadsheets and databases.


327

The VERBOSE command is equivalent to the SET TERSEO 0 command and is maintained for backward compatibility.

WIDTH Use the WIDTH command to set the terminal width for input and output. The syntax of the WIDTH command is: WIDTH n where n is the desired terminal width. You may set the width between 64 and 200. The default is 76. When LINGO generates reports, it limits output lines to the terminal width length. In some reports, lines will be wrapped, so they fall within the line limit. In other reports, lines may be truncated. Since LINGO concatenates variable names in performing set operations, a variable name, such as SHIPMENTS(WAREHOUSE1, CUSTOMER2), may result, which may be truncated in a solution report if too narrow a terminal width is used. The WIDTH command is equivalent to the SET LINLEN command and is maintained for backward compatibility.

8. Tolerances The Tolerances category contains commands for setting system parameters in LINGO.

APISET The APISET command gives you access to all the parameters in the LINDO API, which is the solver library used by LINGO. LINGO allows access to most of the important solver parameters through the SET command and, under Windows, via the LINGO|Options command. However, some of the more advanced parameters may only be accessed through the APISET command. The syntax for this command is: APISET param_id {int|double} param_value where param_id is the parameter’s index and param_value is the value you wish to set the parameter to. You will also need to indicate if the parameter is an integer or double precision quantity. Some examples of the APISET command follow: Example 1: APISET 341 INT 10000 sets the MIP branch limit (LS_IPARAM_MIP_BRANCH_LIMIT=341) to 10000, Example 2: HELP APISET will cause LINGO to display all current APISET settings, and Example 3: APISET DEFAULT removes all custom LINDO API settings, returning to the defaults. You will need to refer to the LINDO API documentation for a list of available parameters and their indices. The LINDO API documentation is available at no charge as part of the LINDO API on the LINDO Systems Web site. The LINGO installation also comes with a macro definition file, Lindo.h, which contains all the parameter indices for the LINDO API.

328 CHAPTER 6 Parameter values set with the APISET command are not stored from one LINGO session to the next. Give the HELP APISET command for a listing of parameters that are currently active. To remove all APISET parameter settings type the command: APISET DEFAULT. If there are some LINDO API parameters you wish to permanently set, you may place a series of APISET commands in an AUTOLG.DAT script file that automatically gets run at the start of each LINGO session.

DBPWD The DBPWD command is used to input a for accessing databases via the @ODBC() function. Any input with this command will not be permanently stored. Therefore, at the start of each session, you will need to reenter your database . The syntax for the command is: DBPWD my_ See the DBUID command below for entering any id required by your database.

DBUID The DBUID command is used to input a id for accessing databases via the @ODBC() function. Any id input with this command will not be permanently stored. Therefore, at the start of each session, you will need to reenter your database id. The syntax for the command is: DBUID my__id See the DBPWD command above for entering any required with your id.

FREEZE The FREEZE command saves your current configuration to LINGO’s configuration file, so it may be automatically restored the next time LINGO starts. Any non-default features of the current configuration are saved to the LINGO.CNF file in LINGO’s main directory. The LINGO.CNF configuration file is a text file, and the curious may examine it by simply opening it in a text editor. All parameters controlled by the SET command, see below, are stored by the FREEZE command. Note:

Be careful when saving a non-default configuration. The saved configuration will automatically be restored next time you start LINGO. Settings of certain parameters will affect the way models are solved, potentially leading to misleading results when used on a different set of models. To restore the default configuration, use the following command sequence: : SET DEFAULT : FREEZE


329

SET The SET command allows you to override LINGO’s default tolerances and settings. All configurable options in LINGO are available through the SET command. The syntax for the SET command is: SET parameter_name|parameter_index [parameter_value] where, parameter_name parameter_index parameter_value

is the name of the parameter to set, is the index of the parameter to set, and is the new value for the parameter that, if omitted, will cause LINGO to display the current value for the specified parameter.

Use the FREEZE command, see above, to save any tolerances modified with the SET command to the configuration file, so they will be automatically restored the next time LINGO starts. You may also enter SET DEFAULT to return all parameters to their default values. Some examples of the SET command follow: Example 1:

SET MXMEMB 128 FREEZE

sets the generator memory limit to 128MB and saves parameter settings to the configuration file, Example 2: SET 5 1.E-7 sets the relative integrality tolerance (RELINT) to 1.e-7, Example 3: SET DEFAULT restores all parameters to their default values, and Example 4: HELP SET causes LINGO to display all parameter settings. The parameters accessible through the SET command are: No. 1 2 3 4 5 6 7 8

Name ILFTOL FLFTOL INFTOL FNFTOL RELINT NOPTOL ITRSLW DERCMP

Default 0.3e-5 0.1e-6 0.1e-2 0.1e-5 0.8e-5 0.1e-6 5 0

9 10 11

ITRLIM TIMLIM OBJCTS

0 0 0

Description Initial linear feasibility tolerance Final linear feasibility tolerance Initial nonlinear feasibility tolerance Final nonlinear feasibility tolerance Relative integrality tolerance NLP optimality tolerance Iteration limit for slow progress Derivatives (0:LINGO chooses, 1:backward analytical, 2:forward analytical, 3:central differences, 4:forward differences) Iteration limit (0:no limit) Solver time limit in seconds (0:no limit) Objective cuts (1:yes, 0:no)

330 CHAPTER 6 12

MXMEMB

32

13 14 15 16 17 18 19 20 21 22 23 24 25 26

CUTAPP ABSINT HEURIS HURDLE IPTOLA IPTOLR TIM2RL NODESL LENPAG LINLEN TERSEO STAWIN SPLASH OROUTE

2 .000001 3 0 0 .1e-4 100 0 0 85 0 1 1 0

27 28 29 30 31 32 33 34

WNLINE WNTRIM STABAR FILFMT TOOLBR CHKDUP ECHOIN ERRDLG

800 400 1 1 1 0 0 1

35 36

USEPNM NSTEEP

0 0

37

NCRASH

0

38

NSLPDR

0

39

SELCON

0

40

PRBLVL

0

41

SOLVEL

0

42 43 44

REDUCE SCALEM PRIMPR

2 1 0

45

DUALPR

0

Memory limit in megabytes for LINGO’s model generator (N/A on some machines) Cuts application (0:root, 1:all, 2:solver chooses) Absolute integrality tolerance IP heuristics (0:none, 100:advanced) Use an IP hurdle value (1:yes, 0:no) IP absolute optimality tolerance IP relative optimality tolerance Seconds before switching to IP relative optimality tolerance 0:LINGO decides, 1:depth first, 2:worst bound, 3:best bound Terminal page length limit (0:none) Terminal page width (0:none) Output level (0:verbose, 1:terse, 2:errors only, 3:no output) Post status window (1:yes, 0:no, Windows only) Display splash screen (1:yes, 0:no, Windows only) Route output to command window (1:yes, 0:no, Windows only) Max command window lines (Windows only) Min command window lines (Windows only) Display status bar (1:yes, 0:no, Windows only) File format (0:lng, 1:lg4, 2:ltx, Windows only) Display toolbar (1:yes, 0:no, Windows only) Check for duplicate model names in data (1:yes, 0:no) Echo command input to terminal (1:yes, 0:no) Route error messages to a dialog box (1:yes, 0:no, Windows only) Allow for unrestricted use of primitive set names (1:yes, 0:no) Use steepest edge variable selection in nonlinear solver (1:yes, 0:no) Run crash procedure to get an initial starting point in nonlinear models (1:yes, 0:no) Compute search directions in nonlinear solver using successive linear programming (1:yes, 0:no) Use selective constraint evaluation in nonlinear solver (1:yes, 0:no) Specify probing level on MILPs (0:LINGO chooses, 1:none, 7:high) Specify linear solver (0:LINGO chooses, 1:primal, 2:dual, 3:barrier) Perform model reduction (2:LINGO chooses, 1:yes, 0:no) Scale the model (1:yes, 0:no) Select primal pricing method (0:LINGO chooses, 1:partial, 2:devex) Select dual pricing method (0:LINGO chooses, 1:Dantzig,

COMMAND-LINE COMMANDS 46

DUALCO

1

47 48

RCMPSN MREGEN

0 2

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85

BRANDR BRANPR CUTOFF STRONG REOPTB REOPTX MAXCTP RCTLIM GUBCTS FLWCTS LFTCTS PLOCTS DISCTS KNPCTS LATCTS GOMCTS COFCTS GCDCTS SCLRLM SCLRDL PRNCLR MULTIS USEQPR GLOBAL LNRISE LNBIGM LNDLTA BASCTS MAXCTR HUMNTM DECOMP GLBOPT GLBDLT GLBVBD GLBUBD GLBBRN GLBBXS

0 0 .1e-8 10 0 0 100 .75 1 1 1 1 1 1 1 1 1 1 1,000 0 1 0 0 0 0 100,000 .1e-5 1 2 0 0 .1e-4 .1e-6 .1e+11 2 5 1

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2:steepest edge) Specify dual computations (0:none, 1:prices only, 2:prices and ranges, 3:prices only on optimizable rows) Use RC format names for MPS I/O (1:yes, 0:no) Select model regeneration (0:only on modifications to model, 1:same as 0 plus whenever model has external references, 2:always) Select branch direction (0:both, 1:up, 2:down) Select branch priority (0:LINGO decides, 1:binary) Cutoff solution values smaller than this Specify strong branch level IP warm start LP (0:LINGO, 1: primal, 2:dual, 3:barrier) IP cold start LP (0:LINGO, 1: primal, 2:dual, 3:barrier) Max top cuts es Relative cuts limit GUB cuts (1:yes, 0:no) Flow cuts (1:yes, 0:no) Lift cuts (1:yes, 0:no) Plant location cuts (1:yes, 0:no) Disaggregation cuts (1:yes, 0:no) Knapsack cover cuts (1:yes, 0:no) Lattice cuts (1:yes, 0:no) Gomory cuts (1:yes, 0:no) Coefficient reduction cuts (1:yes, 0:no) Greatest common divisor cuts (1:yes, 0:no) Syntax coloring line limit (Windows only) Syntax coloring delay in seconds (Windows only) Matching parenthesis coloring (1:yes, 0:no, Windows only) NLP multistart attempts (0:LINGO, n:number of attempts) Use quadratic recognition (1:yes, 0:no) Use global solver on NLPs (1:yes, 0:no) Linearization (0:LINGO, 1:none, 2:low, 3:high) Linearization BigM coefficient Linearization Delta coefficient Basis cuts (1:yes, 0:no) Max tree cuts es Minimum heuristic time limit (seconds) Matrix decomposition (1:yes, 0:no) Global solver optimality tolerance Global solver delta tolerance Global solver variable bound limit Global solver bound use (0:no, 1:all, 2:some) Global solver branch selection (see below) Global solver box selection (0:depth first, 1:worst bound)

332 CHAPTER 6 86 87

GLBREF SUBOUT

3 2

88 89

NLPVER DBGCLD

N/A 0

90

DBGWRM

0

91 92

LCRASH BCROSS

0 1

93 94 95 96 97

LOWMEM FILOUT DBGLVL UNARYM LINEAR

0 0 15 1 0

98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113

LOPTOL SECORD NONNEF BIGMVL KILLSC TATSLV KBESTS LCORES LCORE1 LCORE2 LCORE3 LCORE4 SCALEW REFRAQ SPSMSZ SPSCON

.1e-6 0 1 1.e8 0 0 1 1 1 2 3 4 1.e12 0 2 1

114 115

SPSEED SPMTHD

1,031 0

116 117

SPXVAL SPMAXS

1 40,000

118 119 120

PRECIS LOOPOP HEUCUT

7 0 0

Global solver reformulation level (0:none, 3:high) Fixed variable reduction (0:none, 1:max, 2:not when using global or multistart solvers, 3:linear variables only) No longer ed Debugging cold start solver (0:LINGO, 1:primal, 2:dual, 3:barrier) Debug warm start solver (0:LINGO, 1:primal, 2:dual, 3:barrier) Use aggressive crashing for NLPs (0:no, 1:yes) Perform a basis crossover on LPs when using barrier solver (0:no, 1:yes) Opt for less memory usage (0:no, 1:yes) Fill out workbook output ranges (0:no, 1:yes) Debugger output level (1:low, 15:high) Unary minus priority (0:low, 1:high) Assume model is linear to reduce memory consumption(0:no, 1:yes) Linear optimality tolerance Use second order derivatives for NLPs (0:no, 1:yes) Variables default to being non-negative (0:no, 1:yes) BigM coefficient threshold value Kill scripts on interrupts (0:no, 1:yes) @SOLVE time limit in seconds Number of K-Best MIP solutions to generate Number of concurrent LP solvers to run LP Solver in Core 1 (1:prm1,2:dual,3:barrier,4:prim2) LP Solver in Core 2 LP Solver in Core 3 LP Solver in Core 4 Scaling warning threshold Basis refactor frequency (0:LINGO chooses, iteration count) Default SP sample size per stage Apply SP sampling to continuous random variables only (0:no, 1:yes) SP sampling random number seed SP solution method (0:LINGO chooses, 1:deterministic equivalent, 2:nested Benders, 3:augmented Lagrangian) Calculate all SP expected value statistics (0:no, 1:yes) Max scenarios allowed in an SP before auto sampling takes effect Precision in digits for standard solution reports Perform loop optimization during model compiles (0:no, 1:yes) Heuristics cutoff criterion (0:LINGO chooses, 1:time, 2:iterations)

COMMAND-LINE COMMANDS 121

NUMBIN

0

122 123 124 125

SPBIGM NSLPSV FORCEB NTHRDS

1.E8 0 0 1

126

MTMODE

0

127

BNPBLK

2

128 129

BNPHEU REPROD

1 0

333

The NUMBIN parameter determines the number of bins used in histogram charts. Set this parameter to 0 to have LINGO automatically determine a sensible bin count. The default setting for NUMBIN is 0. Stochastic solver Big M coefficient. Use SLP solver for nonlinear models (0:no, 1:yes) Enforce variable bounds in calc and data sections (0:no, 1:yes) Max number of executions threads (0:use all cores, 1:singlethreaded, N>1:use up to N threads Multithreading mode (0:LINGO chooses, 1:prefer parallel, 2:parallel exclusively, 3:prefer concurrent, 4:concurrent exclusively) Branch-and-price (BNP) blocks (0:use row names, 1: specified, 2:off, >2:max number of blocks) BNP block-finding heuristic (1:GP1, 2:GP2) Favor reproducibility over speed (0:no, 1:yes)

1. ILFTOL and 2. FLFTOL Due to the finite precision available for floating point operations on digital computers, LINGO can’t always satisfy each constraint exactly. Given this, LINGO uses these two tolerances as limits on the amount of violation allowed on a constraint while still considering it “satisfied”. These two tolerances are referred to as the initial linear feasibility tolerance (ILFTOL) and the final linear feasibility tolerance (FLFTOL). The default values for these tolerances are, respectively, 0.000003 and 0.0000001. ILFTOL is used when the solver first begins iterating. ILFTOL should be greater than FLFTOL. In the early stages of the solution process, being less concerned with accuracy can boost the performance of the solver. When LINGO thinks it has an optimal solution, it switches to the more restrictive FLFTOL. At this stage in the solution process, one wants a relatively high degree of accuracy. Thus, FLFTOL should be smaller than ILFTOL. One instance where these tolerances can be of use is when LINGO returns a solution that is almost, but not quite, feasible. You can this by checking the values in the Slack or Surplus column in the model’s solution report. If there are only a few rows with small, negative values in this column, then you have a solution that is close to being feasible. Loosening (i.e., increasing the values of) ILFTOL and FLFTOL may help you get a feasible solution. This is particularly true in a model where scaling is poor (i.e., very large and very small coefficients are used in the same model), and the units of measurement on some constraints are such that minor violations are insignificant. For instance, suppose you have a budget constraint measured in millions of dollars. In this case, a violation of a few pennies would be of no consequence. Short of the preferred method of rescaling your model, loosening the feasibility tolerances may be the most expedient way around a problem of this nature.

334 CHAPTER 6 3. INFTOL and 4. FNFTOL The initial nonlinear feasibility tolerance (INFTOL) and the final nonlinear feasibility tolerance (FNFTOL) are both used by the nonlinear solver in the same manner the initial linear and final linear feasibility tolerances are used by the linear solver. For information on how and why these tolerances are useful, refer to the section immediately above. Default values for these tolerances are, respectively, 0.001 and 0.000001. 5. RELINT RELINT, the relative integrality tolerance, is used by LINGO as a test for integrality in integer programming models. Due to round-off errors, the “integer” variables in a solution may not have values that are precisely integral. The relative integrality tolerance specifies the relative amount of violation from integrality that is acceptable. Specifically, if I is the closest integer value to X, X will be considered an integer if: |X - I| <= Relative Integrality Tolerance. |X| The default value for the relative integrality tolerance is .000008. Although one might be tempted to set this tolerance to 0, doing so may result in feasible models being reported as infeasible. 6. NOPTOL While solving a model, the nonlinear solver is constantly computing a gradient. The gradient gives the rate of improvement of the objective function for small changes in the variables. If the gradient's rate of improvement computation for a given variable is less-than-or-equal-to NOPTOL, the nonlinear optimality tolerance, further adjustments to the variable's value are not considered to be beneficial. The default value for the nonlinear optimality tolerance is .0000001. Decreasing this tolerance towards a limit of 0 will tend to make the solver run longer and may lead to better solutions to poorly formulated or poorly scaled models. 7. ITRSLW LINGO’s nonlinear solver uses the ITRSLW, slow progress iteration limit, as a means of terminating the solution process if little or no progress is being made in the objective value. Specifically, if the objective function’s value has not improved significantly in n iterations, where n is the value of ITRSLW, the nonlinear solver will terminate the solution process. Increasing this tolerance’s value will tend to force the solver to run longer and may be useful in models that have relatively “flat” objective functions around the optimal solution. The default value for ITRSLW is 5 iterations. Refer to the description of ITRLIM below for a definition of iterations. 8. DERCMP Use this parameter to set the style of derivative computation. Set DERCMP to 0 (Solver Decides) to allow LINGO to select the method, 1 for backward analytical derivatives, 2 for forward analytical derivatives, 3 for numerical derivatives using central differences, and 4 for numerical derivatives using forward differences. LINGO defaults to the Solver Decides setting, which presently involves using backward analytical derivatives. However, we suggest you try the various derivative options to see which works best for your particular models.


335

9. ITRLIM Use this tolerance to place an upper limit on the number of iterations the solver will perform. An iteration is the fundamental operation performed by the solver. At the risk of oversimplification, it is a process that involves forcing a variable, currently at a zero value, to become nonzero until some other variable is driven to zero, improving the objective as we go. In general, larger models will take longer to perform an iteration, and nonlinear models will take longer than linear models. The default iteration limit is 0, meaning no limit is imposed on the iteration count. If the solver hits this limit, it returns to normal command mode. If the model contains integer variables, LINGO will restore the best integer solution found so far. You may need to be patient, however, because the solver may have to perform a fair amount of work to reinstall the current best solution after it hits a runtime limit. Note:

Some caution is required when interrupting the solver. There must be an incumbent solution available if you hope to interrupt the solver and have it return a valid solution. You can always tell if an incumbent solution is available by examining the Best Obj field in the Extended Solver Status box of the solver status window. If this field is blank, then an incumbent solution does not exist, and the solution returned after an interrupt will be invalid. If, on the other hand, this field contains a numeric value, then you should be able to interrupt and return to a valid, if not globally optimal, solution.

10. TIMLIM Use this tolerance to place a limit on the number of seconds the solver runs. If the solver hits this limit, it will stop and return with the best solution found so far. The default limit is 0, meaning no time limit is imposed on the solver. If the solver hits this limit, it returns to normal command mode. If the model contains integer variables, LINGO will restore the best integer solution found so far. You may need to be patient, however, because the solver may have to perform a fair amount of work to reinstall the current best solution after it hits a runtime limit. Note:

Some caution is required when interrupting the solver. There must be an incumbent solution available if you hope to interrupt the solver and have it return a valid solution. You can always tell if an incumbent solution is available by examining the Best Obj field in the Extended Solver Status box of the solver status window. If this field is blank, then an incumbent solution does not exist, and the solution returned after an interrupt will be invalid. If, on the other hand, this field contains a numeric value, then you should be able to interrupt and return to a valid, if not globally optimal, solution.

11. OBJCTS LINGO generates twelve different types of constraint cuts when solving mixed integer linear programs. Using the options listed below, these various classes of cuts can be enabled by setting their parameter value to 1, or disabled by setting their parameter value to 0.

336 CHAPTER 6 The available cut classes are as follows: Index 11 57 58 59 60 61 62 63 64 65 66 76

Parameter Name OBJCTS GUBCTS FLWCTS LFTCTS PLOCTS DISCTS KNPCTS LATCTS GOMCTS COFCTS GCDCTS BASCTS

Cut Type Objective cuts GUB Flow Lift Plant location Disaggregation Knapsack cover Lattice Gomory Coefficient reduction Greatest common divisor Basis cuts

By default, all cut classes are enabled with the exception of objective cuts (OBJCTS). Occasionally, on some poorly formulated models, disabling one or more of the cut forms can help in finding feasible solutions. Cuts are not generated for nonlinear integer models. Thus, these options will not affect performance on nonlinear models. 12. MXMEMB Use this parameter to set an upper limit on the amount of memory, in megabytes, that LINGO allocates as workspace for its model generator. When LINGO starts up, it sets aside a fixed amount of memory to use as a generator workspace. The default workspace size is 32Mb. You can determine the size of the current workspace and the amount of memory allotted in this workspace by issuing the MEM command. Large models may run out of generator memory when attempting to solve them. In this case, you will receive the error message, “The model generator ran out of memory.” To avoid this error, increase the value of MXMEMB and issue the FREEZE command to preserve the change. You must then restart LINGO. Note:

Changes in LINGO’s generator memory limit are not established until you restart the program.

The model generator is distinct from the actual solver engines. Memory allocated to the generator will not be available to the solver engines. Thus, you shouldn’t allocate any more memory to the generator than is required. If you set MXMEMB to 0, LINGO will allocate all available memory when it starts up. This is not a recommended practice.


337

Setting LINGO’s generator memory limit abnormally high can result in poor performance of LINGO and the operating system. By setting aside excessive amounts of memory for the model generator, both LINGO and the operating system may have to resort to swapping of virtual memory to and from the hard drive. Accessing the hard drive for memory swaps can slow down your machine dramatically.

13. CUTAPP Use this parameter to control the nodes in the solution tree where the branch-and-bound solver adds constraint cuts in linear integer models. You have the following three options: CUTAPP Setting 0 1 2

Cuts Application at ... Root only All nodes Solver decides

Under the Root Only option, the solver appends cuts only at the first node, or root node, in the solution tree. With the All Nodes option, cuts are appended at each node of the tree. Under the Solver Decides option, the solver dynamically decides when it is best to append cuts at a node. The default is to let the solver decide when to append cuts. In general, this will offer superior performance. There may be instances, however, where one of the other two options prevails. 14. ABSINT Use this parameter to specify an absolute integrality tolerance. This tolerance is used by LINGO as a test for integrality in integer programming models. Due to round-off errors, the "integer" variables in a solution may not have values that are precisely integer. The absolute integrality tolerance specifies the absolute amount of violation from integrality that is acceptable. Specifically, if X is an "integer" variable and I is the closest integer to X, then X would be accepted as being integer valued if: |X - I| <= Absolute Integrality Tolerance. The default value for the absolute integrality tolerance is .000001. Although one might be tempted to set this tolerance to 0, this may result in feasible models being reported as infeasible. 15. HEURIS Use this parameter to control the level of integer programming heuristics used by the integer solver. These heuristics use the continuous solution at each node in the branch-and-bound tree to attempt to quickly find a good integer solution. If an integer solution better than the incumbent is found, then it is used to fix or tighten global and local variable bounds. Heuristics are only applied to linear models. Requesting heuristics on nonlinear models will result in no benefits. HEURIS may be set anywhere from 0 (none) to 100 (highest level), with 3 being the default. 16. HURDLE If you know the objective value of a solution to a model, you can enter it as a hurdle tolerance. This value is used in the branch-and-bound solver to narrow the search for the optimum. More specifically, LINGO will only search for integer solutions where the objective is better than the hurdle value. This comes into play when LINGO is searching for an initial integer solution. LINGO can ignore branches in the search tree with objective values worse than the hurdle value, because a better solution exists

338 CHAPTER 6 (i.e., the hurdle) on some alternate branch. Depending on the problem, a good hurdle value can greatly reduce solution time. Once LINGO finds an initial integer solution, however, the Hurdle tolerance no longer has an effect. Note:

Be sure when entering a hurdle value that a solution exists that is at least as good or better than your hurdle. If such a solution does not exist, LINGO will not be able to find a feasible solution to the model.

The default hurdle value is None. In other words, a hurdle value is not used by the solver. To clear an existing hurdle value, type SET HURDLE NONE. 17. IPTOLA Use this parameter to specify the absolute optimality tolerance. This tolerance is a positive value r, indicating to the branch-and-bound solver that it should only search for integer solutions with objective values at least r units better than the best integer solution found so far. In many integer programming models, there are huge numbers of branches with roughly equivalent potential. This tolerance can help to keep the branch-and-bound solver from being distracted by branches that can’t offer a solution significantly better than the incumbent solution. In general, you shouldn’t have to set this tolerance. Occasionally, particularly on poorly formulated models, you might need to increase this tolerance slightly from zero to improve performance. In most cases, you should experiment with the relative optimality tolerance rather than the absolute optimality tolerance in order to improve performance. The default value for the absolute optimality tolerance is 0. 18. IPTOLR Use this parameter to specify the relative optimality tolerance. This tolerance is a value r, ranging from 0 to 1, indicating to the branch-and-bound solver that it should only search for integer solutions with objective values at least 100*r% better than the best integer solution found so far. The end results of modifying the search procedure in this way are twofold. First, on the positive side, solution times can be improved tremendously. Second, on the negative side, the final solution obtained by LINGO may not be the true optimal solution. You will, however, be guaranteed the solution is within 100*r% of the true optimum. Typical values for the relative optimality tolerance would be in the range .01 to .05. In other words, you would be happy to get a solution within 1% to 5% of the true optimal value. On larger integer models, the alternative of getting a solution within a few percentage points of the true optimum after several minutes of runtime, as opposed to the true optimum after several days, makes the use of an optimality tolerance quite attractive. Note:

Generally speaking, the relative integrality tolerance is the tolerance that will most likely improve runtimes on integer models. You should be sure to set this tolerance whenever possible.

The default for the relative optimality tolerance is 1e-5.


339

19. TIM2RL If an integer programming model is relatively easy to solve, then we would like to have the solver press on to the true optimal solution without immediately resorting to a relative optimality tolerance (discussed above). On the other hand, if, after running for a while, it becomes apparent that the optimal solution won’t be immediately forthcoming, then you might want the solver to switch to using a relative optimality tolerance. TIM2RL, the time to relative tolerance, can be used in this manner. This tolerance is the number of seconds before the branch-and-bound solver begins using the relative optimality tolerance. For the first n seconds, where n is the value of the time to relative tolerance, the branch-and-bound solver will not use the relative optimality tolerance and will attempt to find the true optimal solution to the model. Thereafter, the solver will use the relative optimality tolerance in its search. The default value for the time to relative tolerance is 100 seconds. 20. NODESL The branch-and-bound solver has a great deal of freedom in deciding how to span the branch-and-bound solution tree. NODESL, the node selection option, allows you to control the order in which the solver selects branch nodes in the tree. The four choices available for NODESL are as follows: NODESL Setting 0 1 2 3

Branch Selection LINGO Decides –This is the default option. LINGO makes an educated guess as to the best node to branch on. Depth First – LINGO spans the branch-and-bound tree using a depth first strategy. Worst Bound – LINGO picks the node with the worst bound. Best Bound – LINGO picks the node with the best bound.

In general, LINGO Decides will offer the best results. Experimentation with the other options may be beneficial with some classes of models. 21. LENPAG The LENPAG parameter sets the length of the page or screen size in lines. For instance, setting LENPAG to 25 will cause the display to pause after 25 lines and await a carriage return before displaying the next 25 lines. This is convenient when you wish to page through long reports and not have them scroll off the top of the screen. When LENPAG is set to 0, paging is turned off entirely. LINGO will no longer stop output to wait for a carriage return. Entering SET LENPAGE 0 at the top of any command script is helpful in that you generally want command scripts to run uninterrupted. 22. LINLEN When LINGO generates reports, it limits output lines to a certain width. In some reports, lines will be wrapped so that they fall within the line length limit. In other reports, lines may be truncated. Since LINGO concatenates variable names in performing set operations, a variable name such as SHIPMENTS( WAREHOUSE1, CUSTOMER2) may result, which may be truncated in a solution report if too narrow an output width is used. You can control this line width limit through the LINLEN parameter. You may set it anywhere between 64 and 200, with the default being 85.

340 CHAPTER 6 23. TERSEO You can use the TERSEO parameter to control the amount of output LINGO generates. There are four settings available: TERSEO 1 2

3 4

Description Verbose—Causes LINGO to display the maximum amount of output, including full solution reports. Terse—Less output than Verbose, with full solution reports suppressed. This is a good output level if you tend to solve large models. LINGO also suppresses Export Summary Reports generated when exporting data to spreadsheets or databases. Errors Only—All output is suppressed, with the exception of error messages Nothing—LINGO suppresses all output. This level may be useful when taking advantage of the programming capabilities in LINGO, in which case, you will add statements to your model to generate all required output.

The default setting for TERSEO is 1, or verbose mode.


341

24. STAWIN (Windows Only) If the STAWIN parameter is set to 1, LINGO displays a solver status window whenever you issue the GO command. This window resembles the following:

The solver status window is useful for monitoring the progress of the solver and the dimensions of your model. It is updated every n seconds, where n is the value in the Update interval field in the lower right corner of the window. LINGO defaults to displaying the solver status window. This option applies only to Windows versions of LINGO. For a detailed description of the various fields in the solver status window, see Chapter 1, Getting Started with LINGO. 25. SPLASH (Windows Only) If the SPLASH parameter is set to 1, LINGO will display its splash screen each time it starts up. The splash screen lists the release number of LINGO and the software’s copyright notice. Setting SPLASH to 0 disables the splash screen. The default is for the splash screen to be displayed. This option applies only to Windows versions of LINGO.

342 CHAPTER 6 26. OROUTE (Windows Only) Set this parameter to 1 to send reports generated by LINGO to the command window, or 0 to send them to individual report windows. Since you can log all output to the command window in a log file, routing reports to the command window can be a useful way of logging all reports to disk. This may also be a desirable option when you are using LINGO as part of an automated system where you need LINGO to run without input. The default is for LINGO to display reports in individual windows. This option is available only on Windows versions of LINGO. 27. WNLINE (Windows Only) and 28. WNTRIM (Windows Only) When LINGO sends output to the command window, it places new lines at the bottom of the window. All previous output is scrolled up to make way for the new output. The total number of output lines that can be stored in the command window is limited. When LINGO hits this limit, it begins deleting lines from the top of the command window. You can control this feature by setting the WNLINE and WNTRIM parameters. The WNLINE parameter sets the maximum number of lines allowed in the command window. When LINGO removes lines from the top of the command window, it stops once there are n lines left in the command window, where n is the value of the WNTRIM parameter. In general, output to the command window will become slower as you increase the maximum and minimum line counts. The default values for WNLINE and WNTRIM are, respectively, 800 and 400. Minimum values are 200 and 100, while there are no upper limits. These options are relevant only to Windows versions of LINGO. 29. STABAR (Windows Only) If the STABAR parameter is set to 1, LINGO for Windows displays a status bar along the bottom of the main frame window. Among other things, the status bar displays the time of day, location of the cursor, menu tips, and the current status of the program. To remove the status bar from the screen, set STABAR to 0. The default is for LINGO to display the status bar. This option applies only to Windows versions of LINGO. 30. FILFMT (Windows Only) Use FILFMT to set the default file format LINGO uses when opening a new document. The options are: FILFMT

File Type

Description

0

LNG

LINGO text

1

LG4

LINGO binary

2

LTX

LINDO text

The LG4 format is the default file format for Windows versions of LINGO. This is a binary format that is readable only by LINGO. This format enables you to have custom formatting and fonts in your models, and allows you to use LINGO as an OLE server and container. Files written in LG4 format are useful only on Windows hardware.


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The LNG and LTX formats are text based. Given this, LNG and LTX files may be read into other applications. However, these formats don’t custom formatting and embedded objects. In general, LNG files use LINGO syntax, while LTX files use LINDO syntax. This option applies only to Windows versions of LINGO. 31. TOOLBR (Windows Only) In Windows versions, LINGO can display a row of buttons that act as shortcuts to various commands contained in the LINGO menu. This row of buttons is known as the toolbar. Set TOOLBR to 1 to display the toolbar or 0 to remove it. The default is for LINGO to display the toolbar. This option applies only to Windows versions of LINGO. 32. CHKDUP Prior to release 4.0, LINGO allowed you to use primitive set names in the equations of a model. Primitive set names in a model’s equations returned the index of the set member. Starting with release 4.0, LINGO required you to use the @INDEX function (see Chapter 7, LINGO's Operators and Functions) to get the index of a primitive set member. If you would like to test your LINGO models from releases prior to 4.0 for instances where primitive set appear in the model’s equations, set CHKDUP to 1. Whenever you run a model, LINGO will issue an error message if duplicate names appear as set and as variables in the model. 33. ECHOIN When you run a LINGO command script with the TAKE command, the commands LINGO processes are normally not displayed. If you would like the commands echoed to your screen, set the ECHOIN parameter to 1. This can be a useful feature when you are trying to develop and debug a LINGO command script. 34. ERRDLG (Windows Only) Set the ERRDLG parameter to 1 and LINGO will display error messages issued by the solver in a modal dialog box. This dialog box must be cleared before LINGO proceeds with any other operation. In some instances, you may have LINGO embedded in other applications, where it may not be desirable, or possible, to have s clearing the error dialog boxes. By setting ERRDLG to 0, LINGO will route the solver’s error messages to the report window, where they will be displayed and no intervention will be required to clear the messages. Note that this option allows you to route only those error messages generated by LINGO’s solver to the report window. Error messages displayed by LINGO’s interactive front-end (error codes 1000 and above) will always be posted in dialog boxes. The default is for solver errors to be displayed in dialog boxes. This option applies only to Windows versions of LINGO.

344 CHAPTER 6 35. USEPNM In many instances, you will need to get the index of a primitive set member within its set. Prior to release 4 of LINGO, you could do this by using the primitive set member’s name directly in the model’s equations. This can create problems when you are importing set from an external source. In this case, you will not necessarily know the names of the set beforehand. When one of the imported primitive set happens to have the same name as a variable in your model, unintended results can occur. More specifically, LINGO would not treat the variable as optimizable. In fact, LINGO would treat it as if it were a constant equal to the value of the index of the primitive set member! In short, different primitive set names could potentially lead to different results. Therefore, starting with release 4.0 of LINGO, models such as the following are no longer permitted: MODEL: SETS: DAYS /MO TU WE TH FR SA SU/; ENDSETS INDEX_OF_FRIDAY = FR; END

If you want the index of FR in the DAYS set, you should use the @INDEX function (see Chapter 7, LINGO's Operators and Functions): INDEX_OF_FRIDAY = @INDEX(DAYS, FR);

If you are unable to update your models for some reason and you would like to allow for the direct use of primitive set names, you can enable the USEPNM parameter by setting it to 1. The default is for LINGO to disable USEPNM. 36. NSTEEP Setting the NSTEEP parameter to 1 causes LINGO’s nonlinear solver to use steepest-edge variable selection. When LINGO is not in steepest-edge mode, the nonlinear solver will tend to select variables that offer the highest absolute rate of improvement to the objective, regardless of how far other variables may have to move per unit of movement in the newly introduced variable. The problem with this strategy is that other variables may quickly hit a bound, resulting in little gain to the objective. With the steepest-edge option, the nonlinear solver spends a little more time in selecting variables by looking at what rate the objective will improve relative to movements in the other nonzero variables. Thus, on average, each iteration will lead to larger gains in the objective. In general, the steepest-edge option will result in fewer iterations. However, each iteration will take longer. LINGO defaults to not using the steepest-edge option. 37. NCRASH If you set NCRASH to 1, LINGO’s nonlinear solver will invoke a heuristic for generating a “good” starting point when you solve a model. If this initial point is relatively good, subsequent solver iterations should be reduced along with overall runtimes. LINGO defaults to not crashing an initial solution.


345

38. NSLPDR If you set NSLPDR to 1, LINGO’s nonlinear solver will use successive linear programming (SLP) to compute new search directions. This technique uses a linear approximation in search computations in order to speed iteration times. In general, the number of total iterations will tend to rise when SLP directions are used, but on some models overall runtimes may improve. LINGO defaults to not using SLP directions. 39. SELCON If you set SELCON to 1, LINGO’s nonlinear solver will only evaluate constraints on an as needed basis. Thus, not every constraint will be evaluated at each iteration. This generally leads to faster solution times, but can also lead to problems in models with undefined functions in certain regions. LINGO may not evaluate a constraint for many iterations only to find that it has moved into a region where the constraint is no longer defined. In this case, there may not be a valid point for the solver to retreat to and the solution process terminates with an error. Turning off selective constraint evaluation eliminates these errors. LINGO defaults to not using selective constraint evaluation. 40. PRBLVL On a mixed-integer linear program, LINGO can perform an operation known as probing. Probing involves taking a close look at the integer variables in a model and deducing tighter variable bounds and right-hand side values. In many cases, probing can tighten an integer model sufficiently, thereby speed overall solution times. In other cases, however, probing may not be able to do much tightening and the overall solution time will increase due to the extra time spent probing. You can choose from seven successive levels of probing ranging from 1 to 7. Level 1 disables probing completely, while level 7 involves the highest degree of probing. Setting this option to 0 lets LINGO select the level of probing. LINGO defaults to 0. 41. SOLVEL This option allows you to choose the type of algorithm invoked by LINGO’s linear solver. At present, LINGO offers the following four options: SOLVEL Value 0 1 2 3

Linear Solver Algorithm LINGO chooses Primal simplex Dual simplex Barrier (only available as an option)

In general, it is difficult to say what algorithm will be fastest for a particular model. A rough guideline is that primal simplex tends to do better on sparse models with fewer rows than columns; the dual does well on sparse models with fewer columns than rows; and the barrier works best on densely structured models or very large models. The barrier solver is available only as an additional option to the LINGO package. LINGO defaults to 0, LINGO chooses.

346 CHAPTER 6 42. REDUCE When this parameter is set to 1, LINGO’s linear solver tries to identify and remove extraneous variables and constraints from the formulation before solving. In certain cases, this can greatly reduce the size of the final model to be solved. Setting REDUCE to 1 enables reduction, while 0 disables it. Setting REDUCE to 2 allows LINGO to choose whether or not to enable reduction. LINGO defaults to this last option. 43. SCALEM Setting SCALEM to 1 enables the scaling option in LINGO’s linear solver. This option rescales the coefficients in the model’s matrix, causing the ratio of the largest to smallest coefficients to be reduced. By doing this, LINGO reduces the chances of round-off error, which leads to greater numerical stability and accuracy in the linear solver. LINGO defaults to using scaling. 44. PRIMPR Setting this parameter to 2 causes LINGO’s primal simplex linear solver to use devex pricing techniques. If this parameter is set to 1, the primal simplex solver will use partial pricing. If this parameter is set to 0, LINGO chooses the primal simplex pricing method. LINGO defaults to choosing the primal pricing method. 45. DUALPR If DUALPR is set to 2, LINGO’s dual simplex solver will use steepest edge pricing. If DUALPR is 1, the dual solver will use Dantzig pricing methods. If DUALPR is 0, LINGO chooses the most appropriate pricing method. In Dantzig pricing mode, the dual simplex solver will tend to select variables that offer the highest absolute rate of improvement to the objective, regardless of how far other variables may have to move per unit of movement in the newly introduced variable. The problem with this strategy is that other variables may quickly hit a bound, resulting in little gain to the objective. With the steepest-edge option, the solver spends a little more time selecting variables by looking at the total improvement in the objective by adjusting a particular variable. Thus, on average, each iteration will lead to larger gains in the objective. In general, the steepest-edge option will result in fewer iterations. However, each iteration will take longer. LINGO defaults to choosing the pricing method for the dual solver. 46. DUALCO The DUALCO parameter is used to set the level of dual computations performed by the solver. Setting DUALCO to 0 will cause LINGO to not compute dual values and ranges. This is the fastest option, but is suitable only if you don’t need this information. In fact, the RANGE command will not execute when DUALCO is 0. When DUALCO is 1, LINGO will compute dual values, but not ranges. When DUALCO is 2, LINGO computes both dual prices and ranges. Setting DUALCO to 3 causes LINGO to compute the dual values on optimizable rows only (i.e., fixed rows are excluded) and forgo range computations, LINGO defaults to a DUALCO value of 1. Note:

Range computations can take some time, so, if speed is a concern, you don’t want to enable range computations unnecessarily.


Note:

347

The barrier crossover option must be enabled if you plan to do range analysis. Range computations cannot be performed if the final solution is not a basic solution.

47. RCMPSN Setting RCMPSN to 1 causes LINGO to convert all variable and row names to RC notation when performing MPS file format I/O. Refer to the RMPS command on page 286 for a discussion of why this option is useful. By default, LINGO disables the use of RC format names. 48. MREGEN The MREGEN parameter controls the frequency with which LINGO regenerates a model. With MREGEN set to 0, LINGO regenerates a model only when a change has been made to the model’s text since the last generation took place. When MREGEN is 1, LINGO regenerates whenever a change is made to the model text or if it contains references to external data sources (e.g., text files, databases, or spreadsheets). If MREGEN is 2, then LINGO always regenerates the model each time information regarding the generated model is needed. Commands that will trigger a model generation are GO, GEN, GENL, STATS, RMPS, FRMPS, SMPS, and PICTURE. LINGO defaults to a MREGEN value of 2. 49. BRANDR LINGO uses a branch-and-bound solution procedure when solving integer programming models. One of the fundamental operations involved in the branch-and-bound algorithm is branching on variables. Branching involves forcing an integer variable that is currently fractional to either the next greatest integer value or to the next lowest integer value. As an example, suppose there is a general integer variable that currently has a value of 5.6. If LINGO were to branch on this variable, it would have to choose whether to set the variable first to 6 or 5. The BRANDR parameter controls how LINGO makes this branching decision. There are three possible settings for BRANDR: BRANDR Value 0 1 2

Preferred Branching Direction Both up and down Up Down

The default option, Both up and down, involves LINGO making an intelligent guess as to whether it should branch up or down first on each individual variable. If the Up option is selected, LINGO will always branch up to the next highest integer first. If Down is selected, LINGO will always branch down first. In most cases, the Both up and down option will result in the best performance. Occasionally, models will benefit from use of one of the other two options.

348 CHAPTER 6 50. BRANPR When branching on variables, the branch-and-bound procedure can give priority to branching on the binary variables first, or it can make an intelligent guess as to the next best variable to branch on, regardless of whether it is binary or general. There are two possible settings for BRANPR: BRANPR Value 0 1

Branching Priority LINGO decides Binary variables first

Select the Binary variables first option to have LINGO give branching priority to the binary variables. Select LINGO Decides to have LINGO select the next integer variable for branching based on an intelligent guess regardless of whether it is binary or general. The default for this option is LINGO Decides, which should generally give the best results. However, on occasion, the Binary option may prevail. 51. CUTOFF On occasion, due to round-off error, some of the values returned by LINGO’s solver will be very small (less than 1e-10). In reality, the true values of these variables are either zero or so small as to be of no consequence. These tiny values can be distracting when interpreting a solution report. The CUTOFF parameter can be used to suppress small solution values. Any solution value less-than-or-equal-to CUTOFF will be reported as being zero. The default value for CUTOFF is 1e-9. 52. STRONG The strong branch option uses a more intensive branching strategy during the first n levels of the branch-and-bound tree, where n is the value of the STRONG parameter. During these initial levels, LINGO picks a subset of the fractional variables as branching candidates. LINGO then performs a tentative branch on each variable in the subset, selecting as the final candidate the variable that offers the greatest improvement in the bound on the objective. Although strong branching is useful in tightening the bound quickly, it does take additional computation time. So, you may want to try different settings to determine what works best for your model. The default setting is 10 levels.


349

53. REOPTB The warm start option controls the linear solver that is used by the branch-and-bound solver at each node of the solution tree when a previous solution is present to use as a “warm start”. The cold start option, discussed below, determines the solver to use when a previous solution does not exist. There are four possible settings for REOPTB: REOPTB Value 0 1 2 3

Warm Start Solver LINGO Decides – LINGO chooses the most appropriate solver. Primal – The primal solver will be used exclusively. Dual – The dual solver will be used exclusively. Barrier – LINGO uses the barrier method, assuming you have purchased a license for the barrier solver. Otherwise, the dual solver will be used.

In general, LINGO Decides will yield the best results. The barrier solver can’t make use of a pre-existing solution, so Barrier usually won’t give good results. In general, Dual will be faster than Primal for reoptimization in branch-and-bound. 54. REOPTX The cold start option controls the linear solver that is used by the branch-and-bound solver at each node of the solution tree when a previous solution is not present to use as a “warm start”. The warm start option, discussed above, determines the solver to use when a previous solution does exist. There are four possible settings for REOPTX: REOPTX Value 0 1 2 3

Warm Start Solver LINGO Decides – LINGO chooses the most appropriate solver. Primal – The primal solver will be used exclusively. Dual – The dual solver will be used exclusively. Barrier – LINGO uses the barrier method, assuming you have purchased a license for the barrier solver. Otherwise, the dual solver will be used.

In general, LINGO Decides will yield the best results. However, experimentation with the other options may be fruitful.

350 CHAPTER 6 55. MAXCTP The integer pre-solver makes iterative es through a model determining appropriate constraint cuts to append to the formulation. In general, the marginal benefits of each additional declines. At some point, additional es will only add to total solution times. Thus, LINGO imposes a limit on the maximum number of es. LINGO applies constraint cuts at both the top, or root, node of the branch-and-bound tree, and at all subsequent nodes within the tree. The MAXCTP parameter limits the maximum number of cuts at the top node, while the MAXCTR parameter sets the cut limit on all subsequent nodes in the tree. The default limit is 100 es. 56. RCTLIM Most integer programming models benefit from the addition of some constraint cuts. However, at some point, additional cuts take more time to generate than they save in solution time. For this reason, LINGO imposes a relative limit on the number of constraint cuts that are generated. The default limit is set to .75 times the number of true constraints in the original formulation. You may override this relative limit by changing the setting of RCTLIM. Constraint Cut Types LINGO generates twelve different types of constraint cuts when solving mixed integer linear programs. Using options listed below, these various classes of cuts can be enabled by setting their parameter value to 1, or disabled by setting their parameter value to 0. The available cut classes are as follows: Index 11 57 58 59 60 61 62 63 64 65 66 76

Parameter Name OBJCTS GUBCTS FLWCTS LFTCTS PLOCTS DISCTS KNPCTS LATCTS GOMCTS COFCTS GCDCTS BASCTS

Cut Type Objective cuts GUB Flow Lift Plant location Disaggregation Knapsack cover Lattice Gomory Coefficient reduction Greatest common divisor Basis cuts

By default, all cut classes are enabled. Occasionally, on some poorly formulated models, disabling one or more of the cut forms can help in finding feasible solutions. Cuts are not generated for nonlinear integer models. Thus, these options will not affect performance on nonlinear models.


351

67. SCLRLM (Windows Only) The LINGO editor in Windows is “syntax aware.” In other words, when it encounters LINGO keywords, it displays them in blue. Comments are displayed in green, and all remaining text is displayed in black. Syntax coloring can take a long time if you have very large files. The SCLRLM parameter sets the maximum acceptable file size for syntax coloring. Files with line counts exceeding this parameter will not be syntax colored. Setting this parameter to 0 will disable the syntax coloring feature. The default limit is 1000 lines. This option applies only to Windows versions of LINGO. 68. SCLRDL (Windows Only) The LINGO editor in Windows is “syntax aware”. In other words, when it encounters LINGO keywords it displays them in blue. Comments are displayed in green, and all remaining text is displayed in black. The SCLRDL parameter sets the number of seconds LINGO waits after the last keystroke was typed before recoloring modified text. s on slower machines may want to set this higher to avoid having syntax coloring interfere with typing. s on faster machines may want to decrease this value, so text is recolored more quickly. The default is 0 seconds (i.e., LINGO recolors modified text immediately). This option applies only to Windows versions of LINGO. 69. PRNCLR (Windows Only) The LINGO editor in Windows displays matching parentheses in red when you place the cursor immediately following a parenthesis. The PRNCLR parameter allows you to disable this feature. Setting PRNCLR to 0 will disable parenthesis matching, while setting it to 1 will enable it. This option applies only to Windows versions of LINGO. 70. MULTIS LINGO exploits the convex nature of linear models to find globally optimal solutions. However, we aren’t as fortunate with nonlinear models. With nonlinear programming (NLP) models, LINGO’s default NLP solver uses a local search procedure. This can lead to LINGO stopping at locally optimal points, perhaps missing a global point lying elsewhere. Refer to Chapter 15, On Mathematical Modeling, for more information on how and why this can happen. A strategy that has proven successful in overcoming this problem is to restart the NLP solver several times from different initial points. It is not uncommon for a different starting point to lead to a different local solution point. The idea is that, if we restart from enough unique points, saving the best local solution as we go, then we have a much better chance of finding the true global solution. The MULTIS parameter allows you to set the number of times you would like the NLP solver to resolve your model, starting each time from an intelligently generated, new starting point. We refer to this feature as multistart. The default value for MULTIS, 0, entails restarting 5 times on small NLPs and disabling multistart on larger models. Setting MULTIS to 1 disables multistart on all NLPs. Setting MULTIS to any value greater than 1 will cause the NLP solver to restart that number of times on all NLPs. We have found that setting MULTIS around 5 tends to be adequate for most models. Highly nonlinear models may require a larger setting.

352 CHAPTER 6 Keep in mind, however, that multistart will dramatically increase runtimes. Thus, one should avoid using it unnecessarily on convex models that will converge to a global point in a single without any additional prodding. The following example illustrates the usefulness of multistart. Consider the simple, yet highly nonlinear, model: MODEL: MIN = X * @COS( 3.1416 * X); @BND( 0, X, 6); END

The graph of the objective function is as follows:

The objective function has three local, minimal points over the feasible range. These points are summarized in the following table: Point 1 2 3

X 1.09 3.03 5.02

Objective -1.05 -3.02 -5.01


353

Clearly, the third local point is also the globally best point, and we would like the NLP solver to converge to this point. Below, we attempt this by loading the model, turning off the multistart option, and then solving: : take wavy.lng : look all MODEL: 1] MIN = X * @COS( 3.1416 * X); 2] @BND( 0, X, 6); END : set multis 1 !set solver attempts to 1 only (i.e., disable ms) Parameter MULTIS

Old Value 0

New Value 1

: go Local optimal solution found at step: Objective value: Variable X Row 1



Unfortunately, as you can see, we converged to the least preferable of the local minimums. Below, we will do the same as in the previous run. However, this time, we will set the number of multistarts to five: : take wavy.lng : look all MODEL: 1] MIN = X * @COS( 3.1416 * X); 2] @BND( 0, X, 6); END : set multis 5 Parameter MULTIS

Old Value 0

New Value 5

: go Local optimal solution found at step: Objective value: Variable X Row 1



The extra four restarts allowed LINGO to find the global optimal point.

354 CHAPTER 6 71. USEQPR The USEQPR parameter controls the Quadratic Recognition option. This option consists of an algebraic preprocessor that automatically determines if an arbitrary nonlinear model is actually a quadratic programming (QP) model. If a model is found to be a convex QP, then it can be ed to the faster quadratic solver. Note that the QP solver is not included with the base version of LINGO, but comes as part of the Barrier option. LINGO defaults to using quadratic recognition. You may enable this option with the command: SET USEQPR 1.

72. GLOBAL Many nonlinear models are non-convex and/or non-smooth (for more information see Chapter 15, On Mathematical Modeling). Nonlinear solvers that rely on local search procedures, as does LINGO’s default nonlinear solver, will tend to do poorly on these types of models. Typically, they will converge to a local, sub-optimal point that may be quite distant from the true, global optimal point. Global solvers overcome this weakness through methods of range bounding (e.g., interval analysis and convex analysis) and range reduction techniques (e.g., linear programming and constraint propagation) within a branch-and-bound framework to find the global solutions to non-convex models. LINGO has a global solver capability that is enabled through the GLOBAL parameter. Setting GLOBAL to 1 will enable the global solver on nonlinear models, while setting it to 0 (the default) will not. The following example illustrates the power of the global solver on a non-smooth model. Consider the following model: model: sets: projects: baths, sqft, beds, cost, est; endsets data: projects, p1 p2 p3 p4 p5 p6 p7 ; enddata

beds, 5 2 1 4 4 3 3

baths, 4 1 1 3 2 1 2

sqft, 6200 820 710 4300 3800 2200 3400

cost = 559608 151826 125943 420801 374751 251674 332426

min = @max( projects: @abs( cost - est)); @for( projects: est = a0 + a1 * beds + a2 * baths + a3 * sqft ); end

Model: COSTING


355

This model estimates the cost of home construction jobs based on historical data on the number of bedrooms, bathrooms, and square footage. The objective minimizes the maximum error over the sample project set. Both the @MAX() and @ABS() functions in the objective are non-smooth, and, as a result, can present problems for LINGO’s default, local search NLP solver. Running the model under the default settings with the global solver disabled, we get the following result: Local optimal solution found at step: Objective value: Variable A0 A1 A2 A3

Value 37441.55 27234.51 23416.53 47.77956

91 3997.347 Reduced Cost 0.000000 0.000000 0.000000 0.000000

Enabling the global solver with the SET GLOBAL 1 command and re-optimizing yields the substantially better solution: Global optimal solution found at step: Objective value: Variable A0 A1 A2 A3

Value 46814.64 22824.18 16717.33 53.74674

186 1426.660

Reduced Cost 0.000000 0.000000 0.000000 0.000000

Note that the maximum error has been reduced from 3,997 to 1,426! This example illustrates the power of the global solver. Unfortunately, there is a drawback. You will find the global solver runs considerably slower than the default local solver, and may be swamped when trying to solve larger models. Therefore, the preferred option is to always try to create smooth, convex models, so that the faster, default local solver can successfully solve them. Keep in mind that the global solver s most, but not all, of the functions available in the LINGO language. The following is a list of the nonlinear functions not currently ed by the global solver:     Note:

All probability distributions — cumulative, inverse and pdf, with the exception of the normal distribution, which is fully ed @PFS() — Poisson finite source @PPL() — Poisson linear loss @( )— supplied function

The global solver will not operate on models containing one or more uned nonlinear operations that reference optimizable quantities; the default NLP solver will be called in this case.

The global solver is disabled by default.

356 CHAPTER 6 73-75. LNRISE, LNBIGM, LNDLTA The LNRISE, LNBIGM, and LNDLTA parameters control the linearization option in LINGO. Many nonlinear operations can be replaced by linear operations that are mathematically equivalent. The ultimate goal is to replace all the nonlinear operations in a model with equivalent linear ones, thereby allowing use of the faster and more robust linear solvers. We refer to this process as linearization. The LNRISE parameter determines the extent to which LINGO will attempt to linearize models. The available options are: LNRISE Setting 0 1 2 3

Linearization Level Solver Decides None Math Math and Logic

Under the None option, no linearization occurs. With the Math option, LINGO linearizes the mathematical functions: @ABS(), @MAX(), @MIN(), @SMAX(), and @SMIN() along with any products of binary and continuous variables. The Math and Logicoption is equivalent to the Math option, plus LINGO will linearize all logical operators (#LT#, #LE#, #EQ#, #GT#, #GE#, and #NE#). Under the Solver Decides option, LINGO will do maximum linearization if the number of variables doesn’t exceed 12. Otherwise, LINGO will not perform any linearization. LINGO defaults to the Solver Decides setting. The LNDLTA parameter controls the Delta Coefficient, which is a tolerance indicating how closely you want the additional constraints added as part of linearization to be satisfied. Most models won’t require any changes to this parameter. However, some numerically challenging formulations may benefit from increasing Delta slightly. LINGO defaults to a Delta of 1.e-6. When LINGO linearizes a model, it adds forcing constraints to the mathematical program generated to optimize your model. These forcing constraints are of the form: f( x) ≤ M • y where M is the BigM Coefficient and y is a 0/1 variable. The idea is that, if some activity in the variables is occurring, then the forcing constraint will drive y to take on the value of 1. Given this, if we set the BigM value to be too small, we may end up with an infeasible model. Therefore, the astute reader might conclude that it would be smart to make BigM quite large, thereby minimizing the chance of an infeasible model. Unfortunately, setting BigM to a large number can lead to numerical stability problems in the solver resulting in infeasible or sub-optimal solutions. So, getting a good value for the BigM Coefficient may take some experimentation. The default value for BigM is 100,000.


357

As an example of linearization, consider the following model: model: sets: projects: baths, sqft, beds, cost, est; endsets data: projects, p1 p2 p3 p4 p5 p6 p7 ; enddata

beds, 5 2 1 4 4 3 3

baths, 4 1 1 3 2 1 2

sqft, 6200 820 710 4300 3800 2200 3400

cost = 559608 151826 125943 420801 374751 251674 332426

min = @max( projects: @abs( cost - est)); @for( projects: est = a0 + a1 * beds + a2 * baths + a3 * sqft ); end

Model: COSTING This model estimates the cost of home construction jobs based on historical data on the number of bedrooms, bathrooms, and square footage. The objective minimizes the maximum error over the sample project set. Both the @MAX() and @ABS() functions in the objective are non-smooth nonlinear functions, and, as a result, can present problems for LINGO’s default, local search NLP solver. Running the model under the default settings with linearization disabled, we get the following result: Local optimal solution found at step: Objective value: Variable A0 A1 A2 A3

Value 37441.55 27234.51 23416.53 47.77956

91 3997.347 Reduced Cost 0.000000 0.000000 0.000000 0.000000

358 CHAPTER 6 Enabling linearization and re-optimizing yields the substantially better solution: Global optimal solution found at step: Objective value: Variable A0 A1 A2 A3

Value 46814.64 22824.18 16717.33 53.74674

186 1426.660

Reduced Cost 0.000000 0.000000 0.000000 0.000000

Note that the maximum error has been reduced from 3,997 to 1,426! Linearization will substantially increase the size of your model. The sample model above, in unlinearized form, has a mere 8 rows and 11 continuous variables. On the other hand, the linearized version has 51 rows, 33 continuous variables, and 14 binary variables! Although linearization will cause your model to grow in size, you will tend to get much better solution results if the model can be converted entirely to an equivalent linear form. Note:

Linearization will be of most use when a nonlinear model can be 100% linearized. If LINGO can only linearize a portion of your model, then you may actually end up with a more difficult nonlinear model.

The linearization option is set to Solver Decides by default. 76. BASCTS Please refer to the Constraint Cut Types section above for information on this parameter. 77. MAXCTR This parameter controls the number of es the branch-and-bound solver makes at each node of the tree for cut generation. There is one exception in that MAXCTR does not control the number of es at the root node of the tree. You must use MAXCTP, see above, to control the number of es at the root node. The default value for MAXCTR is 2 es. 78. HUMNTM This parameter sets the minimum amount of time spent in heuristics at each node of the branch-andbound tree. The default value for HUMNTM is 0 seconds. 79. DECOMP Many large scale linear and mixed integer problems have constraint matrices that are totally decomposable into a series of block structures. If total decomposition is possible, LINGO can solve the independent problems sequentially and report a solution for the original model, resulting in dramatic speed improvements. Setting DECOMP to 1 enables the decomposition feature. LINGO defaults to not using matrix decomposition. 80. GLBOPT The GLBOPT tolerance specifies by how much a new solution must beat the objective value of the incumbent solution in order to become the new incumbent in the global solver. The default value for GLBOPT is 1. e-5.


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81. GLBDLT The GLBDLT tolerance specifies how closely the additional constraints, added as part of the global solver’s convexification process, must be satisfied. The default value for GLBDLT is 1. e-7. 82. GLBVBD The GLBVBD tolerance sets the default variable bounds while the global solver is running. If this parameter is set to d, then variables will not be permitted to assume values outside the range of [-d, d]. Setting this parameter as tightly as possible in the Value Field restricts the global solver from straying into uninteresting regions and will reduce run times. You may also need to set the GLBUBD tolerance (see below) to control how the global solver uses the bound. The default value for GLBVBD is 1. e +10. 83. GLBUBD The GLBUBD tolerance controls how the global solver’s variable bound tolerance, GLBVBD (see above), is applied. There are three choices available: 0:None, 1:All, and 2:Selected. Selecting None removes the variable bound entirely and is not recommended. The All setting applies the bound to all variables. Finally, the Selected setting causes the global solver to apply the bound after an initial solver to find the first local solution. The bound will only be applied to a variable if it does not cut off the initial local solution. LINGO defaults to the Selected setting. 84. GLBBRN The GLBBRN tolerance specifies the branching direction for variables when the global solver initially branches on them. Six options are available: GLBBRN Setting 0 1 2 3 4 5

Branching Direction Absolute Width Local Width Global Width Global Distance Absolute Violation Relative Violation

The default setting for branching is 5, or Relative Violation. 85. GLBBXS The GLBBXS parameter specifies the strategy to use for choosing between all active nodes in the global solver’s branch-and-bound tree. The choices are: 0:Depth First and 1:Worst Bound. The default is 1, or Worst Bound. 86. GLBREF The GLBREF option sets the degree of algebraic reformulation performed by the global solver. Algebraic reformulation is critical for construction of tight, convex sub-regions to enclose the nonlinear and nonconvex functions. The available settings are: 0:None, 1:Low, 2:Medium, and 3:High. The default is 3, or High.

360 CHAPTER 6 87. SUBOUT The SUBOUT option is used to control the degree to which fixed variables are substituted out of the ultimate math program ed to the solver engines. For example, consider the model: MAX= 20*X + 30*Y + 12*Z; X = 2*Y; X + Y + Z <= 110; Y = 30;

If we run the GEN command, we see that LINGO is able to reduce this model down to the equivalent, but smaller model: MAX= 12 * Z + 2100; Z <= 20;

From the third constraint of the original model it is obvious that Y is fixed at the value 30. Plugging this value for Y into the first constraint, we can conclude that X has a value of 60. Substituting these two fixed variables out of the original formulation yields the reduced formulation above. In most cases, substituting out fixed variables yields a smaller, more manageable model. In some cases, however, you may wish to avoid this substitution. An instance in which you might want to avoid substitution would be when equations have more than one root. When m multiple roots are present, reduction may select a suboptimal root for a particular equation. On the other hand, the global and multistart solvers are adept at handling equations containing multiple roots. Thus, when using these solvers one may wish to forgo fixed variable reduction. The available options are: SUBOUT Setting

Reduction Degree

0

None

1

Always

2

Not with global and multistart

3

Linear only

Selecting None disables all fixed variable reduction. Selecting Always enables reduction. When Not with global and multistart is selected, LINGO disables reduction whenever either the global or multistart solvers are selected, otherwise reduction is performed. With the Linear Only option, LINGO will not substitute a variable out unless it is a linear variable. Note:

You should be careful when turning off fixed variable reduction. If the model generator is unable to substitute out fixed variables, you may end up turning a linear model into a more difficult nonlinear model.

LINGO defaults to the Linear Only setting for fixed variable reduction.


361

88. NLPVER The NLPVER option is no longer ed. 89. DBGCLD and 90. DBGWRM These two parameters give you control over the linear solver that is used by the DEBUG command for model debugging. The available choices are: DBGCLD/DBGWRM

Debug Linear Solver

0

Solver Decides — LINGO selects the solver it believes is the most appropriate,

1

Primal — the primal simplex solver will be used,

2

Dual — the dual simplex solver will be used, and

3

Barrier — the barrier solver will be used (requires a barrier solver license).

DBGCLD selects the solver for cold starts (starting without an existing basis in memory) and DBGWRM selects the solver for warm starts (restarting from an existing basis). LINGO defaults to Solver Decides for both the cold and warm debug solver. 91. LCRASH The LCRASH parameter controls the use of aggressive crashing techniques on nonlinear programs. Crashing is a heuristic process designed to find a good starting point for a model. The available choices are: 0 for none and 1 to perform aggressive crashing. The default setting is 0 or off. 92. BCROSS The BCROSS parameter controls whether or not the barrier solver performs a basis crossover on linear programs. Barrier solvers do not normally return basic solutions. For example, if alternate optima exist, the barrier method will return a solution that is, loosely speaking, the “average” of all alternate optima. The basis crossover process converts a non-basic barrier solver solution to a basic (i.e., corner point) solution. The available choices are: 0 for no crossover and 1 (the default) to perform a crossover. 93. LOWMEM The LOWMEM option may be used to guide LINGO’s memory usage. Enabling this option ( SET LOWMEM 1) causes LINGO to opt for less memory usage when solving a model. The downside is that opting for less memory may result in longer runtimes. LINGO defaults to disabling the LOWMEM option. 94. FILOUT LINGO can export a model’s solution to databases using the ODBC (Open DataBase Connectivity) standard. Solutions exported to a database are sent to tables within the database. The tables may contain more space for values than you are actually exporting. In other words, there may be records at

362 CHAPTER 6 the end of a table that will not be receiving exported values from LINGO. The Fill Out ODBC Tables option determines how these extra cells and records are treated. When the Fill Out Ranges and Tables option is enabled (SET FILOUT 1), LINGO overwrites the extra values with null entries. Conversely, when the option is not enabled, LINGO leaves any extra values untouched. The Fill Out ODBC Tables option is disabled by default. 95. DBGLVL The DBGLVL option gives you control over the output level of the model debugging command, DEBUG. The debugger is very useful in tracking down problems in models that are either infeasible or unbounded. Possible output levels range from 1 (minimum output) to 15 (maximum output). In general, you will want to generate as much output as possible. The only reason to restrict the amount of output would be to speed debugging times on large models. The default setting for the debugger output level is 15. 96. UNARYM The UNARYM option is used to set the priority of the unary minus operator. The two available options are High (SET UNARYM 1) are Low (SET UNARYM 0). There are two theories as to the priority that should be assigned to the unary minus (i.e., negation) operator in mathematical expressions. On the one hand, there is the Excel practice that the unary minus operator should have the highest priority, in which case, the expression 3^2 would evaluate to +9. On the other hand, there is the mathematicians’ preference for asg a lower priority to unary minus than is assigned to exponentiation, in which case, 3^2 evaluates to 9. Note that regardless which relative priority is used, one can force the desired result through the use of parenthesis. LINGO defaults to the Excel approach of setting a higher priority (High) on negation than on exponentiation. 97. LINEAR The LINEAR option can be enabled (SET LINEAR 1) to minimize memory usage on models that are entirely linear. When this option is in effect, the model generator can take steps to dramatically reduce overall memory consumption without sacrificing performance. In fact, if all your models are linear, we recommend that you enable this option permanently as the default for your installation. The one restriction is that models must prove to be entirely linear. If a single nonlinearity is detected, you will receive an error message stating that the model is nonlinear and model generation will cease. At which point, you should clear this option and attempt to solve the model again. By default, the LINEAR option is disabled. 98. LOPTOL The LOPTOL parameter allows you to control the setting for the linear optimality tolerance. This tolerance is used to determine whether a reduced cost on a variable is significantly different from zero. You may wish to loosen this tolerance (make it larger) on poorly scaled and/or large models to improve performance. The default setting for the LOPTOL parameter is 1.e-7.


363

99. SECORD The SECORD option determines if the nonlinear solver will use second order derivates. If used (SET SECORD 1), second order derivatives will always be computed analytically, as opposed to using numerical differences. Computing second order derivatives will take more time, but the additional information they provide may lead to faster runtimes and/or more accurate solutions. LINGO defaults to not using second order derivatives. 100. NONNEG When enabled (SET NONNEG 1), the NONNEG option tells LINGO to place a default lower bound of 0 on all variables. In other words, unless otherwise specified, variables will not be allowed to go negative. Should you want a variable to take on a negative value, you may always override the default lower bound of 0 using the @BND() function. If this option is disabled, then LINGO’s default assumption is that variables are unconstrained and may take on any value, positive or negative. Unconstrained variables are also referred to as be being free. By default, LINGO enables the non-negative option, thereby setting a default lower bound of 0 on all variables. 101. BIGMVL Many integer programming models have constraints of the form: f(x) ≤ M * z where f(x) is some function of the decision variables, M is a large constant term, and z is a binary variable. These types of constraints are called forcing constraints and are used to force the binary variable, z, to 1 when f(x) is nonzero. In many instances, the binary variable is multiplied by a fixed cost term in the objective; a fixed cost that is incurred when a particular activity, represented by f(x), occurs. The large constant tem, M, Is frequently referred to as being a BigM coefficient. Setting BigM too small can lead to infeasible or suboptimal models. Therefore, the BigM value will typically have to be rather large in order to exceed the largest activity level of f(x). When BigM is large, the solver may discover that by setting z slightly positive (within normal integrality tolerances), it can increase f(x) to a significant level and thereby improve the objective. Although such solutions are technically feasible to tolerances, they are invalid in that the activity is occurring without incurring its associated fixed cost. The BIGMVL parameter, or BigM threshold, is designed to avoid this problem by allowing LINGO to identify the binary variables that are being set by forcing constraints. Any binary variable with a coefficient larger than the BigM threshold will be subject to a much tighter integrality tolerance. The default value for the BigM Threshold is 1.e8. 102. KILLSC LINGO allows the input of scripts in the calc section. These scripts are useful for running multiple models, where the outputs of one model feed into subsequent models as input. Models are solved in calc sections with the @SOLVE command. Time limits can be placed on @SOLVE's via the TATSLV parameter (see below). If a time limit is hit while @SOLVE is running, LINGO will interrupt the solve and either continue executing the script with the next command, or terminate all processing. When the KILLSC option is set to 0 (default), processing continues with the next statement. Setting KILLSC to 1

364 CHAPTER 6 causes LINGO to terminate all processing whenever the @SOLVE time limit is hit, and LINGO will subsequently return to command-prompt level. 103. TATSLV LINGO allows the input of scripts in the calc section. These scripts are useful for running multiple models, where the outputs of one model feed into subsequent models as input. Models are solved in calc sections with the @SOLVE command. Time limits can be placed on @SOLVE's via the TATSLV parameter. If a time limit is hit while @SOLVE is running, LINGO will interrupt the solver and either continue executing the script with the next command, or terminate all processing based on the setting for the KILLSC parameter (see above). LINGO defaults to placing no time limit on @SOLVE commands. 104. KBESTS The KBESTS parameter is used to set the number of solutions desired as part of the K-Best solutions feature of LINGO's mixed integer solver. Whenever this value is greater than 1, say K, LINGO will return up to K unique solutions to the model. These solutions will have the property that they are the next best solutions available in of their objective values. Less than K solutions may be returned if a sufficient number of feasible solutions do not exist. Please refer to section K-Best Solutions Example for an example of the use of the K-Best feature. The default value for this parameter is 1, meaning that LINGO will find only one solution to integer models, i.e, the K-Best feature is disabled by default. 105. LCORES The LCORES parameter may be used to perform parallel solves of linear programs on multiple-cored machines. One of four different linear solvers is chosen for each core. Assignment of solvers to cores is controlled by the LCORE1 - LCORE4 parameters (see below). LINGO will take the solution from the solver that finishes first and then interrupt the remaining solver threads. The idea behind this approach is that different linear solvers will have relatively better or worse performance on different classes of models. However, it may be difficult to predict beforehand the solver that is most likely to outperform. So, by enabling multi-core solves, you guarantee that you will always get top performance, even without knowledge beforehand of which solver is likely to run the fastest. Note:


For the LCORES parameter, you have the following choices: 1, 2, 3, or 4. When the default 1 option is selected, the multi-core feature is disabled, and LINGO will run only one solver on linear programs, namely the one specified as part of the SOLVEL option detailed above. When either option 2, 3, or 4 is selected, LINGO will run linear solvers in the requested number of cores. The choice of the actual solvers used is controlled by the LCORE1 - LCORE4 parameters (see below). 106-109. LCORE1 - LCORE4 The LCORE1, LCORE2, LCORE3 and LCORE4 parameters are used in conjunction with the LCORES parameter to perform parallel solves of linear programs on multiple-cored machines. One of four different linear solvers is chosen for each core, with assignments controlled by the LCORE1 - LCORE4 parameters. LINGO will take the solution from the solver that finishes first and then interrupt the remaining solver threads.


365

The LCORES parameter gives the number of parallel solves that are to be performed on linear programs, while LCORE1 - LCORE4 control the selection of the actual LP solver to use in each core. Parameters LCORE1 - LCORE4 are meaningful only when LCORES is greater than 1. In addition, if LCORES= , then only the parameters LCORE1 - LCORE are meaningful. When the default 1 option is selected for LCORES, the multi-core feature is disabled, and LINGO will run only one solver on linear programs, namely the one specified as part of the SOLVEL option detailed above. When either option 2, 3, or 4 is selected for LCORES, LINGO will run linear solvers in the requested number of cores. The choice of the actual solvers used is controlled by the LCORE1 - LCORE4 parameters. The idea behind this approach is that different linear solvers will have relatively better or worse performance on different classes of models. However, it may be difficult to predict beforehand the solver that is most likely to outperform. So, by enabling multi-core solves, you guarantee that you will always get top performance, even without knowledge beforehand of which solver is likely to run the fastest. Note:


For each of the LCORE1-4 parameters, you have the following choices: LCORE(i) Setting LP Solver Used in Core i 1

Primal1 — Primal simplex algorithm 1

2

Dual — Dual simplex algorithm

3

Barrier — Barrier/Interior point solver (available as a option)

4

Primal2 — Primal simplex algorithm 2, installed as part of the Barrier option

366 CHAPTER 6 As an example, the following session runs an LP model in two cores (LCORES=2), with the barrier solver in core 1 (LCORE1=3) and the dual simplex solver in core2 (LCORE2=2): : set lcores 2

!run in 2 cores

Parameter LCORES

Old Value 1

: set lcore1 3 Parameter LCORE1 : set lcore2 2

New Value 2

!barrier in core 1 Old Value 1

New Value 3

!dual simplex in core 2

Parameter LCORE2

Old Value 2

: take lp.lng : set terseo 1

!load the model !minimal output

Parameter TERSEO

Old Value 0

: go !solve the model First returning solver: BARRIER Global optimal solution found. Objective value: Infeasibilities: Total solver iterations:

New Value 2

New Value 1

1272282. 0.000000 27

:

Once optimization is complete, LINGO will display a line indicating the solver that finished first. In the solution report excerpt above, we see that the barrier solver was the first to completion. 110. SCALEW After LINGO generates a model, it checks all the nonzero coefficients in the model and computes the ratio of the largest to smallest coefficients. This ratio is an indicator of how well the model is scaled. When the ratio gets to be too high, scaling is considered to be poor, and numerical difficulties may result during the solution phase. If the scaling ratio exceeds the value of the SCALEW parameter, LINGO will display error message 205. The default value for SCALEW is 1e12. Instead of simply increasing the SCALEW setting to eliminate error 205, we strongly suggest that you attempt to rescale the units of your model so as to reduce the largest-to-smallest coefficient ratio.


367

111. REFRAQ The REFRAQ parameter allows you to control how frequently the linear solver refactors the basis matrix. The options are either to set REFRAQ to 0, thereby letting LINGO determine the frequency, or to set REFRAQ to some positive integer quantity. If an integer value, N, is selected, then the linear solver will refactor every N iterations. Numerically tough and/or poorly scaled models may benefit from more frequent refactoring. However, refactoring too frequently will cause the solver to slow down. The default setting for the REFRAQ is 0, which will typically result in refactoring about once every 100 iterations. 112. SPSMSZ The SPSMSZ parameter is used to control the default sample size for random variables in stochastic programming (SP) whose outcomes are determined via sampling. In many SP models, LINGO will generate a set of sample values for the some or all of the random variables. This is particularly true when you have one or more continuous random variables, which have an infinite number of possible outcomes. In such a case, sampling is the only viable alternative. One way to specify the desired sample size for each stage is via the @SPSAMPSIZE function used directly in the model's text. If, on the other hand, all or most stages should have the same sample size, then you can use the SPSMSZ parameter to control the default sample size. Any stage which has not had its sample size specified with @SPSAMPSIZE will default to a sample size equal to the SPSMSZ parameter. Note:

In general, we prefer larger sample sizes to smaller ones in order to reduce sampling error. However, SP models can become quite large if sample sizes aren't kept reasonably small. This is particularly true for multiperiod models. For example, suppose we have a model with just one random variable and ten periods/stages. If the default sample size is set to 3, then there will be 3^10=59,049 possible scenarios. With this many scenarios, it would only take a handful of decision variables to end up with an underlying deterministic equivalent model approaching one million variables.

The SPSMSZ parameter defaults to a value of 2. 113. SPSCON The SPSCON parameter is used to control whether LINGO samples continuous distributions only for the random variables in stochastic programs, or if it samples both continuous and discrete random variables. Obviously, continuous random variables must be subject to sampling, given the infinite number of possible outcomes. On the other hand, we have a choice when it comes to discretely distributed random variables. If this option is enabled, then LINGO will generate one outcome for each density point in the discrete distribution of the random variable. These outcomes will be weighted by their corresponding probabilities.

368 CHAPTER 6 Note:

If there are many density points in the distributions for the discrete random variables, the deterministic equivalent generated to solve the SP may become exceedingly large, in which case, disabling this option may be required. Once this option is disabled, discrete random variables will be sampled in the same way as the continuous random variables.

The SPSCON option defaults to being on. 114. SPSEED The SPSEED parameter is used to establish the seed for the random number generator used in generating samples for random variables in stochastic programming (SP) models. In many SP models, LINGO will generate a set of sample values for the random variables. This is particularly true when you have one or more continuous random variables, which leads to an infinite number of possible outcomes. In such a case, sampling is the only viable alternative. The SPSEED parameter defaults to a value of 1031. Note:

The seed parameter has no effect when running demo versions of LINGO. Demo versions always use the default seed value, regardless of this parameter's setting.

115. SPMTHD The SPMTHD, or SP Solver Method, option is used to select the solution method for a stochastic programming (SP) model. Presently, the available choices are: SPMTHD Setting

SP Solver Method

0

Solver Decides — LINGO decides the most appropriate method for solving the SP model

1

Deterministic Equivalent — LINGO generates and directly solves the deterministic equivalent (DE) of the SP model

2

Nested Benders Decomposition — The DE for realistic SP models can get to be quite large, in that the core model is replicated once for every scenario. Fortunately, the DE displays strong block-angular structure, which makes it adaptable to decomposition techniques, such as nested Bender decomposition (NBD). Note that the model must be entirely linear to successfully use the NBD option.

The default setting for the SP Solver Method option is 0, or Solver Decides. 116. SPXVAL The SPXVAL, or Calculate All Expected Value Statistics, option controls whether LINGO displays information regarding the expected values for a number of statistics when solving stochastic programming (SP) models. To illustrate, when solving the SPGAS.LG4 model when this option is enabled, you will see the following expected values at the top of the solution report:

COMMAND-LINE COMMANDS Expected value of: Objective (EV): 1400.000 Wait-and-see model's objective (WS): 1326.667 Perfect information (EVPI = |EV - WS|): 73.33333 Policy based on mean outcome (EM): 1479.444 Modeling uncertainty (EVMU = |EM - EV|): 79.44444

These values are a guide as to how the stochastic nature of the model is impacting the objective value. The following is a brief description of these expected values: Expected Value of Objective (EV) - is the expected value for the model's objective over all the scenarios, and is the same as the reported objective value for the model. Expected Value of Wait-and-See Model's Objective (WS) - reports the expected value of the objective if we could wait and see the outcomes of all the random variables before making our decisions. Such a policy would allow us to always make the best decision regardless of the outcomes for the random variables, and, of course, is not possible in practice. For a minimization, it's true that WS <= EV, with the converse holding for a maximization. Technically speaking, WS is a relaxation of the true SP model, obtained by dropping the nonanticipativity constraints. Expected Value of Perfect Information (EVPI) - is the absolute value of the difference between EV and WS. This corresponds to the expected improvement to the objective were we to obtain perfect information about the random outcomes. As such, this is a expected measure of how much we should be willing to pay to obtain perfect information regarding the outcomes of the random variables. Expected Value of Policy Based On Mean Outcome (EM) - is the expected true objective value if we (mistakenly) assume that all random variables will always take on exactly their mean values. EM is computed using a two-step process. First, the values of all random variables are fixed at their means, and the resulting deterministic model is solved to yield the optimal values for the stage 0 decision variables. Next, a) the stage 0 variables are fixed at their optimal values from the previous step, b) the random variables are freed up, c) the nonanticipativity constraints are dropped, and d) this wait-and-see model is solved. EM is the objective value from this WS model. Expected Value of Modeling Uncertainty (EVMU) - is the absolute value of the difference EV - EM. It is a measure of what we can expect to gain by taking into uncertainty in our modeling analysis, as opposed to mistakenly assuming that random variables always take on their mean outcomes.

369

370 CHAPTER 6 Note:


Note:


The SPXVAL option is enabled by default. 117. SPMAXS The SPMAXS, or Max Scenarios Limit, parameter is used to establish a limit on the maximum number of scenarios in a stochastic programming (SP) model before forcing automatic sampling of the random variables. The SPMAXS parameter defaults to a value of 40,000 scenarios. 118. PRECIS LINGO defaults to displaying seven significant digits for values in standard solution reports. In some cases, you may desire more or less than seven digits of precision, in which case, you will need to adjust the PRECIS parameter. For example, suppose we have a very simple model consisting of the single expression: X = 1/3;

The default solution report for this model using seven digits of precision would contain: Variable X Row 1


Increasing the Precision parameter to 16 and resolving yields the following solution report with 16 significant digits: Variable X Row 1


The default value for the PRECIS parameter is seven significant digits.


371

119. LOOPOP The LOOPOP, or Loop optimization, parameter is used to either enable or disable LINGO's loop optimization step. Loop optimization reformulates expressions containing set looping functions in order to make them more efficient, while maintaining mathematical equivalency. The end goal of loop optimization is to minimize the number of es through the inner loop of any nested loops in an expression. As an example, consider the following transportation model fragment, that just contains constraints for satisfying customer demand: MODEL: ! A transportation problem fragment; SETS: WAREHOUSE /1..50/ : CAPACITY; CUSTOMER /1..5000/ : DEMAND; ROUTES( WAREHOUSE, CUSTOMER) : COST, VOLUME; ENDSETS ! The demand constraints; @FOR( CUSTOMER( J): @SUM( ROUTES( I, J): VOLUME( I, J)) >= DEMAND( J) ); END

Transportation Model Fragment with Inefficient Constraints In the demand constraints expression there are two loops  an outer @FOR() over the CUSTOMER set and an inner @SUM() over the ROUTES set. As written, the inner loop must be executed 50*50*5000=1.25 million times. Note that a valid reformulation of these demand constraints would be: ! The demand constraints; @FOR( CUSTOMER( J): @SUM( WAREHOUSE( I): VOLUME( I, J)) >= DEMAND( J) );

With the expression rewritten in this manner, the inner loop will now only be executed 50*5000 times, for a 98% reduction in total es. LINGO's loop optimizer seeks out such inefficient loops and, invisible to the , rewrites them before the model gets ed to the generator. Of course, the end result of such reformulations are faster generation times. In fact, the speedup from reformulation can be quite dramatic in many cases. In this particular example, the reformulated version ran over 400 times faster in the model generator. LINGO defaults to disabling the LOOPOP feature.

372 CHAPTER 6 120. HEUCUT The HEUCUT, or heuristics cutoff criterion, option controls the criterion for terminating integer programming heuristics. The possible settings for HEUCUT are: HEUCUT Value

Cutoff Criterion

0

LINGO decides

1

Time

2

Iterations

Under the Time setting, LINGO terminates heuristics after a certain amount of elapsed time. The Iterations option terminates heuristics after a certain number of iterations. In general, the Time setting results in the fastest performance. However, due to shifting computational loads on a machine, solution paths may change under the Time setting from one solve to the next, potentially resulting in non-reproducible solutions. If reproducibility of a runs is a concern, then the Iterations option should be selected. Under the Solver Decides setting, LINGO chooses the most appropriate strategy. The default setting for HEUCUT is 1, Solver Decides. 121. NUMBIN - Number of Histogram Bins The NUMBIN parameter determines the number of bins used in histogram charts. Set this parameter to 0 to have LINGO automatically determine a sensible bin count. The default setting for NUMBIN is 0. 122. SPBIGM - SP Big M Coefficient The SPBIGM parameter is used by the Stochastic Programming (SP) solver in constructing forcing constraints that may be required in the deterministic models generated by the solver. Forcing constraints are generally added to force binary variables to 1 when some activity occurs. For example, suppose x is a continuous variable and z is a binary variable. A forcing constraint that would drive z to 1 whenever x is positive would be: xM*z where M is some large number. M in this case is the SPBIGM parameter. There are two things to note about M. First, if M is too small, a forcing constraint can become infeasible, making the entire model infeasible. In general, M should be at least as large as the largest possible value for x so as not to introduce an infeasibility. However, from an algorithmic point of view, M should not be too large. If M is unrealistically large, it will be tougher for the integer solver to converge, plus it also introduces the potential round off error. The default setting for M should be sufficient for most models. However, if your SP model is infeasible for no known reason, you may want to try increasing M. On the other hand, if your SP is running slow, you may want to try reducing M. The SPBIGM parameter defaults to a value of 1.e8.


373

123. NSLPSV - SLP Solver If the NSLPSV parameter is enabled, LINGO uses a successive linear programming (SLP) algorithm for its nonlinear solver. This technique uses a linear approximation of the true nonlinear model within successive, small regions. This then allows the use of the fast linear solver for optimizing each linear subregion. In general, the SLP solver will not be well suited for highly nonlinear models. However, it may offer better performance than the standard nonlinear solver when a model has few nonlinear variables, such that the model is "mostly linear". LINGO defaults to not enabling the SLP Solver option. 124. FORCEB - Enforce Bounds in CALC and DATA If the Enforce Bounds In CALC and DATA option is enabled, it causes LINGO to check the values of any variables input in calc and data sections against their bounds. Normally, variable bounds apply only to optimizable variables that appear in the model's constraints and not to variables fixed to constant values in calc and data sections. If you would also like LINGO to check variables fixed in calc and data sections against their bounds, then you will need to enable this option. By default, this option is disabled. 125. NTHRDS - Number of Execution Threads The NTHRDS option controls the maximum number of threads, or processors, to utilize at any given time. Interesting settings for the thread count are:   

0 — LINGO determines the number of processors on the system and sets the thread limit to this number. 1 — Only one processor will be used, causing LINGO to run in single-threaded mode. N — Here, N is some positive integer greater than 1 and less-than-or-equal-to 32, in which case, LINGO will set the maximum number of threads to N.

LINGO defaults to running in single-threaded mode with a thread limit of 1. By default, this option is disabled. Note:

Setting the maximum thread count parameter, NTHRDS, to N does not guarantee than N cores will be fully utilized throughout the entire solve. There may be serial sections inside parallel code that get executed from time to time, causing only one core to be utilized. Concurrent algorithms may also have fewer than N different serial algorithms to launch, meaning fewer than N cores will be occupied.

374 CHAPTER 6 126. MTMODE - Multithread Mode The MTMODE parameter controls the multithreading mode that LINGO operates in. LINGO offers multicore extensions to its model generator and solvers. The multicore extensions are of two types: concurrent and parallel. Concurrent algorithms run two or more different serial algorithms on multiple copies of the same model, using a separate thread for each algorithm, terminating as soon as the winner thread finishes. These “different algorithms” may in fact be the same algorithm type, but using different strategies and/or parameters. Parallel algorithms, on the other hand, parallelize computationally intensive portions of the serial algorithm to distribute the workload across multiple threads. The following multicore extensions are currently available in LINGO: LINGO Component

Model Class

Barrier Solver BNP Solver Global Solver Integer Solver Linear Solver Model Generator Multistart Solver Stochastic Solver

Linear Programs Mixed Integer Programs Nonlinear Programs Mixed Integer Programs Linear Programs All Nonlinear Programs Stochastic Programs

Parallel Option Yes Yes Yes Yes No Yes Yes Yes

Concurrent Option No No No Yes Yes No No No

The available settings for the Mode, or multithreading mode, parameter are as follows:      

Note:

-1 — Solver Decides — The best available multithreading strategy, either parallel or concurrent, will be selected for each step of the solution process. 0 — Off in Solver — Multithreading will be disabled in the solvers, but allowed in the model generator if the number of threads is greater than 1. 1 — Prefer Parallel — If a parallel option is available for a particular solution step, then it will be selected, otherwise, a concurrent option will be selected when available. 2 — Parallel Only - If a parallel option is available for a particular solution step, then it will be selected, otherwise, the step will be executed in single-thread mode. 3 — Prefer Concurrent - If a concurrent option is available for a particular solution step, then it will be selected, otherwise, a parallel option will be selected when available. 4 — Concurrent Only - If a concurrent option is available for a particular solution step, then it will be selected, otherwise, the step will be executed in single-thread mode. If the maximum thread count parameter, NTHRDS, is set to 1, then the multithreading mode setting will be ignored, and LINGO will execute in single-threaded mode.


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Setting the maximum thread count parameter, NTHRDS, to N does not guarantee than N cores will be fully utilized throughout the entire solve. There may be serial sections inside parallel code that get executed from time to time, causing only one core to be utilized. Concurrent algorithms may also have fewer than N different serial algorithms to launch, meaning fewer than N cores will be occupied.

127. BNPBLK - BNP Block Determination The BNPBLK parameter controls how the the branch-and-price (BNP) solver determines the block structure of the model. The BNP solver is a mixed integer programming solver for solving models with block structures like the following: Minimize:  c(k) * x(k) Subject To:

 A(k) * x(k)

= d x(k) in X(k), for all k

(linking constraints) (decomposition structure)

where d, c(k) and x(k) are vectors and A(k) is a matrix with appropriate dimensions. x(k) contains decision variables and X(k) denotes a linear feasible domain for x(k). The BNP solver is a hybrid of branch-and-bound, column generation, and Lagrangean relaxation methods. It can help to find either the optimal solution or a better lower bound (the Lagrangean bound) for a minimization problem. Based on the decomposition structure, the solver divides the original problem into several subproblems, or blocks, and solves them (almost) independently, exploiting parallel processing if multiple cores are available. BNP may perform better than the default MIP solver if: a) the number of linking constraints is small, b) the number of blocks is large and they are of approximately the same size, and c) the number of available processors (or cores) is large, e.g., 4 or more. Also, there may be some models for which BNP finds a good solution and good bound more quickly than the default MIP algorithm, although it may take longer to prove optimality. The Blocks option for the BNP solver controls the number of subproblems, or blocks, that the model will be partitioned into. Possible setting for the Blocks parameter are:    

-1 - Row Names - Row names are constructed in such a way as to specify each row's block (an example is given below). 0 - Off - This will disable the BNP solver, in which case, the standard MIP solver will be used to solve all mixed integer linear programs. 1 - Specified - The explicitly specifies each row's block using the @BLOCKROW function. N - A positive integer, greater-than-or-equal-to 2, indicating the number of independent blocks to try and partition the model into via one of the graph partitioning algorithms provided by LINGO. The actual heuristic used is chosen with the Heuristic parameter.

The default setting for Blocks is 0, or Off, i.e., the BNP solver will not be used on integer programming models.

376 CHAPTER 6 Note:

The BNP solver can run the independent subproblems on separate threads to improve performance. So, if your machine has multiple cores, be sure to set the thread limit to allow for multithreading. Refer to the NTHRDS parameter above.

For more information on block determination, the BNP solver and the BNPBLK parameter, refer to the section BNP Solver in Chapter 5. 128. BNPHEU - BNP Block Finding Heuristic The Block Heuristic parameter controls the heuristic used to partition the model into blocks. You may currently select from two graph partitioning algorithms named simply GP1 and GP2; simple set BNPHEU to 1 for GP1 and 2 for GP2. For more information on block finding heuristics, the BNP solver and the BNPHEU parameter, refer to the section BNP Solver in Chapter 5. The default setting for BNPHEU is 1, i.e., the GP1 graph partitioning algorithm. 129. REPROD - Favor Reproducibility The REPROD parameter allows you to indicate if your preference is for reproducible solutions across multiple runs, or if your preference is for faster average runtimes. A number of solver steps have been found to perform better when using elapsed time as a measure of work performed. The problem with this approach is that the actual amount of work that gets performed over a fixed time interval will vary slightly across runs depending on the machine's load factor. Many models have alternate optimal solutions, with equal objectives but different variable values. LINGO is indifferent as to which solution is selected; it just wants to find a feasible solution with the best objective value, regardless of the variables values. The end result of this variability in work performed is that a different solution path may be selected from one run to the next, in which case, you may end up with a different alternate optimum from a previous run. On the other hand, enabling the Favor reproducibility over performance option causes LINGO to use fixed measures of work (such as iteration counts), allowing solutions to be reproducible across runs when using the same machine. By default, LINGO disables the Favor reproducibility over performance option. However, if reproducibility of results is important for your application, then you should enable this option.


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9. Miscellaneous The Miscellaneous category contains various LINGO commands that don’t fall into one of the other eight command categories.

! Place an exclamation mark in a command and LINGO ignores the remainder of the line following the exclamation mark.

QUIT Issue the QUIT command to close the LINGO application. Be sure to save any changes made to your model before quitting.

TIME Displays the current elapsed time since the start of the current LINGO session as illustrated in the following example: : TIME Cumulative HR:MIN:SEC = :

2:22:39.54

7 LINGO’s Operators and Functions LINGO provides the mathematical modeler with a number of functions and operators. For our purposes, we have broken them down into the following categories:          

Standard Operators - Arithmetic, logical, and relational operators such as +, -, =, and <=. Mathematical - Trigonometric and general mathematical functions. Financial - Common financial functions used to determine present values. Probability - Functions used to determine a wide range of probability and statistical answers. Poisson and Erlang queuing functions are among those provided. Variable Domain - Functions used to define the range of values (domain) a variable can take on (e.g., lower and upper bounds or integer restrictions). Set Handling - Functions useful for manipulating sets. Set Looping - Looping functions used to perform an operation over a set (e.g., to compute the sum, maximum, or minimum of a set of numbers). Interface - Functions used to create links to external data sources. Distributions - Probability distributions, including cumulative, inverse and pdf functions. Miscellaneous - Miscellaneous functions are listed under this heading.

Select a link above for an in-depth description of the operators and functions available in LINGO.

Standard Operators LINGO has three types of standard operators: 1. 2. 3.

Arithmetic, Logical, and Relational.

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380 CHAPTER 7

Arithmetic Operators Arithmetic operators work with numeric operands. LINGO has five binary (two-operand) arithmetic operators, shown here: Operator ^ * / + -

Interpretation Exponentiation Multiplication Division Addition Subtraction

Since these are binary operators, they require two arguments—one immediately to the left of the operator and one immediately to the right. The only unary (one-operand) arithmetic operator in LINGO is negation (-). In this case, the operator applies to the operand immediately to the right of the negation sign. These operators should be familiar to all readers. The priority of the operators is given in the following: Priority Level Highest

Lowest

Operator(s) - (negation) ^ */ +-

Operators with the highest priority are evaluated first, in order from left to right. As an example, consider the expression: 4+6/2 The division operator (/) has higher priority than the addition operator (+). Thus, it is evaluated first, leaving: 4 + 3. Evaluating the remaining addition gives a final result of 7. The order of evaluation of the operators can be controlled with parentheses. LINGO evaluates the equation in the innermost parentheses first and works out from there. If we recast the expression from above as: (4 + 6) / 2 we will now get a final result of 5, instead of 7. The 4 and 6 are added first because they appear in parentheses. The resulting sum of 10 is divided by 2, giving the final result of 5. Note:

LINGO follows the Excel convention of asg the highest priority to the negation operator. Given this, LINGO evaluates -3^2 as positive 9. Some s may prefer to give the unary minus operator a lower priority so that -3^2 evaluates to minus 9. You can do this by setting the Unary Minus Priority option to Low via the Model Generator tab of the LINGO|Options command. Once you set the unary minus operator’s priority is set to low its priority will be lower than multiplication and division, but higher than addition and subtraction.

OPERATORS AND FUNCTIONS

381

Logical Operators Logical operators were used in Chapter 2, Using Sets, when we introduced set looping functions. In LINGO, logical operators are primarily used in conditional expressions on set looping functions to control which of a set are to be included or excluded in the function. They also play a role in building set hip conditions. Logical operators return either TRUE or FALSE as a result. LINGO uses the value 1 to represent TRUE, and the value 0 to represent FALSE. LINGO considers an argument to be FALSE if, and only if, it is equal to 0. Thus, for example, arguments of 1, 7, -1, and .1234 would all be considered TRUE. LINGO has nine logical operators, which are all binary with the single exception of the #NOT# operator, which is unary. LINGO’s logical operators and their return values are listed below: Logical Operator #NOT# #EQ# #NE# #GT# #GE# #LT# #LE# #AND# #OR#

Return Value TRUE if the operand immediately to the right is FALSE, else FALSE. TRUE if both operands are equal, else FALSE. TRUE if both operands are not equal, else FALSE. TRUE if the left operand is strictly greater than the right operand, else FALSE. TRUE if the left operand is greater-than-or-equal-to the right operand, else FALSE. TRUE if the left operand is strictly less than the right operand, else FALSE. TRUE if the left operand is less-than-or-equal-to the right operand, else FALSE. TRUE only if both arguments are TRUE, else FALSE. FALSE only if both its arguments are FALSE, else TRUE.

The priority ranking of the logical operators is: Priority Level Highest Lowest

Operator(s) #NOT# #EQ# #NE# #GT# #GE# #LT# #LE# #AND# #OR#

Relational Operators In LINGO, relational operators are used in a model to specify whether the left-hand side of an expression should be equal to, less-than-or-equal-to, or greater-than-or-equal-to the right-hand side. Relational operators are used to form the constraints of a model. Relational operators are distinct from the logical operators #EQ#, #LE#, and #GE#, in that they tell LINGO the optimal solution of the model must satisfy the direction of the relational operator. Logical operators, on the other hand, merely report whether or not a condition is satisfied. Relational operators have the lowest priority of all the operators.

382 CHAPTER 7 The three relational operators are described below: Relational Operator = <= >=

Interpretation The expression to the left must equal the one on the right. The expression to the left must be less-than-or-equal-to the expression on the right The expression to the left must be greater-than-or-equal-to the expression on the right

LINGO will also accept “<” for less-than-or-equal-to, and “>” for greater-than-or-equal-to. Note:

LINGO does not directly strictly less than and strictly greater than relational operators. In general, it would be unusual to find a good formulation that requires such a feature. However, if you want A to be strictly less than B: A < B, then convert this expression to an equivalent less-than-or-equal-to expression as follows: A + e  B, where e is a small constant term whose value is dependent upon how much A must be “less than” B in order for you to consider them to be “not equal”.

Operator Priority Table The following table combines all three types of operatorsarithmetic, logical, and relationalinto a single table showing their relative priority rankings. Priority Level Highest

Lowest Note:

Operator(s) #NOT# -(negation) ^ * / + #EQ# #NE# #GT# #GE# #LT# #LE# #AND# #OR# <= = >=

LINGO follows the Excel convention of asg the highest priority to the negation operator. Given this, LINGO evaluates -3^2 as positive 9. Some s may prefer to give the unary minus operator a lower priority so that -3^2 evaluates to minus 9. You can do this by setting the Unary Minus Priority option to Low via the Model Generator tab of the LINGO|Options command. Once you set the unary minus operator’s priority is set to low its priority will be lower than multiplication and division, but higher than addition and subtraction.

OPERATORS AND FUNCTIONS Note:

In the absence of parentheses, all operators of the same priority are processed from left to right. Thus, 4^3^2 evaluates to 4096. Be forewarned that for the exponentiation operator, "^", this differs from the convention that some mathematicians follow, namely, to have the exponentiation operator evaluated from right to left, i.e., 4^3^2 would evaluate to 262144. When in doubt, use parentheses to enforce your intentions, e.g., 4^(3^2) unambiguously evaluates to 262144.

Mathematical Functions LINGO offers a number of standard, mathematical functions. These functions return a single result based on one or more scalar arguments. These functions are listed below: @ABS(X) Returns the absolute value of X. @ACOS( X) Returns the inverse cosine, or arccosine, of X, where X is an angle in radians. @ACOSH( X) Returns the inverse hyperbolic cosine of X, where X is an angle in radians. @ASIN( X) Returns the inverse sine, or arcsine, of X, where X is an angle in radians. @ASINH( X) Returns the inverse hyperbolic sine of X, where X is an angle in radians. @ATAN( X) Returns the inverse tangent, or arctangent, of X, where X is an angle in radians. @ATAN2( Y, X) Returns the inverse tangent of Y/X. @ATANH( X) Returns the inverse hyperbolic tangent of X, where X is an angle in radians. @COS(X) Returns the cosine of X, where X is an angle in radians. @COSH( X) Returns the hyperbolic cosine of X, where X is an angle in radians. @EXP(X) Returns e (i.e., 2.718281 ...) raised to the power X. @FLOOR(X) Returns the integer part of X. To be specific, if X  0, @FLOOR returns the largest integer, I, such that I  X. If X is negative, @FLOOR returns the most negative integer, I, such that I  X.

383

384 CHAPTER 7 @INT( X) This returns the integer part of X. To be specific, if X ≥ 0, @INT returns the largest integer, I, such that I ≤ X. If X is negative, @INT returns the largest negative integer, I, such that X ≥ I. @INTEGRAL( PROCEDURE, X, XL, XU, Y) This function performs numeric integration using a variant of Simpson's Rule. Argument PROCEDURE is the name of a procedure which calculates the function to be integrated. The function's value should be placed in Y in the specified procedure. The integral will be taken with respect to X over the interval [XL,XU]. At present, @INTEGRAL may only be used in calc sections. Example 1: In this example, we illustrate how to compute the integral of the standard normal distribution over the interval [-5,1.645]. MODEL: ! Takes the integral of the standard normal distribution over the interval [-5,1.645]; PROCEDURE NORMAL: Y = @EXP( -X * X / 2) / (( 2 * @PI()) ^ 0.5); ENDPROCEDURE CALC: PROB = @INTEGRAL( NORMAL, X, -5, 1.645, Y); ENDCALC END

@LGM(X) Returns the natural (base e) logarithm of the gamma function of X (i.e., log of (X - 1)!). It is extended to noninteger values of X by linear interpolation. @LOG(X) Returns the natural logarithm of X. @LOG10(X) Returns the base-10 logarithm of X. @MOD( X,Y) Returns the value of X modulo Y, or, in other words, the remainder of an integer divide of X by Y. @PI() Returns the value of PI, i.e., 3.14159265.... @POW( X,Y) Returns the value of X raised to the Y power. @ROUND( X, N) The @ROUND function rounds X to the closest number to X having N digits. For example, if X is equal to 2.576 and N is 2, then @ROUND( X, N) will return 2.58. If X is equal to -2.576 and N is 2, then @ROUND( X, N) will return --2.58.

OPERATORS AND FUNCTIONS @ROUNDDOWN( X, N) The @ROUNDDOWN function rounds X down (towards 0) to the closest number to X having N digits. For example, if X is equal to 2.576 and N is 2, then @ROUNDDOWN( X, N) will return 2.57. If X is equal to -2.576 and N is 2, then @ROUNDUP( X, N) will return -2.57. @ROUNDUP( X, N) The @ROUNDUP function rounds X up (away from 0) to the closest number to X having N digits. For example, if X is equal to 2.576 and N is 2, then @ROUNDUP( X, N) will return 2.58. If X is equal to -2.576 and N is 2, then @ROUNDUP( X, N) will return -2.58. @SIGN(X) Returns -1 if X < 0, 0 if X = 0 and +1 if X > 0. @SIN(X) Returns the sine of X, where X is the angle in radians. @SINH( X) Returns the hyperbolic sine of X, where X is an angle in radians. @SMAX (X1, X2, ..., XN) Returns the maximum value of X1, X2, ..., and XN. @SMIN(X1, X2, ..., XN) Returns the minimum value of X1, X2, ..., and XN. @SQR( X) Returns the value of X squared. @SQRT( X) Returns the square root of X. @TAN(X) Returns the tangent of X, where X is the angle in radians. @TANH( X) Returns the hyperbolic tangent of X, where X is an angle in radians.

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386 CHAPTER 7

Financial Functions LINGO currently offers two financial functions. One computes the present value of an annuity. The other returns the present value of a lump sum. @FPA(I, N) This returns the present value of an annuity. That is, a stream of $1 payments per period at an interest rate of I for N periods starting one period from now. I is not a percentage, but a fraction representing the interest rate (e.g., you would use .1 to represent 10%). To get the present value of an annuity stream of $X payments, multiply the result by X. @FPL(I, N) This returns the present value of a lump sum of $1 N periods from now if the interest rate is I per period. I is not a percentage, but a fraction representing the interest rate (e.g., you would use .1 to represent 10%). To get the present value of a lump sum of $X, multiply the result by X.

Probability Functions LINGO has a number of probability related functions. There are examples that make use of most of these functions in Chapter 12, Developing More Advanced Models, and in Appendix A, Additional Examples of LINGO Modeling. @NORMINV( P, MU, SIGMA) This is the inverse of the normal cumulative distribution. Given a probability, P, the mean of the of the normal distribution, MU, and its standard deviation, SIGMA, this function returns the value Z such that the probability of a normally distributed random variable being less-than-or-equal to Z is P. This function is being replaced by the @PNORMINV function, documented below in the Distributions section. @NORMSINV( P) This is the inverse of the standard normal cumulative distribution. Given a probability, P, this function returns the value Z such that the probability of a normally distributed random variable with a mean of 0 and a standard deviation of 1 being less-than-or-equal to Z is P. @PBN(P, N, X) This is the cumulative binomial probability. It returns the probability that a sample of N items, from a universe with a fraction of P of those items defective, has X or less defective items. It is extended to noninteger values of X and N by linear interpolation. This function is being replaced by the @PBINOCDF function, documented below in the Distributions section. @PCX(N, X) This is the cumulative distribution function for the Chi-squared distribution with N degrees of freedom. It returns the probability that an observation from this distribution is less-than-or-equal-to X. This function is being replaced by the @PCHISCDF function, documented below in the Distributions section.


387

@PEB(A, X) This is Erlang’s busy probability for a service system with X servers and an arriving load of A, with infinite queue allowed. The result of @PEB can be interpreted as either the fraction of time all servers are busy or the fraction of customers that must wait in the queue. It is extended to noninteger values of X by linear interpolation. The arriving load, A, is the expected number of customers arriving per unit of time multiplied by the expected time to process one customer. @PEL(A, X) This is Erlang’s loss probability for a service system with X servers and an arriving load of A, no queue allowed. The result of @PEL can be interpreted as either the fraction of time all servers are busy or the fraction of customers lost due to all servers being busy when they arrive. It is extended to noninteger values of X by linear interpolation. The arriving load, A, is the expected number of customers arriving per unit of time multiplied by the expected time to process one customer. @PFD(N, D, X) This is the cumulative distribution function for the F distribution with N degrees of freedom in the numerator and D degrees of freedom in the denominator. It returns the probability that an observation from this distribution is less-than-or-equal-to X. This function is being replaced by the @PFDSTCDF function, documented below in the Distributions section. @PFS(A, X, C) This returns the expected number of customers waiting for or under repair in a finite source Poisson service system with X servers in parallel, C customers, and a limiting load A. It is extended to noninteger values of X and C by linear interpolation. A, the limiting load, is the number of customers multiplied by the mean service time divided by the mean repair time. @PHG(POP, G, N, X) This is the cumulative hypergeometric probability. It returns the probability that X or fewer items in the sample are good, given a sample without replacement of N items from a population of size POP where G items in the population are good. It is extended to noninteger values of POP, G, N, and X by linear interpolation. This function is being replaced by the @PHYPGCDF function, documented below in the Distributions section. @PPL(A, X) This is the linear loss function for the Poisson distribution. It returns the expected value of MAX(0, Z-X), where Z is a Poisson random variable with mean value A. @PPS(A, X) This is the cumulative Poisson probability distribution. It returns the probability that a Poisson random variable, with mean value A, is less-than-or-equal-to X. It is extended to noninteger values of X by linear interpolation. This function is being replaced by the @PPOISCDF function, documented below in the Distributions section. @PSL(X) This is the unit normal linear loss function. It returns the expected value of MAX(0, Z-X), where Z is a standard normal random variable. In inventory modeling, @PSL(X) is the expected amount that demand exceeds a level X, if demand has a standard normal distribution.

388 CHAPTER 7 @PSN(X) This is the cumulative standard normal probability distribution. A standard normal random variable has mean 0.0 and standard deviation 1.0 (the bell curve, centered on the origin). The value returned by @PSN is the area under the curve to the left of the point on the ordinate indicated by X. This function is being replaced by the @PNORMCDF function, documented below in the Distributions section. @PTD(N, X) This is the cumulative distribution function for the t distribution with N degrees of freedom. It returns the probability that an observation from this distribution is less-than-or-equal-to X. This function is being replaced by the @PSTUTCDF function, documented below in the Distributions section. @QRAND(SEED) The @QRAND function produces a sequence of “quasi-random” uniform numbers in the interval (0, 1). @QRAND is only permitted in a data section. It will fill an entire attribute with quasi-random numbers. Generally, you will be filling two-dimensional tables with, say, m rows and n variables. m represents the number of scenarios, or experiments, you want to run. n represents the number of random variables you need for each scenario or experiment. Within a row, the numbers are independently distributed. Among rows, the numbers are “super uniformly” distributed. That is, the numbers are more uniformly distributed than you would expect by chance. These numbers are generated by a form of “stratified sampling”. For example, suppose m = 4 and n = 2. Even though the numbers are random, you will find that there will be exactly one row in which both numbers are in the interval (0, .5), exactly one row in which both numbers are in (.5, 1), and two rows in which one number is less than .5 and the other is greater than .5. Using @QRAND allows you to get much more accurate results for a given number of random numbers in a Monte Carlo model. If you want 8 ordinary random numbers, then use @QRAND(1,8) rather than @QRAND(4,2). An example of @QRAND follows: MODEL: DATA: M = 4; N = 2; SEED = 1234567; ENDDATA SETS: ROWS /1..M/; COLS /1..N/; TABLE( ROWS, COLS): X; ENDSETS DATA: X = @QRAND( SEED); ENDDATA END

Example of @QRAND function If you don’t specify a seed value for @QRAND, then LINGO will use the system clock to construct a seed value.


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@RAND(SEED) This returns a pseudo-random number between 0 and 1, depending deterministically on SEED.

Variable Domain Functions The default assumption for a variable is that it is continuous with a lower bound of 0. Variable domain functions place additional restrictions on the values that variables can assume. The functions and their effects are as follows: @BIN( variable) Restricts variable to being a binary (0/1) integer value. @BND( lower_bound, variable, upper_bound) Limits variable to being greater-than-or-equal-to lower_bound and less-than-or-equal-to upper_bound. @CARD( 'card_set_name', variable|N) Use @CARD to restrict a a set of variables to have a cardinality of N. See section Cardinality for more information. @FREE( variable) Removes the default lower bound of zero on variable, allowing it to take any positive or negative value. @GIN( variable) Restricts variable to integer values (e.g., 0,1,2, ...). @PRIORITY( variable, priority) Sets the branching priority for integer variable variable to the value priority. Variables with higher priorities will tend to be branched on sooner than variables with lower priorities. @SEMIC( lower_bound, variable, upper_bound,) Restricts variable to being either 0 o in the range of [lower_bound, upperbound]. Refer to section Semicontinuous Variables for more information. @SOS{1|2|3}( 'sos_set_name', variable) Use the @SOS1, @SOS2 and @SOS3 functions to set up special ordered sets of variables. Refer to section SOS Variables for more information. You may use the @FOR function to apply variable domain functions to all the elements of an attribute. Variable domain functions are discussed in detail in Using Variable Domain Functions.

390 CHAPTER 7

Set Handling Functions LINGO offers several functions that assist with handling sets. The @IN function determines if a set element is contained in a set. The @INDEX function returns the index of a set element within its set, while @INSERT allows you to dynamically add set to derived set. The @SIZE function returns the number of elements in a set. Finally, the @WRAP function is useful for "wrapping" set indices from one end of a time horizon to another in multiperiod planning models. These are described in more detail below. @IN( set_name, primitive_1_index [, primitive_2_index ...]) This returns TRUE if the set member referenced by the primitive set member index tuple (primitive_1_index, primitive_2_index, ...) is contained in the set_name set. As the following example shows, the @IN operator is useful for generating complements of subsets in set hip conditions. Example 1: For example, to derive a set of open plants based on a subset of closed plants, your sets section might resemble the following: SETS: PLANTS / SEATTLE, DENVER, CHICAGO, ATLANTA/:; CLOSED( PLANTS) /DENVER/:; OPEN( PLANTS) | #NOT# @IN( CLOSED, &1):; ENDSETS

The OPEN set is derived from the PLANTS set. We use a hip condition containing the @IN function to allow only those plants not contained in the CLOSED set to be in the OPEN set. Example 2: In this example, we illustrate how to determine if the set element (B, Y) belongs to the derived S3 set. In this case, (B, Y) is indeed a member of S3, so X will be set to 1. Note that, in order to get the index of the primitive set elements B and Y, we made use of the @INDEX function, which is discussed next. SETS: S1 / A B C/:; S2 / X Y Z/:; S3( S1, S2) / A,X A,Z B,Y C,X/:; ENDSETS X = @IN( S3, @INDEX( S1, B), @INDEX( S2, Y));


391

@INDEX( set_name, set_member) This returns the index of a set member set_member in the set set_name. If the specified set member does not belong to the set @INDEX will return 0. Unlike @IN, which requires you to specify the indices for the set elements, @INDEX allows you to refer to set member names directly. Example 1: In this example, we illustrate how to get the index of set member (R1, C3) in the derived S3 set. In this case, (R1, C3) is the third member SRXC, so NDX will be set to 3. SETS: ROWS /R1..R27/; COLS /C1..C3/; RXC( ROWS, COLS): XRNG; ENDSETS ! return the index of (r1,c3) in the rxc set; NDX = @INDEX( RXC, R1, C3);

LINGO allows you to omit the set name argument if the set member belongs to a primitive set. This is to maintain compatibility with earlier releases of LINGO. As the following example illustrates, it's good practice to always specify a set name in the @INDEX function: Example 2: A model's set elements can come from external sources that the modeler may have little control over. This can potentially lead to confusion when using the @INDEX function. Consider the sets section: SETS: GIRLS /DEBBIE, SUE, ALICE/; BOYS /BOB, JOE, SUE, FRED/; ENDSETS

Now, suppose you want to get the index of the boy named Sue within the BOYS set. The value of this index should be 3. Simply using @INDEX( SUE) would return 2 instead of 3, because LINGO finds SUE in the GIRLS set first. In this case, to get the desired result, you must specify the BOYS set as an argument and enter @INDEX( BOYS, SUE). Note:

The set member argument to @INDEX is considered to be a text literal. The following example illustrates how this can potentially lead to unexpected results.

392 CHAPTER 7 Example 3: In the model below, we loop over the GIRLS set using a set index variable named G. Given that @INDEX considers the set member argument (G in this case) to be a text literal, each element of XINDEX will be set to 0; this is because the GIRLS set does not contain a set member called G. On the other hand, the elements of XIN will all be set to 1, because @IN treats G as a set index variable, as opposed to a text literal. SETS: GIRLS /DEBBIE, SUE, ALICE/: XINDEX, XIN; BOYS /BOB, JOE, SUE, FRED/; ENDSETS !XINDEX will be 0 because GIRLS does not contain the member 'G'; !XIN will be 1 because G is treated as a set index variable as opposed to the text literal 'G'; @FOR( GIRLS( G): XINDEX( G) = @INDEX( GIRLS, G); XIN( G) = @IN( GIRLS, G); );

@INSERT( set_name, primitive_1_index [, primitive_2_index ...]) This function may be used to dynamically add to derived sets. Each of the primitive set forming the new derived set member must have been included in their respective primitive sets. The following example illustrates:

OPERATORS AND FUNCTIONS MODEL: SETS: PRIMITIVE; LATTICE( PRIMITIVE, PRIMITIVE); ENDSETS DATA: PRIMITIVE = 1..10; ENDDATA CALC: ! Starting at 1,1, generate the lattice of points reachable by the two integer vectors; X1 = 1; Y1 = 3; X2 = 7; Y2 = 5; @INSERT( LATTICE, 1,1); ! insert the seed; @SET( 'TERSEO', 3); !minimal output; ! Now generate all the points reachable, directly or indirectly from the seed via the two vectors within a finite region; @FOR( LATTICE( I, J) | I + I #LE# @SIZE( PRIMITIVE): I1 = I + X1; J1 = J + Y1; @IFC( #NOT# @IN( LATTICE, I1, J1): @IFC( @IN( PRIMITIVE, I1) #AND# @IN( PRIMITIVE, J1): @INSERT( LATTICE, I1, J1); ); ); I1 = I + X2; J1 = J + Y2; @IFC( #NOT# @IN( LATTICE, I1, I1): @IFC( @IN( PRIMITIVE, I1) #AND# @IN( PRIMITIVE, J1): @INSERT( LATTICE, I1, J1); ); ); ); ENDCALC DATA: !display the lattice set; @TEXT() = @TABLE( LATTICE); ENDDATA END

Model: LATTICE

393

394 CHAPTER 7 In this model, we want to find all the points on a 10-by-10 grid that are reachable, either directly or indirectly, from point (1,1) using combinations of the two vectors (1,3) and (7,5). The grid is represented by the 2-dimensional set LATTICE. Initially, LATTICE is empty, but we will add using @INSERT if we discover they are reachable from the seed (1,1). First, we place the seed into the LATTICE with the following insert statement: @INSERT( LATTICE, 1, 1); ! insert the seed;

The following doubly-nested loop iterates through the entire 10x10 grid: @FOR( LATTICE( I, J) | I + I #LE# @SIZE( PRIMITIVE): @FOR( VECTORS( I2, J2): I3 = I + I2; J3 = J + J2; @IFC( #NOT# @IN( LATTICE, I3, J3): @IFC( @IN( PRIMITIVE, I3) #AND# @IN( PRIMITIVE, J3): @INSERT( LATTICE, I3, J3); ); ); ); );

Each of the two vectors are then added individually to the current point, with the resulting point tested to see if a) if it has not already been added to LATTICE, and b) if it lies within the 10x10 grid. If the new point es both these tests, it gets added to the lattice using an @INSERT statement: @INSERT( LATTICE, I3, J3);

At the end of the model we use the @TABLE output function to display the lattice we found: 1 2 3 4 5 6 7 8 9 10

1 X

2

3

4

5

6

7

8

9

10

X X X

X X

Here we see that, in addition to the seed, the following points are of the lattice: (2,4), (3,7), (4,10), (8,6) and (9,9).


395

@SIZE( set_name) This returns the number of elements in the set_name set. Using the @SIZE function is preferred to explicitly listing the size of a set in a model. This serves to make your models more data independent and, therefore, easier to maintain should the size of your sets change. To view an example of the @SIZE function, refer to the PERT/M example in the Sparse Derived Set Example - Explicit List section of Using Sets. @WRAP( INDEX, LIMIT) This allows you to "wrap" an index around the end of a set and continue indexing at the other end of the set. That is, when the last (first) member of a set is reached in a set looping function, use of @WRAP will allow you to wrap the set index to the first (last) member of the set. This is a particularly useful function in cyclical, multiperiod planning models. Formally speaking, @WRAP returns J such that J = INDEX - K * LIMIT, where K is an integer such that J is in the interval [1,LIMIT]. Informally speaking, @WRAP will subtract or add limit to index until it falls in the range 1 to LIMIT. For an example on the use of the @WRAP function in a staff scheduling model, refer to the Primitive Set Example section in Using Sets.

Set Looping Functions Set looping functions operate over an entire set and, with the exception of the @FOR function, produce a single result. The syntax for a set looping function is: @function(setname [ (set_index_list) [ | conditional_qualifier]] : expression_list); @function corresponds to one of the set looping functions listed below. setname is the name of the set you want to loop over. The set_index_list is optional, and is used to create a list of indices, which correspond to the parent primitive sets that form the set setname. As LINGO loops through the of the set setname, it will set the values of the indices in the set_index_list to correspond to the current member of the set setname. The conditional_qualifier is optional and may be used to limit the scope of the set looping function. When LINGO is looping over each member of the set setname, it evaluates the conditional_qualifier. If the conditional_qualifier evaluates to true, then @function is performed for the set member. Otherwise, it is skipped. The expression_list is a list of expressions to be applied to each member of the set setname. When using the @FOR function, the expression_list may contain multiple expressions, separated by semicolons. These expressions will be added as constraints to the model. When using the remaining three set looping functions (@SUM, @MAX, and @MIN), the expression_list must contain one expression only. If the set_index_list is omitted, all attributes referenced in the expression_list must be defined on the set setname.

396 CHAPTER 7 The available set looping functions are listed below: @FOR (setname [ (set_index_list) [ | cond_qualifier]]: exp_list) This generates the expressions contained in exp_list for all of the set setname. @MAX(setname [ (set_index_list) [ | cond_qualifier]]: expression) This returns the maximum value of expression taken over the set setname. @MIN(setname [ (set_index_list) [ | cond_qualifier]]: expression) This returns the minimum value of expression taken over the set setname. @PROD( setname [ ( set_index_list) [ | cond_qualifier]]: expression) This returns the product of an expression over the setname set. @SUM(setname [ (set_index_list) [ | cond_qualifier]]: expression) This returns the sum of expression over the set setname. Set looping functions are discussed in more detail in Chapter 2, Using Sets.


397

Interface Functions Interface functions allow you to link your model to external data sources such as text files, databases, spreadsheets and external applications. With the exception of @FILE, interface functions are valid only in sets and data sections, and may not be used in calc and model sections.The interface functions currently available in LINGO are listed below. @FILE ( 'filename') The @FILE function allows you to include data from external text files anywhere in your model, where filename is the name of the file to include text form. This is particularly useful for incorporating data stored in text files in your sets and data sections. When this function is encountered in a model, LINGO will continue to take text from this file until it encounters either the end-of-file or a LINGO end-of-record mark (~). For subsequent @FILE references in the same model that use the same file name, LINGO resumes taking input from the file at the point where it left off. Nesting of @FILE function calls (embedding an @FILE in a file which is itself called by @FILE) is not allowed. For more information on use of the @FILE function, refer to Interfacing with External Files. @ODBC ( ['data_source'[, 'table_name'[, 'col_1'[, 'col_2' ...]]]]) The @ODBC function is used to open ODBC links between LINGO and databases. You can use @ODBC in the sets section to retrieve set from a database, or in the data section to import data and/or export solutions. The data_source is the name of the ODBC data source you ed with the ODBC . The table_name is the name of the table in the data_source you want to open a link to. Finally, col_i is the column in the table table_name that you wish to link to. The @ODBC function is discussed in detail in Interfacing with Databases. @OLE ( 'workbook_file'[, range_name_list]) The @OLE function is used to move data and solutions back and forth from Excel using OLE based transfers. You can use @OLE in the sets section to retrieve set from Excel, or in the data section to import data and/or export solutions. OLE transfers are direct memory transfers and do not make use of intermediate files. When using @OLE for exports, LINGO loads Excel, tells Excel to load the desired spreadsheet, and sends ranges of data containing solution values to the sheet. You must have Excel 5, or later, to use the @OLE function. The @OLE function is valid only in data and sets sections. @OLE can export two-dimensional ranges (rectangular ranges that lie on a single worksheet in Excel), but cannot export three-dimensional ranges (ranges which traverse more than one worksheet in Excel) or discontinuous ranges. The workbook_file argument is the name of the workbook to link to. The range_name_list is the list of named ranges in the sheet to link to. For more information on use of the @OLE function, refer to Interfacing with Spreadsheets.

398 CHAPTER 7 @POINTER ( N) This function is strictly for use with the LINGO Dynamic Link Library (DLL) under Windows. @POINTER allows you to transfer data directly through shared memory locations. For more information on the use of the @POINTER function, refer to Interfacing with Other Applications. @TEXT ( [['filename'], ‘a’]) The @TEXT function is used in the data section of a model to export solutions to text files, where filename is the name of the file you want to export the solution to. If filename is omitted, the solution data will be sent to the standard output device (in most cases this corresponds to the screen). If you specify a file name and you wish to append output to the file instead of overwriting the file, include a second argument of ‘a’. For additional documentation on the @TEXT function, see Interfacing with External Files.

Distributions Distribution functions are available for an extensive number of probability distributions. LINGO s the probability density functions (PDF) for each distribution, as well as their cumulative (CDF) and inverse functions. ed distributions are listed below: Continuous Distribution Functions

Description

Parameters and Domain

@PBETACDF( A, B, X)

Cumulative Beta

A = alpha > 0 B = beta > 0

@PBETAINV( A, B, X)

Inverse Beta

X ∈ (0,1) A = alpha > 0 B = beta > 0

Beta PDF

X ∈ [0,1] A = alpha > 0 B = beta > 0

@PBETAPDF( A, B, X)

@PCACYCDF( L, C, X)

Cumulative Cauchy

@PCACYINV( L, S, X)

Inverse Cauchy

@PCACYPDF( L, S, X)

Cauchy PDF

@PCHISCDF( DF, X)

Cumulative Chi-Square

X ∈ (0,1) L = location S = scale > 0 X a real L = location S = scale > 0 X ∈ [0,1] L = location S = scale > 0 X a real DF = degrees of freedom

OPERATORS AND FUNCTIONS = a positive integer @PCHISINV( DF, X)

@PCHISPDF( DF, X)

Inverse Chi-Square

X≥ 0 DF = degrees of freedom = a positive integer

Chi-Square PDF

X ∈ [0,1] DF = degrees of freedom = a positive integer

@PEXPOCDF( L, X)

Cumulative Exponential

X≥ 0 L = lambda > 0

@PEXPOINV( L, X)

Inverse Exponential

X≥ 0 L = lambda > 0

@PEXPOPDF( L, X)

Exponential PDF

X ∈ [0,1] L = lambda > 0

@PFDSTCDF( DF1, DF2, X)

Cumulative F-Distribution

X≥ 0 DF1,DF2 = degrees of freedom = a positive integer

@PFDSTINV( DF1, DF2, X)

Inverse F-Distribution

X≥ 0 DF1,DF2 = degrees of freedom = a positive integer

@PFDSTPDF( DF1, DF2, X)

F-Distribution PDF

X ∈ [0,1] DF1,DF2 = degrees of freedom = a positive integer

Cumulative Gamma

X≥ 0 SC = scale > 0 SH = shape > 0

Inverse Gamma


@PGAMMPDF( SC, SH, X)

Gamma PDF

X ∈ [0,1] SC = scale > 0 SH = shape > 0

@PGMBLCDF( L, S, X)

Cumulative Gumbel

@PGMBLINV( L, S, X)

Inverse Gumbel

@PGAMMCDF( SC, SH, X)

@PGAMMINV( SC, SH, X)

X≥ 0 L = location S = scale > 0 X a real L = location S = scale > 0 X ∈ [0,1]

399

400 CHAPTER 7 @PGMBLPDF( L, S, X)

Gumbel PDF

@PLAPLCDF( L, S, X)

Cumulative Laplace

@PLAPLINV( L, S, X)

Inverse Laplace

@PLAPLPDF( L, S, X)

Laplace PDF

@PLGSTCDF( L, S, X)

Cumulative Logistic

@PLGSTINV( L, S, X)

Inverse Logistic

@PLGSTPDF( L, S, X)

Logistic PDF

@PLOGNCDF( M, S, X)

Cumulative Lognormal

@PLOGNINV( M, S, X)

Inverse Lognormal

@PLOGNPDF( M, S, X)

Lognormal PDF

@PNORMCDF( M, S, X)

Cumulative Normal

@PNORMINV( M, S, X)

Inverse Normal

@PNORMPDF( M, S, X)

Normal PDF

@PPRTOCDF( SC, SH, X)

Cumulative Pareto

@PPRTOINV( SC, SH, X)

Inverse Pareto

L = location S = scale > 0 X a real L = location S = scale > 0 X a real L = location S = scale > 0 X ∈ [0,1] L = location S = scale > 0 X a real L = location S = scale > 0 X a real L = location S = scale > 0 X ∈ [0,1] L = location S = scale > 0 X a real M = mu S = sigma > 0 X>0 M = mu S = sigma > 0 X ∈ [0,1] M = mu S = sigma > 0 X>0 M = mu S = sigma > 0 X a real M = mu S = sigma > 0 X ∈ [0,1] M = mu S = sigma > 0 X a real SC = scale > 0 SH = shape > 0 X ≥ SC SC = scale > 0

OPERATORS AND FUNCTIONS SH = shape > 0 @PPRTOPDF( SC, SH, X)

Pareto PDF

X ∈ [0,1] SC = scale > 0 SH = shape > 0 X ≥ SC

@PSMSTCDF( A, X)

Cumulative Symmetric Stable

A = alpha  [0.2,2] X a real

@PSMSTINV( A, X)

Inverse Symmetric Stable

@PSMSTPDF( A, X)

Symmetric Stable PDF

@PSTUTCDF( DF, X)

Cumulative Student's t

@PSTUTINV( DF, X)

Inverse Student's t

A = alpha  [0.2,2] X Î [0,1] A = alpha  [0.2,2] X a real DF = degrees of freedom = a positive integer X a real DF = degrees of freedom = a positive integer

@PSTUTPDF( DF, X)

Student's t PDF

@PTRIACDF( L, U, M, X)

Cumulative Triangular

@PTRIAINV( L, U, M, X)

Inverse Triangular

X ∈ [L,U] L = lower limit U = upper limit M = mode

@PTRIAPDF( L, U, M, X)

Triangular PDF

X ∈ [0,1] L = lower limit U = upper limit M = mode

Cumulative Uniform

X ∈ [L,U] L = lower limit U = upper limit

Inverse Uniform

X ∈ [L,U] L = lower limit U = upper limit

Uniform PDF

X ∈ [0,1] L = lower limit U = upper limit

@PUNIFCDF( L, U, X)

@PUNIFINV( L, U, X)

@PUNIFPDF( L, U, X)

X ∈ [0,1] DF = degrees of freedom = a positive integer X a real L = lower limit U = upper limit M = mode

401

402 CHAPTER 7 @PWEIBCDF( SC, SH, X)

@PWEIBINV( SC, SH, X)

@PWEIBPDF( SC, SH, X)

Cumulative Weibull

X ∈ [L,U] SC = scale > 0 SH = shape > 0

Inverse Weibull


Weibull PDF

X ∈ [0;1] SC = scale > 0 SH = shape > 0 X≥ 0

Discrete Distribution Functions

Description

Parameters

@PBTBNCDF( N, A, B, X)

Cumulative Beta Binomial

N = trials ∈ {0,1,...} A = alpha ∈ (0,+inf) B = beta ∈ (0,+inf) X ∈ [0,

@PBTBNINV( N, A, B, X)

Beta Binomial Inverse

N = trials ∈ {0,1,...} A = alpha ∈ (0,+inf) B = beta ∈ (0,+inf) X ∈ [0,+inf)

@PBTBNPDF( N, A, B, X)

Beta Binomial PDF

N = trials ∈ {0,1,...} A = alpha ∈ (0,+inf) B = beta ∈ (0,+inf) X ∈ [0,+inf)

@PBINOCDF( N, P, X)

Cumulative Binomial

N = trials ∈ {0,1,...} P = probability of success ∈ [0,1] X ∈ {0,1,...,N}

@PBINOINV( N, P, X)

Inverse Binomial

N = trials ∈ {0,1,...} P = probability of success ∈ [0,1] X ∈ [0,1]

OPERATORS AND FUNCTIONS @PBINOPDF( N, P, X)

Binomial PDF

N = trials ∈ {0,1,...} P = probability of success ∈ [0,1] X ∈ {0,1,...,N}

@PGEOMCDF( P, X)

Cumulative Geometric

P = probability of success ∈ (0,1] X ∈ {0,1,...}

@PGEOMINV( P, X)

Inverse Geometric

@PGEOMPDF( P, X)

Geometric PDF

@PHYPGCDF( N, D, K, X)

Cumulative Hypergeometric

P = probability of success ∈ (0,1] X ∈ [0,1] P = probability of success∈ (0,1] X ∈ {0,1,...} N = population ∈ {0,1,...} D = number defective ∈ {0,1,...,N} K = sample size ∈ {0,1,...,N} X ∈ {max(0,D+KN),...,min(D,K)}

@PHYPGINV( N, D, K, X)

Inverse Hypergeometric

N = population ∈ {0,1,...} D = number defective ∈ {0,1,...,N} K = sample size ∈ {0,1,...,N} X ∈ [0,1]

@PHYPGPDF( N, D, K, X)

Hypergeometric PDF

N = population ∈ {0,1,...} D = number defective ∈ {0,1,...,N} K = sample size ∈ {0,1,...,N} X ∈ {max(0,D+KN),...,min(D,K)}

@PLOGRCDF( P, X)

Cumulative Logarithmic

P = p-factor ∈ (0,1) X ∈ {0,1,...,N}

@PLOGRINV( P, X)

Inverse Logarithmic

P = p-factor ∈ (0,1)

403

404 CHAPTER 7 X ∈ [0,1] @PLOGRPDF( P, X)

Logarithmic PDF

@PNEGBCDF( R, P, X)

Cumulative Negative Binomial

P = p-factor ∈ (0,1) X ∈ {0,1,...,N} R = number of failures ∈ (0,+inf) P = probability of success ∈ (0,1) X ∈ {0,1,...}

@PNEGBINV( R, P, X)

Inverse Negative Binomial

R = number of failures ∈ (0,+inf) P = probability of success ∈ (0,1) X ∈ [0,1]

@PNEGBPDF( R, P, X)

Negative Binomial PDF

R = number of failures ∈ (0,+inf) P = probability of success ∈ (0,1) X ∈ {0,1,...}

@PPOISCDF( L, X)

Cumulative Poisson

L = lambda ∈ (0,+inf) X ∈ {0,1,...,N}

@PPOISINV( L, X)

Inverse Poisson

L = lambda ∈ (0,+inf) X ∈ [0;1]

@PPOISPDF( L, X)

Poisson PDF

L = lambda ∈ (0,+inf) X ∈ {0,1,...,N}

Report Functions Report functions are used to construct reports based on a model’s results, and are valid on both calc and data sections. Combining report functions with interface functions in a data section allows you to export the reports to text files, spreadsheets, databases, or your own calling application. Note:

The interested reader will find an exhaustive example of the use of report functions in the sample model TRANSOL.LG4 in the Samples subfolder. This model makes extensive use of many of the report functions to mimic the standard LINGO solution report.


405

@DUAL ( variable_or_row_name) The @DUAL function outputs the dual value of a variable or a row. For example, consider a model with the following data section: DATA: @TEXT( 'C:\RESULTS\OUTPUT.TXT') = @WRITEFOR( SET1( I): X( I), @DUAL( X( I)), @NEWLINE(1)); ENDDATA

When this model is solved, the values of attribute X and their reduced costs will be written to the file C:\RESULTS\OUTPUT.TXT. Output may be routed to a file, spreadsheet, database or memory location. The exact destination will depend on the export function used on the left-hand side of the output statement. If the argument to the @DUAL function is a row name, then the dual price on the generated row will be output. @FORMAT ( value, format_descriptor) @FORMAT may be used in @WRITE and @WRITEFOR statements to format a numeric or string value for output as text, where value is the numeric or string value to be formatted, and format_descriptor is a string detailing how the number is to be formatted. The format descriptor is interpreted using C programming conventions. For instance, a format descriptor of ‘12.2f’ would cause a numeric value to be printed in a field of 12 characters with 2 digits after the decimal point. For a string values, such as a set member name, a format descriptor of '12s' would cause the string to be right justified in a field of 12 characters, while '-12s' would cause the string to be left justified in a field of 12. You can refer to a C reference manual for more details on the available formatting options. The following example uses the @FORMAT function to place a shipping quantity into a field of eight characters with no trailing decimal value: DATA: @TEXT() = @WRITE( ' From To Quantity', @NEWLINE(1)); @TEXT() = @WRITE( '--------------------------', @NEWLINE(1)); @TEXT() = @WRITEFOR( ROUTES( I, J) | X( I, J) #GT# 0: 3*' ', WAREHOUSE( I), 4*' ', CUSTOMER( J), 4*' ', @FORMAT( X( I, J), '8.0f'), @NEWLINE( 1)); ENDDATA

The report will resemble the following: From To Quantity -------------------------WH1 C1 2 WH1 C2 17 WH1 C3 1 WH2 C1 13 WH2 C4 12 WH3 C3 21

406 CHAPTER 7 This next example, GRAPHPSN.LG4, graphs the standard normal function, @PSN, over the interval [ –2.4, 2.4]. The @FORMAT function is used to print the X coordinates with one trailing decimal point. ! Graphs @PSN() over a specified interval around 0; DATA: ! height of graph; H = 49; ! width of graph; W = 56; ! interval around 0; R = 2.4; ENDDATA SETS: S1 /1..H/: X, FX; ENDSETS @FOR( S1( I): ! X can be negative; @FREE( X); ! Compute x coordinate; X( I) = -R + ( I - 1)* 2 * R / ( H - 1); ! Compute y coordinate = @psn( x); FX( I) = @PSN( X( I)); ); DATA: ! Print the header; @TEXT() = @WRITE( 'Graph of @PSN() on the interval [-', R,',+',R,']:',@NEWLINE(1)); @TEXT() = @WRITE( '| 0 ',(W/2-5)*'-', ' 0.5 ',(W/2-5)*'-', '1.0 X(i)',@NEWLINE(1)); ! Loop to print the graph over; @TEXT() = @WRITEFOR( S1( I): '| ', ( W * FX( I) + 1/2) * '*', @IF( X( I) #LT# 0, '', ' '), ( W ( W * FX( I) + 1/2) + 3)*' ', @FORMAT( X(I), '.1f'),@NEWLINE(1)); !Trailer; @TEXT() = @WRITE( '| 0 ',(W/2-5)*'-', ' 0.5 ',(W/2-5)*'-', '1.0',@NEWLINE(1)); ENDDATA

Model: GRAPHPSN

OPERATORS AND FUNCTIONS Here is how the graph will appear when the model is solved: Graph of @PSN() on the interval [-2.4,+2.4]: | 0 ----------------------- 0.5 -----------------------1.0 | | * | * | * | * | ** | ** | ** | *** | **** | ***** | ***** | ****** | ******** | ********* | ********** | ************ | ************** | *************** | ***************** | ******************* | ********************* | ************************ | ************************** | **************************** | ****************************** | ******************************** | *********************************** | ************************************* | *************************************** | ***************************************** | ****************************************** | ******************************************** | ********************************************** | *********************************************** | ************************************************ | ************************************************** | *************************************************** | *************************************************** | **************************************************** | ***************************************************** | ****************************************************** | ****************************************************** | ****************************************************** | ******************************************************* | ******************************************************* | ******************************************************* | ******************************************************* | ******************************************************** | 0 ----------------------- 0.5 -----------------------1.0

X(i) -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4

407

408 CHAPTER 7 @ITERS () The @ITERS function returns the total number of iterations required to solve the model. @ITERS is available only in the data and calc sections, and is not allowed in the constraints of a model. For example, the following output statement writes the iteration count to the standard output device: DATA: @TEXT() = @WRITE('Iterations= ', @ITERS()); ENDDATA

@NAME ( var_or_row_reference) Use @NAME to return the name of a variable or row as text. @NAME is available only in the data and calc sections, and is not allowed in the constraints of a mode. The following example prints a variable name and its value: DATA: @TEXT() = @WRITEFOR( ROUTES( I, J) | X( I, J) #GT# 0: @NAME( X), ' ', X( I, J), @NEWLINE(1)); ENDDATA

The report will resemble the following: X( X( X( X( X( X(

WH1, WH1, WH1, WH2, WH2, WH3,

C1) C2) C3) C1) C4) C3)

2 17 1 13 12 21

@NEWLINE ( n) Use @NEWLINE to write n new lines to the output device. @NEWLINE is available only in the data and calc sections, and is not allowed in the constraints of a model. See the example immediately below in the @RANGED section. @OBJBND () @OBJBND returns the bound on the objective value. @RANGED ( variable_or_row_name) @RANGED outputs the allowable decrease on a specified variable’s objective coefficient or on a specified row’s right-hand side. @RANGED is available only in the data and calc sections, and is not allowed in the constraints of a model. For example, consider a model with the following data section: DATA: @TEXT( 'C:\RESULTS\OUTPUT.TXT') = @WRITEFOR( SET( I): X( I), @RANGED( X( I), @NEWLINE(1)); ENDDATA

OPERATORS AND FUNCTIONS When this model is solved, the values of attribute X and the allowable decreases on its objective coefficients will be written to the file C:\RESULTS\OUTPUT.TXT. If @RANGED is ed a row name it will output the allowable decrease on the right-hand side value for the row. Output may be routed to a file, spreadsheet, database, or memory location. The exact destination will depend on the export function used on the left-hand side of the output statement. Range computations must be enabled in order for @RANGED to function properly. For more information on the interpretation of allowable decreases, refer to the LINGO|Range command. @RANGEU ( variable_or_row_name) @RANGEU outputs the allowable increase on a specified variable’s objective coefficient or on a specified row’s right-hand side. For example, consider a model with the following data section: DATA: @TEXT( 'C:\RESULTS\OUTPUT.TXT') = @WRITEFOR( SET( I): X, @RANGEU(X), @NEWLINE(1)); ENDDATA

When this model is solved, the values of X and the allowable increases on its objective coefficients will be written to the file C:\RESULTS\OUTPUT.TXT. If @RANGEU is ed a row name it will output the allowable increase on the right-hand side value for the row. Output may be routed to a file, spreadsheet, database, or memory location. The exact destination will depend on the export function used on the left-hand side of the output statement. Range computations must be enabled in order for @RANGED to function properly. For more information on the interpretation of allowable increases, refer to the LINGO|Range command.

409

410 CHAPTER 7 @STATUS () This returns the final status of the solution process using the following codes: @STATUS() Code

Interpretation

0

Global Optimum — The optimal solution has been found.

1

Infeasible — No solution exists that satisfies all constraints.

2

Unbounded — The objective can be improved without bound.

3

Undetermined — The solution process failed.

4

Feasible — A feasible solution was found that may, or may not, be the optimal solution.

5

Infeasible or Unbounded — The preprocessor determined the model is either infeasible or unbounded. If you need to narrow the result down to either infeasible or unbounded, then you will need to turn off presolving and run the model again.

6

Local Optimum — Although a better solution may exist, a locally optimal solution has been found.

7

Locally Infeasible — Although feasible solutions may exist, LINGO was not able to find one.

8

Cutoff — The objective cutoff level was achieved.

9

Numeric Error — The solver stopped due to an undefined arithmetic operation in one of the constraints.

In general, if @STATUS does not return a code of 0, 4, 6 or 8, the solution is of little use and should not be trusted. The @STATUS function is available only in the data and calc sections. The @STATUS function is not allowed in the constraints of a model.

OPERATORS AND FUNCTIONS For example, the following output statement uses @STATUS to print a message to let the know if the solution is globally optimal, or not: DATA: @TEXT() = @WRITE( @IF( @STATUS() #EQ# 0, 'Global solution found', 'WARNING: Solution *not* globally optimal!'); ENDDATA

For additional examples of the use of the @STATUS function, refer to Interfacing with Other Applications. @STRLEN( string) Use @STRLEN to get the length of a specified string. This can be a useful feature when formatting reports. As an example, @STRLEN( ‘123’) would return the value 3. @STRLEN is available only in the data and calc sections, and is not allowed in the constraints of a model. @TABLE( ‘attr|set’) The @TABLE function is used to display either an attribute’s values or a set’s in tabular format. The @TABLE function is available only in the data section of a model. You can refer to either QUEENS8.LG4 or PERT.LG4 for examples of @TABLE. These models can be found in the SAMPLES folder off the main LINGO folder. For instance, QUEENS8.LG4 is a model for positioning eight queens on a chessboard so that no one queen can attack another. At the end of this model you will find the following data section: DATA: @TEXT() = ' The final chessboard:'; @TEXT() = @TABLE( X); ENDDATA

Here we are using the @TABLE function to display the X attribute in the standard output window via the @TEXT interface function (see below for more on @TEXT). The X attribute is an 8-by-8 table of 0’s and 1’s indicating if a queen is positioned in a given square of the chessboard, or not. The output generated by @TABLE in this instance follows: The final chessboard: E1 E2 E3 E4 E5 E1 0 1 0 0 0 E2 0 0 0 0 1 E3 0 0 0 0 0 E4 0 0 0 1 0 E5 1 0 0 0 0 E6 0 0 0 0 0 E7 0 0 0 0 0 E8 0 0 1 0 0

E6 0 0 0 0 0 0 1 0

E7 0 0 1 0 0 0 0 0

E8 0 0 0 0 0 1 0 0

Note that all eight queens (indicated by the 1’s on the board) are safe from attack.

411

412 CHAPTER 7 In addition to displaying attributes, @TABLE can also display sets. When displaying a set, @TABLE will print the letter X in the table cell if the corresponding set member exists, otherwise it will leave the table cell blank. The PERT.LG4 sample model is a project scheduling model. A project consists of many tasks, and some tasks must be completed before others can begin. The list of all the tasks that must precede certain other tasks is called the precedence relations. At the end of PERT4.LG4 there is the following data section which uses @TABLE to display the precedence relations set, PRED: DATA: !Use @TABLE() to display the precedence relations set, PRED; @TEXT() = @TABLE( PRED); ENDDATA When we run the model, @TABLE displays the following table: DESIGN FORECAST SURVEY PRICE SCHEDULE COSTOUT TRAIN DESIGN FORECAST SURVEY PRICE X SCHEDULE COSTOUT X TRAIN

X

X X X

X

X

Whenever one task must precede another, an ‘X’ appears in the particular cell of the table. So, for instance, the DESIGN task must precede the FORECAST and SURVEY tasks. If a line of a table exceeds the page width setting in Lingo, it simply gets wrapped around. So, if you want to display wide tables without line wraps, you may need to increase the page width. Note:

Currently, @TABLE can only send tables to the standard solution window or to text files. In other words, it is not possible to send tables to Excel, databases or to applications calling the LINGO DLL.


413

@TABLE can also display sets and attributes of more than two dimensions. In fact, @TABLE allows you great control over how multidimensional objects get displayed. Specifically, four forms of @TABLE are ed: 

@TABLE( attr/set) – This is the simplest form of @TABLE. If the object is of one dimension, it will be displayed as a column. Otherwise, the first n-1 dimensions will be displayed on the vertical axis, while the n-th dimension will be displayed on the horizontal axis of the table. An example follows: @TABLE( X)



@TABLE( attr/set, num_horz_indices) – In this form, a second argument, num_horz_indices, is supplied. This argument is an integer quantity equal to the number of the object’s dimensions to display along the horizontal axis. In which case, dimensions (n – num_horz_indices) to n will be displayed along the horizontal axis of the table. The following example displays dimension 2 and 3 of a 3dimensional set along the horizontal axis: @TABLE( MY_3_DIM_SET, 2)



@TABLE( attr/set, prim_set1,...,prim_setn) – Here we specify the exact ordering of the object’s n dimensions. In which case, the first n-1 dimensions specified will be displayed along the vertical axis, while the last dimension specified will be displayed along the horizontal axis. Here’s an example that displays a 4-dimensional attribute with dimensions 3, 4 and 1 on the vertical, and dimension 2 on the horizontal: @TABLE( MY_4_DIM_ATTRIBUTE, 3, 4, 1, 2)



@TABLE( attr/set, prim_set1,...,prim_setn, num_horz_indices) – In this final form, we again specify the ordering of all the indices, but a final integer argument is added to specify the number of dimensions to be displayed on the horizontal axis. The following example displays a 4-dimensional attribute with dimensions 3 and 4 on the vertical, and dimensions 1 and 2 on the horizontal: @TABLE( MY_4_DIM_ATTRIBUTE, 3, 4, 1, 2, 2)

@WRITE(obj1[, …, objn]) Use @WRITE to output one or more objects. @WRITE is available only in the data and calc sections, and is not allowed in the constraints of a model. In a data section, output from @WRITE may be routed to a file, spreadsheet, or database. The exact destination will depend on the interface function used on the left-hand side of the output statement. @WRITE is valid only in the data sections of a model. @WRITE may also be used to display computations that are a function of variable values. As an example, the output statement below prints the ratio of the two variables X and Y: DATA: @TEXT() = @WRITE( 'The ratio of X to Y is: ', X / Y); ENDDATA

414 CHAPTER 7 @WRITEFOR( setname[ ( set_index_list) [ | cond_qualifier]]: obj1[, …, objn]) Use @WRITEFOR to output one or more objects across a set. @WRITEFOR is available only in the data and calc sections, and is not allowed in the constraints of a model @WRITEFOR operates like the other set looping functions in that you may, or may not, specify an index list and a conditional qualifier. The ability to use a conditional qualifier allows great flexibility in specifying exactly what you wish to display in your reports. @WRITEFOR may also be used to display computations that are a function of variable values. Using @WRITEFOR in conjunction with export functions in data sections allows you to route output to a file, spreadsheet, or database. The exact destination will depend on the export function used on the left-hand side of the output statement. As an example, the output statement below prints the number of units to ship, the warehouse name, and the customer name for each route with nonzero volume: DATA: @TEXT() = @WRITEFOR( ROUTES( I, J) | X( I, J) #GT# 0: 'Ship ', X( I, J), ' units from warehouse ', WAREHOUSE( I), ' to customer ', CUSTOMER( J), @NEWLINE( 1)); ENDDATA

The resulting report would appear as follows: Ship Ship Ship Ship Ship Ship

2 units from warehouse WH1 to customer C1 17 units from warehouse WH1 to customer C2 1 units from warehouse WH1 to customer C3 13 units from warehouse WH2 to customer C1 12 units from warehouse WH2 to customer C4 21 units from warehouse WH3 to customer C3

Text Replication Operator (*) The text replication operator (*) may be used inside either the @WRITE or @WRITEFFOR functions to repeat a string a specified number of times. The operator should be preceded by a numeric value and then followed by a string (e.g., 3*’text’), which will cause the string to be printed n times, where n is the numeric value. In the following example, the text replication operator is used twice to produce a simple graph of onduty staff during the seven days of the week: DATA: LEAD = 3; @TEXT() = 'Staff on duty graph:'; @TEXT() = @WRITEFOR( DAY( D): LEAD*' ', DAY( D), ' ', ON_DUTY( D), ' ', ON_DUTY( D)*'+', @NEWLINE(1) ); ENDDATA


415

The graph would appear as follows, with one plus sign displayed for each staff member on duty: Staff on duty graph: MON 20 ++++++++++++++++++++ TUE 16 ++++++++++++++++ WED 12 ++++++++++++ THU 16 ++++++++++++++++ FRI 19 +++++++++++++++++++ SAT 14 ++++++++++++++ SUN 13 +++++++++++++

Date, Time and Calendar Functions @TIME() The @TIME function returns the total runtime, in seconds, required so far to generate and solve the model. @TIME is available only in the data and calc sections, and is not allowed in the constraints of a model. For example, the following output statement writes the solution time to the standard output device: CALC: @WRITE('Solve time in seconds =', @TIME()); ENDCALC

@YMD2STM( year, month, day, hour, minute, second) The @YMD2STM function takes a date and time as input and returns the equivalent scalar time, where scalar time is the number of seconds since the start of the base date of 1 Jan 2000. Note that dates prior to 1 Jan 2000 will return negative scalar time values. The first five arguments must be integral values, while the number of seconds can be fractional. If the seconds value is fractional, it will be rounded to the nearest 10 microseconds (or .00001 seconds). Hours are reported using military time and can range from 0 to 23. An example follows: CALC:

!Convert two dates to scalar time; STM1 = @YMD2STM( 2013, 1, 24, 13, 30, 10.5); STM2 = @YMD2STM( 1787, 9, 17, 0, 0, 0); ENDCALC

Here, we compute the scalar time for the two dates and times: 1/23/2013 13:30:10.5 and 9/17/1787 00:00:00. The solution report lists the following values: Variable

STM1 STM2

Value

412349410.50 -6699196800.0

There are two things to note: First, STM1 has a fractional value of .5, given that its time includes .5 fractional seconds. Second, STM2 is negative reflecting the fact that its date if prior to the base date to 1 Jan 2000.

416 CHAPTER 7 @STM2YR( stm) / @STM2MON( stm) / @STM2DAY( stm) @STM2HR( stm) / @STM2MIN( stm) / @STM2SEC( stm) These six functions are the inverses of the @STM2YR function presented above. Each function takes a scalar time value as input and returns, respectively, the year, month, day, hour, minute or second of the point in time represented by the scalar time value. Note that @STM2SEC can return fractional seconds. The following example takes a date and time, converts it to scalar time, and then converts the scalar time back to the original date and time: SETS: MONTHS /JAN..DEC/; DAYS /SUN..SAT/; ENDSETS CALC: !Convert a date to scalar time; STM = @YMD2STM( 1787, 9, 17, 12, 0, 0); !Convert the scalar time back to the original date and time; YR = @STM2YR( STM); MO = @STM2MON( STM); DA = @STM2DAY( STM); HR = @STM2HR( STM); MN = @STM2MIN( STM); SC = @STM2SEC( STM); DOW = @STM2DWK( STM); !Write out the original date; @WRITE( 'Date: ', DA, ' ', MONTHS( MO), ' ', YR, @NEWLINE( 1); ); !And the original date; @WRITE( 'Time: ', HR, ':', @FORMAT( MN, '%02g'), ':', @FORMAT( SC, '%02g'), @NEWLINE( 1); ); !And the day of the week; @WRITE( 'Day: ', DAYS( DOW), @NEWLINE( 2); ); ENDCALC


417

The output from this sample follows: Date: 17 SEP 1787 Time: 12:00:00 Day: MON

@STMNOW() The @STMNOW returns the current date and time as a scalar time value, where scalar time is the number of hours since the base date of 1 Jan 2000. The returned may be fractional to indicate fractions of an hour. The returned value can be broken down into the corresponding year, month, day, hour, minute and second by calling the @STM2 inverse functions described above. An example for displaying the current date and time follows: MODEL: SETS: MONTHS /JAN..DEC/; DAYS /SUN..SAT/; ENDSETS CALC: T = @STMNOW(); YR = @STM2YR( T); MO = @STM2MON( T); DA = @STM2DAY( T); HR = @STM2HR( T); MN = @STM2MIN( T); SC = @STM2SEC( T); DWK = @STM2DWK( T); @SET( 'TERSEO', 1); @WRITE( ' The current date and time is:', ' ', DAYS( DWK), ' ', DA, ' ', MONTHS( MO), ' ', YR, ', ', HR, ':', @FORMAT( MN, '%02g'), ':', @FORMAT( SC, '%04.1f'), @NEWLINE( 2) ); ENDCALC END

418 CHAPTER 7 Here, we retrieve the current date and time as a scalar time value using @STMNOW. We then convert the scalar time to a calendar date and time and then write out the result. The following is a sample of output from the model: The current date and time is: MON 11 FEB 2013, 15:54:56.7

@STM2DWK( stm) The @STM2DWK function takes a scalar time and returns the index of its day of the week, with Sunday returning 1 and Saturday returning 7. An example of @STM2DWK can be found immediately above.

Miscellaneous Functions @BLOCKROW( block_index, row_name) The @BLOCKROW function is used in conjunction with the branch-and-price (BNP) solver. Models suited for BNP have 2 or more mostly independent subproblems known as blocks. @BLOCKROW is used to assign rows to their respective blocks, where block_index is the index of the block and row_name is the name of the row to assign to block block_index. Block indices may be any nonnegative, integer value, with 0 corresponding to the block of linking rows. Refer to the BNP Solver section in Chapter 5 for more information on the use of @BLOCKROW. @IF (logical_condition, true_result, false_result) The @IF function evaluates logical_condition and, if true, returns true_result. Otherwise, it returns false_result. For example, consider the following simple model that uses @IF to compute fixed production costs: MIN = COST; COST = XCOST +YCOST; XCOST = @IF( X #GT# 0, 100, 0) + 2 * X; YCOST = @IF( Y #GT# 0, 60, 0) + 3 * Y; X + Y >= 30;

Model: IFCOST We produce two products—X and Y. We want to minimize total cost, subject to producing at least 30 total units of X and Y. If we produce X, there is a fixed charge of 100 along with a variable cost of 2. Similarly, for Y, these respective values are 60 and 3. We use the @IF function to determine if either of the products are being produced in order to apply the relevant fixed cost. This is accomplished by testing to see if their production levels are greater than 0. If so, we return the fixed cost value. Otherwise, we return zero. Experienced modelers know that, without the benefit of an @IF function, modeling fixed costs requires invoking some “tricks” using binary integer variables. The resulting models are not as intuitive as models constructed using @IF. However, the caveat is that the @IF function is not a linear function. At best, the graph of an @IF function will be piecewise linear. In our current example, the @IF functions are piecewise linear with a discontinuous break at the origin.


419

It is always best to try and keep a model linear (see Chapter 15, On Mathematical Modeling). Barring this, it is best for all functions in a nonlinear model to be continuous. The @IF function violates both these conditions. Thus, models containing @IF functions may be tough to solve to global optimality. Fortunately, LINGO has two options that can help overcome the difficult nature of models containing @IF functions—linearization and global optimization. To illustrate the difficulty in solving models with discontinuous functions such as @IF, we will solve our example model with both linearization and global optimization disabled. When we do this, we get the following solution: Local optimal solution found at iteration: Objective value: 160.0000 Variable COST XCOST YCOST X Y

42

Value 160.0000 160.0000 0.000000 30.00000 0.000000

This solution involves producing only X at a total cost of 160. Given that producing only Y and not X will result in a lower total cost of 150, this is clearly a locally optimal point. In order to find the globally optimal point, we must resort to either the linearization or global optimization features in LINGO. Briefly, linearization seeks to reformulate a nonlinear model into a mathematically equivalent linear model. This is desirable for two reasons. First, linear models can always be solved to global optimality. Secondly, linear models will tend to solve much faster than equivalent nonlinear models. Unfortunately, linearization can not always transform a model into an equivalent linear state. In which case, it may be of no benefit. Fortunately, our sample model can be entirely linearized. To enable the linearization option, run the LINGO|Options command and set the Linearization Degree to High on the General Solver tab. Global optimization breaks a model down into a series of smaller, local models. Once this series of local models has been solved, a globally optimal solution can be determined. To enable global optimization, run the LINGO|Options command, select the Global Solver tab, then click on the Global Solver checkbox. Note that the global solver is an add-on option to LINGO. The global solver feature will not be enabled for some installations. Run the Help|About LINGO command to determine if your installation has the global solver capability enabled. Whether using the linearization option or the global solver, LINGO obtains the true, global solution: Global optimal solution found at iteration: Objective value: 150.0000 Variable COST XCOST YCOST X Y

Value 150.0000 0.000000 150.0000 0.000000 30.00000

6

420 CHAPTER 7 Note:

Starting with release 9.0, the false branch of the @IF function may contain arithmetic errors without causing the solver to trigger an error. This makes the @IF function useful in avoiding problems when the solver strays into areas where certain functions become undefined. For instance, if your model involves division by a variable, you might use @IF as follows: @IF( X #GT# 1.E-10, 1/X, 1.E10).

@WARN (‘text’, logical_condition) This displays the message ‘text’ if the logical_condition is met. This feature is useful for ing the validity of a model’s data. In the following example, if the has entered a negative interest rate, the message “INVALID INTEREST RATE” is displayed: ! A model of a home mortgage; DATA: ! Prompt the for the interest rate, years, and value of mortgage. We will compute the monthly payment; YRATE = ?; YEARS = ?; LUMP = ?; ENDDATA ! Number of monthly payment; MONTHS = YEARS * 12; ! Monthly interest rate; (1 + MRATE) ^ 12 = 1 + YRATE; ! Solve next line for monthly payment; LUMP = PAYMENT * @FPA(MRATE, MONTHS); ! Warn them if interest rate is negative @WARN('INVALID INTEREST RATE', YRATE #LT# 0);

@ (_determined_arguments) The can supply this in an external DLL or object code file. For a detailed example on the use of @, please see the Defined Functions section in Chapter 11, Interfacing with Other Applications.

8 Interfacing with External Files It can be cumbersome and impractical to try to maintain your data in a LINGO model file. In most cases, your model’s data will reside externally in text files, spreadsheets, and databases. Also, a solution generated by LINGO is of little use if you can’t export it to other applications. For these reasons, LINGO has many methods to assist you in moving information in and out of the application. The primary focus of this chapter is to illustrate how to move data in and out of LINGO through the use of text based ASCII files. In Chapter 9, Interfacing with Spreadsheets, we will look at using spreadsheets. In Chapter 10, Interfacing with Databases, we will illustrate the use of databases for maintaining your model’s data.

Cut and Paste Transfers Perhaps the simplest and most straightforward way to move data in and out of an application in Windows is by using cut and paste commands. Windows maintains an information buffer called the clipboard. Applications that the cut command can move information into the clipboard. Applications that the paste command can move information from the clipboard into their memory. Thus, cut and paste offers a simple, but effective, technique for moving small amounts of data from one application to another.

Pasting in Data from Excel You should be able to paste data into LINGO from any application that s a cut command. For illustration purposes, we will show how to paste data from an Excel worksheet into a LINGO model. Recall our staff-scheduling model from Chapter 2, Using Sets, which is reproduced here with the data for the REQUIRED attribute omitted from the data section: SETS: DAYS / MON TUE WED THU FRI SAT SUN/: REQUIRED, START; ENDSETS DATA: REQUIRED = ; ENDDATA MIN = @SUM(DAYS(I): START(I)); @FOR(DAYS(J): @SUM(DAYS(I) | I #LE# 5: START(@WRAP(J - I + 1, 7))) >= REQUIRED(J) );

421

422 CHAPTER 8 Suppose your staffing requirements data is maintained in an Excel worksheet resembling the following:

To paste the staffing requirements data from Excel into the LINGO model above, follow these steps: 1. 2. 3. 4. 5.

Select the range containing the data (C3:I3) by placing the cursor on the C3 cell, press and hold down the left mouse button, drag the mouse to cell I3, then release the mouse button. Select the Copy command from Excel’s Edit Menu. Click once on the LINGO model window. Place the LINGO cursor directly to the right of the data statement: REQUIRED =. Select the Paste command from LINGO’s Edit menu.

The data should now appear in the LINGO model as follows: DATA: REQUIRED = 20 16 13 16 19 14 12; ENDDATA

You may need to adjust the font of the data to your liking. You can use the Edit|Select Font command in LINGO to accomplish this. Your model now has the required data and is ready to be solved. Note that LINGO also has features that allow you to import data directly from Excel. See Chapter 9, Interfacing with Spreadsheets, for more information.

Pasting Data Out to Microsoft Word Suppose you went ahead and solved the previous staffing model. LINGO will present you with a new Window containing the solution to your model. Now, suppose you would like to get a copy of the solution into MS Word for a report you are writing. You can do this by following these steps: 1. 2. 3. 4. 5.

Select the solution report window in LINGO by clicking on it once. Select all the text in the window by issuing the Edit|Select All command in LINGO. Place the solution into the clipboard by selecting the Edit|Copy command in LINGO. Activate MS Word by clicking once on the window containing the report you are writing. Paste the solution from the clipboard into the report by issuing the Edit|Paste command in MS Word.

INTERFACING WITH EXTERNAL FILES

423

Text File Interface Functions LINGO has several interface functions that perform input and output operations. There are interface functions for dealing with text files, spreadsheets, and databases. There is even an interface function that lets you data back and forth from other applications. In this chapter, we are focusing on interfacing with text files, so we will investigate the @FILE function for importing the contents of external text files and the @TEXT function for exporting solutions to text files.

Including External Files with @FILE The @FILE interface function in LINGO allows you to include data from external text files anywhere in your model. This is particularly useful for incorporating data stored in text files into your sets and data sections. The syntax for the @FILE function is: @FILE(‘filename’) where filename is the name of the file to include text from. When this function is encountered in a model, LINGO will continue to take text from this file until it encounters either the end-of-file mark or a LINGO end-of-record mark (~). For subsequent @FILE references in the same model that use the same file name, LINGO resumes taking input from the file at the point where it left off. Nesting of @FILE function calls (embedding an @FILE in a file that is itself called by @FILE) is not allowed.

424 CHAPTER 8

Using @FILE in a Transportation Model As an example, we will use the Wireless Widgets transportation model developed in Chapter 1, Getting Started with LINGO. It is reproduced in its original form below: ! A 6 Warehouse 8 Vendor Transportation Problem; SETS: WAREHOUSES /WH1 WH2 WH3 WH4 WH5 WH6/: CAPACITY; VENDORS /V1 V2 V3 V4 V5 V6 V7 V8/ : DEMAND; LINKS(WAREHOUSES, VENDORS): COST, VOLUME; ENDSETS ! The objective; MIN = @SUM(LINKS(I, J): COST(I, J) * VOLUME(I, J)); ! The demand constraints; @FOR(VENDORS(J): @SUM(WAREHOUSES(I): VOLUME(I, J)) = DEMAND(J)); ! The capacity constraints; @FOR(WAREHOUSES(I): @SUM(VENDORS(J): VOLUME(I, J)) <= CAPACITY(I)); ! Here is the data; DATA: CAPACITY = 60 55 51 43 41 52; DEMAND = 35 37 22 32 41 32 43 38; COST = 6 2 6 7 4 2 5 9 4 9 5 3 8 5 8 2 5 2 1 9 7 4 3 3 7 6 7 3 9 2 7 1 2 3 9 5 7 2 6 5 5 5 2 2 8 1 4 3; ENDDATA

Model: WIDGETS Note that data appears two places in the model. First, there are the lists of warehouses and vendors in the sets section. Second, there is data on capacity, demand, and shipping costs in the data section.

INTERFACING WITH EXTERNAL FILES In order to completely isolate the data from our model, we would like to move it to an external text file, and modify the model so it will draw the data from the text file using the @FILE function. The following modified version of the model has all the data removed. Changes are represented in bold type: ! A 6 Warehouse 8 Vendor Transportation Problem; SETS: WAREHOUSES / @FILE('WIDGETS2.LDT')/: CAPACITY; VENDORS / @FILE('WIDGETS2.LDT')/ : DEMAND; LINKS(WAREHOUSES, VENDORS): COST, VOLUME; ENDSETS ! The objective; MIN = @SUM(LINKS(I, J): COST(I, J) * VOLUME(I, J)); ! The demand constraints; @FOR(VENDORS(J): @SUM(WAREHOUSES(I): VOLUME(I, J)) = DEMAND(J)); ! The capacity constraints; @FOR(WAREHOUSES(I): @SUM(VENDORS(J): VOLUME(I, J)) <= CAPACITY(I)); ! Here is the data; DATA: CAPACITY = @FILE('WIDGETS2.LDT'); DEMAND = @FILE('WIDGETS2.LDT'); COST = @FILE('WIDGETS2.LDT'); ENDDATA

Model: WIDGETS2

425

426 CHAPTER 8 The model is now set to draw all data from the file WIDGETS2.LDT. The contents of this data file appear below: !List of warehouses; WH1 WH2 WH3 WH4 WH5 WH6 ~ !List of vendors; V1 V2 V3 V4 V5 V6 V7 V8 ~ !Warehouse capacities; 60 55 51 43 41 52 ~ !Vendor requirements; 35 37 22 32 41 32 43 38 ~ !Unit 6 2 6 4 9 5 5 2 1 7 6 7 2 3 9 5 5 2

shipping costs; 7 4 2 5 9 3 8 5 8 2 9 7 4 3 3 3 9 2 7 1 5 7 2 6 5 2 8 1 4 3

File: WIDGETS2.LDT Note:

We use the convention of placing the extension of .LDT on all LINGO data files.

Sections of the data file between end-of-record marks (~) are called records. If an included file has no end-of-record marks, LINGO reads the whole file as a single record. Notice that, with the exception of the end-of-record marks, the model text and data appear just as they would if they were in the model itself. Also, notice how the end-of-record marks in the include file work along with the @FILE function calls in the model. The first call to @FILE opens WIDGETS2.LDT and includes the first record. The second call includes the second record, and so on. The last record in the file does not need an end-of-record mark. When LINGO encounters an end-of-file, it includes the last record and closes the file. If you end the last record in an include file with an end-of-record mark, LINGO will not close the file until it is done solving the current model. This could cause problems if multiple data files are opened in the model—files that remain open can cause the limit on open files to be exceeded. When using the @FILE function, think of the contents of the record (except for any end-of-record mark) as replacing the text @FILE(‘filename’) in the model. This way, you can include a whole statement, part of a statement, or a whole series of statements in a record. For example, the first two records of the WIDGETS2.LDT file in the above example: !List of warehouses; WH1 WH2 WH3 WH4 WH5 WH6 ~ !List of vendors; V1 V2 V3 V4 V5 V6 V7 V8 ~


427

are included in the model in the sets section as follows: WAREHOUSES / @FILE('WIDGETS2.LDT')/: CAPACITY; VENDORS / @FILE('WIDGETS2.LDT')/ : DEMAND;

The net effect of these @FILE calls is to turn the model statements into: WAREHOUSES / WH1 WH2 WH3 WH4 WH5 WH6/: CAPACITY; VENDORS / V1 V2 V3 V4 V5 V6 V7 V8/ : DEMAND;

Comments in the include file are ignored. The maximum number of include files a model can simultaneously reference is 16.

Writing to Files Using @TEXT The @TEXT interface function is used for exporting solutions to text files. The @TEXT function can export both set and attribute values. The syntax of the @TEXT function is: @TEXT( ['filename', [‘a’]]) where filename is the name of the file you want to export the solution to. If filename is omitted, the solution data will be sent to the standard output device (this is typically the screen). If the second argument of ‘a’ is present, then LINGO will append output to the file, otherwise it will create a new file for subsequent output, erasing any existing file. The @TEXT function may only appear on the lefthand side of a data statement in the data section of a model. We refer to data statements that use interface functions to generate output as output operations. Output operations are only performed when the solver finishes running a model. The operations are run in the sequence that they were encountered in the model. Here are some examples of using @TEXT: Example 1: @TEXT('RESULTS.TXT') = X; Sends the value(s) for X to the file RESULTS.TXT. Any existing version of the file is overwritten Example 2: @TEXT() = DAYS, START; In this example, we are exporting the DAYS set and the START attribute. We routed the output to the screen by omitting the filename argument. Example 3: @TEXT() = @WRITEFOR( DAYS( D) | START( D) #GT# 0: DAYS( D), ' ', START( D)); In this example, we use the @WRITEFOR reporting function to loop over the of the DAYS set. Contrary to the previous example, we only print information for those days where START( D) is greater than 0. Now, let’s turn to a more detailed example of the use of @TEXT in our staff scheduling model.

428 CHAPTER 8

Example - Using @TEXT for Staff-Scheduling Let’s once again make use of the staff scheduling model from Chapter 2, Using Sets. However, this time we will modify it to use the @TEXT function and write the solution to a file. The model follows with the critical change listed in bold type: SETS: DAYS / MON TUE WED THU FRI SAT SUN/: REQUIRED, START; ENDSETS DATA: REQUIRED = 20 16 13 16 19 14 12; @TEXT('OUT.TXT') = DAYS, START; ENDDATA MIN = @SUM(DAYS(I): START(I)); @FOR(DAYS(J): @SUM(DAYS(I) | I #LE# 5: START(@WRAP(J - I + 1, 7))) >= REQUIRED(J) );

We have added the one output operation: @TEXT('OUT.TXT') = DAYS, START;

which writes the values of the DAYS set and the values of the START attribute to the file OUT.TXT. Once you solve the model, LINGO will run this output operation, the file OUT.TXT will be generated, and it will contain the of the DAYS set and the optimal values for the START attribute: MON TUE WED THU FRI SAT SUN

8.0000000 2.0000000 0.0000000 6.0000000 3.0000000 3.0000000 0.0000000

File: OUT.TXT


429

You may now import the data from OUT.TXT into other applications. For instance, if you want to import the data into MS Access, you could use the File|Get External Data|Import command in Access to read the data into a table. We defined a small table in Access called Start, and imported the data in this manner yielding the following:

To import the data into an Excel sheet, you must first use the File|Open command on the OUT.TXT file to get the data into a spreadsheet by itself, as we have done here:

Once the results are imported into a spreadsheet, you may cut and paste them to any other sheet you desire. Before we move on, suppose you are not interested in all the output generated by the standard LINGO solution report. Suppose, in the case of this example, all you want to see is the objective value and the values for the DAYS set and the START attribute. Here's how you can do it. First, add the additional output operation, shown here in bold, to your data section, so it looks like: DATA: @TEXT('OUT.TXT') = DAYS, START; @TEXT() = DAYS, START; ENDDATA

The new output operation causes the values of DAYS and START to be sent to the screen (since we omitted a file name). Next, you will need to suppress the normal LINGO solution report. In Windows versions of LINGO, select the LINGO|Options command, click on the Interface tab in the Options

430 CHAPTER 8 dialog box, check the Terse output checkbox, then press the OK button (on platforms other than Windows, enter the TERSE command). Now, solve your model and you will be presented with the following, abbreviated report: Global optimal solution found at step: Objective value: MON TUE WED THU FRI SAT SUN

8 22.00000

8.0000000 2.0000000 0.0000000 6.0000000 3.0000000 3.0000000 0.0000000

In the example above, we simply listed the names of set DAYS and attribute START on the right-hand side of our output operation. This causes LINGO to display all values for DAYS and START. Suppose we'd like more control over what values do and do not get displayed. In particular, suppose we are only interested in viewing those days in which the value for START is nonzero. We can do this by using the @WRITEFOR report function, which allows us to provide a condition to test before printing output: DATA: @TEXT( ) = @WRITEFOR( DAYS( D) | START( D) #GT# 0: DAYS( D), @FORMAT( START( D), '6.1f')); ENDDATA

Note how we now only display the days where START > 0 in our new report: Global optimal solution found at iteration: Objective value: MON TUE THU FRI SAT

15 22.00000

8.0 2.0 6.0 3.0 3.0

Another feature of this last example to note is the use of the @FORMAT function, which we used to display the nonzero start values in a field of six columns with one trailing decimal point. @WRITEFOR also allows us to form arithmetic expressions of the variable values. Here's an example that we could add to our staff model to compute the number of staff on duty each day: DATA: @TEXT( ) = @WRITE( 'Day On-Duty'); @TEXT( ) = @WRITE( 14*'-'); @TEXT( ) = @WRITEFOR( DAYS( D): DAYS(D), @FORMAT( @SUM( DAYS( D2) | D2 #LE# 5: START( @WRAP( D - D2 + 1, 7))), '11.1f')); ENDDATA


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Here's the report generated by these output operations: Day On-Duty -------------MON 20.0 TUE 16.0 WED 13.0 THU 16.0 FRI 19.0 SAT 14.0 SUN 12.0

This previous example also illustrates the use of the @WRITE function. The @WRITE function is similar to the @WRITEFOR function with the exception that it does not accept a set to loop on, and, therefore, is used to write single occurrences of text output. As with @WRITEFOR, @WRITE accepts expressions of variables. Here's an example that calculates the maximum number of employees starting on a particular day. DATA: @TEXT() = @WRITE( 'Max start = ', @MAX( DAYS: START)); ENDDATA

which yields the following output: Max start = 8.

LINGO Command Scripts A LINGO command script is any text file containing a series of LINGO commands. In addition to understanding the syntax of LINGO models, using command scripts requires some understanding of LINGO’s command language (covered in Chapter 6, Command-line Commands). You can think of these commands as a macro language that allows you to automate the running of frequently used commands and/or models. To run a command script in Windows versions of LINGO, use the File|Take Commands command. In other versions of LINGO, use the TAKE command. In both cases, you will be prompted for the name of a file that contains your command script. Once you have input the file name, LINGO will begin to execute the commands in this file. Execution will continue until either a QUIT command is encountered, causing LINGO to terminate, or an end-of-file is encountered, causing LINGO to return to normal input mode.

432 CHAPTER 8

A Command Script Example Once again, we will make use of the staff-scheduling model introduced on page 63 to illustrate the use of a command script. Suppose, instead of one hot dog stand, our operations have expanded and we now have three hot dog stands: Pluto Dogs, Mars Dogs, and Saturn Dogs. Our staffing requirements at the three sites are: Site Pluto Mars Saturn

Mon 20 10 8

Tue 16 12 12

Wed 13 10 16

Thu 16 11 16

Fri 19 14 18

Sat 14 16 22

Sun 12 8 19

Running staffing models for all three sites is cumbersome and prone to error. We would like to automate the process by constructing a script file that runs all three staffing models automatically. To do this, we construct the following script file: ! Have LINGO echo input to the screen; SET ECHOIN 1 ! Suppresses the standard solution report; SET TERSEO 1 ! Begins input of a new model; MODEL: SETS: DAYS / MON TUE WED THU FRI SAT SUN/: REQUIRED, START; ENDSETS DATA: REQUIRED = @FILE('PLUTO.LDT'); @TEXT('PLUTO.TXT') = START; ENDDATA MIN = @SUM(DAYS(I): START(I)); @FOR(DAYS(J): @SUM(DAYS(I) | I #LE# 5: START(@WRAP(J - I + 1, 7))) >= REQUIRED(J) ); @FOR(DAYS: @GIN(START)); END ! Solve Pluto Dogs model; GO ! Alter model for Mars; ALTER ALL 'PLUTO'MARS' ! Solve Mars model; GO ! Alter model for Saturn; ALTER ALL 'MARS'SATURN'


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! Solve Saturn model; GO ! Restore parameters; SET TERSEO 0 SET ECHOIN 0

Command Script: DOGS.LTF We use two SET commands to set two of LINGO’s parameters. First, we set ECHOIN to 1, which causes LINGO to echo all command script input to the screen. This can be useful when you are trying to debug a script file. Next, we set TERSEO to 1. This causes LINGO to go into terse output mode, which suppresses the default solution report each time we solve a model. Next, we include the MODEL: command to put LINGO into model input mode. It is important here to the MODEL: statement is a command. When LINGO encounters this command in a script file, it reads all subsequent text in the file as model text until it encounters the END command. This model then becomes the current model in memory. The key feature to note in our model is the data section: DATA: REQUIRED = @FILE('PLUTO.LDT'); @TEXT('PLUTO.TXT') = START; ENDDATA

We use the @FILE function to include the staffing requirements from an external file and we use the @TEXT function to send the values of the START attribute to a file. After the END statement, we have a GO command to solve the model for the Pluto stand. We then include an ALTER command to change all occurrences of PLUTO with MARS. This command will change the data section to (changes in bold): DATA: REQUIRED = @FILE('MARS.LDT'); @TEXT('MARS.TXT') = START; ENDDATA

Assuming we have the staffing requirements for the Mars stand in the file MARS.LDT, our model is then ready to run again. However, this time it will solve for the START values for the Mars hot dog stand. We include commands to do the same for the Saturn location as well. Finally, we have two SET commands to restore the modified parameters. You can run this command script by issuing the File|Take Commands command in Windows versions of LINGO, or you can use the TAKE command in other versions. Once the command script has been executed, you will find the three solution files: PLUTO.TXT, MARS.TXT, and SATURN.TXT. These files will contain the optimal values for the START attribute for the three locations.

The AUTOLG.DAT Script File LINGO has an option that allows you to automatically execute a command script each time LINGO starts. To do this, simply name the command script AUTOLG.DAT and place it in LINGO’s working directory. Each time LINGO starts, it will automatically execute the commands in this script file.

434 CHAPTER 8

Specifying Files in the Command-line When a Windows version of LINGO starts up, it checks the command-line for the presence of the following three commands: Command -Tfilename -Ofilename -Lfilename

Action at Runtime LINGO executes a File|Take Commands command on the script file filename. If an AUTOLG.DAT file is present in LINGO’s working directory, it will be queued for execution before filename. LINGO performs a File|Open command on filename, reading the file into a standard window. LINGO executes a File|Log Output command, causing all output that would normally have been sent to the command window to be routed to filename.

As an example, suppose we have the following command script: ! Input a small model MODEL: MAX = 20 * X + 30 * Y; X <= 50; Y <= 60; X + 2 * Y <= 120; END ! Terse output mode SET TERSEO 1 ! Solve the model GO ! Open a file DIVERT SOLU.TXT ! Send solution to the file SOLUTION ! Close solution file RVRT ! Quit LINGO QUIT

Command Script: TEST.LTF This script file inputs a small model, solves it, and then writes a solution report out to the file SOLU.TXT. Let’s suppose the script file is titled TEST.LTF. We can instruct LINGO to automatically execute this file by adding the following command to LINGO’s command-line: -tTEST.LTF. To do this under Windows, you will first need to create a shortcut icon for LINGO. Click the right mouse button on your desktop, and then select the New command followed by the Shortcut command. Press


435

the Browse button and then select the LINGO application file, which is found under the name LINGO.EXE in your main LINGO directory. You should now have a LINGO shortcut icon on your desktop that looks like:

To edit the command-line, you must right click on this icon and then select the Properties command. You will then see the dialog box:

In the Target edit box, add the command -tTEST.LTF. If you want LINGO to run without opening up a window, you can also select the Minimized option from the Run list box. Now, click the Apply button followed by the OK button.

436 CHAPTER 8 You can now run LINGO and have it execute the script file by double clicking the shortcut icon on the desktop. Once you have done this, the solution file, SOLU.TXT, should contain: Variable X Y Row 1 2 3 4



File: SOLU.TXT

RunLingo Windows versions of LINGO include a utility program called RunLingo. RunLingo may be used to process LINGO script files, however, RunLingo does not include the front-end graphical interface found in the interactive version of LINGO. RunLingo is invoked from the command line, where you may enter a script file name. RunLingo writes its output to the standard output device. RunLingo can prove useful in a production environment, where you want LINGO to operate quietly in the background as part of a larger planning system. The following example illustrates RunLingo being invoked from the command line to process a small transportation model contained in the script file TRAN.LTF (also included in the main LINGO folder).

INTERFACING WITH EXTERNAL FILES C:\LINGO14>runlingo tran.ltf LINGO/WIN32 14.0.1.29 (29 Jan 13) LINDO API 8.0.1053.301 (Jan 30 2013 23:27:41) Copyright (C) 2011-2013 LINDO Systems Inc. Licensed material, all rights reserved. Copying except as authorized in license agreement is prohibited. License location: C:\LINGO14\lndlng14.lic Config location: C:\LINGO14\LINGO.CNF lindo Licensed for commercial use. Branch-and-bound solver enabled. Nonlinear solver enabled. Barrier solver enabled. Global solver enabled. Integer solver enabled. Stochastic solver enabled. Conic solver enabled. Default parameter values restored. Parameter ECHOIN

Old Value 0

New Value 1

: MODEL: ? ! A 3 Warehouse, 4 Customer ? Transportation Problem; ? SETS: ? WAREHOUSE / WH1, WH2, WH3/ : CAPACITY; ? CUSTOMER / C1, C2, C3, C4/ : DEMAND; ? ROUTES( WAREHOUSE, CUSTOMER) : COST, VOLUME; ? ENDSETS ? ! The objective; ?

[OBJ] min = @SUM( ROUTES: COST * VOLUME);

? ! The demand constraints; ? @FOR( CUSTOMER( J): [DEM] ? @SUM( WAREHOUSE( I): VOLUME( I, J)) >= ? DEMAND( J)); ? ! The supply constraints; ? @FOR( WAREHOUSE( I): [SUP] ? @SUM( CUSTOMER( J): VOLUME( I, J)) <= ? CAPACITY( I)); ? ! Here are the parameters; ? DATA: ? CAPACITY =

30, 25, 21 ;

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438 CHAPTER 8 ? DEMAND = ? COST = ? ? ? ENDDATA ? END : set terseo 1 Parameter TERSEO

15, 17, 22, 12; 6, 2, 6, 7, 4, 9, 5, 3, 8, 8, 1, 5;

Old Value 0

New Value 1

: go Compiling model ... Structural analysis, 1 ... Scalarizing model ... Generating nonzero matrix ... Solving... Global optimal solution found. Objective value: Infeasibilities: Total solver iterations:

161.0000 0.000000 6

: nonz volume Global optimal solution found. Objective value: Infeasibilities: Total solver iterations:

161.0000 0.000000 6

: quit C:\LINGO14>

Variable

Value

Reduced Cost

VOLUME( WH1, C1) VOLUME( WH1, C2) VOLUME( WH1, C3)

2.000000 17.00000 1.000000

0.000000 0.000000 0.000000

VOLUME( WH2, C1) VOLUME( WH2, C4) VOLUME( WH3, C3)

13.00000 12.00000 21.00000

0.000000 0.000000 0.000000


439

Redirecting Input and Output In most Unix environments, it is possible to redirect all screen output from LINGO to a text file. You can also redirect all input from the keyboard to an input file. This is accomplished by specifying the files in the command-line using the following syntax: LINGO < input_file > output_file Upon execution of this command, LINGO will take its input from input_file, and a text file will be created called output_file, which contains everything that would have appeared on the screen had LINGO been started normally. Path names may be included in the names of the input and output files. Exploiting this capability allows you to use LINGO as a “black box” in larger turnkey applications. If done properly, the will not be aware LINGO is running in the background. As an example, we could run the script file from the previous section, TEST.LTF, generating the same solution file, SOLU.TXT, with the command: LINGO < TEST.LTF > CAPTURE.TXT

The file CAPTURE.TXT is used to capture all screen output.

Managing LINGO Files In order to help keep track of your files, you may want to adopt a file naming standard, at least as far as the file name extensions are concerned. Here are some suggestions for file name extensions: Extension .LG4 .LNG .LTF .LDT .LRP

Description model files (Windows only) model files in text format script files included data files report files

440 CHAPTER 8 Note:

If you use the default LINGO format (.LG4) for a model, the file will be saved in a special binary format. Files in this format will only be readable by Windows versions of LINGO, and will appear garbled if read into other applications. This binary format allows LINGO models to function as both Object Linking and Embedding (OLE) containers and servers, allows for embedding of objects (e.g., bitmaps), and permits custom formatting of model text. Thus, you will generally want to use the .LG4 format. Cases where you must use a text format are: 1. a model that must be transferred to another platform or application, 2. a LINGO script file, or 3. a LINGO data file to be included with @FILE. LINGO anticipates this by saving any file with an extension other than .LG4 as text. You can force LINGO to always use the .LNG text format for models by checking the .LNG box on the Interface tab of the LINGO|Options dialog box. Any special formatting of the text (e.g., bold fonts) will be lost when a model is saved in this format. Also, models saved as text may not function as OLE containers or servers. On platforms other than Windows, LINGO always saves files in text format.

9 Interfacing With Spreadsheets As we have mentioned, it can be cumbersome and impractical to try to maintain your data in a LINGO model file. This is particularly true if you have more than just a small handful of data—as is the case with most practical models. Spreadsheets are adept at handling small to moderate amounts of data. Spreadsheets are also very useful tools for manipulating and presenting the results generated by your model. For these reasons, LINGO has a number of features that allow the to import data from spreadsheets and export solutions back out to spreadsheets. These features include real-time Object Linking and Embedding (OLE) links to Excel, OLE automation links that can be used to drive LINGO from Excel macros, and embedded OLE links that allow you to import the functionality of LINGO into Excel. At present, all of these features are ed only under Windows versions of LINGO.

Importing Data from Spreadsheets LINGO provides the @OLE function for importing data from spreadsheets. This function is only available under Windows versions of LINGO. @OLE performs direct OLE transfers of data between LINGO and Excel.

Using @OLE to Import Data from Excel @OLE is an interface function for moving data back and forth from Excel using OLE based transfers. OLE transfers are direct memory transfers and do not make use of intermediate files. When using @OLE, LINGO loads Excel, tells Excel to load the desired spreadsheet, and requests ranges of data from the sheet. You must have Excel 5, or later, to be able to use the @OLE function. The @OLE function may be used in the data and init sections to import data. @OLE can read both set and set attributes—set are expected in text format, while set attributes are expected in numeric format. Primitive sets require one cell of data for each set member (i.e., one set member per cell). You will need n cells of values to initialize each n-dimensional derived set member, where the first n cells contain the first set member, the second n cells contain the second set member, and so on. @OLE can read one or two-dimensional ranges (ranges on a single worksheet in Excel), but cannot read discontinuous ranges or three-dimensional ranges (ranges that traverse more than one worksheet in Excel). Ranges are read left-to-right, top-to-bottom.

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442 CHAPTER 9

Importing in the Data and Init Sections with @OLE The syntax for using @OLE to import data in both the data and init sections is: object_list = @OLE([‘spreadsheet_file’] [, range_name_list]); The object_list is a list of model objects, optionally separated by commas, which are to be initialized from the spreadsheet. Object_list may contain any combination of set names, set attributes, and scalar variables. The spreadsheet_file is the name of the Excel spreadsheet file to retrieve the data from. If the name is omitted, LINGO defaults to using whatever workbook is currently open in Excel. The range_name_list is the list of named ranges in the sheet to retrieve the data from. The ranges must contain exactly one element for each member in the object_list. There are three options available in how you specify the ranges. First, you can omit the range arguments entirely. In which case, LINGO defaults to using a list of range names identical to the list of object names in object_list. Second, you can specify a single range name to retrieve all the data from. In which case, all the objects in object_list must be defined on the same set, and LINGO reads the data as if it was a table. When specifying a single range name for multiple objects, all the objects must be of the same data type. Furthermore, you can’t mix set (text) with set attributes (numeric). Finally, you can specify one range name for each object in object_list. In which case, the objects do not have to be defined on the same set and can be of differing data types. Examples of these three methods for using @OLE follow: Example 1: COST, CAPACITY = @OLE(); In this example we specify no arguments to the @OLE() function. In which case, LINGO will supply the default arguments. Given that no workbook name was specified, LINGO will use whatever workbook is currently open and active in Excel. In addition, no range names were specified, so LINGO defaults to using the names of the model objects. Thus, COST and CAPACITY are initialized to the values found, respectively, in the ranges COST and CAPACITY in the currently open workbook in Excel. Note:

If you do not specify a workbook name in @OLE() function references, LINGO defaults to using whatever workbook is currently open in Excel. Therfore, you will need to open Excel and load the relevant workbook prior to solving your model. Example 2: COST, CAPACITY = @OLE( 'SPECS.XLS', 'DATATABLE'); In this example, we are specifying a single range to initialize both COST and CAPACITY. Assuming the range DATATABLE has two columns, LINGO initializes COST to the data in column 1 and CAPACITY to the data in column 2. Note, in order for this method to work, both COST and CAPACITY must be defined on the same set. Furthermore, they must both be either sets or set attributes—mixed types aren’t allowed using this form. Example 3: COST, CAPACITY = @OLE( 'SPECS.XLS', 'COST01', 'CAP01'); In this example, we are specifying individual ranges to initialize both COST and CAPACITY. COST will be initialized with the data in the COST01 range and CAPACITY will receive the values in the CAP01 range.

INTERFACING WITH SPREADSHEETS

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As a final note, it is important to know that derived sets may be imported from either a single range or from n ranges, where n is the dimension of the derived set. Here are two examples to illustrate: Example 4: ARCS, COST = @OLE( 'TRAN.XLS', 'ARCS', 'COST'); You might find something similar to this example in a transportation model. ARCS is the set of shipping arcs and COST is an attribute for storing the cost of shipping one unit down each arc. Typically, ARCS would be a 2-dimensional derived set of from-to coordinate pairs. In this example, we've chosen to bring the list of arcs in from a single workbook range. Suppose there are 10 shipping arcs in our model, then the ARCS range would have 20 cells, with the first column of 10 cells being the ship-from points and the second column of 10 cells being the ship-to points. Since we are using the single range method to input a derived set, the two columns are adjacent and are both contained in the single range call ARCS. This is in contrast to the following example… Example 5: ARCS, COST = @OLE( 'TRAN.XLS', 'FROM', 'TO', 'COST'); …where we use two ranges to store the two-dimensional ARCS set. The first range, FROM, would contain 10 ship-from points, while the second range, TO, would contain 10 ship-to points. These two ranges may lie in different areas of the workbook and need not be adjacent.

444 CHAPTER 9

Importing in a Transportation Model with @OLE We will now make use of the Wireless Widgets transportation model introduced in Chapter 1, Getting Started with LINGO, to illustrate in more detail the use of the @OLE function. The model is reproduced below with changes listed in bold type: ! A 6 Warehouse 8 Vendor Transportation Problem; SETS: ! Import warehouses and vendors from Excel; WAREHOUSES: CAPACITY; VENDORS : DEMAND; LINKS(WAREHOUSES, VENDORS): COST, VOLUME; ENDSETS ! The objective; MIN = @SUM(LINKS(I, J): COST(I, J) * VOLUME(I, J)); ! The demand constraints; @FOR(VENDORS(J): @SUM(WAREHOUSES(I): VOLUME(I, J)) = DEMAND(J)); ! The capacity constraints; @FOR(WAREHOUSES(I): @SUM(VENDORS(J): VOLUME(I, J)) <= CAPACITY(I)); DATA: ! Import the data from Excel; WAREHOUSES, VENDORS, CAPACITY, DEMAND, COST = @OLE('\LINGO\SAMPLES\WIDGETS.XLS', 'WAREHOUSES', 'VENDORS', 'CAPACITY', 'DEMAND', 'COST'); ENDDATA

Model: WIDGETS3 Instead of explicitly listing the data in the text of the model, we are now importing it entirely from the WIDGETS.XLS spreadsheet. Below is an illustration of the WIDGETS.XLS:


445

In addition to inputting the data into this sheet, we also had to define range names for the cost, capacity, demand, vendor name, and warehouse name regions. Specifically, we defined the following range names: Name Capacity Cost Demand Vendors Warehouses

Range K5:K10 C5:J10 C11:J11 C4:J4 B5:B10

To define a range name in Excel: 1) select the range by dragging over it with the mouse with the left button down, 2) release the mouse button, 3) select the Insert|Name|Define command, 4) enter the desired name, and 5) click the OK button. We use the following instance of the @OLE function in the data section of our model to import the data from Excel: WAREHOUSES, VENDORS, CAPACITY, DEMAND, COST = @OLE('\LINGO\SAMPLES\WIDGETS.XLS', 'WAREHOUSES', 'VENDORS', 'CAPACITY', 'DEMAND', 'COST');

446 CHAPTER 9 Note that because the model objects are all either primitive sets or set attributes, and they have the same names as their corresponding spreadsheet ranges, we could have dropped the range name arguments and used the equivalent, but shorter, version: WAREHOUSES, VENDORS, CAPACITY, DEMAND, COST = @OLE('\LINGO\SAMPLES\WIDGETS.XLS');

As an aside, note that we used a single @OLE function call to read all the data for this model. This is not a requirement, however. For clarity, you may choose to use multiple @OLE function callsperhaps one for each model object. When we solve this model, LINGO will load Excel (assuming it isn't already running), load the WIDGETS worksheet, and then pull the values for the CAPACITY, COST, and DEMAND attributes from the worksheet, along with the of the WAREHOUSES and VENDORS sets. Excerpts from the solution appear below: Global optimal solution found. Objective value: Total solver iterations: Variable VOLUME(WH1, V2) VOLUME(WH1, V5) VOLUME(WH2, V4) VOLUME(WH2, V8) VOLUME(WH3, V2) VOLUME(WH3, V3) VOLUME(WH3, V7) VOLUME(WH4, V6) VOLUME(WH4, V8) VOLUME(WH5, V1) VOLUME(WH5, V2) VOLUME(WH6, V6) VOLUME(WH6, V7)

Value 19.00000 41.00000 32.00000 1.000000 12.00000 22.00000 17.00000 6.000000 37.00000 35.00000 6.000000 26.00000 26.00000

664.0000 15

Reduced Cost 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

Exporting Solutions to Spreadsheets LINGO allows you to place interface functions in a model’s data section to automatically perform exports to Excel each time your model is solved. In this chapter, we will focus on how to place interface functions in your model to export your model’s results to Excel. In the previous section, we saw how the @OLE interface function can be used to import data from Excel. As we will demonstrate here, this function may also be used to send solutions back out to Excel. As with imports, @OLE uses OLE technology to create real time links with Excel.


447

Using @OLE to Export Solutions to Excel @OLE is an interface function for moving data back and forth from Excel using OLE based transfers. OLE transfers are direct memory transfers and do not make use of intermediate files. When using @OLE for exports, LINGO loads Excel, tells Excel to load the desired spreadsheet, and sends ranges of data containing solution values to the sheet. @OLE can export one and two-dimensional ranges (rectangular ranges that lie on a single spreadsheet in Excel), but cannot export discontinuous or three-dimensional ranges (ranges that traverse more than one spreadsheet in Excel). In order to export solutions with @OLE, you place calls to @OLE in the data section of your model. These @OLE export instructions are executed each time your model is solved.

Syntax Form 1 The first form of syntax for using @OLE to export data is: @OLE( ['spreadsheet_file’] [, range_name_list]) = object_list; The object_list is a comma delimited list of sets, set attributes, and/or scalar variables to be exported. The 'spreadsheet_file’is the name of the workbook file to export the values to. If the name is omitted, LINGO defaults to using whatever workbook is currently open in Excel. The range_name_list is the list of named ranges in the sheet to export solution values to. The ranges must contain exactly one cell for each exported value for each object in the object_list. Primitive sets and set attributes export one value per element. Derived sets, on the other hand, export one value for each dimension of the set. Thus, a two-dimensional set exports two values per set member. As an example, consider the following model and its solution: SETS: S1: X; S2(S1, S1): Y; ENDSETS DATA: S1,X = M1,1 M2,2 M3,3; S2,Y = M1,M2,4 M3,M1,5; ENDDATA Variable X(M1) X(M2) X(M3) Y(M1, M2) Y(M3, M1)

Value 1.000000 2.000000 3.000000 4.000000 5.000000

X and Y, both set attributes, export one numeric value per element. More specifically, X exports 1,2, and 3; while Y exports 4 and 5. S1, a primitive set, exports one text value per element, or the values: M1, M2, and M3. S2, on the other hand, is a two-dimensional derived set. Thus, it exports two text values per element. In this case, S2 has two , so it exports the four values M1, M2, M3, and M1.

448 CHAPTER 9 There are three options available for how you specify the range names. First, you can explicitly specify one receiving range for each model object in object_list. Secondly, you can omit the range arguments entirely. In that case, LINGO supplies the range names with default values that are the same as the names of the model objects. Finally, you can specify a single range name to export all the solution values to. In this final case, all the variables in object_list must be defined on the same set, and LINGO exports the data in a tabular format. Examples of these three methods for using @OLE to export solutions follow: Example 1:

@OLE('\XLS\DEVELOP.XLS', 'BUILD_IT', 'HOW_BIG') = BUILD, SQ_FEET; Here, an individual range for receiving each model object is specified. Thus, the values of BUILD will be placed in the range BUILD_IT and SQ_FEET in the range HOW_BIG. When specifying individual ranges, model objects are not required to be defined on the same set.

Example 2: @OLE('\XLS\DEVELOP.XLS') = BUILD, SQ_FEET; In this case, we omitted the range name argument. Thus, LINGO defaults to using the model object names for the range names. So, LINGO exports BUILD and SQ_FEET to ranges of the same name in the DEVELOP.XLS Excel sheet. Example 3: @OLE('\XLS\DEVELOP.XLS', 'SOLUTION') = BUILD, SQ_FEET; Here we have specified a single range, SOLUTION, to receive both model objects. Assuming that the receiving range has two columns and our model objects are both one-dimensional, the values of BUILD will be placed in column 1 and SQ_FEET in column 2. In order for this method to work, BUILD and SQ_FEET must be defined on the same set. Note:

The major difference to notice between using @OLE for exports, as opposed to imports, is the side of the statement the @OLE function appears on. When the @OLE function appears on the left of the equals sign, you are exporting. When it appears on the right, you are importing. So, always : @OLE( … ) = object_list; object_list = @OLE( … );

 

Export, and Import.

Another way to this convention is that the left-hand side of the expression is receiving the data, while the right-hand side is the source. For those familiar with computer programming languages, this follows the convention used in assignment statements.

Syntax Form 2 As with @TEXT, you may also use the @WRITEFOR function in conjunction with @OLE to give you more control over the values that are exported, which brings us to our second form of syntax for exporting to Excel: @OLE( ['spreadsheet_file’], range_name_list) = @WRITEFOR( setname [ ( set_index_list) [ | conditional_qualifier]] : output_obj_1[,…,output_obj_n]); One thing to note that differs from the previous syntax is that the range name list is now required when exporting via the @WRITEFOR function. The range name list can be a single-cell range, a single multiple-cell range, or a list of multiple-cell ranges.


449

In the case of a single cell range, the i-th output object will be written to the (i-1)-th column to the right of the named cell. Note that single-cell ranges act dynamically in that all values will be written to the workbook even though, of course, they lie outside the single-cell range. When all output is written, the original single-cell range will be at the upper left corner of the table of output values. In the case of a single multiple-cell range, LINGO creates a table of all the output values, where output object i forms the i-th column of the table. This table is then written to the output range. Items are written from upper-left to lower-right. In general, your output range will have one column for each output object. If not, the columns will get scrambled on output. If a list of multiple cell ranges is specified, then you must specify one range name for each output object. Each output object will be written to its output range in upper-left to lower-right direction. @WRITEFOR functions like any other set looping function in that, as a minimum, you will need to specify the set to loop over. Optionally, you may also specify an explicit set index list and a conditional qualifier. If a conditional qualifier is used, it is tested for each member of the looping set and output will not occur for any that don't the test. It's this feature of being able to base output on the results of a condition that distinguish this second style of syntax. The list of output objects, of course, specifies what it is you want to output. As with the first form of syntax, the output objects may be labels, set and variable values. However, you have additional latitude in that the output objects may now consist of complex expressions of the variable values (e.g., you could compute the ratio of two variables). This is a useful feature when you need to report statistics and quantities derived from the variable values. By placing these calculations in the data section, as opposed to the model section, you avoid adding unnecessary complications to the constraints of the model. In general, you can do everything in the second form of syntax that you can do in the first, and more. However, the first form has an advantage in that it can be very concise. Some examples follow: Example 1:

@OLE( 'RESULTS.XLS', 'A1') = @WRITEFOR( DAYS( D) | START( DAYS) #GT# 0: DAYS( D), START( D));

Here, our target is the single cell A1. Starting at A1 we will write two columns. The first column will contain the names of the DAYS set for which the attribute START is nonzero. The second column will contain the START values. Assuming that there are five days that have nonzero values, then range A1:A5 will contain the names of the days and B1:B5 will contain the start values. Example 2:

@OLE( 'RESULTS.XLS', 'SKED') = @WRITEFOR( DAYS( D) | START( DAYS) #GT# 0: DAYS( D), START( D));

Here, our target is the multiple-cell range SKED. Assuming SKED is a two-column range, column one will receive the DAYS set and column 2 will receive the START values.

450 CHAPTER 9 Example 3:

@OLE( 'RESULTS.XLS', 'DAYS', 'START') = @WRITEFOR( DAYS( D) | START( DAYS) #GT# 0: DAYS( D), START( D));

In this example, we specify one named range for each output object. In which case, each output object will be written to its corresponding range. Note:

When exporting to workbooks, receiving ranges that are larger than the number of exported values can be filled out by either erasing the contents of the extra cells or leaving the extra cells untouched. The default is to leave the extra cells untouched. If you would like to erase the contents of the extra cells, you'll need to enable the Fill Out Ranges and Tables option.


451

Exporting in a Transportation Model with @OLE In a previous section of this chapter, @OLE Importing in a Transportation Model, we used the Wireless Widgets transportation model to demonstrate the use of the @OLE function for importing data from Excel. At the time, we did not use @OLE to export the solution back to the spreadsheet file. We will now extend the model in order to have it also export the solution back to the spreadsheet. The model is reproduced below with changes in the data section listed in bold type: ! A 6 Warehouse 8 Vendor Transportation Problem; SETS: ! Import warehouses and vendors from Excel; WAREHOUSES: CAPACITY; VENDORS : DEMAND; LINKS(WAREHOUSES, VENDORS): COST, VOLUME; ENDSETS ! The objective; MIN = @SUM(LINKS(I, J): COST(I, J) * VOLUME(I, J)); ! The demand constraints; @FOR(VENDORS(J): @SUM(WAREHOUSES(I): VOLUME(I, J)) = DEMAND(J)); ! The capacity constraints; @FOR(WAREHOUSES(I): @SUM(VENDORS(J): VOLUME(I, J)) <= CAPACITY(I)); DATA: ! Import the data from Excel; WAREHOUSES, VENDORS, CAPACITY, DEMAND, COST = @OLE('\LINGO\SAMPLES\WIDGETS.XLS', 'WAREHOUSES', ' VENDORS', 'CAPACITY', 'DEMAND', 'COST'); ! Export the solution back to Excel; @OLE('\LINGO\SAMPLES\WIDGETS.XLS', 'VOLUME') = VOLUME; ENDDATA

Model: WIDGETS5 We now use the @OLE function to send the decision variables contained in the VOLUME attribute back to the Excel file WIDGETS.XLS with the statement: @OLE('\LINGO\SAMPLES\WIDGETS.XLS', 'VOLUME') = VOLUME;

Note, since the attribute name is identical to the range name, we could have omitted the range name in the @OLE function and simply used the following: @OLE('\LINGO\SAMPLES\WIDGETS.XLS') = VOLUME;

452 CHAPTER 9 We will now need to add a range titled VOLUME for receiving the solution values in the WIDGETS spreadsheet. Here is the range as it appears after adding it to the sheet:

We have also used the Insert|Name|Define command in Excel to assign the range name VOLUME to the receiving range of C16:J21. To define a range name in Excel: 1. 2. 3. 4. 5.

select the range by dragging over it with the mouse with the left button down, release the mouse button, select the Insert|Name|Define command, enter the desired name (VOLUME in this case), and click the OK button.


453

When we solve this model, LINGO will load Excel (assuming it isn't already running), load the WIDGETS worksheet, and then pull the data for WAREHOUSES, VENDORS, CAPACITY, COST, and DEMAND from the worksheet. Once the solver has found the optimal solution, LINGO will send the values for the VOLUME attribute back to the worksheet storing them in the range of the same name and the updated range will appear as follows:

Export Summary Reports Whenever you use the @OLE function to export solutions to spreadsheets, you will receive a summary of the results of the export process. This summary is called the export summary report. These reports will be found at the top of the solution report window. There will be one export summary report for each @OLE function in the model used for exporting solutions. Here is the export summary report obtained when we exported the solution of the Wireless Widgets transportation model to Excel using @OLE: Export Summary Report --------------------Transfer Method: OLE BASED Workbook: \LINGO\SAMPLES\WIDGETS.XLS Ranges Specified: 1 VOLUME Ranges Found: 1 Range Size Mismatches: 0 Values Transferred: 48

The Transfer Method field lists the type of transfer used. @OLE transfers will be listed as “OLE BASED”.

454 CHAPTER 9 The Workbook field lists the name of the workbook the export was performed on. The Ranges Specified field lists the total number of ranges specified in the export function followed by a listing of the range names. The Ranges Found figure lists the number of ranges actually found in the sheet from the total number specified. In general, the spreadsheet range should have one cell for each data value being exported. If a range has too few or too many cells, it gets counted in the Range Size Mismatches field. The Values Transferred field lists the total number of data values actually transferred into all the specified ranges.

OLE Automation Links from Excel LINGO allows you to place a LINGO command script in a range in an Excel spreadsheet and then the script to LINGO by means of OLE Automation. This allows you to setup a client-server relationship between Excel and LINGO. To illustrate this feature, we will once again make use of the staff-scheduling model introduced in the Primitive Set Example — Staff-Scheduling section in Chapter 2, Using Sets. This illustration assumes the reader is moderately familiar with the use of Excel Visual Basic macros. If needed, you can refer to the Excel documentation for more background.


455

Consider the following Excel spreadsheet:

Spreadsheet: STAFOLE1.XLS We have placed the staffing requirements in the range C16:I16 and assigned the name REQUIREMENTS to this range. We have also assigned the name START to the range C18:I18. LINGO will be sending the solution to the START range. We have also included two graphs in the sheet to help visualize the solution. The graph on the left shows how many employees to start on each day of the week, while the graph on the right compares the number on duty to the number required for each day.

456 CHAPTER 9 Note that our spreadsheet has a second tab at the bottom titled Model. Select this tab and you will find the following:

Spreadsheet: STAFOLE1.XLS This page contains the command script we will use to solve the staffing model. For more information on command scripts, refer to page 431. In line 1, we turn on terminal echoing, so LINGO will echo the command script to the command window as it is read. Lines 2 through 21 contain the text of the model, which should be familiar by now. Note, in the data section, we are using two @OLE functions—the first to import the data from the spreadsheet and the second to export the solution back to the spreadsheet. The data is read from the range named REQUIRED, and the solution is written to the START range on the first tab of the sheet. In line 22, we use the GO command to solve the model. We have also assigned the range name MODEL to the range that contains this script (Model!A1:A23).


457

Given that we have our LINGO command script contained in our spreadsheet, the next question is how we it to LINGO to run it. This is where OLE Automation comes in. If you recall, the first tab of our sheet (the tab labeled Data) had a Solve button. We added this button to the sheet and attached the following Excel Visual Basic macro to it: Sub LINGOSolve() Dim iErr As Integer iErr = LINGO.RunScriptRange("MODEL") If (iErr > 0) Then MsgBox ("Unable to solve model") End If End Sub

We use OLE Automation to call the LINGO exported method RunScriptRange, ing it the range name MODEL. This, of course, is the name of the range that contains the command script. The RunScriptRange routine calls Excel to obtain the contents of the range and begins processing the commands contained therein. Processing continues until either a QUIT command is encountered or there are no further commands remaining in the range. RunScriptRange will return a value of 0 if it was successfully able to queue the script for processing. If RunScriptRange was not successful, it will return one of the following error codes: Error Code 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Description Invalid argument Unable to open log file Null script Invalid array format Invalid array dimension Invalid array bounds Unable to lock data Unable to allocate memory Unable to configure script reader LINGO is busy OLE exception Unable to initialize Excel Unable to read Excel range Unable to find Excel range

We have also added the following Auto_Open macro to the sheet: Dim LINGO As Object Sub Auto_Open() Set LINGO = CreateObject("LINGO.Document.4") End Sub

An Auto_Open macro is automatically executed each time a sheet is opened. We declare LINGO as an object and attach the LINGO object to the LINGO application with the CreateObject function.

458 CHAPTER 9 Now, go back to the first tab on the workbook and press the Solve button. After a brief pause, you should see the optimal solution installed, so the sheet resembles:

Spreadsheet: STAFOLE1.XLS The optimal number of employees to start on each day of the week is now contained in the START range (C18:I18), and the graphs have been updated to reflect this solution.

Embedding LINGO Models in Excel LINGO is capable of functioning as an OLE server. This means you can embed a LINGO model in any application that can function as an OLE container. Excel is one such application. Embedding a LINGO model into Excel is convenient in that the LINGO model is always immediately available once the spreadsheet is opened. You don’t have to worry about also starting LINGO and finding the correct LINGO model that corresponds to the spreadsheet.


459

To embed a LINGO document in an Excel file, select the Excel command Insert|Object. You will be presented with a list of embeddable objects available on your system. Select the LINGO Document object from this list as shown here:

Click the OK button and a blank LINGO model window will be embedded within the spreadsheet. You can enter text directly into this window just as you would in LINGO, or you can paste it in from another application. When you save the Excel sheet, the embedded LINGO model will automatically be saved with it. Similarly, whenever you read the sheet back into Excel, the embedded LINGO model will be restored, as well.

460 CHAPTER 9 To illustrate this feature, we will continue with the staffing model introduced in Chapter 2, Using Sets. The spreadsheet will contain the data for the model, and it will also contain an embedded LINGO model to perform the optimization and install the solution back into the spreadsheet. This example may be found in the spreadsheet file SAMPLES\STAFOLE2.XLS. If you load this sheet into Excel, you should see the following:

Spreadsheet: STAFOLE2.XLS As in the previous example, the staffing requirements are in the range C16:I16. This range has been titled REQUIRED. The range C18:I18 has been assigned the name START and will receive the solution after the model is solved. In the upper right-hand corner of the sheet, we have defined a graph to help visualize the solution.


461

In the upper left corner, there is a region labeled <Embedded LINGO Model>. This region contains a LINGO model that will solve the staffing model and place the solution values into the spreadsheet. If you double-click on this region, you will be able to see the model:

Spreadsheet: STAFOLE2.XLS Note, when this LINGO model is active, the LINGO menus and toolbar replace the Excel menus and toolbar. Thus, when working with an embedded LINGO model in Excel, you have all the functionality of LINGO available to you. When you deselect the LINGO model, the Excel menus and toolbar will automatically become available once again. This begins to illustrate the power of embedded OLE—it allows the to seamlessly combine the features of two or more applications together as if they were a single, integrated application.

462 CHAPTER 9 You can drag the lower right-hand corner of the LINGO model region to expose the contents of the entire model: SETS: DAYS / MON TUE WED THU FRI SAT SUN/: REQUIRED, START; ENDSETS DATA: REQUIRED = @OLE('C:\LINGO\SAMPLES\STAFOLE2.XLS'); @OLE('C:\LINGO\SAMPLES\STAFOLE2.XLS') = START; ENDDATA MIN = @SUM(DAYS: START); @FOR(DAYS(J): @SUM(DAYS(I) | I #LE# 5: START(@WRAP(J - I + 1, 7))) >= REQUIRED(J)); @FOR(DAYS: @GIN(START));

Once again, we are making use of our familiar staff-scheduling model. The main feature to note is that we are using two instances of the @OLE function. The first instance gets the staffing requirements from the spreadsheet. The second sends the solution back to the START range.


463

To solve the model, double-click on the region containing the LINGO model. The LINGO command menus will become visible along the top of the screen. Select the LINGO|Solve command. After LINGO finishes optimizing the model, it will return the solution to the sheet and we will have:

Spreadsheet: STAFOLE2.XLS

464 CHAPTER 9

Embedding Excel Sheets in LINGO Just as you can embed a LINGO model into Excel, you can reverse the process and embed an Excel sheet in a LINGO model. To illustrate this, load the STAFFOLE model from the SAMPLES subdirectory into LINGO. Once again, this is our familiar staff-scheduling model and it appears as follows:

Model: STAFFOLE


465

This model reads data from and writes the solution to the Excel file STAFFOLE.XLS. To make our lives easier, it would be nice to embed this spreadsheet in the LINGO model to avoid having to load it into Excel each time we need to work with our model. To do this, select the Edit|Insert New Object command in LINGO. You will be presented with the dialog box:

Click on the Create from File button, enter the spreadsheet file name in the File field, click on the Link checkbox, and then press the OK button.

466 CHAPTER 9 Your LINGO model should now appear as:

The data spreadsheet is now embedded at the top of the LINGO document. You can easily edit the spreadsheet by double-clicking on it. When you save the LINGO model, the link to the Excel file will be saved as well.


467

At this point, go ahead and optimize the model by selecting the LINGO|Solve command. When LINGO has finished solving the model, you should see the following:

The optimal solution now appears in the embedded spreadsheet and the graphs of the solution have been updated.

468 CHAPTER 9

Summary We have demonstrated a number of intriguing methods for combining the modeling features of LINGO with the data management features of spreadsheets. The concise, structured nature of LINGO’s modeling language makes reading and understanding even large models relatively easy. Whereas, it can often be a challenge to discern the underlying mathematical relationships of large, spreadsheet models with formulas spread throughout numerous cells on multiple tabbed sheets. LINGO models are also easier to scale than spreadsheet models. As the dimensions of your problem change, spreadsheet models can require the insertion or deletion of rows or columns and the editing of cell formulas. Whereas, if your data is separate from the model, your LINGO formulation will generally require little or no modification. Combining the power of model expression with LINGO and the ease of data handling in spreadsheets gives the mathematical modeler the best of both worlds.

10 Interfacing with Databases Spreadsheets are good at managing small to moderate amounts of data. Once your models start dealing with large amounts of data, database management systems (DBMSs) are, unquestionably, the tool of choice. Also, in many business modeling situations, you will find most, if not all, of the data is contained in one or more databases. For these reasons, LINGO s links to any DBMS that has an Open DataBase Connectivity (ODBC) driver (effectively all popular DBMSs). ODBC defines a standardized interface to DBMSs. Given this standardized interface, LINGO can access any database that s ODBC. LINGO has one interface function for accessing databases. This function’s name is @ODBC. The @ODBC function is used to import data from and export data to any ODBC data source. @ODBC is currently available only in Windows versions of LINGO.

469

470 CHAPTER 10

ODBC Data Sources Windows versions of LINGO include a variation of the standard transportation model that retrieves all data from a database, and writes a solution back to the same database. This file can be found in the file SAMPLES\TRANDB.LG4. The contents of this model are displayed below: MODEL: ! A 3 Plant, 4 Customer Transportation Problem; ! Data is retrieved from an either an Access database or an Oracle database an ODBC link. You *MUST* use the ODBC to one of the supplied databases under the name "Transportation" in order to get this model to run. Refer to Chapter 10 for more details.; TITLE Transportation; SETS: PLANTS: CAPACITY; CUSTOMERS: DEMAND; ARCS(PLANTS, CUSTOMERS): COST, VOLUME; ENDSETS ! The objective; [OBJ] MIN = @SUM(ARCS: COST * VOLUME); ! The demand constraints; @FOR(CUSTOMERS(C): @SUM(PLANTS(P): VOLUME(P, C)) >= DEMAND(C)); ! The supply constraints; @FOR(PLANTS(P): @SUM(CUSTOMERS(C): VOLUME(P, C)) <= CAPACITY(P)); DATA: ! Import the data via ODBC; PLANTS, CAPACITY = @ODBC(); CUSTOMERS, DEMAND = @ODBC(); ARCS, COST = @ODBC(); ! Export the solution via ODBC; @ODBC() = VOLUME; ENDDATA END

Model: TRANDB You will note that in the data section of this model, we use the @ODBC function to establish a link to an ODBC data source to retrieve all the data and to export the final solution. The technical details of the @ODBC function are discussed below. Right now, we will focus on how you set up an ODBC data source that can be accessed by LINGO.

INTERFACING WITH DATABASES

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An ODBC data source is a database that 1) resides in a DBMS for which you have an ODBC driver, and 2) has been ed with the ODBC . Databases in an ODBC enabled DBMS do not qualify as an ODBC data source until they have been ed with the ODBC . The ODBC is a Windows Control utility. ing a database with the ODBC is a straightforward process. In the following two sections, we will illustrate the registration process for a Microsoft Access database and for an Oracle database.

Creating an ODBC Data Source from an Access Database When you installed LINGO, a Microsoft Access database for the transportation model above was included as part of the installation. This file is contained in the directory SAMPLES under the name TRANDB.MDB. To this database as an ODBC data source for our transportation model, you must start the ODBC by doing the following: 1) 2) 3) 4)

double-click on the My Computer icon on your desktop, find the Control icon and double-click on it, double-click on the istrative Tools icon, and search for the Data Sources (ODBC) icon and double-click on it.

You should now see the ODBC dialog box shown below:

472 CHAPTER 10 To install the TRANDB.MDB database as a data source, do the following: 1.

Click the Add button in the ODBC dialog box to reveal the dialog box below:

2.

We are installing an Access data source, so select the Microsoft Access Driver option and press the Finish button.

3.

In the next dialog box:


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assign the data source the name Transportation in the Data Source Name field. In the Description field, enter “Datasource for a LINGO transportation model”. Press the Select button and enter the name of the database “LINGO\SAMPLES\TRANDB.MDB” (this assumes LINGO has been installed in the LINGO directory—your installation may differ). The dialog box should now resemble the one below:

474 CHAPTER 10 4.

Press the OK button and you should see the Transportation data source has been added to the list of ODBC data sources:

5.

Click the OK button to close the ODBC .


475

You should now be able to start LINGO and solve the TRANDB.LG4 model. LINGO knows to go to the Transportation data source for data because the model’s title (input with the TITLE statement) is Transportation. If you solve this model, you should see the following results: Global optimal solution found at step: Objective value: Model Title: Transportation Variable Value CAPACITY(PLANT1) 30.00000 CAPACITY(PLANT2) 25.00000 CAPACITY(PLANT3) 21.00000 DEMAND(CUST1) 15.00000 DEMAND(CUST2) 17.00000 DEMAND(CUST3) 22.00000 DEMAND(CUST4) 12.00000 COST(PLANT1, CUST1) 6.000000 COST(PLANT1, CUST2) 2.000000 COST(PLANT1, CUST3) 6.000000 COST(PLANT1, CUST4) 7.000000 COST(PLANT2, CUST1) 4.000000 COST(PLANT2, CUST2) 9.000000 COST(PLANT2, CUST3) 5.000000 COST(PLANT2, CUST4) 3.000000 COST(PLANT3, CUST1) 8.000000 COST(PLANT3, CUST2) 8.000000 COST(PLANT3, CUST3) 1.000000 COST(PLANT3, CUST4) 5.000000 VOLUME(PLANT1, CUST1) 2.000000 VOLUME(PLANT1, CUST2) 17.00000 VOLUME(PLANT1, CUST3) 1.000000 VOLUME(PLANT1, CUST4) 0.0000000 VOLUME(PLANT2, CUST1) 13.00000 VOLUME(PLANT2, CUST2) 0.0000000 VOLUME(PLANT2, CUST3) 0.0000000 VOLUME(PLANT2, CUST4) 12.00000 VOLUME(PLANT3, CUST1) 0.0000000 VOLUME(PLANT3, CUST2) 0.0000000 VOLUME(PLANT3, CUST3) 21.00000 VOLUME(PLANT3, CUST4) 0.0000000 Row OBJ 2 3 4 5 6 7 8

TRANDB Solution

Slack or Surplus 161.0000 0.0000000 0.0000000 0.0000000 0.0000000 10.00000 0.0000000 0.0000000

6 161.0000 Reduced Cost 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 2.000000 0.0000000 9.000000 1.000000 0.0000000 7.000000 11.00000 0.0000000 5.000000 Dual Price 1.000000 -6.000000 -2.000000 -6.000000 -5.000000 0.0000000 2.000000 5.000000

476 CHAPTER 10 As an interesting exercise, you may wish to redirect TRANDB.LG4 to use a second database that was provided as part of your installation. The second database is called TRANDB2.MDB and is also located in the SAMPLES directory. The main difference between the files is the dimension of data. TRANDB.MDB involves only 3 plants and 4 customers, while TRANDB2.MDB contains 50 plants and 200 customers. If you return to the ODBC and TRANDB2. MDB under the name “Transportation2”, then you can redirect LINGO to use the new data source by changing the model title from “Transportation” to “Transportation2” in the following line in the LINGO model: TITLE Transportation2;

The new model will have 10,000 variables as opposed to the 12 variables generated with the smaller data source. This ability to easily run different size data sets illustrates the usefulness of writing data independent, set-based models.

Creating an ODBC Data Source from an Oracle Database LINGO installs an SQL script that can be run by the SQL Plus utility in Oracle to build a small database for the TRANDB.LG4 transportation model. The script file is named TRANDB.SQL and may be found in the SAMPLES folder off the main LINGO directory. Here is the procedure you will need to follow to set up this data source: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Start up the SQL Plus utility provided with Oracle by going to the Start menu in Windows, select the Run command, type “SQLPLUSW”, and then click on the OK button. Enter your Oracle ID and . If you have a default installation of Oracle, the ID “sys” and the “change_on_install” should be valid. A host name is required only if Oracle is running remotely. Run the TRANDB.SQL script by typing “@\LINGO\SAMPLES\TRANDB.SQL” to the SQL Plus system prompt. Exit SQL Plus by entering “EXIT” to the prompt. Start up the ODBC as described in the previous section. When selecting an ODBC driver, be sure to use the "Microsoft ODBC for Oracle" driver. Do not use the "Oracle ODBC Driver", because it does not provide all the necessary functionality. Assign the data source the name “Transportation”. Load TRANDB.LG4 into LINGO and solve it. Enter your Oracle ID and as prompted.

If you would like to avoid entering your Oracle ID and each time LINGO accesses a database, you may set them once at the start of your session with the File|Database Info command. For more information, see the File|Database Info section above in Chapter 5, Windows Commands.


477

Creating an ODBC Data Source from an SQL Server Database LINGO installs an SQL script that can be run by the SQL Analyzer utility in SQL Server to build a small database for the TRANDB.LG4 transportation model. The script file is named TRANDB_SQL.SQL and may be found in the SAMPLES folder off the main LINGO directory. Here is the procedure you will need to follow to set up this data source: 1.

Start up the SQL Analyzer utility provided with SQL Server by going to the Start menu in Windows, selecting the Programs command, selecting the Microsoft SQL Server program group, and then clicking on Query Analyzer.

2.

Point the Query Analyzer to the machine running SQL Server.

3.

Run the File|Open command and select the script file TRANDB_SQL.SQL from the LINGO samples folder.

4.

Run the script by issuing the Query|Execute command.

5.

Exit the Query Analyzer.

6.

Start up the ODBC as described in the section above: Creating an ODBC Data Source from an Access Database.

7.

Press the Add button to add a new ODBC data source: Select the SQL Server ODBC drive, assign the data source the name "Transportation", and select the appropriate server machine.

8.

You should now be able to load TRANDB.LG4 into LINGO and solve it.

Note:

If you plan to use LINGO’s ODBC interface to write solutions to a table in SQL Server, then the table must contain a field that is declared as a unique identifier. Failure to do so will result in an error message claiming that the table is read-only.

Importing Data from Databases with @ODBC To import a model’s data from an ODBC data source, we use the @ODBC function in the model’s data section. @ODBC allows us to import both text formatted set and numerical set attribute values. The syntax for using @ODBC to import data inside a data section is: object_list = @ODBC([‘data_source’[, ‘table_name’ [,‘column_name_1’[, ‘column_name_2’ …]]]]);

478 CHAPTER 10 The object_list is a list, optionally separated by commas, containing model objects (i.e., attributes, sets, or variables) that are to be initialized from the ODBC data source. Object_list may contain up to one set and/or multiple set attributes. All set attributes in object_list must be defined on the same set. If object_list contains a set, then all attributes in object_list must be defined on this set. The data_source argument is the name of the ODBC data source that contains the data table. The table_name argument is the name of the data table within the data source that contains the data. Finally, the column_name arguments are the names of the data columns, or fields, in the data table table_name to retrieve the initialization data from. Set attributes and primitive sets require one column name each to retrieve their data from. Derived sets require one column name for each dimension of the set. Thus, a two-dimensional derived set would require two columns of data to initialize its . If the data_source argument is omitted, the model's title is used in its place (see the discussion of the TITLE statement in Chapter 1, Getting Started with LINGO). If table_name is omitted, the name of any set in the object_list is used in its place. If there is no set in object_list, then the name of the set that the attributes in object_list are defined on is used. If the column_name arguments are omitted, LINGO will choose default names based on whether the corresponding object in object_list is a set attribute, a primitive set, or a derived set. When the object to be initialized is a set attribute or a primitive set, LINGO will use the name of the object as the default column name. When the object is a derived set, LINGO will generate one default column name for each dimension of the derived set, with each name being the same as the parent set that the given dimension is derived from. As an example, a two-dimensional set named LINKS derived from the two primitive sets SOURCE and DESTINATION would default to being initialized from the two columns titled SOURCE and DESTINATION. Keep in mind that LINGO expects to find set in text format in the database, while set attributes are expected to be in numeric format. Some examples of using @ODBC to import data in a model's data section are: Example 1:

SHIPPING_COST = @ODBC( 'TRANSPORTATION', 'LINKS','COST');

LINGO initializes the attribute SHIPPING_COST from the column COST contained in the data table LINKS found in the ODBC data source TRANSPORTATION. Example 2:

VOLUME, WEIGHT = @ODBC( 'TRUCKS', 'CAPACITY');

The database column names are omitted, so, assuming VOLUME and WEIGHT are set attributes, LINGO defaults to using the attribute names (VOLUME and WEIGHT) as the database column names. Therefore, LINGO initializes the attributes VOLUME and WEIGHT from the columns also titled VOLUME and WEIGHT contained in the data table named CAPACITY found in the ODBC data source TRUCKS. Example 3: REQUIRED, DAYS = @ODBC(); In this example, we will assume a) we have titled the model PRODUCTION, b) REQUIRED is a derived set derived from the two primitive sets JOB and WORKSTATION ( e.g., REQUIRED( JOB, WORKSTATION)), and c) DAYS is a set attribute defined on the set REQUIRED. All arguments to the @ODBC function have been omitted, so LINGO supplies PRODUCTION as the data source name; the set name REQUIRED as the data


479

table name; and the three data column names JOB, WORKSTATION, and DAYS. Had we wanted to be more specific, we could have explicitly included all the arguments to the @ODBC function with the equivalent statement: REQUIRED, DAYS = @ODBC( 'PRODUCTION', 'JOB', 'WORKSTATION', 'DAYS');

Importing Data with ODBC in a PERT Model We will now modify the project scheduling model, PERT introduced in Chapter 2, Using Sets, to demonstrate the use of @ODBC to import the set names of the project's tasks from a Microsoft Access database. The modified model appears below, with changes listed in bold type: SETS: TASKS: TIME, ES, LS, SLACK; PRED(TASKS, TASKS); ENDSETS DATA: TASKS = @ODBC('PERTODBC', 'TASKS', 'TASKS'); PRED = @ODBC('PERTODBC', 'PRECEDENCE', 'BEFORE', 'AFTER'); TIME = @ODBC('PERTODBC'); ENDDATA @FOR(TASKS(J)| J #GT# 1: ES(J) = @MAX(PRED(I, J): ES(I) + TIME(I)) ); @FOR(TASKS(I)| I #LT# LTASK: LS(I) = @MIN(PRED(I, J): LS(J) - TIME(I)); ); @FOR(TASKS(I): SLACK(I) = LS(I) - ES(I)); ES(1) = 0; LTASK = @SIZE(TASKS); LS(LTASK) = ES(LTASK);

Model: PERTODBC

480 CHAPTER 10 With the statement: TASKS = @ODBC('PERTODBC', 'TASKS', 'TASKS');

we now fetch the of the TASKS set from our ODBC data source, as opposed to explicitly listing them in the model. Specifically, we get the of the TASKS set from the data column, or field, TASKS contained in the table named TASKS from the ODBC data source PERTODBC. Here is the data table as it appears in Access:

Access Database: PERTODBC.MDB Next, we use the statement: PRED = @ODBC('PERTODBC', 'PRECEDENCE', 'BEFORE', 'AFTER');

to fetch the of the PRED set from an ODBC data source, as opposed to explicitly listing them in the model. More specifically, we pull the of the PRED set from the data columns BEFORE and AFTER contained in the table named PRECEDENCE from the ODBC data source PERTODBC. Here is the data table showing the precedence relations as it appears in Access:

Access Database: PERTODBC.MDB Note that the PRECEDENCE set is a two-dimensional set. Thus, we must supply two database columns containing the set .


481

In order to retrieve the values for the task times, we create the ODBC link: TIME = @ODBC('PERTODBC');

Note that we only specified the ODBC data source name—the table and column names have been omitted. In which case, LINGO supplies default values for the table and column. The object being initialized, TIME, is a set attribute. Thus, LINGO supplies its parent set name, TASKS, as the default table name. For the default column name, LINGO supplies the set attribute’s name, TIME. Had we wanted to be specific, however, we could have explicitly entered all arguments to @ODBC with: TIME = @ODBC('PERTODBC', 'TASKS', 'TIME');

Exporting Data with @ODBC As is the case with most interface functions, @ODBC can export data as well as import it. Specifically, you can use the @ODBC function in the data section of a model to export set and attribute values to ODBC data sources. In order to export solutions with @ODBC, you place calls to @ODBC in the data section of your model. These @ODBC export instructions are executed each time your model is solved. The first form of syntax for using @ODBC to export data is: @ODBC( ['data_source'[, 'table_name'[, 'column_name_1'[,…, 'column_name_n']]]]) = object_list; Note:

When importing, @ODBC appears on the right of the equality sign. When exporting, the @ODBC function appears on the left of the equals sign.

The object_list is a list, optionally separated by commas, containing model objects (i.e., attributes, sets, or variables) that are to be exported to the ODBC data source. Object_list may contain up to one set and/or multiple set attributes. All set attributes in object_list must be defined on the same set. If object_list contains a set, then all attributes in object_list must be defined on this set. The data_source argument is the name of the ODBC data source containing the data table that will receive the exported values. The table_name argument is the name of the data table within the data source that will receive the data. Finally, the column_name arguments are the names of the receiving columns, or fields, in the data table table_name. Set attributes and primitive sets require one receiving column name each. Derived sets require one receiving column name for each dimension of the set. Thus, a twodimensional derived set would require two receiving columns in a data table. If the data_source argument is omitted, the model's title is used in its place (see the discussion of the TITLE statement in Chapter 1, Getting Started with LINGO). If table_name is omitted, the name of any set in the object_list is used in its place. If there is no set in object_list, then the name of the set where the attributes in object_list are defined is used. If the column_name arguments are omitted, LINGO will choose default names based on whether the corresponding object in object_list is either a set attribute, a primitive set, or a derived set. When the object to be initialized is a set attribute or a primitive set, LINGO will use the name of the object as the default column name. When the object is a derived set, LINGO will generate one default column name

482 CHAPTER 10 for each dimension of the derived set, with each name being the same as the parent set that the given dimension is derived from. As an example, a two-dimensional set named LINKS derived from the SOURCE and DESTINATION primitive sets would default to being exported to the two columns titled SOURCE and DESTINATION. Keep in mind that set are exported as text, while set attributes are exported as double precision floating point values. Some examples of using @ODBC to export data values to an ODBC data source are: Example 1:

@ODBC( 'TRANSPORTATION', 'LINKS', 'VOLUME') = VOLUME;

LINGO sends the values of the VOLUME attribute to the column also titled VOLUME in the data table LINKS in the ODBC data source TRANSPORTATION. Example 2: @ODBC() = NUMBER_WORKING; All arguments to the @ODBC function have been omitted and will default to the model's title for the data source, the attributes parent set for the data table, and the attribute's name for the column name. So, assuming we have used the TITLE statement to name this model SCHEDULING, and the attribute NUMBER_WORKING is defined on the set SCHEDULES, then LINGO exports the attribute NUMBER_WORKING to the column also titled NUMBER_WORKING in the data table SCHEDULES in the ODBC data source SCHEDULING. The first form of syntax will generally be sufficient for most database export operations. However, there may be times when you need to export only portions of the attributes, or you need to export quantities computed from the attribute values. Our second form of syntax uses the @WRITEFOR reporting function to handle these more general cases: @ODBC( 'data_source', 'table_name', 'column_name_1'[,…, 'column_name_n']) = @WRITEFOR( setname [ ( set_index_list) [ | conditional_qualifier]] : output_obj_1[,…, output_obj_n]); @WRITEFOR functions like any other set looping function in that, as a minimum, you will need to specify the set to loop over. Optionally, you may also specify an explicit set index list and a conditional qualifier. If a conditional qualifier is used, it is tested for each member of the looping set and output will not occur for any that don't the test. It's this feature of being able to base output on the results of a condition that distinguish this second style of syntax. The list of output objects, of course, specifies what it is you want to output. As with the first form of syntax, the output objects may be labels, set and variable values. However, you have additional latitude in that the output objects may now consist of complex expressions of the variable values (e.g., you could compute the ratio of two variables). This is a useful feature when you need to report statistics and quantities derived from the variable values. By placing these calculations in the data section, as opposed to the model section, you avoid adding unnecessary complications to the constraints of the model. In general, you can do everything in the second form of syntax that you can do in the first, and more. However, the first form has an advantage in that it can be very concise.


483

Some examples of using @WRITEFOR for ODBC exports follow: Example 1:

@ODBC( 'TRANSPORTATION', 'SOLUTION', 'FROM', 'TO', 'VOLUME') = @WRITEFOR( LINKS( I, J) | VOLUME( I, J) #GT# 0: WAREHOUSE( I), CUSTOMER( J), VOLUME( I, J));

In this example, we exploit the ability to specify a conditional expression to weed zero shipments out of the export. The nonzero values of the VOLUME attribute are sent to the SOLUTION table in the TRANSPORTATION data source. The shipping warehouse set name is placed in column FROM, the receiving customer set name goes to column TO, and the shipping volume for the arc in placed in the VOLUME column. Example 2:

@ODBC( 'STAFFREP', 'STATS', 'RATIO') = @WRITEFOR( DEPARTMENTS( D): ON_DUTY( D) / NEEDS( D));

Here, we make use of @WRITEFOR's ability to perform computations to compute a ratio of two variables. Specifically, the ratio of on-duty staff to staffing needs by department is placed into the column RATIO of table STATS in data source STAFFREP. Note:

When exporting to data tables, receiving columns that are longer than the number of exported values can be filled out by either erasing the contents of the extra cells or leaving the extra cells untouched. The default is to leave the extra cells untouched. If you would like to erase the contents of the extra cells, you'll need to enable the Fill Out Ranges and Tables option. If this option is enabled, extra text fields will be blanked out, while extra numeric fields will be zeroed out.

484 CHAPTER 10

Exporting Data with ODBC in a PERT Model Continuing from the PERT example used in the Importing Data with ODBC in a PERT Model section above, we can add modifications to export the solution values of the earliest and latest start times (ES and LS) back out to the PERTODBC data source. We will put these into a blank table titled SOLUTION. We will also export the of the TASKS set in order to label our table. The data column, or field, that receives the TASKS should be formatted as text, while the columns receiving the ES and LS attributes should be declared as numeric. Here is a look at the blank table we will be exporting to:

Access Database: PERTODBC.MDB


485

After modifying the model to export the data back to the TASKS table, we have (with the relevant changes in bold): SETS: TASKS: TIME, ES, LS, SLACK; PRED(TASKS, TASKS); ENDSETS DATA: TASKS = @ODBC('PERTODBC', 'TASKS', 'TASKS'); PRED = @ODBC('PERTODBC', 'PRECEDENCE', 'BEFORE', 'AFTER'); TIME = @ODBC('PERTODBC'); @ODBC('PERTODBC', 'SOLUTION', 'TASKS', 'EARLIEST START', 'LATEST START') = TASKS, ES, LS; ENDDATA @FOR(TASKS(J)| J #GT# 1: ES(J) = @MAX(PRED(I, J): ES(I) + TIME(I)) ); @FOR(TASKS(I)| I #LT# LTASK: LS(I) = @MIN(PRED(I, J): LS(J) - TIME(I)); ); @FOR(TASKS(I): SLACK(I) = LS(I) - ES(I)); ES(1) = 0; LTASK = @SIZE(TASKS); LS(LTASK) = ES(LTASK);

Model: PERTODBC With the data statement: @ODBC('PERTODBC', 'SOLUTION', 'TASKS', 'EARLIEST START', 'LATEST START') = TASKS, ES, LS;

we are sending the set TASKS to the text column TASKS, and the ES and LS attributes to the numeric columns EARLIEST START and LATEST START. The data table is called SOLUTION, while the ODBC data source name is PERTODBC.

486 CHAPTER 10 Once the model has been solved, the updated data table will resemble:

Access Database: PERTODBC.MDB At the top of the solution report window, you will also notice an export summary report. There will be one report for each @ODBC statement in the model used for exporting data. This report lists details as to the operation of the @ODBC export. In the case of our PERT model, you should see the following report: Export Summary Report --------------------Transfer Method: ODBC BASED ODBC Data Source: PERTODBC Data Table Name: TASKS Columns Specified: 3 TASKS EARLIEST LATEST LINGO Column Length: 7 Database Column Length: 7

The Transfer Method will always list “ODBC BASED” when doing ODBC exports. Next, the data source and table names are listed along with the number of columns specified and the column names. The LINGO Column Length field lists the number of elements in each attribute. The Database Column Length lists the length of the receiving columns in the database. In general, the LINGO Column Length will agree with the Database Column Length. If not, LINGO must either truncate its output or it will have insufficient data to fill out the columns in the database. Export summary reports are not displayed when LINGO is in terse output mode. To place LINGO in terse output mode, click on the Terse Output checkbox on the Interface tab of the LINGO|Options dialog box.


487

This version of the PERT model and its ing database are contained in the SAMPLES directory. Feel free to run the model to experiment with it if you like—you will find it under the name PERTODBC. The ing Access database file, PERTODBC.MDB, is also in the SAMPLES subdirectory and you will need to it with the ODBC as described above in ODBC Data Sources. Note that we exported start and finish times for all the tasks in the project. If we were dealing with a large project there could be thousands of tasks to consider. With such an abundance of tasks, we might be interested in reporting only those tasks that lie on the critical path. We'll modify our PERTODBC example one last time to accomplish this using the @WRITEFOR reporting function. For those unfamiliar with the concept of a critical path, it is the subset of tasks such that if any are delayed the entire project will be delayed. Generally, and somewhat counterintuitive to what one would expect, the set of tasks on the critical path will tend to be quite small compared to the total number of tasks in a large project. A task is considered to be on the critical path when its earliest start time is equal to its latest start time (i.e., there is no slack with respect to when the task must be started). Below, we have modified PERTODBC to export only those tasks on the critical path. MODEL: SETS: TASKS: TIME, ES, LS, SLACK; PRED( TASKS, TASKS); ENDSETS DATA: TASKS = @ODBC( 'PERTODBC', 'TASKS', 'TASKS'); PRED = @ODBC( 'PERTODBC', 'PRECEDENCE', 'BEFORE', 'AFTER'); TIME = @ODBC( 'PERTODBC'); @ODBC( 'PERTODBC', 'SOLUTION', 'TASKS', 'EARLIEST START', 'LATEST START') = @WRITEFOR( TASKS( I) | ES( I) #EQ# LS( I): TASKS( I), ES( I), LS( I)); ENDDATA @FOR( TASKS( J)| J #GT# 1: ES( J) = @MAX( PRED( I, J): ES( I) + TIME( I)) ); @FOR( TASKS( I)| I #LT# LTASK: LS( I) = @MIN( PRED( I, J): LS( J) - TIME( I)); ); @FOR( TASKS( I): SLACK( I) = LS( I) - ES( I)); ES( 1) = 0; LTASK = @SIZE( TASKS); LS( LTASK) = ES( LTASK); END

488 CHAPTER 10 We specify a conditional expression to test for the earliest start time equaling the latest start time, thereby restricting the list of exported tasks to those that lie on the critical path. Note that if you limit the number of output records there won't be enough records to completely fill our output table. We can have LINGO fill out any extra fields with blanks and zeroes by enabling the Fill Out Ranges and Tables option. Doing this, the solution table will resemble the following after solving the model:

Note that that LINGO nulled out all the extra records.

11 Interfacing with Other Applications Although LINGO has a convenient, interactive interface and a large library of callable functions that make it easy to set up and solve models, there may be times when you would like to bundle LINGO's functionality into your own applications, or call functions from within your LINGO models that were written in an external programming language. LINGO makes use of the Dynamic Link Library (DLL) standard under Windows to provide you with a “hook” to access LINGO's functionality from within your own custom applications. This gives you the ability to build application specific front-ends to LINGO that allow naïve s to input data and view solutions in a simple, familiar way without having to worry about the details of LINGO modeling. Your application handles the details of driving LINGO in the background, invisible to the . LINGO also allows you to provide custom functions in the DLL format that you can call from within any model with the @ function. In the following section, The LINGO Dynamic Link Library, we document how to call the LINGO DLL in order to add LINGO's functionality to your own applications. Following this, in the section Defined Functions, we will show you how to build a function in an external programming language and call it from a LINGO model with the @ function.

The LINGO Dynamic Link Library Windows versions of LINGO include a callable DLL. The ability to call a DLL is a standard feature of all Windows development environments (e.g., Visual Basic, Delphi, and Visual C++). The LINGO DLL is supplied in either 32-bit of 64-bit configurations and, thus, will run under all current releases of Windows (Win32 and Win64). The interface to the LINGO DLL is relatively simple and gives you the ability to run a LINGO command script from within your application. Given that you can access all the major features of LINGO from a command script, the LINGO DLL interface is very powerful. You will, however, need to familiarize yourself with LINGO's command language in order to build useful command scripts. For more details on the commands available in the command language, see Command-line Commands. For an example of a script file, see A Command Script Example.

489

490 CHAPTER 11 When LINGO is installed a number of examples on calling the DLL are installed, too. These examples may be found in the Programming Samples folder below the main LINGO folder. You will find examples for each of the following development environments:  Visual C/C++  Visual Basic  Excel  FORTRAN  ASP .NET  C# .NET  VB .NET  Delphi  Java In this chapter, we will walk through detailed examples on calling the LINGO DLL using both Visual C/C++ and Visual Basic. s of other development environments will also be interested in these programming examples. Many of the ideas presented carry over to other development environments.

Staff-Scheduling Example Using the LINGO DLL To illustrate interfacing with the LINGO DLL, we will again make use of the Pluto Dogs staff-scheduling example introduced on page 54. We will create examples using both the Visual C++ and Visual Basic programming languages. We will construct a dialog box that resembles the following:

INTERFACING WITH OTHER APPLICATIONS

491

The enters the staffing requirements in the cells of the Needs column and presses the Solve button. The application then extracts the staffing requirements from the dialog box, es them along with a model to the LINGO DLL for solution, and places the results back into the dialog box for viewing. Specifically, it shows the number of employees to start on each given day in the Start column, the number of staff on duty in the On Duty column, and the total number of staff required in the Total cell. As an example, here is how the dialog box will appear after a sample run:

492 CHAPTER 11

The Model The model ed to LINGO to solve this problem is the familiar staffing model we’ve used before with some modifications, and appears below with important changes in bold: MODEL: SETS: DAYS / MON TUE WED THU FRI SAT SUN/: NEEDS, START, ONDUTY; ENDSETS [OBJECTIVE] MIN = @SUM(DAYS(I): START(I)); @FOR(DAYS(TODAY): ! Calculate number on duty; ONDUTY(TODAY) = @SUM(DAYS(D)| D #LE# 5: START(@WRAP(TODAY - D + 1, @SIZE(DAYS)))); ! Enforce staffing requirement; ONDUTY(TODAY) >= NEEDS(TODAY); @GIN(START); ); DATA: NEEDS = @POINTER(1); @POINTER(2) = START; @POINTER(3) = ONDUTY; @POINTER(4) = OBJECTIVE; @POINTER(5) = @STATUS(); ENDDATA END

Model: STAFFPTR Since the LINGO script processor will read the model, it must begin with the MODEL: command and end with the END command. The script processor treats all text between the MODEL: and END keywords as model text, as opposed to script commands. We have added the ONDUTY attribute to compute the number of employees on duty each day. We compute these figures for the purpose of ing them back to the calling application, which, in turn, posts them in the dialog box in the On Duty column. We have named the objective row OBJECTIVE for the purpose of returning the objective’s value to the calling application, so it may place the value in the Total cell.

The @POINTER Function The @POINTER function in data, init and calc sections acts as direct memory link between the calling application and the LINGO solver. This ability to do direct memory transfers of data in and out of LINGO is very powerful. It allows for the fast transfer of data without the hassle of building and parsing disk based files.


493

When you call the LINGO DLL, you can it a list of memory pointers. @POINTER( n) refers to the n-th pointer in the ed pointer list. The @POINTER function only makes sense in a model when accessing LINGO through the DLL or OLE interfaces. If the @POINTER function appears on the right-hand side of a data statement, as in the case of the following: NEEDS = @POINTER( 1);

then LINGO will initialize the objects(s) on the left-hand side of the statement from the values in memory beginning at the location referenced by the pointer on the right-hand side of the statement. On the other hand, when the @POINTER function appears on the left-hand side of a data statement, LINGO will export the values of the objects(s) on the right-hand side of the data statement to the memory location referenced by the pointer on the left-hand side of the statement. The @POINTER function reads and writes all numeric data (i.e., attribute values) using double precision floating point format. BASIC and C/C++ developers know this as the double data type, while FORTRAN developers refer to it as either REAL*8 or DOUBLE PRECISION. Set are ed as strings of ASCII text, with each member separated by a line feed character, with the list being terminated with an null, or ASCII 0. When setting aside memory locations for @POINTER, you must be sure to allocate adequate space. LINGO assumes adequate space and/or supplied values are available at the ed memory locations. If you do not allocate sufficient space or values, you may experience memory protection faults or, even worse, erroneous results without warning. So, reviewing the model's data section: DATA: NEEDS = @POINTER( 1); @POINTER( 2) = START; @POINTER( 3) = ONDUTY; @POINTER( 4) = OBJECTIVE; @POINTER( 5) = @STATUS(); ENDDATA

we see the staffing requirements attribute, NEEDS, is read from the first memory location. The START, ONDUTY, and OBJECTIVE values are written to the second, third, and fourth memory locations, respectively. Finally, the value for the @STATUS function is written to the fifth memory location. See the following section for more details on the @STATUS function.

494 CHAPTER 11

The @STATUS Function The @STATUS function returns the status of the solution process using the following codes: @STATUS Code 0 1 2 3 4 5 6 7 8 9

Interpretation Global Optimum - The optimal solution has been found, subject to current tolerance settings. Infeasible - No solution exists that satisfies all constraints. Unbounded - The objective can be improved without bound. Undetermined - The solution process failed. Feasible - A feasible solution was found that may, or may not, be the optimal solution. Infeasible or Unbounded - The preprocessor determined the model is either infeasible or unbounded. Turn off presolving and re-solve to determine which. Local Optimum - Although a better solution may exist, a locally optimal solution has been found. Locally Infeasible - Although feasible solutions may exist, LINGO was not able to find one. Cutoff - The objective cutoff level was achieved. Numeric Error - The solver stopped due to an undefined arithmetic operation in one of the constraints.

In general, if @STATUS does not return a code of 0, 4, 6, or 8, the solution is of little use and should not be trusted. In fact, if @STATUS does not return 0, 4, 6, or 8, in many cases LINGO will not even export data to the @POINTER memory locations. The @POINTER function reads and writes all data using double precision floating point format. BASIC and C/C++ developers know this as the double data type, while FORTRAN developers refer to it as either REAL*8 or DOUBLE PRECISION. When setting aside memory locations for @POINTER, you must be sure to allocate adequate space. LINGO assumes adequate space and/or supplied values are available at the ed memory locations. If you do not allocate sufficient space or values, you may experience memory protection faults or, even worse, erroneous results without warning.

Functions Exported by the LINGO DLL The LINGO DLL exports twelve functions.The exported functions are contained in the file Lingo14\Lingd14.Dll. The library file Lingo14\Programming Samples\Lingd14.lib may be used to import these functions into your own custom application. Below is a list of the functions exported by the LINGO DLL along with a brief description of each routine’s functionality. The definitions are written using C language conventions. Refer to the programming samples in the following sections for specific examples of the use of each of these routines using Visual C and Visual Basic.


495

void LSclearPointersLng( pLSenvLINGO pL) This clears out the list of @POINTER() pointers to memory transfer areas established through calls to LSsetPointerLng(). Arguments: pL

Pointer to a LINGO environment created by a previous call to LScreateEnvLng().

int LScloseLogFileLng( pLSenvLINGO pL) This closes LINGO’s log file that was opened previously by a call to LSopenLogFileLng(). Arguments: pL


Return Value: Returns 0 if no error occurred. Otherwise, one of the nonzero error codes listed below in the section LINGO DLL Error Codes is returned.

pLSenvLINGO CALLTYPE LScreateEnvLng() This creates a LINGO environment object. All other LINGO DLL routines require a valid pointer to a LINGO environment object. You should free this object at the end of your application by calling LSdeleteEnvLng(). LScreateEnvLng() will search the application's startup directory and the system path for a valid LINGO license file. If you would prefer to LINGO the license key directly, refer to the following function, LScreateEnvLicenseLNG(). Return Value: Returns 0 if an error occurred, otherwise, a pointer to a LINGO environment object is returned.

pLSenvLINGO CALLTYPE LScreateEnvLicenseLng( char* pcLicenseKey, int* pnErr) This creates a LINGO environment object. All other LINGO DLL routines require a valid pointer to a LINGO environment object. You should free this object at the end of your application by calling LSdeleteEnvLng(). LScreateEnvLicenseLng() requires that you the license key directly. If you would prefer to have LINGO search the application's startup directory and the system path for a valid LINGO license file, refer to the previous function, LScreateEnvLng().

496 CHAPTER 11 Arguments: pcLicenseKey

pnError

Pointer to a text string containing a LINGO license key. The license key may be found at the bottom of your LINGO license file, typically stored in the main LINGO directory under the file name LNDLNG??.LIC, where ?? is LINGO's version number. The license key is delimited in the license file by a greater-than (>) character. Pointer to an integer that returns 0 if no problem or an error code listed in section LINGO DLL Error Codes.

Return Value: Returns 0 if an error occurred, otherwise, a pointer to a LINGO environment object is returned.

int LSdeleteEnvLng( pLSenvLINGO pL) This deletes a LINGO environment object previously created through a call to LScreateEnvLng(), which frees up the system memory allocated to the LINGO object. Arguments: pL



int LSexecuteScriptLng( pLSenvLINGO pL, char* pcScript) This routine is the main workhorse of the LINGO DLL that processes LINGO command scripts. The script may be contained entirely in memory, or it may contain one or more TAKE commands to load scripts from disk. Arguments: pL pcScript

Pointer to a LINGO environment created by a previous call to LScreateEnvLng(). Pointer to a character string containing a LINGO command script. Each line must be terminated with a linefeed character (ASCII 10), and the entire script must be terminated with a NULL (ASCII 0).



497

int LSgetCallbackInfoLng( pLSenvLINGO pL, int nObject, void* pResult) You may establish a function in your application that the LINGO solver calls back to at regular intervals to keep you abreast of its progress. We refer to this type of routine as being a callback function and you may refer to section Callback Functions for more information. Your callback function may then call LINGO through LSgetCallbackInfoLng() to request specific information from the solver while it is processing a model. Arguments: pL


nObject Index 0

Index of the information item you’re seeking. Current possibilities are: Name LS_IINFO_VARIABLES_LNG

Type Int

1

LS_IINFO_VARIABLES_INTEGER_LNG

Int

2

LS_IINFO_VARIABLES_NONLINEAR_LNG

Int

3

LS_IINFO_CONSTRAINTS_LNG

Int

4

LS_IINFO_CONSTRAINTS_NONLINEAR_LNG

Int

5

LS_IINFO_NONZEROS_LNG

Int

6

LS_IINFO_NONZEROS_NONLINEAR_LNG

Int

7

LS_IINFO_ITERATIONS_LNG

Int

8

LS_IINFO_BRANCHES_LNG

Int

9

LS_DINFO_SUMINF_LNG

Double

10 11

LS_DINFO_OBJECTIVE_LNG LS_DINFO_MIP_BOUND_LNG

Double Double

12

LS_DINFO_MIP_BEST_OBJECTIVE_LNG

Double

pResult

Information Item Total number of variables Number of integer variables Number of nonlinear variables Total number of constraints Number of nonlinear constraints Total nonzero matrix elements Number of nonlinear nonzero matrix elements Number of iterations Number of branches (IPs only) Sum of infeasibilities Objective value Objective bound (IPs only) Best objective value found so far (IPs only)

Pointer to where you want LINGO to store the results of your query. LINGO will place the result at this address. You must

498 CHAPTER 11 allocate four bytes of memory for ints and eight bytes of memory for doubles beforehand. Return Value: Returns 0 if no error occurred. Otherwise, one of the nonzero error codes listed below in the section LINGO DLL Error Codes is returned.

int LSgetCallbackVarPrimalLng( pLSenvLINGO pL, char* pcVarName, double* pdPrimals) You may establish a function in your application that the LINGO solver calls back to at regular intervals to keep you abreast of its progress. We refer to this type of routine as being a callback function, and you may refer to section Callback Functions for more information. Your callback function may then call LINGO through LSgetCallbackVarPrimal() to request the current values for attributes and/or scalar variables. This will allow you to keep your s posted on the values of the model's variables in the current best solution found so far. This routine will only return values successfully when solving integer models, or when solving models with the global solver. Arguments: pL


pcVarName Pointer to a character string containing the name of the scalar variable or attribute. The name must be terminated with either a blank or a NULL (ASCII 0). pdPrimals

Points to where LINGO should write the variable value(s). You must ensure that enough space is available for the full set of values to avoid memory protection faults.

Return Value: Returns 0 if no error occurred. Otherwise, one of the nonzero error codes listed in section LINGO DLL Error Codes is returned.

int LSopenLogFileLng( pLSenvLINGO pL, char *pcLogFile) This creates a file for LINGO to write a log to while processing your script. In general, you should always try to create a log file (at least in the early stages of your project) to assist with debugging. If an error is occurring and you are not exactly sure why, then it is always a good idea to refer to the log file for a clue. Arguments: pL pcLogFile

Pointer to a LINGO environment created by a previous call to LScreateEnvLng(). Pointer to a character string containing the pathname for your log file.



499

int LSsetCallbackErrorLng( pLSenvLINGO pL, lngCBFuncError_t pcbf, void* pData) Use this routine to specify a callback function that LINGO will call whenever an error is encountered. This allows your application to keep close watch on any unusual conditions that might arise while processing your script. Arguments: pL


pcbf

Pointer to your callback routine.

pData

This is a specified pointer. You may use it to point to any data you might need to reference from your callback function. LINGO merely es the value of this pointer through to your callback function. You may set this pointer to NULL if it is not required.

Return Value: Returns 0 if no error occurred. Otherwise, one of the nonzero error codes listed below in the section LINGO DLL Error Codes is returned. The callback function you supply must use the standard call convention and must have the following interface: int MyErrCallback( pLSenvLINGO pL, void* pData, int nErrorCode, char* pcErrorText); Your computer will most likely crash if you don’t follow this interface specification exactly. The LINGO error code is reported through the nErrorCode argument, along with the error text in pcErrorText. You should set aside at least 200 bytes for the error text. The list of LINGO error codes can be found in Appendix B, Error Messages.

int LSsetCallbackSolverLng( pLSenvLINGO pL, lngCBFuncError_t pcbf, void* pData) Use this routine to specify a callback function that LINGO will call at frequent intervals when solving a model. Arguments: pL


pcbf

Pointer to your callback routine.

pData

This is a specified pointer. You may use it to point to any data you might need to reference from your callback function. LINGO merely es the value of this pointer through to your callback function. You may set this pointer to NULL if it is not required.


500 CHAPTER 11 The callback function you supply must use the standard call convention and must have the following interface: int MySolverCallback( pLSenvLINGO pL, int nReserved, void* pData); Your computer will most likely crash if you don’t follow this interface specification exactly. The nReserved argument is reserved for future use and may be ignored.

int CALLTYPE LSsetPointerLng( pLSenvLINGO pL, double* pdPointer, int* pnPointersNow) Call this routine one or more times to a list of memory pointers to LINGO. These pointers are used by LINGO’s @POINTER() function for moving data into and solutions out of the solver. In other words, this allows you to have direct memory links with LINGO for fast and convenient data transfer. Arguments: pL


pdPointer

Pointer to a memory transfer location to be used by an instance of @POINTER().

pnPointersNow

This is a pointer to an integer variable in which LINGO returns the current number of entries in the @POINTER() pointer list. Thus, on the first call to LSsetPointersLng(), this argument will return a value of 1. The pointer list may be cleared at any time by a call to LSclearPointersLng().



501

LINGO DLL Error Codes Most of the LINGO DLL functions return an error code. Below is a list of all possible codes: Value 0 1

Name LSERR_NO_ERROR_LNG LSERR_OUT_OF_MEMORY_LNG

2 3

LSERR_UNABLE_TO_OPEN_LOG_FILE_LNG LSERR_INVALID_NULL_POINTER_LNG

4

LSERR_INVALID_INPUT_LNG

5

LSERR_INFO_NOT_AVAILABLE_LNG

6

LSERR_UNABLE_TO_COMPLETE_TASK_LNG

7

LSERR_INVALID_LICENSE_KEY_LNG

8

LSERR_INVALID_VARIABLE_NAME_LNG

1000

LSERR_JNI_CALLBACK_NOT_FOUND_LNG

Descriptions No error. Out of dynamic system memory. Unable to open the log file. A NULL pointer was ed to a routine that was expecting a non-NULL pointer. An input argument contained invalid input. A request was made for information that is not currently available. Unable to successfully complete the specified task. The license key ed to LScreateEnvLicenceLng() was invalid. A variable name ed to LSgetCallbackVarPrimal() was invalid. A valid callback function was not found in the calling Java application.

ing DLLs The main LINGO DLL requires many additional component DLLs. All the required DLLs are installed as part of the normal LINGO installation in the main LINGO folder. If you need a specific list of all the ing DLLs required by the LINGO DLL, then you can open Lingd14.Dll in the Dependency Walker utility available at no charge from the following website: http://www.dependencywalker.com. The Dependency Walker program is a useful tool that lists all the required DLLs for an application, their current location on your system, and flags any DLLs that are not currently available.

Staff Scheduling Using the LINGO DLL and Visual C++ In this section, we will illustrate the use of Microsoft Visual C/C++ to build an application that interfaces with the LINGO DLL to solve the staff-scheduling problem presented above. This section assumes the reader is well versed in the use of Visual C++. Visual Basic s may want to jump ahead to the Visual Basic Example section.

502 CHAPTER 11 If you would rather skip the details involved in constructing this project and would prefer to experiment with the completed application, it can be found under the path \LINGO14\Programming Samples\VC++\STAFF1\STAFF.EXE.

Building the Application in C++ We will be working from a code base generated using Visual C++’s AppWizard facility. The generated code is for a dialog based application. In other words, the interface to the application will consist of a single dialog box. This dialog box will be modified, so it resembles the one illustrated above with fields for staffing requirements, daily employee starts, and so on. The source code for this project may be found in the Programming Samples\STAFF1 subdirectory or you can create your own project using Visual C/C++ 6.0 as follows: 1. 2. 3. 4. 5. 6.

Start Visual C++ and issue the File|New command. Select the Projects tab, give the project a name, select MFC AppWizard(Exe) as the type of application, and click the OK button. Click the dialog based radio button and then the Next button. Clear the About box button, check the OLE Automation button, check the OLE Controls button, and press the Next button. OLE is required for some of the features in the LINGO DLL. Click the Next button again. Click the Finish button.

INTERFACING WITH OTHER APPLICATIONS After completing these steps, the following summary of the project should be shown:

Click the OK button, and AppWizard will generate the skeleton code base.

503

504 CHAPTER 11 Use the resource editor to modify the application’s dialog box, so it resembles the following:

Next, use the ClassWizard in Visual C++ to associate member variables with each of the fields in the dialog box. From the View menu, select the ClassWizard command and then select the Member Variables tab. At this point, the LINGO DLL import library must be added to the project in order to make the LINGO DLL available to the code. Do this by running the Project|Add to Project|Files command and select the file \LINGO14\Programming Samples\LINGD14.LIB for addition to the project.


505

After doing that, add the definitions of the LINGO DLL routines to the project. Simply include the Lingd14.h header file at the top of the dialog class header file as follows (code changes listed in bold): // staffDlg.h : header file // #include "lingd14.h" #if !defined(AFX_STAFFDLG_H__74D746B7_CA4D_11D6_AC89_00010240D2AE__INCLUD ED_) #define AFX_STAFFDLG_H__74D746B7_CA4D_11D6_AC89_00010240D2AE__INCLUDED_ #if _MSC_VER > 1000 #pragma once #endif // _MSC_VER > 1000 ///////////////////////////////////////////////////////////////////// //////// // CStaffDlg dialog . . .

All the groundwork has now been laid down and we’re ready to begin writing the actual code to call LINGO to solve the staffing model. Go to the Resource View of the project, open the dialog box resource, and then double click on the Solve button. You will be prompted to create a handler function for the button, which should be named OnSolve. Now, edit the stub version of OnSolve, so it contains the following code: void CStaffDlg::OnSolve() { int nError, nPointersNow; CString csScript, cs; double dNeeds[7], dStart[7], dOnDuty[7], dStatus, dTotal; // Get 's staffing requirements from our dialog box UpdateData(); // Load staffing requirements into the LINGO transfer array. // LINGO uses double precision for all values. dNeeds[ 0] = (double) m_nNeedsMon; dNeeds[ 1] = (double) m_nNeedsTue; dNeeds[ 2] = (double) m_nNeedsWed; dNeeds[ 3] = (double) m_nNeedsThu; dNeeds[ 4] = (double) m_nNeedsFri; dNeeds[ 5] = (double) m_nNeedsSat; dNeeds[ 6] = (double) m_nNeedsSun;

506 CHAPTER 11 // create the LINGO environment object pLSenvLINGO pLINGO; pLINGO = LScreateEnvLng(); if ( !pLINGO) { AfxMessageBox("Unable to create LINGO Environment"); return; } // Open LINGO's log file nError = LSopenLogFileLng( pLINGO, "LINGO.log"); if ( nError) goto ErrorExit; // memory transfer pointers to LINGO // @POINTER(1) nError = LSsetPointerLng( pLINGO, dNeeds, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(2) nError = LSsetPointerLng( pLINGO, dStart, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(3) nError = LSsetPointerLng( pLINGO, dOnDuty, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(4) nError = LSsetPointerLng( pLINGO, &dTotal, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(5) nError = LSsetPointerLng( pLINGO, &dStatus, &nPointersNow); if ( nError) goto ErrorExit; // Here is the script we want LINGO to run csScript = "SET ECHOIN 1\n"; csScript = csScript + "TAKE \\LINGO14\\SAMPLES\\STAFFPTR.LNG\n"; csScript = csScript + "GO\n"; csScript = csScript + "QUIT\n"; // Run the script dStatus = -1.e0; nError = LSexecuteScriptLng( pLINGO, (LPCTSTR) csScript); // Close the log file LScloseLogFileLng( pLINGO); // Any problems? if ( nError || dStatus) {


507

// Had a problem AfxMessageBox("Unable to solve!"); } else { // Everything went ok ... load results into the dialog box m_csStartMon.Format( "%d", (int) dStart[0]); m_csStartTue.Format( "%d", (int) dStart[1]); m_csStartWed.Format( "%d", (int) dStart[2]); m_csStartThu.Format( "%d", (int) dStart[3]); m_csStartFri.Format( "%d", (int) dStart[4]); m_csStartSat.Format( "%d", (int) dStart[5]); m_csStartSun.Format( "%d", (int) dStart[6]); m_csOnDutyMon.Format( m_csOnDutyTue.Format( m_csOnDutyWed.Format( m_csOnDutyThu.Format( m_csOnDutyFri.Format( m_csOnDutySat.Format( m_csOnDutySun.Format(

"%d", "%d", "%d", "%d", "%d", "%d", "%d",

(int) (int) (int) (int) (int) (int) (int)

dOnDuty[0]); dOnDuty[1]); dOnDuty[2]); dOnDuty[3]); dOnDuty[4]); dOnDuty[5]); dOnDuty[6]);

m_csCost.Format( "%g", dTotal); UpdateData( FALSE); } goto Exit; ErrorExit: cs.Format( "LINGO Errorcode: %d", nError); AfxMessageBox( cs); return; Exit: LSdeleteEnvLng( pLINGO); }

The first section of OnSolve is straightforward and deals with extracting the ’s staffing requirements from the dialog box. Note that the data is stored in a double precision array rather than as integers. This is because these values will be ed to LINGO, which only es values in double precision format. Our first call to LINGO creates the LINGO environment object with the following code: // create the LINGO environment object pLSenvLINGO pLINGO; pLINGO = LScreateEnvLng(); if ( !pLINGO) { AfxMessageBox("Unable to create LINGO Environment"); return; }

508 CHAPTER 11 The pLSenvLINGO data type is defined in the LINGO header file, lingd14.h. Then, a log file for LINGO is established with the following call: // Open LINGO's log file nError = LSopenLogFileLng( pLINGO, "LINGO.log"); if ( nError) goto ErrorExit;

As mentioned above, opening a log file for LINGO is good practice, at least when you’re debugging the application. If something should go wrong, the log file will generally contain a helpful clue. Our next step is to LINGO the physical addresses it will use to resolve the @POINTER() function references used in the data section of the model (refer to the Staff Scheduling Example Using the LINGO DLL section above for more details). This is done as follows: // memory transfer pointers to LINGO // @POINTER(1) nError = LSsetPointerLng( pLINGO, dNeeds, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(2) nError = LSsetPointerLng( pLINGO, dStart, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(3) nError = LSsetPointerLng( pLINGO, dOnDuty, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(4) nError = LSsetPointerLng( pLINGO, &dTotal, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(5) nError = LSsetPointerLng( pLINGO, &dStatus, &nPointersNow); if ( nError) goto ErrorExit;

In summary, when LINGO is called to solve the model, the staffing needs are ed to LINGO in the dNeeds array via the @POINTER( 1) reference. Solution information is ed from LINGO back to the application in the dStart, dOnDuty, dTotal, and dStatus structures via @POINTER() references 2 through 5, respectively. If any of this is unfamiliar, review The @POINTER() Function section under the Staff Scheduling Example Using the LINGO DLL section above. Next, the following code is used to build the command script: // Here is the script we want LINGO to run csScript = "SET ECHOIN 1\n"; csScript = csScript + "TAKE \\LINGO14\\SAMPLES\\STAFFPTR.LNG\n"; csScript = csScript + "GO\n"; csScript = csScript + "QUIT\n";


509

The script consists of four commands, which are each terminated with a new line character (ASCII 10). The end of the script is terminated with a NULL character (ASCII 0). These commands and their functions are: Command SET ECHOIN 1 TAKE

GO QUIT

Function Causes LINGO to echo input to the log file. This is a useful feature while building and debugging an application. Loads the model from a disk file. The TAKE command may be used to load model files, as in this example. It may also be used to run nested command scripts contained in files. Calls the solver to optimize the model. Closes down LINGO’s script processor and returns control to the calling application.

For more information on scripting commands, refer to Chapter 6, Command-line Commands. At this point, the script is ready to be ed off to LINGO for processing with the following call: // Run the script dStatus = -1.e0; nError = LSexecuteScriptLng( pLINGO, (LPCTSTR) csScript);

Note that dStatus is initialized to –1. LINGO returns the model status through memory transfer location number 5 (i.e., @POINTER( 5)) to the dStatus variable. LINGO will only return a status value if it was able to solve the model. If an unexpected error were to occur, LINGO might not ever reach the solution phase. In that case, dStatus would never be set. Initializing dStatus to a negative value tests for this situation. Given that LINGO returns only non-negative status codes, a negative status code would indicate a problem. This method of error trapping is effective, but not particularly elegant. Another method that involves specifying an error callback routine is demonstrated below. Now, LINGO’s log file may be closed down by calling LScloseLogFileLng(): // Close the log file LScloseLogFileLng( pLINGO);

Next, the following code tests to see if LINGO was able to find an optimal solution: // Any problems? if ( nError || dStatus) // Had a problem AfxMessageBox("Unable to solve!"); } else {

Note that the code returned in nError pertains only to the mechanical execution of the script processor. It has nothing to do with the status of the model’s solution, which is returned in dStatus via the use of the @POINTER() and @STATUS() functions (see the Staff Scheduling Example Using the LINGO DLL section above). A model may actually be infeasible or unbounded, and the error code returned by LSexecuteScript() will give no indication. Thus, it is important to add a mechanism to return a solution’s status to the calling application, as done here with the @STATUS() -> @POINTER(5) ->

510 CHAPTER 11 dStatus link. The end result in the sample code is that “Unable to solve” is printed if either error condition occurs. A more -friendly application would offer more specific information regarding the error condition. As a final step, in order to avoid memory leaks in your application, to free up LINGO’s environment when through: Exit: LSdeleteEnvLng( pLINGO);

If everything has been entered correctly, you should now be able to build and run the project.

Visual Basic Staff Scheduling Using the LINGO DLL In this section, we will illustrate the use of Microsoft Visual Basic to build an application that interfaces with the LINGO DLL to solve the staff-scheduling problem presented above. This section assumes the reader is well versed in the use of Visual Basic. If you would rather skip the details involved in constructing this project and would prefer to experiment with the completed appliction, it can be found under the path \LINGO14\Programming Samples\VBasic\STAFFVB.EXE.

Building the Application You can build the project as follows: 1. 2. 3.

Start Visual Basic and then issue the File|NewProject command. Select Standard Exe and click the OK button. Use the resource editor to format the project’s form until it resembles the following:


511

Now, add handler code for the two buttons in the form. The Exit button is easy. All the Exit button does when the clicks it is exit the application. So, double click on the Exit button and enter the following code: Private Sub Exit_Click() End End Sub

A little more work will be required to setup the Solve button. When the Solve button is pressed, the application will retrieve the staffing requirements from the form, them along with the model to the LINGO script processor (LGVBSCRIPT) to solve the model, retrieve the solution, and post the solution in the form. First, we must declare the external LINGO functions. Do this by adding the \LINGO14\Programming Samples\LINGD14.BAS module to the project using the Project|Add Module command in VB. This module contains the definitions of all the exported function in the LINGO DLL, and makes them available to our project. Now, add the handler code for the Solve button. Go to the form, double click on the Solve button, and enter the following code: Private Sub Solve_Click() ' ' ' '

Calls the LINGO DLL to solve the staffing model in STAFFPTR.LNG. Staffing requirements are taken from the dialog box.

' Get the staffing needs from the dialog box Dim dNeeds(7) As Double For i = 1 To 7 dNeeds(i) = Needs(i - 1).Text Next i ' Create the LINGO environment object Dim pLINGO As Long pLINGO = LScreateEnvLng() If pLINGO = 0 Then MsgBox ("Unable to create LINGO Environment.") GoTo FinalExit End If ' Open LINGO's log file Dim nError As Long nError = LSopenLogFileLng(pLINGO, "LINGO.log") If nError <> 0 Then GoTo ErrorExit ' memory transfer pointers to LINGO Dim dStart(7) As Double, dOnDuty(7) As Double

Dim dTotal As Double, dStatus As Double

512 CHAPTER 11 ' @POINTER(1) nError = LSsetPointerLng(pLINGO, dNeeds(1), nPointersNow) If nError <> 0 Then GoTo ErrorExit ' @POINTER(2) nError = LSsetPointerLng(pLINGO, dStart(1), nPointersNow) If nError <> 0 Then GoTo ErrorExit ' @POINTER(3) nError = LSsetPointerLng(pLINGO, dOnDuty(1), nPointersNow) If nError <> 0 Then GoTo ErrorExit ' @POINTER(4) nError = LSsetPointerLng(pLINGO, dTotal, nPointersNow) If nError <> 0 Then GoTo ErrorExit ' @POINTER(5) nError = LSsetPointerLng(pLINGO, dStatus, nPointersNow) If nError <> 0 Then GoTo ErrorExit ' Build LINGO's command script (commands ' are terminated with an ASCII 10 Dim cScript As String ' Causes LINGO to echo input cScript = "SET ECHOIN 1" & Chr(10) ' Read in the model file cScript = cScript & _ "TAKE \LINGO14\SAMPLES\STAFFPTR.LNG" & Chr(10) ' Solve the model cScript = cScript & "GO" & Chr(10) ' Quit LINGO DLL cScript = cScript & "QUIT" & Chr(10) ' Mark end of script with a null byte cScript = cScript & Chr(0) ' Run the script dStatus = -1# nError = LSexecuteScriptLng(pLINGO, cScript) ' Close the log file LScloseLogFileLng (pLINGO) ' Problems? If nError > 0 Or dStatus <> 0 Then MsgBox ("Unable to solve!") GoTo ErrorExit End If


513

' Place Start values in dialog box For i = 1 To 7 Start(i - 1).Caption = dStart(i) Next i ' Place On Duty values in dialog box For i = 1 To 7 OnDuty(i - 1).Caption = dOnDuty(i) Next i ' Put Total staffing in dialog box Total.Caption = dTotal LSdeleteEnvLng (pLINGO) GoTo FinalExit: ErrorExit: MsgBox ("LINGO Error Code: " & nError&) LSdeleteEnvLng (pLINGO) FinalExit: End Sub

The first section of the Solve_Click procedure is straightforward and deals with extracting the ’s staffing requirements from the dialog box. Note that the data is stored in a double precision array, rather than as integers. This is because these values will be ed to LINGO, which only es numeric values in double precision format. The first call to LINGO creates the LINGO environment object with the following code: ' Create the LINGO environment object Dim pLINGO As Long pLINGO = LScreateEnvLng() If pLINGO = 0 Then MsgBox ("Unable to create LINGO Environment.") GoTo FinalExit End If

Next, a log file for LINGO is established with the call: ' Open LINGO's log file Dim nError As Long nError = LSopenLogFileLng(pLINGO, "LINGO.log") If nError <> 0 Then GoTo ErrorExit

As mentioned above, opening a log file for LINGO is good practice, at least when you’re debugging the application. If something should go wrong, the log file will generally contain a helpful clue.

514 CHAPTER 11 The next step is to LINGO the physical addresses it will use to resolve the @POINTER() function references used in the data section of the model (refer to the Staff Scheduling Example Using the LINGO DLL section above for more details). This is done as follows: ' memory transfer pointers to LINGO Dim dStart(7) As Double, dOnDuty(7) As Double Dim dTotal As Double, dStatus As Double ' @POINTER(1) nError = LSsetPointerLng(pLINGO, dNeeds(1), nPointersNow) If nError <> 0 Then GoTo ErrorExit ' @POINTER(2) nError = LSsetPointerLng(pLINGO, dStart(1), nPointersNow) If nError <> 0 Then GoTo ErrorExit ' @POINTER(3) nError = LSsetPointerLng(pLINGO, dOnDuty(1), nPointersNow) If nError <> 0 Then GoTo ErrorExit ' @POINTER(4) nError = LSsetPointerLng(pLINGO, dTotal, nPointersNow) If nError <> 0 Then GoTo ErrorExit ' @POINTER(5) nError = LSsetPointerLng(pLINGO, dStatus, nPointersNow) If nError <> 0 Then GoTo ErrorExit

In summary, when LINGO is called to solve the model, the staffing needs are ed to LINGO in the dNeeds array via the @POINTER( 1) reference. Solution information is ed from LINGO back to the application in the dStart, dOnDuty, dTotal, and dStatus structures via @POINTER() references 2 through 5, respectively. If any of this is unfamiliar, review The @POINTER() Function section under the Staff Scheduling Example Using the LINGO DLL section above. Next, the following code is used to build the command script: ' Build LINGO's command script (commands ' are terminated with an ASCII 10 Dim cScript As String ' Causes LINGO to echo input cScript = "SET ECHOIN 1" & Chr(10) ' Read in the model file cScript = cScript & _ "TAKE \LINGO14\SAMPLES\STAFFPTR.LNG" & Chr(10) ' Solve the model cScript = cScript & "GO" & Chr(10) ' Quit LINGO DLL cScript = cScript & "QUIT" & Chr(10) ' Mark end of script with a null byte cScript = cScript & Chr(0)


515

The script consists of four commands, which are each terminated with a new line character (ASCII 10). The end of the script is terminated with a NULL character (ASCII 0). These commands and their functions are: Command SET ECHOIN 1 TAKE

GO QUIT

Function Causes LINGO to echo input to the log file. This is a useful feature while building and debugging an application. Loads the model from a disk file. The TAKE command may be used to load model files, as in this example. It may also be used to run nested command scripts contained in files. Calls the solver to optimize the model. Closes down LINGO’s script processor and returns control to the calling application.

At this point, the script is ready to be ed off to LINGO for processing with the following call: ' Run the script dStatus = -1# nError = LSexecuteScriptLng(pLINGO, cScript)

Note that dStatus is initialized to –1. LINGO returns the model status through memory transfer location number 5 (i.e., @POINTER( 5)) to the dStatus variable. LINGO will only return a status value, if it was able to solve the model. If an unexpected error were to occur, LINGO might not ever reach the solution phase. In that case, dStatus would never be set. Initializing dStatus to a negative value tests for this situation. Given that LINGO returns only non-negative status codes, a negative status code would indicate a problem. This method of error trapping is effective, but not particularly elegant. Another method that involves specifying an error callback routine is demonstrated below. Now, LINGO’s log file may be closed down by calling LScloseLogFileLng(): ' Close the log file LScloseLogFileLng (pLINGO)

Next, the following code tests to see if LINGO was able to find an optimal solution: ' Problems? If nError > 0 Or dStatus <> 0 Then MsgBox ("Unable to solve!") GoTo ErrorExit End If

Note that the code returned in nError pertains only to the mechanical execution of the script processor. It has nothing to do with the status of the model’s solution, which is returned in dStatus via the use of the @POINTER() and @STATUS() functions (see the Staff Scheduling Example Using the LINGO DLL section above). A model may actually be infeasible or unbounded, and the error code returned by LSexecuteScriptLng()will give no indication. Thus, it is important to add a mechanism to return a solution’s status to the calling application, as done here with the @STATUS() -> @POINTER(5) -> dStatus link. The end result in the sample code is that “Unable to solve” is printed if either error condition occurs. A more -friendly application would offer more specific information regarding the error condition.

516 CHAPTER 11 As a final step, in order to avoid a memory leak in the application, to free up LINGO’s environment when through: LSdeleteEnvLng (pLINGO) GoTo FinalExit:

If everything has been entered correctly, you should now be able to run the project.

ing Set with @POINTER In the previous examples, when using the @POINTER function, we only ed attribute values back and forth. You may also set using @POINTER, with the primary difference being that the set are ed as ASCII text, as opposed to double precision numeric values. In addition, each set member is separated by a line feed (ASCII 10), with the end of the list of set denoted with a null (ASCII 0) character. In order to illustrate ing set via @POINTER, we will modify the knapsack problem discussed above in section Binary Integer Example - The Knapsack Problem. You will recall that the data in a knapsack problem consists of the list of potential items to include in the knapsack, their weight, their utility/value, and the knapsack's capacity. We will all this data to LINGO via the @POINTER function, including the set of potential items. After solving the model, we will create a new set, derived from the original set of items, which contains only those items that are included in the optimal solution. Finally, we will use the @POINTER function to the optimal set of items back to our calling program so that we can display them for the . We will develop this example using the C programming language, however, the concepts should carry over in a straightforward manner to all other development environments. The code for this example may be found in the Programming Samples\VC++\Knapsack folder off the main LINGO directory.


The Model Here is a copy of the knapsack model our application will load into LINGO: MODEL: SETS: ITEMS: INCLUDE, WEIGHT, RATING; ENDSETS DATA: ITEMS = @POINTER( 1); WEIGHT = @POINTER( 2); RATING = @POINTER( 3); KNAPSACK_CAPACITY = @POINTER( 4); ENDDATA SUBMODEL SACK: MAX = @SUM( ITEMS: RATING * INCLUDE); @SUM( ITEMS: WEIGHT * INCLUDE) <= KNAPSACK_CAPACITY; @FOR( ITEMS: @BIN( INCLUDE)); ENDSUBMODEL CALC: !keep output to a minimum; @SET( 'TERSEO', 1); !solve the model; @SOLVE( SACK); !fix the INCLUDE attribute to it's optimal value; @FOR( ITEMS( I): INCLUDE( I) = INCLUDE( I)); ENDCALC SETS: !construct a set of the optimal items; ITEMSUSED( ITEMS) | INCLUDE( &1) #GT# .5:; ENDSETS DATA: !send optimal items set back to caller; @POINTER( 5) = ITEMSUSED; !along with the solver status; @POINTER( 6) = @STATUS(); ENDDATA END

Model: SACK

517

518 CHAPTER 11 Note:

Some of the model features in this example take advantage of the scripting capability in LINGO models. Scripting in models is discussed in more detail in Chapter 13, Programming LINGO.

In the model's sets section: SETS: ITEMS: INCLUDE, WEIGHT, RATING; ENDSETS

We define the ITEMS set, which will store the set of potential items for the knapsack. Each item has the following attributes:   

INCLUDE  a binary variable indicating whether or not the item is to be included in the optimal knapsack. WEIGHT  the item's weight RATING  the items utility, or value.

In the data section, we ask LINGO to import all the data for the model via the @POINTER function: DATA: ITEMS = @POINTER( 1); WEIGHT = @POINTER( 2); RATING = @POINTER( 3); KNAPSACK_CAPACITY = @POINTER( 4); ENDDATA

Note that in addition to strictly numeric values, we are also importing the set ITEMS. This set will be ed from our calling application as an ASCII string. Next, we have the actual knapsack model. We partition the model as a submodel within our main model (the concept of submodels is discussed further in Chapter 13): SUBMODEL SACK: MAX = @SUM( ITEMS: RATING * INCLUDE); @SUM( ITEMS: WEIGHT * INCLUDE) <= KNAPSACK_CAPACITY; @FOR( ITEMS: @BIN( INCLUDE)); ENDSUBMODEL

This submodel contains three expressions. First, there's the objective that maximizes total utility of the selected items. Second, there is a constraint that forces total weight to not exceed capacity. Finally, via the @BIN function, we force the INCLUDE attribute to be either 0 or 1, given that fractional solution do not make sense for this model.


519

The next section is a calc section where we perform three steps: CALC: !keep output to a minimum; @SET( 'TERSEO', 1); !solve the model; @SOLVE( SACK); !fix the INCLUDE attribute to it's optimal value; @FOR( ITEMS( I): INCLUDE( I) = INCLUDE( I)); ENDCALC

The first step sets the TERSEO function to minimize output from LINGO. Next, we use the @SOLVE function to solve the knapsack submodel. Finally,. we loop over the ITEMS set, fixing the INCLUDE attribute to their optimal values in the SACK submodel. The reason we need to fix their values is that we will need them in the following sets section where we use a set hip condition to generate the optimal set of items in the knapsack (set hip conditions will reject variables that aren't fixed in value). Next, we derive the set ITEMSUSED from the original set of all potential items, however, we only include those items that have a nonzero value in the optimal solution: SETS: !construct a set of the optimal items; ITEMSUSED( ITEMS) | INCLUDE( &1) #GT# .5:; ENDSETS

As our last step, we construct a data section to send the set of optimal items back to the calling program via @POINTER: DATA: !send optimal items set back to caller; @POINTER( 5) = ITEMSUSED; !along with the solver status; @POINTER( 6) = @STATUS(); ENDDATA

520 CHAPTER 11

The Code Here is a copy of the C code we will use to drive our application: #include <stdlib.h> #include <string.h> #include "..\..\lingd14.h" /* Solves a simple knapsack problem, ing all data to Lingo and retrieving the optimal set of knapsack items for display */ void main() { // input data for model: // potential items in knapsack char pcItems[256] = "ANT_REPEL \n BEER \n BLANKET \n" "BRATWURST \n BROWNIES \n FRISBEE \n SALAD \n" "WATERMELON"; // and their weights double pdItemWeight[8] = { 1, 3, 4, 3, 3, 1, 5,10}; // and their rankings double pdItemRank[8]

= { 2, 9, 3, 8,10, 6, 4,10};

// knapsack size double dSackSize = 15; // other declarations int i, nPointersNow, nError; double dStatus=-1.0; char pcScript[256]; char pcItemsSolution[256]; // create the LINGO environment object pLSenvLINGO pLINGO; pLINGO = LScreateEnvLng(); if ( !pLINGO) { printf( "Can''t create LINGO environment!\n"); goto FinalExit; } // Open LINGO's log file nError = LSopenLogFileLng( pLINGO, "LINGO.log"); if ( nError) goto ErrorExit;

INTERFACING WITH OTHER APPLICATIONS // memory transfer pointers to LINGO // @POINTER(1) - Items set nError = LSsetPointerLng( pLINGO, (void*) pcItems, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(2) - Item weights nError = LSsetPointerLng( pLINGO, (void*) pdItemWeight, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(3) - Item ranks nError = LSsetPointerLng( pLINGO, (void*) pdItemRank, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(4) - Sack size nError = LSsetPointerLng( pLINGO, (void*) &dSackSize, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(5) - Output region for optimal items set nError = LSsetPointerLng( pLINGO, (void*) pcItemsSolution, &nPointersNow); if ( nError) goto ErrorExit; // @POINTER(6) - Variable to receive solution status nError = LSsetPointerLng( pLINGO, &dStatus, &nPointersNow); if ( nError) goto ErrorExit; // Here is the script we want LINGO to run: // Load the model, solve the model, exit. stry( pcScript, "TAKE SACK.LNG \n GO \n QUIT \n"); // Run the script nError = LSexecuteScriptLng( pLINGO, pcScript); if ( nError) goto ErrorExit; // display solution status printf("\nSolution status (should be 0): %d\n", (int) dStatus); // display items in optimal sack printf("\nItems in optimal sack:\n%s\n", pcItemsSolution); // Close the log file LScloseLogFileLng( pLINGO); // All done

521

522 CHAPTER 11 goto NormalExit; ErrorExit: printf("LINGO Error Code: %d\n", nError); NormalExit: LSdeleteEnvLng( pLINGO); FinalExit: ; }

SACK.C There are a couple of interesting features to note in this code pertaining to the ing of set . First off, there is the declaration of the original set : // potential items in knapsack char pcItems[256] = "ANT_REPEL \n BEER \n BLANKET \n" "BRATWURST \n BROWNIES \n FRISBEE \n SALAD \n" "WATERMELON";

The set are merely listed as one long string, separated by line feeds (\n). We also added blank spaces for readability, which LINGO strips out when it parses the names. Note, that since we are working in C, there is an implicit null byte at the end of this string due to the use of double quotes. This terminating null is important, because it lets LINGO know where the end of the list occurs. We a pointer to the set to LINGO with the following call to LSsetPointerLng // @POINTER(1) - Items set nError = LSsetPointerLng( pLINGO, (void*) pcItems, &nPointersNow); if ( nError) goto ErrorExit;

We a pointer to the set to LINGO with the following call to LSsetPointerLng. For receiving the optimal set of items back from LINGO we set aside the following text array: char pcItemsSolution[256];

We let LINGO know to store the solution set in this array with the following call to LSLsetPointerLng: // @POINTER(5) - Output region for optimal items set nError = LSsetPointerLng( pLINGO, (void*) pcItemsSolution, &nPointersNow);

Recall that this statement in the code pairs with the following statement in the model to establish the link for receiving the optimal set of items: !send optimal items set back to caller; @POINTER( 5) = ITEMSUSED;


523

If you have Visual C/C++ 6.0 installed on your machine, you should be able to build the application by going to the \LINGO\Programming Samples\VC++\Knapsack folder and issuing the NMAKE command. Alternatively, you may simply go to the folder and run the sack.exe executable. After running the application, you should see the following: Solution status (should be 0): 0 Items in optimal sack: ANT_REPEL BEER BLANKET BRATWURST BROWNIES FRISBEE

From the solution we see that all items except the salad and watermelon are included in the optimal solution.

Callback Functions In many instances, solving a model can be a lengthy operation. Given this, it may be useful to provide some device to keep the posted as to the progress of the optimization. The standard interactive version of LINGO displays a status window showing the iteration count, objective value, and various other model statistics each time the solver is invoked. In addition, the has the ability to interrupt the solver by clicking on a button in the status window. You can provide similar functionality to s through the LINGO DLL by supplying LINGO with a callback function—so named because the code calls the LINGO solver, and the LINGO solver then calls back to the supplied callback routine. In the next section, the calling sequences used in establishing a callback function are discussed. After that, there are sample callback routines to the Visual C++ and Visual Basic staff scheduling examples illustrated above.

Specifying a Callback Function To specify a callback function, the LINGO exported routine LSsetCallbackSolverLng() needs to be called before calling LINGO’s script processor. The callback function will be called frequently by the LINGO solver. You will recall from above the calling sequence for LSsetCallbackSolverLng:

int LSsetCallbackSolverLng( pLSenvLINGO pL, lngCBFuncError_t pcbf, void* pData) Arguments: pL pcbf

Pointer to a LINGO environment created by a previous call to LScreateEnvLng(). Pointer to your callback routine.

pData This is a specified pointer. You may use it to point to any data you might need to reference from your callback function. LINGO merely es the value of this pointer through to your callback function. You may set this pointer to NULL if it is not required.

524 CHAPTER 11 Return Value: Returns 0 if no error occurred. Otherwise, it returns one of the nonzero error codes listed below in the section LINGO DLL Error Codes. The callback function must use the standard call convention and must have the following interface: int MySolverCallback( pLSenvLINGO pL, int nReserved, void* pData); Your computer will most likely crash if this interface specification is not followed exactly. The nReserved argument is reserved for future use and may be ignored. Once the callback function has control, information regarding the solver’s status can be retrieved from LINGO using the LSgetCallbackInfoLng()function. See the section Functions Exported by the LINGO DLL above for more information on the LSgetCallbackInfoLng() interface.

A Visual C++ Callback Function We will now extend the Visual C++ staffing example presented above by introducing a callback function. The callback function in this example will post a small dialog box each time the solver finds a better integer solution. The dialog box will display the solver’s iteration count, the objective value for the new solution, the bound on the objective, and also include a button to allow the to interrupt the solver if they desire. You can find the complete project for this sample in the directory LINGO14\Programming Samples\VC++\STAFF2. The first step is to design the dialog box that we will post whenever a new integer solution is found. Here is a look at the dialog box for this example:

The box has three edit fields for the iterations, objective, and bound. There are two buttons—one to interrupt the solver and the other to clear the dialog box.


525

The next step is to use the ClassWizard to attach a handler class to the new dialog box. This class was named CNewIPDlg. When a class has been attached to the dialog box, then the ClassWizard must be used to assign member variables to handle the Iterations, Objective, and Bound edit fields. Once this is done, the header file for the dialog box will resemble: // NewIPDlg.h : header file // //////////////////////////////////////////////////////////////// // CNewIPDlg dialog class CNewIPDlg : public CDialog { // Construction public: CNewIPDlg(CWnd* pParent = NULL);

// standard constructor

// Dialog Data //{{AFX_DATA(CNewIPDlg) enum { IDD = IDD_NEW_IP_SOLUTION }; CString m_csBound; CString m_csIteration; CString m_csObjective; //}}AFX_DATA // Overrides // ClassWizard generated virtual function overrides //{{AFX_VIRTUAL(CNewIPDlg) protected: virtual void DoDataExchange(CDataExchange* pDX); //}}AFX_VIRTUAL // Implementation protected: // Generated message map functions //{{AFX_MSG(CNewIPDlg) // NOTE: the ClassWizard will add member functions here //}}AFX_MSG DECLARE_MESSAGE_MAP() };

Callback Dialog Header File (NewIPDlg.h)

526 CHAPTER 11 Here is the code to handle events from the new dialog box: // NewIPDlg.h : header file // ///////////////////////////////////////////////////////////////// // CNewIPDlg dialog class CNewIPDlg : public CDialog { // Construction public: CNewIPDlg(CWnd* pParent = NULL);

// standard constructor

// Dialog Data //{{AFX_DATA(CNewIPDlg) enum { IDD = IDD_NEW_IP_SOLUTION }; CString m_csBound; CString m_csIteration; CString m_csObjective; //}}AFX_DATA // Overrides // ClassWizard generated virtual function overrides //{{AFX_VIRTUAL(CNewIPDlg) protected: virtual void DoDataExchange(CDataExchange* pDX); //}}AFX_VIRTUAL // Implementation protected: // Generated message map functions //{{AFX_MSG(CNewIPDlg) // NOTE: the ClassWizard will add member functions here //}}AFX_MSG DECLARE_MESSAGE_MAP() };

Callback Dialog Handler (NewIPDlg.p) The code in these two files was entirely generated by the ClassWizard, requiring no actual input.


527

Next, the code for the callback function must be added. Here is a copy of the global routine that was added for this purpose: int __stdcall MyCallback( void* pModel, int nReserved, void* pData) { // Callback function called by the LINGO solver // // return value: >= 0 if solver is to continue, else < 0 to interrupt // Get current best IP int nErr; double dBestIP; nErr = LSgetCallbackInfoLng( pModel, LS_DINFO_MIP_BEST_OBJECTIVE_LNG, &dBestIP); if ( nErr) return( 0); // Get best IP value published in dialog box double* pdBestIPShown = (double*) pData; // Is current better than incumbent? if ( dBestIP < *pdBestIPShown) { // Yes ... save its value *pdBestIPShown = dBestIP; // Get iteration count from LINGO int nIterations; LSgetCallbackInfoLng( pModel, LS_IINFO_ITERATIONS_LNG, &nIterations); // Get bound on solution double dBound; LSgetCallbackInfoLng( pModel, LS_DINFO_MIP_BOUND_LNG, &dBound); // Create a dialog to show the current solution value CNewIPDlg dlgNewIP; // Initialize the fields of the dialog dlgNewIP.m_csIteration.Format( "%d", nIterations); dlgNewIP.m_csObjective.Format( "%d", (int) dBestIP); dlgNewIP.m_csBound.Format( "%d", (int) dBound); // Post the dialog and check for a interrupt if ( dlgNewIP.DoModal() == IDCANCEL) return( -1); } return( 0); }

Callback Routine

528 CHAPTER 11 Of course, this routine has the same interface that was described in the previous section: int __stdcall MySolverCallback( pLSenvLINGO pL, int nReserved, void* pData) The following code uses the LSgetCallbackInfo routine to get the value of the current best integer solution: // Get current best IP int nErr; double dBestIP; nErr = LSgetCallbackInfoLng( pModel, LS_DINFO_MIP_BEST_OBJECTIVE_LNG, &dBestIP); if ( nErr) return( 0);

The constant, LS_DINFO_MIP_BEST_OBJECTIVE, is defined in the LINGD11.H header file. We then get the value for the best integer solution that we have currently posted in the dialog box with the statement: // Get best IP value published in dialog box double* pdBestIPShown = (double*)pData;

Note that this statement references the data pointer, pData. This pointer is ed to LINGO when the callback function is established, and it is useful as a means of accessing the data from within the callback function. In this particular case, a single variable is being pointing to. If needed, the pointer could reference a large data structure containing whatever data desired. Next, test to see if the latest integer solution is better than the incumbent solution with the statement: // Is current better than incumbent? if ( dBestIP < *pdBestIPShown)

If the new solution is better, then we get additional information from LINGO on the iteration count and solution bound. Also, an instance of the callback dialog box to display the latest information in is created with the line: // Create a dialog to show the current solution value CNewIPDlg dlgNewIP;

The new data is then loaded into the fields of the dialog box: // Initialize the fields of the dialog dlgNewIP.m_csIteration.Format( "%d", nIterations); dlgNewIP.m_csObjective.Format( "%d", (int) dBestIP); dlgNewIP.m_csBound.Format( "%d", (int) dBound);

As the final step in this callback routine, we display the dialog box, and if the hits the interrupt button we return a –1, indicating that the solver should stop and return with the best answer found so far: // Post the dialog and check for a interrupt if ( dlgNewIP.DoModal() == IDCANCEL) return( -1);


529

At this point, there is one more piece of code to add. We must make a call to the LSsetCallbackSolverLng() routine to LINGO a pointer to our callback routine. A good place to do this is right after creating the LINGO environment in the OnSolve() handler code for our Solve button. The changes are listed below in bold type: // create the LINGO environment object pLSenvLINGO pLINGO; pLINGO = LScreateEnvLng(); if ( !pLINGO) {

}

AfxMessageBox("Unable to create LINGO Environment"); return;

// LINGO a pointer to our callback function nError = LSsetCallbackSolverLng( pLINGO, &MyCallback, &dBestIPShown); if ( nError) goto ErrorExit; // Open LINGO's log file nError = LSopenLogFileLng( pLINGO, "LINGO.log"); if ( nError) goto ErrorExit;

A Visual Basic Callback Function We will now extend the Visual Basic staffing example presented above by introducing a callback function. The callback function in this example will post a small dialog box each time the solver finds a better integer solution. The dialog box will display the solver’s iteration count, the objective value for the new solution, and the bound on the objective. You can find the complete project for this sample in the directory LINGO14\Programming Samples\VBasic\STAFF2. First, we need to construct a callback function and place it in a separate Module file (.bas file) . Here are the contents of the callback function file Module1.bas: Public Function MySolverCallback(ByVal pModel As Long, _ ByVal nReserved As Long, ByRef dBestIP As Double) As Long ' Callback function called by the LINGO DLL ' during model solution. ' ' Return value: >= 0 if solver is to continue, ' else < 0 to interrupt the solver Dim nReturnVal As Long

nReturnVal = 0

' Get current best IP Dim dObj As Double Dim nError As Long nError = LSgetCallbackInfoDoubleLng(pModel, _ LS_DINFO_MIP_BEST_OBJECTIVE_LNG, dObj)

530 CHAPTER 11 ' Check for any error If (nError = LSERR_NO_ERROR_LNG) Then ' Is it better than the best one displayed so far? If (dObj < dBestIP) Then ' Yes ... display this solution ' Save the new best objective value dBestIP = dObj ' Get the iteration count from LINGO Dim nIterations As Long nResult = LSgetCallbackInfoLongLng(pModel, _ LS_IINFO_ITERATIONS_LNG, nIterations) ' Get the objective bound from LINGO Dim dBound As Double nResult = LSgetCallbackInfoDoubleLng(pModel, _ LS_DINFO_MIP_BOUND_LNG, dBound) ' Display the information in a dialog box Dim nButtonPressed Dim cMessage As String cMessage = "Objective:" + Str(dBestIP) _ + Chr(10) + "Bound:" + Str(dBound) _ + Chr(10) + "Iterations:" + Str(nIterations) nButtonPressed = MsgBox(cMessage, vbOKCancel) If (nButtonPressed = vbCancel) Then nReturnVal = -1 End If End If End If MySolverCallback = nReturnVal End Function

VB Callback Module (Module1.bas) Note:

A VB callback function must be placed in a Module file (.bas file) . The callback function will not work if it is placed in a Forms file (.frm file). Also, the callback function and the module file must have different names. If the module file has the same name as the callback function, then the VB AddressOf operator will not be able to return the address of the callback function.

You will recall from the section Specifying a Callback Function above that the callback routine must use the following calling sequence: int __stdcall MySolverCallback( pLSenvLINGO pL, int nReserved, void* pData)


531

An equivalent function definition using VB code is: Public Function MySolverCallback(ByVal pModel As Long, _ ByVal nReserved As Long, ByRef dBestIP As Double) As Long

VB uses the standard call (__stdcall) convention by default, so we need not specify this explicitly. We will make use of the data pointer to the value of the best objective displayed so far in the dBestIP argument. This variable will hold the objective value of the best integer solution found so far. We compare each new objective value to the best one found so far. If the latest is an improvement over the incumbent, then we display a dialog box summarizing the new solution. The following code uses the LSgetCallbackInfo routine to get the value of the current best integer solution: ' Get current best IP Dim dObj As Double Dim nError As Long nError = LSgetCallbackInfoDoubleLng(pModel, _ LS_DINFO_MIP_BEST_OBJECTIVE_LNG, dObj)

In the VB header file for LINGO (LINGD14.BAS), we created two aliases for the LSgetCallbackInfoLng() function: LSgetCallbackInfoDoubleLng() and LSgetCallbackInfoLongLng(). These were for retrieving, respectively, double and long data from LINGO. This is required due to VB not ing the void data type found in C. We use LSgetCallbackInfoDoubleLng() to retrieve the objective value given that it is a double precision quantity. Next, we check for any errors in retrieving the objective value. If none occurred, we check to see if the latest objective is better than the incumbent: ' Check for any error If (nError = LSERR_NO_ERROR_LNG) Then ' Is it better than the best one displayed so far? If (dObj < dBestIP) Then

If the new objective is better than the incumbent, then we save the new objective value, and retrieve the iteration count and objective bound: ' Save the new best objective value dBestIP = dObj ' Get the iteration count from LINGO Dim nIterations As Long nResult = LSgetCallbackInfoLongLng(pModel, _ LS_IINFO_ITERATIONS_LNG, nIterations) ' Get the objective bound from LINGO Dim dBound As Double nResult = LSgetCallbackInfoDoubleLng(pModel, _ LS_DINFO_MIP_BOUND_LNG, dBound)

532 CHAPTER 11 We post a summary of the new solution in a dialog box: ' Display the information in a dialog box Dim nButtonPressed Dim cMessage As String cMessage = "Objective:" + Str(dBestIP) _ + Chr(10) + "Bound:" + Str(dBound) _ + Chr(10) + "Iterations:" + Str(nIterations) nButtonPressed = MsgBox(cMessage, vbOKCancel)

If the pressed the Cancel button, as opposed to the OK button, then we set the return value to –1 before returning, which will cause the LINGO solver to interrupt: If (nButtonPressed = vbCancel) Then nReturnVal = -1 End If End If End If MySolverCallback = nReturnVal End Function

At this point, there is one more piece of code to add. We must make a call to the LssetCallbackSolverLng() routine to LINGO a pointer to our callback routine. This is accomplished at the start of the Solve button handler routine with the call: ' LINGO a pointer to the callback routine Dim nError As Long Dim dBestObj As Double dBestObj = 1E+30 nError = LSsetCallbackSolverLng(pLINGO, _ AddressOf MySolverCallback, dBestObj)

Note the use of the VB AddressOf operator in the call to LSsetCallbackSolverLng(). This operator may be used only in function calls to the address of routines contained in module files. Note:

The AddressOf operator was added to VB starting with release 5.0. Thus, earlier releases of VB won’t be able to exploit the callback feature in LINGO. Also, Visual Basic for Applications (VBA), the VB macro capability supplied with Microsoft Office, does not the AddressOf operator. Thus, VBA applications calling LINGO will also not be able to establish callback routines.


533

Summary The LINGO DLL has a very simple structure. You need only acquaint yourself with a handful of functions in order to access the DLL and add the power of the LINGO solver to your applications. We’ve given brief examples on calling the DLL from Visual Basic and Visual C++. Additional examples are provided in the Programming Samples. Application developers working in other environments should be able to access the LINGO DLL in a fashion similar to the examples given here. Finally, keep in mind that any application you build that makes use of the LINGO DLL is protected under the LINGO License Agreement. Thus, you may not distribute such applications without explicit permission from LINDO Systems. If you would like to make arrangements for distributing your application, please feel free to LINDO Systems regarding available runtime licensing arrangements.

Defined Functions The @ function allows the use of custom functions of your own design in LINGO. In Windows versions of LINGO, you provide a Dynamic Link Library (DLL) that contains your @ function. Most programming languages that Windows should allow you to build a DLL. For platforms other than Windows, you provide LINGO with a compiled C or FORTRAN subroutine containing your @ function. From the perspective of a LINGO modeler, an @ function is a function that can take any number of arguments, but must take at least one. It returns a result calculated by the -written routine. From the perspective of the programmer writing the custom function, an @ function takes only two input arguments and returns a single result. The two input arguments consist of: 1. 2.

an integer specifying the number of arguments encountered in the @ reference in the LINGO model, and a vector containing the values of the arguments in the order in which they were encountered in the @ reference in double precision format (i.e., an 8 byte floating point format).

In other words, although to the LINGO modeler an @ function can appear to take any number of arguments, to the programmer implementing the @ function, only two input arguments are ed. It is possible to use multiple functions with @ by writing and compiling each function as a separate subroutine and taking an argument to @ as the index number of the subroutine that you want to branch to.

534 CHAPTER 11

Installing @ Under Windows When LINGO for Windows starts up, it searches for the DLL file called MY.DLL. LINGO searches for this file in your startup directory. The startup directory is the directory where you installed the LINGO program. If LINGO finds MY.DLL, it loads the DLL into memory and calls the exported MY routine every time a model references an @ function. On platforms other than Windows, you must link a compiled FORTRAN or C subroutine with the LINGO libraries in order to provide a customized @ function. Refer to the REE file for your version of LINGO for technical information on how to link your custom routines with LINGO. In the following section, we illustrate the details of building such a DLL using Microsoft Visual C++.

Visual C++ Example In this section, we will use Microsoft Visual C/C++ to create a 32-bit DLL that contains an @ function to perform the square root function. This is very easy to do if we make use of the AppWizard in Visual C++ to build the base code for the DLL. Or, you will find the code for this example in the \VC++ subdirectory off of your main LINGO directory. To build the base code, start the Visual C++ Developer Studio and do the following: 1. 2. 3. 4.

Issue the File|New command. You should now see a New dialog box. Select the Project Workspace options and then click OK. You will now see a New Project Workspace dialog box. Give the project the name sqroot. In the Type box, select the MFC AppWizard (dll) option. Click on the Create button. A new MFC AppWizard dialog box should appear. Simply click on the Finish button.

INTERFACING WITH OTHER APPLICATIONS 5.

You should now see a New Project Information box containing a summary of the options selected for your project that resembles:

Click the OK button to finish creating the base code for our DLL.

535

536 CHAPTER 11 Now, edit the SQROOT.P file and add the modifications listed below in bold: // sqroot.p : Defines the initialization // routines for the DLL. // #include "stdafx.h" #include "sqroot.h" #ifdef _DEBUG #define new DEBUG_NEW #undef THIS_FILE static char THIS_FILE[] = __FILE__; #endif ///////////////////////////////////////////////////// // CSqrootApp BEGIN_MESSAGE_MAP(CSqrootApp, CWinApp) //{{AFX_MSG_MAP(CSqrootApp) //NOTE-the ClassWizard will add and // remove mapping macros here. // DO NOT EDIT what you see in these // blocks of generated code! //}}AFX_MSG_MAP END_MESSAGE_MAP() CSqrootApp::CSqrootApp() { // The constructor // Remove next line for a "quiet" version // of My.DLL AfxMessageBox("@ DLL installed"); } CSqrootApp theApp; #include <math.h> extern "C" __declspec(dllexport) void My(int* pnNumberOfArgs, double* pdArgs, double* dResult) { // This is an @ routine callable by LINGO. // this particular case we simply take the // square root of the first argument. *dResult = sqrt(*pdArgs); }

File: SQROOT.P

In


537

You should now be able to build the DLL. When Visual C++ completes the build, copy the SQROOT.DLL file to LINGO’s startup directory (the one where LINGO14.EXE or LINGO64-14.EXE is located) and rename SQROOT.DLL to be MY.DLL. Now, start LINGO and you should see the following dialog box confirming the DLL was successfully loaded:

Input a small model to compute the square root of 9 and solve it to get the following results:

If you don’t have a copy of Visual C++, you may experiment with this @ routine by copying the DLL supplied with LINGO into your LINGO startup directory. You can find the SQROOT.DLL file in the \VC++ subdirectory off the main LINGO directory.

12 Developing More Advanced Models In this chapter, we will walk through the development of models in a handful of different application areas. The goal here is twofold. On the one hand, walking through a number of different models should help hone your skills at building LINGO models. Secondly, by applying LINGO to an array of different application areas, we hope to illustrate the potential benefits of mathematical modeling in almost any business situation. Below is a list of the application areas we will be touching upon in this chapter:     

Production Management Logistics Finance Queuing Marketing

The reader interested in additional examples of LINGO modeling should refer to Optimization Modeling with LINGO, by Linus Schrage, published by LINDO Systems. This book contains an exhaustive set of modeling examples from a wide array of application areas. In many cases, you should be able to find a model that will suit your needs with only modest modifications. Interested readers may also refer to Appendix A, Additional Examples of LINGO Modeling, to find additional samples of LINGO models.

539

540 CHAPTER 12

Production Management Models Blending Model

Model Name: Blend

Background In blending problems, two or more raw materials are to be blended into one or more finished goods satisfying one or more quality requirements on the finished goods. Blending problems are of interest in a number of different fields. Some examples include determining an optimal blend for animal feed, finding the least costly mix of ores that produce an alloy with specified characteristics, or deciding on the most efficient combination of advertising media purchases to reach target audience percentages.

The Problem in Words As a production manager at a fuel processing plant, you would like to maximize profit from blending the raw materials butane, catalytic reformate, and naphtha into the two finished products Regular and . These final products must satisfy quality requirements on octane, vapor pressure, and volatility. Each raw material has a known limited availability and cost per unit. You know the minimum required for each finished good, the maximum you will be able to sell, and the profit contribution per unit.

The Model MODEL: TITLE BLEND; SETS: !Each raw material has an availability and cost/unit; RAWMAT/ BUTANE, CATREF, NAPHTHA/: AVAIL, COST; !Each finished good has a min required, max sellable, selling price, and batch size to be determined; FINGOOD/ REGULAR, /: MINREQ, MAXSELL, PRICE, BATCH; !Here is the set of quality measures; QUALMES/ OCTANE, VAPOR, VOLATILITY/; !For each combo of raw material and quality measure there is a quality level; RXQ(RAWMAT, QUALMES): QLEVEL; !For each combination of quality measure and finished good there are upper and lower limits on quality, and a slack on upper quality to be determined; QXF(QUALMES, FINGOOD): QUP, QLOW, QSLACK;

Developing More Advanced Models !For each combination of raw material and finished good there is an amount of raw material used to be solved for; RXF(RAWMAT, FINGOOD): USED; ENDSETS DATA: !Raw material availability; AVAIL = 1000, 4000, 5000; !Raw material costs; COST = 7.3, 18.2, 12.5; !Quality parameters of raw materials; QLEVEL = 120, 60, 105, 100, 2.6, 3, 74, 4.1, 12; !Limits on finished goods; MINREQ = 4000, 2000; MAXSELL = 8000, 6000; !Finished goods prices; PRICE = 18.4, 22; !Upper and lower limits on quality for each finished good; QUP = 110, 110, 11, 11, 25, 25; QLOW = 90, 95, 8, 8, 17, 17; ENDDATA !Subject to raw material availability; @FOR(RAWMAT(R): [RMLIM] @SUM(FINGOOD(F): USED(R, F)) <= AVAIL(R); ); @FOR(FINGOOD(F): !Batch size computation; [BATCOMP] BATCH(F) = @SUM(RAWMAT(R): USED(R, F)); !Batch size limits; @BND(MINREQ, BATCH, MAXSELL); !Quality restrictions for each quality measure; @FOR(QUALMES(Q):

541

542 CHAPTER 12 [QRESUP] @SUM(RAWMAT(R): QLEVEL(R, Q) * USED(R, F)) + QSLACK(Q, F) = QUP(Q, F) * BATCH(F); [QRESDN] QSLACK(Q, F) <= (QUP(Q, F) - QLOW(Q, F)) * BATCH(F); ); ); ! We want to maximize the profit contribution; [OBJECTIVE] MAX = @SUM(FINGOOD: PRICE * BATCH) @SUM(RAWMAT(R): COST(R) * @SUM(FINGOOD(F): USED(R, F))); END

Model: BLEND

The Sets We have three primitive sets in this model raw materials (RAWMAT), finished goods (FINGOOD), and quality measures (QUALMES). From these three primitive sets, we create three derived sets. The first derived set, RXQ, is a dense derived set and is a cross on the raw materials set and the quality measures set. We need this derived set in order to define an attribute to hold the quality values for the raw materials. The second derived set, QXF, is a cross on the quality measures and the finished goods. We need this set because we will be concerned with the levels of the quality measures in the finished goods. The final derived set, RXF, is a cross on the raw materials and finished goods. We need this set because we need to compute the amount of raw material used in each finished good.

The Variables The primary variable that drives this model is the amount of each raw material used in each finished good (USED). We have also added two other variables for computation purposes. First, there is the BATCH variable, which is used to compute the batch size of each finished good. There is also the QSLACK variable, which computes the slack on the upper quality limit for each finished good and each quality measure.

The Objective The objective in this model is to maximize the total profit contribution. This is computed using the following expression: [OBJECTIVE] MAX = @SUM(FINGOOD: PRICE * BATCH) @SUM(RAWMAT(R): COST(R) * @SUM(FINGOOD(F): USED(R, F)));

Developing More Advanced Models

543

Total profit, of course, is total revenue minus total expenses. Total revenue is the sum over the finished goods of the price of each finished good multiplied by the batch size of each finished good, or, in LINGO syntax: @SUM(FINGOOD: PRICE * BATCH)

Total expenses are computed by taking the sum over the raw materials of the price of the raw material multiplied by the amount of each raw material used. This is computed with the expression: @SUM(RAWMAT(R): COST(R) * @SUM(FINGOOD(F): USED(R, F))

The Constraints There are four sets of constraints in this model. Two of them are merely computational. This first computational set of constraints computes the batch size of each finished good with the following expression: !Batch size computation; [BATCOMP] BATCH(F) = @SUM(RAWMAT(R): USED(R, F));

In words, the batch size of finished good F is the sum of all the raw materials used in the production of the finished good. The second set of computational constraints computes the slack on the upper quality limit for each finished good and each quality measure as follows: [QRESUP] @SUM(RAWMAT(R): QLEVEL(R, Q) * USED(R, F)) + QSLACK(Q, F) = QUP(Q, F) * BATCH(F);

In words, the actual quality level plus the slack on the upper quality level is equal to the upper quality level. The first true constraint is on the finished products’ batch sizes, which must fall within minimum and maximum levels. We use the @BND function to set this up as follows: !Batch size limits; @BND(MINREQ, BATCH, MAXSELL);

Note that we could have entered explicit constraints for the batch size limits, but the @BND function is more efficient at handling simple bounds on variables. The final constraint forces the quality level of each finished good for each quality measure to fall within specifications. We do this with the following: [QRESDN] QSLACK(Q, F) <= (QUP(Q, F) - QLOW(Q, F)) * BATCH(F);

In words, the slack must be less than the difference between the upper limit and the lower limit. If this were not true, quality would be beneath the lower limit. The fact that the slack variable cannot be negative guarantees we won’t exceed the quality limit.

544 CHAPTER 12

The Solution Rounding all solution values to the nearest whole integer, total profit is $44,905 with batch sizes of 4,000 Regular and 4,095 . The following matrix shows the recommended quantities of each raw material to use in the Regular and blends: Raw Material Butane Catalytic Reformate Naphtha

Regular 534 1,516 1,950

466 2,484 1,146


Material Requirements Planning

545

Model: MRP

Background Material Requirements Planning, or MRP, is used to generate production schedules for the manufacture of complex products. MRP uses the demand schedule for a finished product, the lead times to produce the finished product, and all the various subcomponents that go into the finished product to work backwards and develop a detailed just-in-time production schedule that meets the demand schedule. MRP’s main focus is finding a feasible just-in-time production schedule to meet demand. MRP does not, however, attempt to optimize the production schedule to minimize total production costs. In many cases, the problems solved by MRP are so complex optimization would be prohibitive.

The Problem in Words Suppose you are a manufacturer of human powered vehicles. You have a given schedule of demands for finished products over time. You need to know when and how many of each component, subcomponent, etc. is needed to meet demand. Your final products are as follows: 1. 2. 3.

Unicycles, made from a seat and a wheel, Bicycles, made from a seat, two wheels, and a chain, Tandems (bicycles built for two), made from two seats, two wheels, and two chains.

Each product is assembled from a set of components. Each component in turn may be assembled from other subcomponents. For simplicity and generality, we will refer to all products, components, and subcomponents as parts. The subcomponents are as follows: 1. 2. 3. 4. 5. 6.

Seats, Wheels, made from a hub and 36 spokes, Chains, made from 84 links, Hubs, Spokes, and Links.

It takes a certain amount of time, called the lead-time, to produce each batch of parts. The component parts for each part must be on hand when you begin production of a part. The sales department has estimated demand for two months (eight weeks) in the future. According to their findings, there will be a demand for 10 unicycles in week eight, and 20 bicycles and 20 tandems in week nine.

546 CHAPTER 12

The Model MODEL: ! Data for this model is read from MRP.LDT; SETS:

! The set of parts; PART: LT; ! LT(i) = Lead time to produce part i; ! The set of time periods; TIME;

! A relationship called USES between pairs of parts; USES( PART, PART): NEEDS; ! Parent part i needs NEEDS(i, j) units of child part j; ! For each part and time period we're interested in; PXT( PART, TIME): ED, TD; ! ED(i, j) = External demand for part i at time j; ! TD(i, j) = Total demand for part i at time j; ENDSETS DATA: ! Load the data from an external file; ! Parts list; PART = @FILE( 'MRP.LDT'); ! Time periods; TIME = @FILE( 'MRP.LDT'); ! Get the parent child relations and the number of parts required; USES, NEEDS = @FILE( 'MRP.LDT'); ! Get the lead times from the file; LT = @FILE( 'MRP.LDT'); ! Get the external demands over time for each part; ED = @FILE( 'MRP.LDT'); ENDDATA ! Set NP = no. of time periods in the problem; NP = @SIZE( TIME); ! For each part P and period T, the total demand =

Developing More Advanced Models external demand + demand generated by parents one lead time in the future; @FOR( PXT( P, T) | T + LT( P) #LE# NP : TD( P, T) = ED( P, T + LT( P)) + @SUM( USES( P2, P): TD( P2, T + LT( P)) * NEEDS( P2, P)); ); DATA: ! Display a table showing the production schedule; @TEXT() = ' The production schedule:'; @TEXT() = @TABLE( TD); ENDDATA END

Model: MRP

The Data An interesting feature of this model is all of the data is imported from an external text file using the @FILE function (for more information on @FILE, see Chapter 8, Interfacing with External Files). More specifically, @FILE reads all the data from a single file, MRP.LDT, that is pictured below: ! Parts list;

U, B, T, S, W, C, H, P, L~

!

! ! ! ! ! ! ! ! !

Unicycles; Bicycles; Tandems; Seats; Wheels; Chains; Hubs; sPokes; Links;

The set of periods; 1..9 ~

! The parent-child use relationships (i.e., Unicycles use Seats, etc.) The number of child parts required in each parent- child relationship, respectively (i.e, Unicycles use 1 Seat, Wheels use 36 sPokes, etc.); U S 1, U W 1, B S 1, B W 2, B C 1, T S 2, T W 2, T C 2, W H 1, W P 36, C L 84~ ! The lead times for each part; 1, 2, 1, 1, 3, 1, 1, 2, 2~

547

548 CHAPTER 12 ! The external demands or master schedule; ! Time period; ! 1 2 3 4 5 6 7 8 9; ! U; 0, 0, 0, 0, 0, 0, 0,10, 0, ! B; 0, 0, 0, 0, 0, 0, 0, 0,20, ! T; 0, 0, 0, 0, 0, 0, 0, 0,20, ! S; 0, 0, 0, 0, 0, 0, 0, 0, 0, ! W; 0, 0, 0, 0, 0, 0, 0, 0, 0, ! C; 0, 0, 0, 0, 0, 0, 0, 0, 0, ! H; 0, 0, 0, 0, 0, 0, 0, 0, 0, ! P; 0, 0, 0, 0, 0, 0, 0, 0, 0, ! L; 0, 0, 0, 0, 0, 0, 0, 0, 0

File: MRP.LDT You will recall the tilde (~) is LINGO’s end-of-record mark. Whenever a tilde is encountered in a data file, LINGO stops reading input from the data file and begins reading where it left off in the model file.

The Sets We have two primitive sets in this modelthe component parts (PARTS) and the time periods (TIME). From these two primitive sets, we create two derived sets. The first derived set, USES, is a sparse set derived from the cross of the PARTS set on itself. We use this set to construct a data table, or input-output matrix (NEEDS), containing the parts usage data, which tells us how many of each of the other parts are required to produce a given part. The set is sparse because not all parts are required in the production of certain other parts (e.g., chains are not required to produce a spoke). The other derived set, PXT, is a dense set formed from the cross on the parts and time periods sets. We need this set because we will be concerned with the demand for each part in each period and the amount of each part to begin producing in each period.

The Variables The only unknown in this model is the total demand (TD) attribute, where TD(p, t) is the total demand for product p in period t. Total demand stems from two sources—external demand (ED) from customers and internal demand for production requirements. We compute TD by incorporating the part’s lead time. Thus, when TD is nonzero in a period, production must begin in that period.

The Formulas The key computation in this model is: ! For each part P and period T, the total demand = external demand + demand generated by parents one lead time in the future; @FOR(PXT(P, T) | T + LT(P) #LE# NP: TD(P, T) = ED(P, T + LT(P)) + @SUM(USES(P2, P): TD(P2, T + LT(P)) * NEEDS(P2, P)); );


549

For each part in each time period, the amount of that part we must begin producing in the period is the amount we will need one lead time away to satisfy 1) external demand, and 2) internal production requirements. The subexpression that gives external demand is simply: ED(P, T + LT(P))

The remainder of the expression: @SUM(USES(P2, P): TD(P2, T + LT(P)) * NEEDS(P2, P));

sums up the amount of the part needed for internal production of all other parts one lead time away. Note, we place a logical condition on the outer @FOR loop in this calculation (shown here in bold): @FOR(PXT(P, T) | T + LT(P) #LE# NP:

Without this condition, the calculation would extend beyond the final period in the TIME set.

The Solution Solving the model, we get the following nonzero values for TD: Variable TD(U, 7) TD(B, 7) TD(T, 8) TD(S, 6) TD(S, 7) TD(W, 4) TD(W, 5) TD(C, 6) TD(C, 7) TD(H, 3) TD(H, 4) TD(P, 2) TD(P, 3) TD(L, 4) TD(L, 5)

Solution: MRP

Value 10.00000 20.00000 20.00000 30.00000 40.00000 50.00000 40.00000 20.00000 40.00000 50.00000 40.00000 1800.000 1440.000 1680.000 3360.000

550 CHAPTER 12 Putting this solution in tabular form, we get the following production schedule: 2

3

Unicycles Bicycles Tandems Seats Wheels Chains 50 Hubs 1,800 1,440 Spokes Links MRP Production Schedule

4

5

6

7 10 20

30

40

20

40

8

20 50

40

40 1,680

3,360

Note the use of the @TABLE output function in the data section at the end of the model that displays the TD attribute in table form similar to the table above: The production schedule: U B T S W C H P L

1 2 3 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 50 40 0 1800 1440 0 0 0 0 1680

5 0 0 0 0 40 0 0 0 3360

6 0 0 0 30 0 20 0 0 0

7 10 20 0 40 0 40 0 0 0

8 0 0 20 0 0 0 0 0 0

9 0 0 0 0 0 0 0 0 0


Assembly Line Balancing

551

Model: ASLBAL

Background In the assembly line balancing problem, tasks are assigned to workstations on an assembly line, so the line’s cycle time is minimized. An assembly line consists of a series of workstations, which each perform one or more specialized tasks in the manufacture of a final product. The cycle time is the maximum time it takes any workstation to complete its assigned tasks. The goal in balancing an assembly line is to assign tasks to stations, so equal amounts of work are performed at each station. Improperly balanced assembly lines will experience bottlenecksworkstations with less work are forced to wait on preceding stations that have more work assigned. The problem is complicated further by precedence relations amongst the tasks, where some tasks must be completed before others may begin (e.g., when building a computer, installing the disk drives must precede putting on the outer casing). The assignment of tasks to workstations must obey the precedence relations.

The Problem in Words For our example, we have eleven tasks (A through K) to assign to four stations (1 through 4). The task precedence diagram looks like this:

F A

B

C

G

J

K

H D

E I

The times to complete the various tasks are given in the table below: Task: Minutes:

A 45

B 11

C 9

D 50

E 15

F 12

G 12

H 12

I 12

J 8

K 9

We need to find an assignment of tasks to workstations that minimize the assembly line's cycle time.

552 CHAPTER 12

The Model MODEL: ! Assembly line balancing model; !This model involves asg tasks to stations in an assembly line so bottlenecks are avoided. Ideally, each station would be assigned an equal amount of work.; SETS: ! The set of tasks to be assigned are A through K, and each task has a time to complete, T; TASK/ A B C D E F G H I J K/: T; ! Some predecessor,successor pairings must be observed(e.g. A must be done before B, B before C, etc.); PRED(TASK, TASK)/ A,B B,C C,F C,G F,J G,J J,K D,E E,H E,I H,J I,J /; ! There are 4 workstations; STATION/1..4/; TXS(TASK, STATION): X; ! X is the attribute from the derived set TXS that represents the assignment. X(I,K) = 1 if task I is assigned to station K; ENDSETS DATA: ! Data taken from Chase and Aquilano, POM; ! There is an estimated time required for each task: A B C D E F G H I J K; T = 45 11 9 50 15 12 12 12 12 8 9; ENDDATA ! The model; ! *Warning* may be slow for more than 15 tasks; ! For each task, there must be one assigned station; @FOR(TASK(I): @SUM(STATION(K): X(I, K)) = 1); ! Precedence constraints; ! For each precedence pair, the predecessor task I cannot be assigned to a later station than its successor task J; @FOR(PRED(I, J): @SUM(STATION(K): K * X(J, K) - K * X(I, K)) >= 0);


553

! For each station, the total time for the assigned tasks must less than the maximum cycle time, CYCTIME; @FOR(STATION(K): @SUM(TXS(I, K): T(I) * X(I, K)) <= CYCTIME); ! Minimize the maximum cycle time; MIN = CYCTIME; ! The X(I,J) assignment variables are binary integers; @FOR(TXS: @BIN(X)); END

Model: ASLBAL

The Sets We have two primitive sets in this modelthe tasks (TASK) and the workstations (STATION). From these two primitive sets, we create two derived sets. The first derived set, PRED, is a sparse derived set and is based on a cross of the TASK set on itself. The of this set are the precedence relations amongst the tasks. For instance, the first member of this set is the pair (A,B), indicating task A must precede task B. The other derived set, TXS, is a dense derived set formed by taking the cross of the task set on the workstation set. We need this set because we will be determining what tasks get assigned to what workstations.

The Variables The decision variables in this model are the of the X attribute that is defined in the TXS set. X(t, s) is a binary variable that is 1 if task t is assigned to station s, otherwise 0. The X attribute is forced to being binary in the expression: ! The X(I,J) assignment variables are binary integers; @FOR(TXS: @BIN(X));

We also introduce the scalar variable, CYCTIME, to represent the entire assembly line’s cycle time, which is computed by taking the maximum cycle time over the workstations.

The Objective The objective in this model is simply to minimize total cycle time for the line and is given as: ! Minimize the maximum cycle time; MIN = CYCTIME;

554 CHAPTER 12

The Constraints We have the following three types of constraints in this model: 1. 2. 3.

each task must be assigned to one station, precedence relations must be observed amongst the tasks, and the line cycle time variable, CYCTIME, must be greater-than-or-equal-to the actual cycle time.

The following expression sums up the assignment variable for each task, and sets the sum to equal 1: ! For each task, there must be one assigned station; @FOR(TASK(I): @SUM(STATION(K): X(I, K)) = 1);

This forces each task to be assigned to a single station. We use the following expression to enforce the precedence relationships amongst the tasks: ! Precedence constraints; ! For each precedence pair, the predecessor task I cannot be assigned to a later station than its successor task J; @FOR(PRED(I, J): @SUM(STATION(K): K * X(J, K) - K * X(I, K)) >= 0);

Suppose task I is a predecessor to task J. If I were incorrectly assigned to a workstation later than J, the sum of the K * X(I, K) would exceed the sum of the K * X(J, K) and the constraint would be violated. Thus, this constraint effectively enforces the predecessor relations. We restrain the cycle time using the following constraints: ! For each station, the total time for the assigned tasks must be less than the maximum cycle time, CYCTIME; @FOR(STATION(K): @SUM(TXS(I, K): T(I) * X(I, K)) <= CYCTIME);

The quantity: @SUM(TXS(I, K): T(I) * X(I, K))

in this constraint computes the cycle time for station K. We use the @FOR statement to make the CYCTIME variable greater-than-or-equal-to the cycle times for all the workstations. If we couple this with the fact that we are minimizing CYCTIME in the objective, CYCTIME will be “squeezed” into exactly equaling the maximum of the cycle times for the workstations. By “squeezing” CYCTIME to the correct value, we avoid using the @MAX function. Had the @MAX function been used, LINGO would have had to resort to its nonlinear solver to handle the piecewise linear @MAX. Avoiding nonlinear models whenever possible is a critical modeling practice.


The Solution Solving the model, we get the following nonzero values for the assignment X variable: Variable X(A, 2) X(B, 3) X(C, 4) X(D, 1) X(E, 3) X(F, 4) X(G, 4) X(H, 3) X(I, 3) X(J, 4) X(K, 4)

Value 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

Solution: ASLBAL Summarizing this solution, we have: Workstation 1 2 3 4

Assigned Tasks D A B, E, H, I C, F, G, J, K

Cycle Time 50 45 50 50

The cycle time for the entire line is 50 minutesthe maximum of the cycle times across all the workstations. We have a well-balanced line in that only workstation 2 has slack time totaling 5 minutes.

555

556 CHAPTER 12

Logistics Models Capacitated Plant Location

Model: CAPLOC

Background The capacitated plant location model is a generalization of the transportation model we introduced in Chapter 1, Getting Started with LINGO. The capacitated plant location problem allows greater latitude of decision making in that the points of origin (plant locations) are variable. Manufacturers and wholesale businesses are likely to encounter problems of this sort in matching existing customer demand to product availability and minimizing transportation costs.

The Problem in Words Your firm has a choice of three locations to operate a manufacturing facility in. Four customers exist with a known demand for your product. Each potential plant location has an associated monthly operating cost, and shipping routes to the demand cities have varying costs. In addition, each potential plant will have a shipping capacity that must not be exceeded. You need to determine what plant(s) to open and how much of a product to send from each open plant to each customer to minimize total shipping costs and fixed plant operating costs.

The Model MODEL: ! Capacitated Plant Location Problem; SETS: PLANTS: FCOST, CAP, OPEN; CUSTOMERS: DEM; ARCS( PLANTS, CUSTOMERS) : COST, VOL; ENDSETS DATA: ! The plant, their fixed costs and capacity; PLANTS, FCOST, CAP = P1 91 39 P2 70 35 P3 24 31; ! Customers and their demands; CUSTOMERS, DEM = C1 15 C2 17 C3 22 C4 12; ! The plant to cust cost/unit shipment matrix; COST = 6 2 6 7 4 9 5 3 8 8 1 5;


557

ENDDATA ! The objective; [TTL_COST] MIN = @SUM( ARCS: COST * VOL) + @SUM( PLANTS: FCOST * OPEN); ! The demand constraints; @FOR( CUSTOMERS( J): [DEMAND] @SUM( PLANTS( I): VOL( I, J)) >= DEM( J) ); ! The supply constraints; @FOR( PLANTS( I): [SUPPLY] @SUM( CUSTOMERS( J): VOL( I, J)) <= CAP( I) * OPEN( I) ); ! Make OPEN binary(0/1); @FOR( PLANTS: @BIN( OPEN)); END

Model: CAPLOC

The Sets We have two primitive sets in this modelthe plants (PLANTS) and the customers (CUSTOMERS). From these two primitive sets, we create a dense derived set, ARCS, which is the cross of the plants and customers sets. We use this set to represent the shipping arcs between the plants and customers.

The Variables There are two sets of decision variables in this model. The VOL attribute, defined on the ARCS set, represents the shipment volume from the plants to the customers along each arc. The OPEN attribute, defined on the PLANTS set, is used to represent the plants that are open. Specifically, OPEN(p) is 1 if plant p is opened, else it is 0. The of the OPEN attribute are set to being binary using the expression: ! Make OPEN binary(0/1); @FOR(PLANTS: @BIN(OPEN));

The Objective The objective in this model is to minimize total costs, which is the sum of the shipping costs and fixed plant costs. This is computed using the following expression: ! The objective; [TTL_COST] MIN = @SUM(ARCS: COST * VOL) + @SUM(PLANTS: FCOST * OPEN);

558 CHAPTER 12 The shipping cost component of the objective is computed with: @SUM(ARCS: COST * VOL)

while the fixed plant costs component is given by: @SUM(PLANTS: FCOST * OPEN)

The Constraints There are two sets of constraints in the model: 1. 2.

each customer must be sent enough product to satisfy demand, and each plant can’t supply more than its capacity.

The following expression guarantees each customer receives the quantity of product demanded: ! The demand constraints; @FOR(CUSTOMERS(J): [DEMAND] @SUM(PLANTS(I): VOL(I, J)) >= DEM(J) );

For each customer, we sum the amount being shipped to that customer and set it to be greater-than-or-equal-to the customer’s demand. To limit shipments from a plant to the plant’s capacity, we use: ! The supply constraints; @FOR(PLANTS(I): [SUPPLY] @SUM(CUSTOMERS(J): VOL(I, J)) <= CAP(I) * OPEN(I) );

For each plant, we sum up the amount being shipped from the plant and set this quantity to be less-than-or-equal-to the plant’s capacity multiplied by the plant’s 0/1 OPEN indicator. Note that, in order for the plant to be able to ship any quantity of product, the OPEN binary variable will be forced to 1 by these constraints.


559

The Solution Solving the model, we get the following solution: Global optimal solution found. Objective value: Extended solver steps: Total solver iterations: Variable OPEN(P1) OPEN(P3) VOL(P1, C1) VOL(P1, C2) VOL(P1, C4) VOL(P3, C3) VOL(P3, C4)

327.0000 4 25 Value 1.000000 1.000000 15.00000 17.00000 3.000000 22.00000 9.000000

Total costs are minimized at 327 by opening plants 1 and 3. From plant 1, we ship 15, 17, and 3 units respectively to customers 1, 2, and 4. From plant 3, we ship 22 units to customer 3, and 9 units to customer 4.

560 CHAPTER 12

Shortest Route Problem

Model: DYNAMB

Background In the shortest route problem, we want to find the shortest distance from point A to point B in a network. We will use an approach called dynamic programming (DP) to solve this problem. Dynamic programming involves breaking a large, difficult problem up into a series of smaller, more manageable problems. By solving the series of smaller problems, we are able to construct the solution to the initial large problem. Typically, the most difficult aspect of DP is not the mathematics involved in solving the smaller subproblems, but coming up with a scheme, or recursion, for decomposing the problem. To find the distance of the shortest path through the network, we will use the following DP recursion: F(i) = min [ D(i, j) + F(j)] j

where F(i) is the minimal travel distance from point i to the final destination point, and D(i, j) is the distance from point i to point j. In words, the minimal distance from node i to the terminal node is the minimum over all points reachable along a single arc from i of the sum of the distance from i to the ading node plus the minimal distance from the ading node to the terminal node.

The Problem in Words Suppose we have the following network of cities:

2

5 8

1

3

6

10 9

4

7

Links are assumed to be one-way. The distances between the cities are given to us. We want to determine the shortest distance between cities 1 and 10.


The Model SETS: ! Dynamic programming illustration (see Anderson, Sweeney & Williams, An Intro to Mgt Science, 6th Ed.). We have a network of 10 cities. We want to find the length of the shortest route from city 1 to city 10.; ! Here is our primitive set of ten cities, where F(i) represents the shortest path distance from city i to the last city; CITIES /1..10/: F; ! The derived set ROADS lists the roads that exist between the cities (note: not all city pairs are directly linked by a road, and roads are assumed to be one way.); ROADS(CITIES, CITIES)/ 1,2 1,3 1,4 2,5 2,6 2,7 3,5 3,6 3,7 4,5 4,6 5,8 5,9 6,8 6,9 7,8 7,9 8,10 9,10/: D; ! D(i, j) is the distance from city i to j; ENDSETS DATA: ! Here are the distances that correspond to the above links; D = 1 5 2 13 12 11 6 10 4 12 14 3 9 6 5 8 10 5 2; ENDDATA ! If you are already in City 10, then the cost to travel to City 10 is 0; F(@SIZE(CITIES)) = 0; ! The following is the classic dynamic programming recursion. In words, the shortest distance from City i to City 10 is the minimum over all cities j reachable from i of the sum of the distance from i to j plus the minimal

561

562 CHAPTER 12 distance from j to City 10; @FOR(CITIES(i)| i #LT# @SIZE(CITIES): F(i) = @MIN(ROADS(i, j): D(i, j) + F(j)) );

Model: DYNAMB

The Sets We have the single primitive set, CITIES, which corresponds to the cities in the network. From this primitive set, we form a single derived set, ROADS, to represent the links between the cities. We specify the of this set. Thus, ROADS is a sparse derived set.

The Variables The F attribute defined on the CITIES set is used to store the distance from each city to the destination city.

The Formulas The recursion, discussed above, is entered in LINGO with the following statement: @FOR(CITIES(i)| i #LT# @SIZE(CITIES): F(i) = @MIN(ROADS(i, j): D(i, j) + F(j)) );

The Solution Solving the model, we get the following values for F: Variable F(1) F(2) F(3) F(4) F(5) F(6) F(7) F(8) F(9) F(10)

Value 19.00000 19.00000 14.00000 20.00000 8.000000 7.000000 12.00000 5.000000 2.000000 0.000000

F(1), the shortest distance from city 1 to city 10, gives us the distance of the shortest path of 19. For the curious reader, this distance corresponds to the path 135810. Refer to model DYNAMB2.LG4 to see how to extend this model to compute the actual path as well as its distance.


Financial Models

Markowitz Portfolio Selection

563

Model: GENPRT

Background In the March, 1952 issue of Journal of Finance, Harry M. Markowitz published an article titled Portfolio Selection. In the article, he demonstrates how to reduce the risk of asset portfolios by selecting assets whose values aren’t highly correlated. At the same time, he laid down some basic principles for establishing an advantageous relationship between risk and return. This has come to be known as diversification of assets. In other words, don’t put all your eggs in one basket. A key to understanding the Markowitz model is to be comfortable with the statistic known as the variance of a portfolio. Mathematically, the variance of a portfolio is:

ij Xi Xj i,j where, Xi

i,j

is the fraction of the portfolio invested in asset i, for ij: the covariance of asset i with asset j, and for i=j: the variance of asset i.

Variance is a measure of the expected fluctuation in return—the higher the variance, the riskier the investment. The covariance is a measure of the correlation of return fluctuations of one stock with the fluctuations of another. High covariance indicates an increase in one stock’s return is likely to correspond to an increase in the other. A covariance close to zero means the return rates are relatively independent. A negative covariance means that an increase in one stock’s return is likely to correspond to a decrease in the other. The Markowitz model seeks to minimize a portfolio’s variance, while meeting a desired level of overall expected return.

The Problem in Words You’re considering investing in three stocks. From historical data, you have calculated an expected return, the variance of the return rate, and the covariance of the return between the different stocks. You want to reduce variability, or risk, by spreading your investment wisely amongst the three stocks. You have a target growth rate of 12%. As an additional safety feature, you decide to invest no more than 75% in any single asset. What percentages of your funds should you invest in the three stocks to achieve this target and minimize the risk of the portfolio?

564 CHAPTER 12

The Model ! GENPRT: Generic Markowitz portfolio; SETS: ASSET/1..3/: RATE, UB, X; COVMAT(ASSET, ASSET): V; ENDSETS DATA: ! The data; ! Expected growth rate of each asset; RATE = 1.3 1.2 1.08; ! Upper bound on investment in each; UB = .75 .75 .75; ! Covariance matrix; V = 3 1 -.5 1 2 -.4 -.5 -.4 1; ! Desired growth rate of portfolio; GROWTH = 1.12; ENDDATA ! The model; ! Min the variance; [VAR] MIN = @SUM(COVMAT(I, J): V(I, J) * X(I) * X(J)); ! Must be fully invested; [FULL] @SUM(ASSET: X) = 1; ! Upper bounds on each; @FOR(ASSET: @BND(0, X, UB)); ! Desired value or return after 1 period; [RET] @SUM(ASSET: RATE * X) >= GROWTH;

Model: GENPRT

The Sets We define a single primitive set, ASSETS, corresponding to the three stocks in the model. From the ASSETS set, we derive the dense set named COVMAT, which is the cross of the ASSETS set on itself. We use the COVMAT set for defining the covariance matrix.

The Attributes We define four attributes in this model. The RATE, UB and V attributes are for storing data. RATE stores the expected return for each asset, UB stores the upper bound on the fraction of the asset allowed in the portfolio, and V stores the covariance matrix. (Note the covariance matrix is symmetric and larger portfolio models would benefit from storing just half of the matrix, rather than the entire matrix as we have done here for simplicity reasons.)


565

The final attribute, X, constitutes the decision variables for the model. Specifically, X(i) is the fraction of the portfolio devoted to asset i.

The Objective The objective in this model is to minimize the portfolio’s risk. As we mentioned above, we use the portfolio’s variance as a measure of risk in the following expression: ! Min the variance; [VAR] MIN = @SUM(COVMAT(I, J): V(I, J) * X(I) * X(J));

The Constraints There are three forms of constraints in the model: 1. 2. 3.

we must be fully invested, we can’t invest too much in any one asset, and expected return should meet, or exceed, our threshold level of 12%.

The following expression enforces the 100% investment requirement: ! Must be fully invested; [FULL] @SUM(ASSET: X) = 1;

In words, the sum of all the weights of all the assets in the portfolio must equal 1. Without this constraint, LINGO will tend to under invest in order to get a lower variance. You can confirm this by dropping the constraint and running the model. To keep the solution from investing too much in any one asset, we use the @BND function: ! Upper bounds on each; @FOR(ASSET: @BND(0, X, UB));

As you recall, the @BND function is the most efficient method for placing simple bounds on variables. We constrain the portfolio’s expected return to be greater-than-or-equal-to our target with the expression: ! Desired value or return after 1 period; [RET] @SUM(ASSET: RATE * X) >= GROWTH;

The left-hand side of this expression is the expected rate of return, and is the sum of the fractions invested in each asset weighted by each asset’s return.

566 CHAPTER 12

The Solution Solving the model, we get the following solution: Local optimal solution found. Objective value: Total solver iterations:

Variable X(1) X(2) X(3) Row VAR FULL RET

0.4173749 13

Value 0.1548631 0.2502361 0.5949008 Slack or Surplus 0.4173749 0.0000000 0.2409821E-01

Solution: GENPRT Total variance is minimized at .4174 when we put 15.5% in asset 1, 25% in asset 2, and 59.5% in asset 3.


Scenario Portfolio Selection

567

Model: PRTSCEN

Background Scenarios are outcomes of events with an influence on the return of a portfolio. Examples might include an increase in interest rates, war in the Middle East, etc. In the scenario-based approach to portfolio selection, the modeler comes up with a set of scenarios, each with a certain probability of occurring during the period of interest. Given this set of scenarios and their probabilities, the goal is to select a portfolio that minimizes risk, while meeting a target return level. In the Markowitz portfolio model, presented above, we used a portfolio’s variance as a measure of risk. As one might imagine, variance is not the only possible measure of risk. Variance is a measure of the fluctuation of return above and below its average. As a statistic, variance weights a scenario that returns 20% above average the same as a scenario that returns 20% below average. If you’re like most investors, you’re probably more worried about the risk that return will be below average. In our scenario-based model, we will expand our options by including two new measures of risk that focus on returns below the target level—downside risk and semi-variance risk. Both semi-variance and downside risk only consider the option that returns will be below the target. Downside risk is a measure of the expected difference between the target and returns below the target, while semi-variance is a measure of the squared expected difference between the target and returns below the target. Therefore, semi-variance puts a relatively higher weight on larger shortfalls.

The Problem in Words Again, you are considering investing in three stocks (ATT, GM, and USX). You have determined there are 12 equally likely scenarios in the forthcoming period. You have come up with predicted rates of return for the stocks under the twelve scenarios. You have a target growth rate of 15% for your portfolio. Your goal is to construct optimal portfolios that minimize expected risk while meeting the expected level of return using three different risk measures—variance, downside, and semi-variance.

The Model ! Scenario portfolio model; SETS: SCENE/1..12/: PRB, R, DVU, DVL; STOCKS/ ATT, GMT, USX/: X; SXI(SCENE, STOCKS): VE; ENDSETS DATA: TARGET = 1.15; ! Data based on original Markowitz example; VE = 1.300 1.225 1.149 1.103 1.290 1.260 1.216 1.216 1.419 0.954 0.728 0.922 0.929 1.144 1.169 1.056 1.107 0.965

568 CHAPTER 12 1.038 1.321 1.133 1.089 1.305 1.732 1.090 1.195 1.021 1.083 1.390 1.131 1.035 0.928 1.006 1.176 1.715 1.908; ! All scenarios happen to be equally likely; PRB= .08333; ENDDATA ! Compute expected value of ending position; AVG = @SUM(SCENE: PRB * R); ! Target ending value; AVG >= TARGET; @FOR(SCENE(S): ! Compute value under each scenario; R(S) = @SUM(STOCKS(J): VE(S, J) * X(J)); ! Measure deviations from average; DVU(S) - DVL(S) = R(S) - AVG ); ! Budget; @SUM(STOCKS: X) = 1; ! Our three measures of risk; [VARI] VAR = @SUM(SCENE: PRB * (DVU+DVL)^2); [SEMI] SEMIVAR = @SUM(SCENE: PRB * (DVL) ^2); [DOWN] DNRISK = @SUM(SCENE: PRB * DVL); ! Set objective to VAR, SEMIVAR, or DNRISK; [OBJ] MIN = VAR;

Model: PRTSCEN

The Sets We define two primitive setsSCENE and STOCKS. The SCENE set corresponds to the set of 12 scenarios, while STOCKS is the set of three candidate stocks. We form a single dense derived set, STXSC, which is the cross of these two sets. We need the STXSC set in order to establish the table of returns for each stock under each scenario.

The Attributes We define four attributes on the scenarios setPRB, R, DVU, and DVL. PRB stores the probabilities of the scenarios. R is the expected rate of return under each scenario for a given allocation amongst the stocks. DVU is the deviation above average return for each scenario for a given stock allocation. Finally, DVL is the deviation below average return for each scenario for a given stock allocation. The X attribute is defined on the stocks set. X denotes the fraction of the portfolio allocated to each stock. The of X must be determined and constitute the decision variables of the model.


569

Finally, on the SXI set, we define the VE attribute, which stores the table containing the returns of each stock under each scenario.

The Objective Once again, the objective in this model is to minimize the portfolio’s risk. The default version of the objective minimizes variance as follows: ! Set objective to VAR, SEMIVAR, or DNRISK; [OBJ] MIN = VAR;

To solve using the other two measures of risk, simply change the name of the variable from VAR to either SEMIVAR or DNRISK.

The Formulas There are six categories of formulas in this model: 1. 2. 3. 4. 5. 6.

computing the expected rate of return (AVG), constraining the expected rate of return to exceed our target, computing the expected rate of return under each scenario (R), computing deviations from average return for each scenario (DVU and DVL), constraining the portfolio to be fully invested, and computing the three measures of risk (VAR, SEMIVAR, and DNRISK).

The following expression computes the expected rate of return for the entire portfolio: ! Compute expected value of ending position; AVG = @SUM(SCENE: PRB * R);

We do this by taking the sum of the expected returns for each scenario (R) weighted by the probabilities of the scenarios (PRB). We constrain expected return to be greater-than-or-equal-to our target using the constraint: ! Target ending value; AVG >= TARGET;

The expected return under scenario S is computed using: R(S) = @SUM(STOCKS(J): VE(S, J) * X(J));

This is just the sum of the return of each stock under the scenario weighted by the fraction of the portfolio in each stock.

570 CHAPTER 12 The deviations from average return for the scenarios are computed with: DVU(S) - DVL(S) = R(S) - AVG

If the expected return from the scenario is greater than the average, DVU will be positive. If the expected return is less than average, DVL will be positive. Given the fact that DVL and DVU impact the objective, there will never be an optimal solution where both DVL and DVU are greater than 0. If both are greater than 0, then a better solution can always be obtained by driving one of the two to zero. By setting things up this way, we have partitioned the deviations into two parts. DVU represents deviations above the average, which only the variance measure is concerned with. On the other hand, DVL represents deviations below average, which both the downside risk and semi-variance measures are computed from. The following constraint forces us to be fully invested and should be familiar from the discussion of the Markowitz model: ! Budget; @SUM(STOCKS: X) = 1;

We compute our three measures of risk using: ! Our three measures of risk; [VARI] VAR = @SUM(SCENE: PRB * (DVU + DVL)^2); [SEMI] SEMIVAR = @SUM(SCENE: PRB * (DVL) ^2); [DOWN] DNRISK = @SUM(SCENE: PRB * DVL);

These are simply the sum of a measure of deviation across the scenarios weighted by the probabilities of the scenarios. The variance measure takes the square of both the above and below average deviations. The semi-variance measure squares just the below average deviation. While, the downside risk measure uses the below average deviations without squaring.

The Solution If we solve the model using the three risk measures in the objective, we get the following three solutions: ATT GMT USX

Variance .530 .357 .113

Semi-variance .575 .039 .386

Downside Risk .511 .489 .000

The fraction of the portfolio devoted to ATT is fairly consistent. However, the fractions of GMT and USX can vary significantly when the risk measure changes.


Options Pricing

571

Model: OPTION

Background A call option is a financial instrument that gives the holder the right to buy one share of a stock at a given price (the exercise price) on or before some specified expiration date. A frequent question is, “How much should one be willing to pay for such an option?” We can easily come up with some broad bounds on what we would be willing to pay. Suppose the stock price is $105 and we have a chance to buy an option with an exercise price of $100. We should certainly be willing to pay up to $5 for the option, because, as a minimum, we could buy the stock for $100 and immediately sell it for $105, realizing a profit of $5. Thus, when the stock price exceeds the exercise price, we should be willing to pay at least the difference in the two prices. When the exercise price exceeds the stock price, buying at the exercise price and selling at the market price would not be an intelligent option. In this instance, we should be willing to “pay” at least $0. In any case, the most we would ever be willing to pay for an option is the current price of the underlying stock. Suppose the option price was more than the stock price. Then, we could get the same expected return at a lower price by buying the stock. The following graph illustrates these bounds on an option’s value as a function of the price of the underlying stock:

Option Value

P Exercise Price

Stock Price

Unfortunately, these bounds are not very tight. In reality, the option value function will resemble the curve P in the graph. The exact shape of this curve is influenced by three additional factors. These are, 1) the time to expiration, 2) the volatility, or variance, in the price movements of the stock, and 3) the interest rate. The underlying formula for curve P eluded researchers for many years until Fischer Black and Myron Scholes derived it in 1973. A Nobel Prize was subsequently awarded for their work in 1997.

572 CHAPTER 12

The Problem in Words You believe that the stock of National Semiconductor (symbol: NSM) is about to go up. To capitalize on this belief, you would like to purchase call options on NSM. Checking the quotes on NSM options with your online brokerage service, you find that NSM call options are trading at 6 5/8 per share. These call options have an exercise price of $40, and expire in 133 days from today. The current price of NSM shares is 40 3/4. Are the options trading at a fair price?

The Model ! Computing the value of an option using the Black & Scholes formula (see "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, May-June, 1973); SETS: ! We have 27 weeks of prices P(t), LOGP(t) is log of prices; WEEK/1..27/: P, LOGP; ENDSETS DATA: ! Weekly prices of National Semiconductor; P = 26.375, 27.125, 28.875, 29.625, 32.250, 35.000, 36.000, 38.625, 38.250, 40.250, 36.250, 41.500, 38.250, 41.125, 42.250, 41.500, 39.250, 37.500, 37.750, 42.000, 44.000, 49.750, 42.750, 42.000, 38.625, 41.000, 40.750; ! The current share price; S = 40.75; ! Time until expiration of the option, expressed in years; T = .3644; ! The exercise price at expiration; K = 40; ! The yearly interest rate; I = .163; ENDDATA SETS: ! We will have one less week of differences; WEEK1(WEEK)| &1 #LT# @SIZE(WEEK): LDIF; ENDSETS ! Take log of each week's price; @FOR(WEEK: LOGP = @LOG(P)); ! and the differences in the logs; @FOR(WEEK1(J): LDIF(J) = LOGP(J + 1) - LOGP(J));


573

! Compute the mean of the differences; MEAN = @SUM(WEEK1: LDIF)/ @SIZE(WEEK1); ! and the variance; WVAR = @SUM(WEEK1: (LDIF - MEAN)^2)/ (@SIZE(WEEK1) - 1); ! Get the yearly variance and standard deviation; YVAR = 52 * WVAR; YSD = YVAR^.5; ! The Black & Scholes option pricing formula; Z = ((I + YVAR/2) * T + @LOG(S/ K))/(YSD * T^.5); ! where VALUE is the expected value of the option; VALUE = S *@PSN(Z) - K *@EXP(- I * T) * @PSN(Z - YSD *T^.5); ! LDIF may take on negative values; @FOR(WEEK1: @FREE(LDIF)); ! The price quoted in the Wall Street Journal for this option when there were 133 days left was $6.625;

Model: OPTION

The Sets and Attributes In this model, there is a single primitive set, WEEK, which corresponds to the 27 weeks of price data. We define the two attributes P and LOGP on this set. P stores the raw price data on NSM, while LOGP stores the logarithm of the prices.

The Formulas It is beyond the scope of this document to get into the theoretical details behind the Black & Scholes pricing model. The interested reader should refer to Black and Scholes (1973) for the details. However, we will briefly discuss some of the mechanics of the key formulas involved in the model. In our model, we compute the option’s value in two key steps. First, we compute Z as follows: Z = ((I + YVAR/2) * T + @LOG(S/ K))/(YSD * T^.5); where, I YVAR T S K YSD

= the yearly interest rate, = variance of the stock’s price, = time until expiration in years, = current share price, = exercise price, and = standard deviation on stock’s price.

574 CHAPTER 12 We then use the value of Z in the following formula to determine the expected value of the option: VALUE = S *@PSN(Z)-K *@EXP(- I * T) * @PSN(Z-YSD *T^.5); where, @PSN(Z) = returns the cumulative standard normal probability, and @EXP(X) = returns ex.

The Solution Solving the model, LINGO comes up with a value of $6.58 for the call option―not too far from the quoted price of $6.63.


Bond Portfolio Optimization

575

Model: PBOND

Background In certain situations, a business or individual may be faced with financial obligations over a future number of periods. In order to defease (i.e., eliminate) this future debt, the debtor can determine a minimal cost mix of current assets (e.g., cash and bonds) that can be used to cover the future stream of payments. This problem is sometimes referred to as the cash flow matching problem or the debt defeasance problem.

The Problem in Words You are the head of your state’s lottery office. Lottery prizes are not paid out immediately, but are parceled out over a 15 year period. You know exactly how much your office needs to pay out in prizes over the next 15 years. You would like to set aside enough money from lottery receipts to invest in secure government bonds to meet this future stream of payments. All remaining lottery receipts will be turned over to the state’s treasurer to help fund the education system. You would like to turn over as many of the receipts as possible to the treasurer, so your plan is to purchase a minimal cost mix of bonds that just meets your future cash needs. Here is the amount of cash you will need (in millions) to make future prize payments: Year Needs

0 $10

1 $11

2 $12

3 $14

4 $15

5 $17

6 $19

7 $20

8 $22

9 $24

10 $26

11 $29

12 $31

There are two bonds currently being offered that you feel are of sufficient quality to guarantee the future stream of prize payments. These bonds are listed below: Bond A B

Years to Maturity 6 13

Price ($M) .98 .965

Coupon ($M) .06 .065

If funds are not invested in bonds, they can be placed into short-term money market funds. You conservatively estimate that short-term rates will be about 4% over the 15 year time horizon. How many of each bond should you buy, and how much additional cash should you allocate to money market funds to minimize your total outlay while still being able to meet all the future prize payments?

576 CHAPTER 12

The Model ! Bond portfolio/cash matching problem. Given cash needs in a series of future periods, what collection of bonds should we buy to meet these needs?; SETS: BOND/A B/ : MATAT, ! Maturity period; PRICE, ! Price; CAMNT, ! Coupon; BUY; ! Amount to buy; PERIOD/1..15/: NEED, !Cash needed each period; SINVEST; !Short term investment each period; ENDSETS DATA: STRTE = .04; !Short term interest rate; MATAT = 6, 13; !Years to maturity; PRICE = .980, .965; !Bond purchase prices; CAMNT = .060, .065; !Bond coupon amounts; NEED = 10, 11, 12, 14, 15, 17, 19, 20, 22, 24, 26, 29, 31, 33, 36; ! Cash needs; ENDDATA !Minimize total investment required to generate the stream of future cash needs; MIN = LUMP; ! First period is slightly special; LUMP = NEED(1) + SINVEST(1) + @SUM(BOND: PRICE * BUY); ! For subsequent periods; @FOR(PERIOD(I)| I #GT# 1: @SUM(BOND(J)| MATAT(J) #GE# I: CAMNT(J) * BUY(J)) + @SUM(BOND(J)| MATAT(J) #EQ# I: BUY(J)) + (1 + STRTE) * SINVEST(I - 1) = NEED(I) + SINVEST(I); ); ! Can only buy integer bonds; @FOR(BOND(J): @GIN(BUY(J)));

Model: PBOND

The Sets We define two primitive setsBOND and PERIOD. The BOND set corresponds to the two bonds, while PERIOD is the set of 15 years in the time horizon.


577

The Attributes We define four attributes on the BONDS setMATAT, PRICE, CAMNT, and BUY. MATAT stores the bonds’ maturities, PRICE the price, CAMNT the coupon amount, and BUY the number of each bond to buy. The NEED and SINVEST attributes are defined on the PERIOD set. NEED stores the cash needs in each period and SINVEST stores the amount in short-term investments in each period.

The Objective The objective of this model is to minimize the initial cash outlay. The initial cash outlay, or lump sum, is stored in the LUMP variable. Thus, our objective is written simply as: ! Minimize the total investment required to generate the stream of future cash needs; MIN = LUMP;

The Formulas There are three categories of formulas in this model: 1. 2. 3.

computing the initial lump sum payment (LUMP), sources=uses constraints, which enforce the condition that all sources of cash in a period must equal uses for cash in a period, and integer restrictions on the BUY variable limiting us to buying only whole numbers of bonds.

The following expression computes the lump outflow of cash in the initial period: LUMP = NEED(1) + SINVEST(1) + @SUM(BOND: PRICE * BUY);

Cash is needed for three purposes in the initial period: 1. 2. 3.

payment of lottery prizes (NEED(1)), allocations to short-term money funds (SINVEST(1)), and bond purchases (@SUM(BOND: PRICE * BUY)).

In the remaining 14 periods, sources of cash must equal uses of cash. We enforce this condition with: ! For subsequent periods; @FOR(PERIOD(I)| I #GT# 1: @SUM(BOND(J)| MATAT(J) #GE# I: CAMNT(J) * BUY(J)) + @SUM(BOND(J)| MATAT(J) #EQ# I: BUY(J)) + (1 + STRTE) * SINVEST(I - 1) = NEED(I) + SINVEST(I); );

578 CHAPTER 12 The sources of cash in a period are threefold: 1.

coupon receipts: @SUM(BOND(J)| MATAT(J) #GE# I: CAMNT(J) * BUY(J))

2.

maturing bonds: @SUM(BOND(J)| MATAT(J) #EQ# I: BUY(J))

3.

maturing short-term investments from the previous period: (1 + STRTE) * SINVEST(I - 1)

These sources must equal the following two uses: 1.

lottery prize payments: NEED(I)

2.

new short-term investments: SINVEST(I)

Finally, to force bond purchases to be whole numbers, we add: ! Can only buy integer bonds; @FOR(BOND(J): @GIN(BUY(J)));


The Solution If we solve the model, we get the following values: Global optimal solution found. Objective value: Extended solver steps: Total solver iterations: Variable LUMP BUY(A) BUY(B) SINVEST(1) SINVEST(2) SINVEST(3) SINVEST(4) SINVEST(5) SINVEST(6) SINVEST(7) SINVEST(8) SINVEST(9) SINVEST(10) SINVEST(11) SINVEST(12) SINVEST(13) SINVEST(14) SINVEST(15)

195.7265 4 27 Value 195.7265 96.00000 90.00000 4.796526 5.598387 5.432322 3.259615 0.000000 90.61000 81.08440 70.17778 56.83489 40.95828 22.44661 0.1944784 65.05226 34.65435 0.4052172E-01

Solution: PBOND You have been able to cover a total future debt of $319 million with a lump payment of $195.73 million. To do this, you buy 96 A bonds, 90 B bonds, put $4.8 million into short-term investment funds, and hold $10 million in cash to pay prizes in the initial period.

579

580 CHAPTER 12

Queuing Models

Erlang Queuing Models

Model: EZQUEUE

Background The telephone, communications, and computer industries have long used queuing models to estimate the performance of a service system in the face of random demand. The two most frequently used models are the Erlang loss and Erlang waiting models. In both cases, customers arrive randomly at a number of identical servers. In the Erlang loss model, there is no queue, so any customer finding all servers busy is lost. In the Erlang waiting model, there is an infinite queue space, so any customer finding all servers busy waits until a server is free. In either case, the major measure of performance is the fraction of customers that find all servers busy. To compute a system’s performance, we must know the load placed on the system per unit of time and the number of servers. The load is a unitless measure of the amount of work arriving per unit of time. For example, if 20 customers arrive each hour and each requires ½ hour of work, then the arriving load is 10 (20 customers per hour multiplied by ½ hour per customer). The most crucial probabilistic assumption in both cases is the number of arrivals per unit of time is Poisson distributed with a constant mean. The held case (with a queue) further requires that service times be exponentially distributed. If the arriving load is denoted AL and the number of servers by NS, then the expected fraction finding all servers busy is given in the loss case by @PEL(AL, NS) and in the held case by @PEB(AL, NS).

The Problem in Words You manage the customer service department of your business. Calls arrive at the rate of 25 customers per hour. Each call requires an average of 6 minutes to process. How many service representatives would be required for no more than 5% of customers to receive a busy signal?

The Model ! Arrival rate of customers/ hour; AR = 25; ! Service time per customer in minutes; STM = 6; ! Service time per customer in hours; STH = STM/ 60; ! Fraction customers finding all servers busy; FB = .05; ! The PEL function finds number of servers needed, NS; FB = @PEL(AR * STH, NS);

Model: EZQUEUE


581

The Solution Feasible solution found at step: Variable AR STM STH FB NS

0

Value 25.00000 6.000000 0.1000000 0.5000000E-01 5.475485

Because we cannot have fractional servers, we need at least six servers to meet our requirement that no more than 5% of customers find all servers busy. Suppose you install a sufficient number of incoming lines, so customers finding all servers busy can wait. Further, you will still use six servers. You want to find the following:    

the fraction of customers finding all servers busy, the average waiting time for customers who wait, the average overall waiting time, and the average number waiting.

The following variation on the previous model computes these four statistics: ! Arrival rate of customers/ hour; AR = 25; ! Service time per customer in minutes; STM = 6; ! Service time per customer in hours; STH = STM/ 60; ! The number of servers; NS = 6; ! The PEL function finds number of servers needed, NS; FB = @PEB(AR * STH, NS); ! The conditional wait time for those who wait; WAITC = 1 / (NS / STH - AR); ! The unconditional wait time; WAITU = FB * WAITC; ! The average number waiting; NWAIT = AR * WAITU;

582 CHAPTER 12 Note how we now use the @PEB function, rather than @PEL, to for the presence of a queue to hold callers finding all lines busy. The solution to the modified model is: Variable AR STM STH NS FB WAITC WAITU NWAIT

Value 25.00000 6.000000 0.1000000 6.000000 0.4744481E-01 0.2857143E-01 0.1355566E-02 0.3388915E-01

the unit of time is an hour, so the expected waiting time for those who wait is .2857 * 60 = 1.7 minutes.


Machine Repairman Models

583

Model: EZMREPAR

Background Another queuing model illustrates service demand from a finite population of s. The underlying assumption is, if a significant fraction of these s are already waiting for service, then the arrival rate of further service demands decreases until more of the s are serviced and returned to the calling community. Models of this class are referred to as Machine Repairman models because the calling population can be viewed as a set of machines where individual machines occasionally break down and need repair. The @PFS ((Poisson Finite Source) function computes the expected number of customers either in repair or waiting for repair, given the number of customers, number of repairmen, and the limiting load of the queue. The following model illustrates the use of the @PFS function to model a computer timesharing system. Each incoming port can be thought of as a customer. Usually, the number of incoming ports is limited. Thus, the finite source assumption is appropriate.

The Problem in Words You manage the ing department in your company. You are contemplating installing a new server to handle the growing computing needs of your department. However, you have your doubts as to whether or not the system proposed by the computing services department will be adequate to meet your current and future needs. Your department has 32 employees who will make continuous use of the server throughout the day. You do some research and determine the mean time between server requests is 40 seconds per . On average, it will take the proposed server 2 seconds to process a request. Will the proposed server provide the horsepower needed by your department?

584 CHAPTER 12

The Model ! Model of a computer timesharing system; ! The mean think time for each (more generally, Mean Time Between Failures in a repair system); MTBF = 40; ! The mean time to process each compute request (more generally, Mean Time To Repair in seconds); MTTR = 2; ! The number of s; N = 32; ! The number of servers/repairmen; NREPR = 1; ! The mean number of s waiting or in service (more generally, the mean number of machines down); NDOWN = @PFS(MTTR * N/ MTBF, NREPR, N); ! The overall request for service rate (more generally, overall failure rate), FR, must satisfy; FR = (N - NDOWN)/ MTBF; ! The mean time waiting for or in service (more generally, the mean time down), MTD, must satisfy; NDOWN = FR * MTD;

Model: EZMREPAR

The Solution Variable MTBF MTTR N NREPR NDOWN FR MTD

Value 40.00000 2.000000 32.00000 1.000000 12.06761 0.4983098 24.21707

Solution: EZMREPAR This would probably be considered a heavily loaded systemon average, about 12 s are waiting for a response from the server. Each request for processing requires an expected elapsed time of over 24 seconds, even though the average request is for only two seconds of processing. As a group, s request service at the rate of almost one request every 2 seconds.


Steady State Queuing Model

585

Model: QUEUEM

Background A useful approach for tackling general queuing models is to use the Rate In = Rate Out Principle (RIRO) to derive a set of steady state equations for a queuing system. RIRO assumes a system can reach a state of equilibrium. In equilibrium, the tendency to move out of a certain state must equal the tendency to move towards that state. Given the steady state equations derived from this assumption, we can solve for the probability that a system is in a given state at any particular moment. The following example assumes a system with multiple servers and “customers” that arrive in batches.

The Problem in Words You operate a motor rebuilding business that services auto repair shops throughout a several state area. Motors arrive on trucks in batches of up to four motors per truck. On average, trucks arrive 1.5 times a day. The probabilities that a truck contains 1, 2, 3 or 4 motors are, respectively, .1, .2, .3 and .4. You presently have five workstations that can each rebuild two motors per day. You would like to introduce a priority service that, in exchange for a higher fee, guarantees your customers a one day turnaround. To do this effectively, at least one workstation must be free 90 percent of the time in order to begin immediate repair of any incoming priority job. Are your current facilities adequate for this level of service?

The Model ! Model of a queue with arrivals in batches. In this particular example, arrivals may show up in batches of 1, 2, 3, or 4 units; SETS: ! Look at enough states so P(i) for large i is effectively zero, where P(i) is the steady state probability of i-1 customers in the system; STATE/ 1..41/: P; ! Potential batch sizes are 1, 2, 3 or 4 , customers and A(i) = the probability that an arriving batch contains i customers; BSIZE/ 1..4/: A; ENDSETS DATA: ! Batch size distribution; A = .1, .2, .3, .4; ! Number of batches arriving per day; LMDA = 1.5; ! Number of servers; S = 5;

586 CHAPTER 12 ! Number of customers a server can process per day; MU = 2; ENDDATA ! LAST = number of STATES; LAST = @SIZE(STATE); ! Balance equations for states where the number of customers in the system is less than or equal to the number of servers; @FOR(STATE(N)| N #LE# S: P(N) * ((N - 1)* MU + LMDA) = P(N + 1) * MU * N + LMDA * @SUM(BSIZE(I)| I #LT# N: A(I) * P(N - I)) ); ! Balance equations for states where number in system is greater than the number of servers, but less than the limit; @FOR(STATE(N)| N #GT# S #AND# N #LT# LAST: P(N) * (S * MU + LMDA) = P(N + 1) * MU * S + LMDA * @SUM(BSIZE(I)| I #LT# N: A(I) * P(N - I)) ); ! Probabilities must sum to 1; @SUM(STATE: P) = 1;

Model: QUEUEM

The Formulas The model computes the probabilities P(i), i = 1 to 41, where P(i) is the probability there are i - 1 motors in the system for repair. We have chosen to stop at 41 because the probability the system would have more than 40 machines waiting for repair is effectively 0. In order to solve for the 41 unknown P(i), we will need 41 equations. One of these equations comes from the fact that the probabilities must all sum to 1: ! Probabilities must sum to 1; @SUM(STATE: P) = 1;


587

The remaining 40 equations are derived from the steady state assumptions that the rate of movement into any state must equal the rate out. We partition these balance equations into two types. In the first case, there are the states where the number of motors in the system is less-than-or-equal-to the number of servers. The balance equations in this case are: @FOR(STATE(N)| N #LE# S: P(N) * ((N - 1)* MU + LMDA) = P(N + 1) * MU * N + LMDA * @SUM(BSIZE(I)| I #LT# N: A(I) * P(N - I)) );

For each state where the number of motors in the system is less-than-or-equal-to the number of servers, we set the rate of flow out of the state: P(N) * ((N - 1)* MU + LMDA)

equal to the rate of flow into the state: P(N + 1) * MU * N + LMDA * @SUM(BSIZE(I)| I #LT# N: A(I) * P(N - I))

The rate of flow out is the rate of flow to states with more motors, LMDA, plus the rate of flow to lesser states, (N - 1) * MU, multiplied by the probability of being in the state, P(N). The rate of flow in is the rate of flow in from higher states, P(N + 1) * MU * N, plus the expected rate of flow in from lower states, LMDA * @SUM(BSIZE(I)| I #LT# N: A(I) * P(N - I)). We generate equivalent balance equations for the higher states where the number of motors exceeds the number of servers with the following: @FOR(STATE(N)| N #GT# S #AND# N #LT# LAST: P(N) * (S * MU + LMDA) = P(N + 1) * MU * S + LMDA * @SUM(BSIZE(I)| I #LT# N: A(I) * P(N - I)) );

588 CHAPTER 12

The Solution An excerpt from the full solution report showing the first 10 values of P is listed below: Variable LMDA S MU LAST P(1) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(10)

Value 1.500000 5.000000 2.000000 41.00000 0.2450015 0.1837511 0.1515947 0.1221179 0.9097116E-01 0.5707410E-01 0.4276028E-01 0.3099809E-01 0.2187340E-01 0.1538003E-01

The probability at least one workstation is free is the probability that four or fewer motors are in the system, in other words, the sum of the first five P(i). In this case, it works out that at least one workstation will be free only about 70% of the time. Experimenting with the model by increasing the number of servers reveals you will need at least seven workstations to provide the level of service required by the new priority plan.


Marketing Models Markov Chain Model

589

Model: MARKOV

Background A standard approach used in modeling random variables over time is the Markov chain approach. Refer to an operations research or probability text for complete details. The basic idea is to think of the system as being in one of a discrete number of states at each point in time. The behavior of the system is described by a transition probability matrix that gives the probability the system will move to a specified other state from some given state. Some example situations are: System Consumer brand switching Inventory System

States Brand of product most recently purchased by consumer Amount of inventory on hand

Cause of Transition Consumer changes mind, advertising Orders for new material, demands

An interesting problem is to determine the long-term, steady state probabilities of the system. If we assume the system can reach equilibrium, then it must be true the probability of leaving a particular state must equal the probability of arriving in that state. You will recall this is the Rate In = Rate Out Principle (RIRO) we used above in building the multi-server queuing model, QUEUEM. If we let:

i = the steady state probability of being in state i, and pij = the transition probability of moving from state i to j, then, by our RIRO assumption, for each state i:

ji pj pji = i (1 - pjj)

Rewriting the above, we get:

i = j j pji

This gives us n equations to solve for the n unknown steady state probabilities. Unfortunately, it turns out this system is not of full rank. Thus, there is not a unique solution. To guarantee a valid set of probabilities, we must make use of one final condition—the sum of the probabilities must be 1.

590 CHAPTER 12

The Problem in Words Your company is about to introduce a new detergent and you’re interested in whether it will clean up in the market. It will be competing against three other existing brands. As a result of a small test market and interviews with consumers familiar with all four detergents, we have derived the following purchase transition matrix:

Previous Purchase:

A B C D

Next Purchase: A B C .75 .1 .05 .4 .2 .1 .1 .2 .4 .2 .2 .3

D .1 .3 .3 .3

Our new detergent is brand A in this matrix. The interpretation of the matrix is, for example, if someone most recently purchased brand A, then with probability .75 his next purchase of this product will also be brand A. Similarly, someone who most recently purchased brand B will next purchase brand D with probability .3. An associate of yours who looked at this matrix said, “Aha, we should expect to get 75% of the market in the long run with brand A.” Do you think your associate is correct?

The Model ! Markov chain model; SETS: ! There are four states in our model and over time the model will arrive at a steady state equilibrium. SPROB(J) = steady state probability; STATE/ A B C D/: SPROB; ! For each state, there's a probability of moving to each other state. TPROB(I, J) = transition probability; SXS(STATE, STATE): TPROB; ENDSETS DATA: ! The transition probabilities. These are probabilities of moving from one state to the next in each time period. Our model has four states, for each time period there's a probability of moving to each of the four states. The sum of probabilities across each of the rows is 1, since the system either moves to a new state or remains in the current one.; TPROB = .75 .1 .05 .1 .4 .2 .1 .3 .1 .2 .4 .3 .2 .2 .3 .3; ENDDATA

Developing More Advanced Models ! !

591

Steady state equations; Only need N equations, so drop last; @FOR(STATE(J)| J #LT# @SIZE(STATE): SPROB(J) = @SUM(SXS(I, J): SPROB(I) * TPROB(I, J)) );

! The steady state probabilities must sum to 1; @SUM(STATE: SPROB) = 1; ! Check the input data, warn the if the sum of probabilities in a row does not equal 1.; @FOR(STATE(I): @WARN('Probabilities in a row must sum to 1.', @ABS(1 - @SUM(SXS(I, K): TPROB(I, K))) #GT# .000001); );

Model: MARKOV

The Sets The primitive STATE set represents the four states of purchasing detergents A, B, C, and D. We build one derived set, SXS that is the cross of the STATE set on itself. The SXS set is used to establish the state transition matrix.

The Attributes We have two attributes. The first, SPROB, is defined on the STATES set and is used to store the steady state probabilities of the system. We will be solving for the values of the SPROB attribute. The second attribute, TPROB, is defined on the two-dimensional SXS set and is used to store the values of the state transition matrix.

The Formulas First off, to ensure data integrity, we use the @WARN function to the probabilities in each row of the state transition matrix sum to 1 using: ! Check the input data, warn the if the sum of probabilities in a row does not equal 1.; @FOR(STATE(I): @WARN('Probabilities in a row must sum to 1.', @ABS(1 - @SUM(SXS(I, K): TPROB(I, K))) #GT# .000001);

Due to the potential for roundoff error, we allow for a modest transgression by using a tolerance factor of .000001. If the probabilities in a row sum up to more than 1.000001 or less than .999999, the will receive a warning. Next, the steady state probabilities must be exhaustive. We guarantee this by setting their sum to 1 with: ! The steady state probabilities must sum to 1; @SUM(STATE: SPROB) = 1;

592 CHAPTER 12 Finally, in addition to this last equation, we need an additional n-1 equations to solve for the n steady state probabilities. We get these equations from the RIRO derivation above using: ! !

Steady state equations; Only need N equations, so drop last; @FOR(STATE(J)| J #LT# @SIZE(STATE): SPROB(J) = @SUM(SXS(I, J): SPROB(I) * TPROB(I, J)) );

The Solution The solution is: Variable SPROB(A) SPROB(B) SPROB(C) SPROB(D)

Value 0.4750000 0.1525000 0.1675000 0.2050000

Solution: MARKOV So, we see the long run share of our new brand A will only amount to about 47.5% of the market, which is considerably less than the conjectured share of 75%.


Cont Analysis

593

Model: CONJNT

Background When deg a product, it is useful to know how much customers value various attributes of that product. This allows us to design the product most preferred by consumers within a limited budget. For instance, if we determine consumers place a very high value on a long product warranty, then we might be more successful in offering a long warranty with fewer color options. The basic idea behind cont analysis is, while it may be difficult to get consumers to accurately reveal their relative utilities for product attributes, it’s easy to get them to state whether they prefer one product configuration to another. Given these rank preferences, you can use cont analysis to work backwards and determine the implied utility functions for the product attributes.

The Problem in Words Your company is about to introduce a new vacuum cleaner. You have conducted customer surveys and you determined the relative preferences for the following warranty/price product configurations (9 being the most preferred): 2 Years 1 Year None

$129 7 3 1

$99 8 4 2

$79 9 6 5

In order to derive an optimal product configuration, you need to know the utility functions for warranty length and price implied by the preferences in the table above.

The Model ! Cont analysis model to decide how much weight to give to the two product attributes of warranty length and price; SETS: ! The three possible warranty lengths; WARRANTY /LONG, MEDIUM, SHORT/ : WWT; ! where WWT(i) = utility assigned to warranty i; ! The three possible price levels (high, medium, low); PRICE /HIGH, MEDIUM, LOW/ : PWT; ! where PWT(j) = utility assigned to price j; ! We have a customer preference ranking for each combination; WP(WARRANTY, PRICE) : RANK; ENDSETS DATA: ! Here is the customer preference rankings running from a least preferred score of 1 to the most

594 CHAPTER 12 preferred of 9. Note that long warranty and low price are most preferred with a score of 9, while short warranty and high price are least preferred with a score of 1; RANK = 7 8 9 3 4 6 1 2 5; ENDDATA SETS: ! The next set generates all unique pairs of product configurations such that the second configuration is preferred to the first; WPWP(WP, WP) | RANK(&1, &2) #LT# RANK(&3, &4): ERROR; ! The attribute ERROR computes the error of our estimated preference from the preferences given us by the customer; ENDSETS ! For every pair of rankings, compute the amount by which our computed ranking violates the true ranking. Our computed ranking for the (i,j) combination is given by the sum WWT(i) + PWT(j). (NOTE: This makes the bold assumption that utilities are additive!); @FOR(WPWP(i, j, k, l): ERROR(i, j, k, l) >= 1 + (WWT(i) + PWT(j)) - (WWT(k) + PWT(l)) ); ! The 1 is required on the right-hand-side of the above equation to force ERROR to be nonzero in the case where our weighting scheme incorrectly predicts that the combination (i,j) is equally preferred to the (k,l) combination. Since variables in LINGO have a default lower bound of 0, ERROR will be driven to zero when we correctly predict (k,l) is preferred to (i,j). Next, we minimize the sum of all errors in order to make our computed utilities as accurate as possible; MIN = @SUM(WPWP: ERROR);

Model: CONJNT

The Sets We have two primitive sets in the model: WARRANTY is the set of warranty lengths and PRICE is the set of prices. We form the derived set WP by taking the cross of WARRANTY and PRICE in order to create an attribute to store the preferences for each (WARRANTY, PRICE) pair.


595

The interesting set in this model is the sparse derived set WPWP: ! The next set generates all unique pairs of product configurations such that the second configuration is preferred to the first; WPWP(WP, WP) | RANK(&1, &2) #LT# RANK(&3, &4): ERROR;

This set is derived from the cross of WP on itself. Each member of this set contains two product configurations. Furthermore, we use a hip condition to limit the set to combinations of product configurations where the first configuration is preferred to the second configuration. We need this set because our model will be testing its proposed utility functions against all unique pairings of configurations. Note that this set grows on the order of n2, where n is the number of product configurations. Thus, it will tend to get big for large numbers of product configurations.

The Attributes The model defines four attributes: WWT, PWT, RANK and ERROR. WWT and PWT are used to store the utilities of the warranty and price configurations, respectively. The model will solve for values of WWT and PWT that minimize total prediction error. These values will then give us the implied utility functions for the two product attributes. RANK is used to store the customer preference rankings. Finally, ERROR stores the error in predicting the preference of one configuration over another given the implied utility values contained in WWT and PWT.

The Objective The objective is quite simplewe want to minimize the total prediction error over all the unique product configurations: MIN = @SUM(WPWP: ERROR);

The Constraints The model has just one class of constraints that computes the error term in predicting the preferences over each product configuration: @FOR(WPWP(i, j, k, l): ERROR(i, j, k, l) >= 1 + (WWT(i) + PWT(j)) (WWT(k) + PWT(l)) );

We need to add a 1 to the right-hand side of the constraint to for the case where we predict that configuration (i,j) is equally preferred to (k,l). Because of the way that we defined the WPWP set, (i,j) will always be preferred to (k,l). Thus, it would be an error to predict they are equally preferred. Note, also, because the lower bound on ERROR is 0 and we are also minimizing ERROR, ERROR(i,j,k,l) will be driven to 0 when the model correctly predicts that (i,j) is preferred to (k,l).

596 CHAPTER 12

The Solution Portions of the solution are reproduced below: Global optimal solution found at step: Objective value: Variable WWT(LONG) WWT(MEDIUM) WWT(SHORT) PWT(HIGH) PWT(MEDIUM) PWT(LOW)

35 0.000000

Value 7.000000 2.000000 0.000000 0.000000 1.000000 4.000000

Solution: CONJNT Short, medium and long warranties rate utilities of 0, 2, and 7, while high, medium and low prices rate utilities of 0, 1, and 4. Note that, given the objective value of zero, this utility weighting scheme exactly predicts preferences for each pair of product configurations.

13 Programming LINGO Up to this point, we have primarily been concerned with self-contained models, which are solved once and don’t receive input from, nor output to, other models. However, in many modeling situations, there may be requirements for one, or more, models to be run in a sequence. Furthermore, the outputs of one model may be required as inputs to a subsequent model. Such a series of models will typically be run in a loop until some termination criterion is met. LINGO provides a number of programming control structures to automate processing in situations where one has multiple, dependent models that one must run in a sequence. In the section immediately following, we will provide a brief introduction to the programming features available in LINGO. We will then illustrate the use of these features in several models.

Programming Features The programming capabilities in LINGO fall into seven categories:       

Model control Flow of control Model generation Output statements Setting parameters Stochastic Programming Utility functions

Model Control In this category we have two types of statements. First are the SUBMODEL and ENDSUBMODEL statements used to identify submodels, which are separate models contained within larger models. We also have the @SOLVE statement for use in solving submodels.

SUBMODEL and ENDSUBMODEL: These two statements are used to bracket a submodel within a larger, composite model. The SUBMODEL statement must be followed immediately by a unique name that follows the normal LINGO naming conventions. The name must then be followed by a colon. You must also place a ENDSUBMODEL statement immediately after the last statement of your submodel. Submodels may only exist in the model section and are not allowed in data, init and calc sections.

597

598 CHAPTER 13 As an example, the following illustrates a submodel for solving a knapsack type model: SUBMODEL Pattern_Gen: [R_OBJ] MAX = @SUM( FG(i): PRICE(i)* Y(i)); [R_WIDTH] @SUM( FG(i): WIDTH(i)*Y(i)) <= RMWIDTH; @FOR( FG(i): @GIN(Y(i))); ENDSUBMODEL

In this example, the submodel is titled Pattern_Gen and contains one objective, R_OBJ, one constraint, R_WIDTH, and one @FOR loop to force the Y variables to be general integers via the @GIN function.

@DEBUG( [SUBMODEL_NAME[, …, SUBMODEL_NAME_N]]) Infeasible or unbounded submodels may be debugged in a calc section with the use of the @DEBUG statement. Refer to the LINGO|Debug command for more information of model debugging. If your model contains submodels, you can choose to debug a particular submodel by specifying its name as an argument to @DEBUG. If desired, you may also specify more than one submodel name, in which case, LINGO will simultaneously debug all the specified models as one combined model. If a submodel name is omitted, LINGO will solve all model statements occurring before the @DEBUG statement and not lying within a submodel section. It is the ’s responsibility to make sure the submodels together make sense, e.g., at most one submodel in an @DEBUG invocation can have an objective function. In the following example, we solve a small submodel and then invoke the debugger if the solution is found to be non-optimal: MODEL: SUBMODEL M: MIN = X + Y; X>4; Y<3; Y>X; ENDSUBMODEL CALC: @SOLVE( M); @IFC( @STATUS() #NE# 0: @DEBUG( M)); ENDCALC END

Note:

Submodels must be defined in the model prior to any references to them via the @DEBUG statement.

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@SOLVE( [SUBMODEL_NAME[, …, SUBMODEL_NAME_N]]) Submodels may be solved in a calc section with the use of the @SOLVE statement. If your model contains submodels, you can choose to solve a particular submodel by specifying its name as an argument to @SOLVE. If desired, you may also specify more than one submodel name, in which case, LINGO will simultaneously solve all the specified models as one combined model. If a submodel name is omitted, LINGO will solve all model statements occurring before the @SOLVE statement and not lying within a submodel section. It is the ’s responsibility to make sure the submodels together make sense, e.g., at most one submodel in an @SOLVE can have an objective function. As an example, to solve the submodel Pattern_Gen listed immediately above, you would add the following statement to a calc section: @SOLVE( Pattern_Gen);

Note:

Submodels must be defined in the model prior to any references to them via the @SOLVE statement.

Charting LINGO has a number of features to generate charts of your data. Chart types currently included are Bar, Bubble, Contour, Curve, Histogram, Line, Network (node based and arc based), Pie, Radar, Scatter and Surface. A description of the functions for each of the various chart types follows.

@CHARTBAR( 'TITLE', 'X-AXIS LABEL', 'Y-AXIS LABEL', 'LEGEND1', ATTRIBUTE1[, ..., 'LEGENDN', ATTRIBUTEN]); @CHARTBAR will generate a bar chart of one or more attributes. The bar chart will be displayed in a new window. @CHARTBAR argument descriptions follow: TITLE - This is the title to display at the top of the chart. The title must be in quotes. X-AXIS LABEL - The x-axis label in quotes. Y-AXIS LABEL - The y-axis label in quotes. LEGENDi - The legend to be used to label attribute i in quotes. ATTRIBUTEi - This is the attribute to chart. At least one attribute argument must appear. If multiple attributes appear, the bars will be grouped together. As an example, consider the following model, CHARTSTAFF, which is a modified version of the staff scheduling example presented in section Primitive Set Example in Chapter 2, Using Sets:

600 CHAPTER 13 MODEL: SETS: DAYS: REQUIRED, START, ONDUTY; ENDSETS DATA: DAYS = MON TUE WED THU FRI SAT SUN; REQUIRED = 23 16 13 16 19 14 12; ENDDATA SUBMODEL MODSTAFF: MIN = @SUM( DAYS( I): START( I)); @FOR( DAYS( J): ONDUTY( J) = @SUM( DAYS( I) | I #LE# 5: START( @WRAP( J - I + 1, 7))); ONDUTY( J) >= REQUIRED( J) ); ENDSUBMODEL CALC: ! Solve the staffing model; @SOLVE( MODSTAFF); ! Bar chart of required vs. actual staffing; @CHARTBAR( 'Staffing Example', !Chart title; 'Day', !X-Axis label; 'Employees', !Y-Axis label; 'Employees Required', !Legend 1; REQUIRED, !Attribute 1; 'Employees On Duty', !Legend 2; ONDUTY !Attribute 2; ); ENDCALC END

Model: CHARTSTAFF Here, we've added a reference to @CHARTBAR to generate a bar chart of staffing requirements vs. staff on duty for each day of the week. Here's is the bar chart that is generated:

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Note that since we specified more than one attribute, the bars for the two attributes for each day of the week are grouped together. LINGO also automatically labeled each bar using the attributes' parent set DAYS. From the chart it's easy to see that staffing requirements are just met on days Monday through Saturday, while on Sunday we have slightly more staffing than is actually required. All arguments to charting function are optional, with the exception of the attribute values. In many cases, if an argument is omitted, LINGO will provide a reasonable default. For example, if the title is omitted as we have done here:

602 CHAPTER 13 ! Bar chart of required vs. actual staffing; @CHARTBAR( , !Chart title will default to file name; 'Day', !X-Axis label; 'Employees', !Y-Axis label; 'Employees Required', !Legend 1; REQUIRED, !Attribute 1; 'Employees On Duty', !Legend 2; ONDUTY !Attribute 2; );

Then LINGO will use the model's file name instead. Note that you must still include the comma separator when omitting an argument. The complete list of charting functions ed is given in the table below. You may also be interested in running the CHARTS.LG4 sample model, which generates examples of all the chart types. Chart Type

Syntax

Notes

Bar

@CHARTBAR( 'Title', 'X-Axis Label', 'Y-Axis Label', 'Legend1', Attribute1[, ..., 'Legendn', Attributen]);

Multiple attributes will result in bars being grouped.

Bubble

@CHARTBUBBLE( 'Title', 'X-Axis Label', 'Y-Axis Label', 'Legend1', X1, Y1, Size1,[, ..., 'Legendn', Xn, Yn, Sizen]);

Each series of bubbles is represented as a triplet of (xlocation, y-location, bubble size).

Contour

@CHARTCONTOUR( 'Title', 'X-Axis Label', 'Y-Axis Label', 'Legend', X, Y, Z);

You can thinks of a contour chart as a 3-dimensional surface chart compressed into two dimensions. Only one series is allowed and it is represented by a grid (x,y,z) coordinates.

Curve

@CHARTCURVE( 'Title', 'X-Axis Label', 'Y-Axis Label', 'Legend1', X1, Y1[, ..., 'Legendn, Xn, Yn]);

Curves are good for representing two-dimensional (x,y) plots, e.g., y=sin(x) . Multiples curves may be drawn in a single chart, with each curve represented as a series of (x,y) coordinates. See the sample model CHARTDISTRO for an example of a curve chart.

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Histogram

@CHARTHISTO( 'Title', 'X-Axis Label', 'Y-Axis', 'Legend', Bins, X);

If the bin count, Bins, is omitted, LINGO will automatically compute a number of bins that results in a resonably distributed histogram. Only one histogram may be displayed per chart.

Line

@CHARTLINE( 'Title', 'X-Axis Label', 'Y-Axis Label', Multiple lines can be displayed 'Legend1', X1[, ..., 'Legendn', Xn]); on a single chart, with each being drawn in a different color.

Netarc

@CHARTNETARC( 'Title', 'X-Axis', 'Y-Axis', 'Legend1', X1, Y1, X2, Y2[, ..., 'Legendn', X1n, Y1n, X2n, Y2n]);

Netnode

@CHARTNETNODE( 'Title', 'X-Axis Label', 'Y-Axis' Netnode charts are network Label, 'Legend1', X1, Y1, I1, J1[, ..., 'Legendn', Xn, Yn, charts, where the network is In, Jn]); represented by a two lists of 2tuples, (x,y) and (i,j), where the (x,y) are the coordinates of each node in the network and the (i,j) give the indices of two nodes from the (x,y) list that exist as an arc in the network. Multiple networks may be displayed on a single chart.

Pie

@CHARTPIE( 'Title', 'Legend', X);

Only one attribute may be displayed in a pie chart.

Radar

@CHARTRADAR( 'Title', 'Legend1', X1[, ..., 'Legendn', Xn]);

Multiple attributes can be displayed, with each being in a different color.

Scatter

@CHARTSCATTER( 'Title', 'X-Axis Label', 'Y-Axis Label', 'Legend1', X1, Y1[, ..., 'Legendn', Xn, Yn]);

Each scatter plot is represented by a list of (x,y) coordinates

Netarc charts are network charts, where the network is represented by a list of 4-tuples, (x1,y1,x2,y2), where (x1,y1)-i is one end point of arc i and (x2,y2)-i is the other end point of arc i. Multiple networks may be displayed on a single chart.

604 CHAPTER 13 giving the location of each point in the scatter. Multiple scatter plots can be displayed in a single chart, with each being in a different color. Surface

@CHARTSURFACE( 'Title', 'X-Axis Label', 'Y-Axis Label', 'Z-Axis Label', 'Legend', X, Y, Z);

Each surface plot is represented by a list of (x,y,z) coordinates, each containing the location of a points on the surface. A sufficient number of points should be included to avoid excessive interpolation between points. Only one surface may be drawn on a chart. See the sample model CHARTSURF below for an example of a surface chart.

As another example, we will show how to create a surface chart by supplying a grid of (x,y,z) points lying on the surface: z = x * sin( y) + y * sin( x) using the following model, CHARTSURF:

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MODEL: ! Creates a surface chart of z = x * sin( y) + y * sin( x); SETS: !Set up a 21x21 grid of (X,Y,Z) points on the surface; SURF1 /1..21/; SURF2( SURF1, SURF1): X, Y, Z; ENDSETS CALC: !First point; XS = @FLOOR( - (@SIZE( SURF1) / 2) + .5); YS = XS; !Generate remaining points and compute z; @FOR( SURF2( I, J): X( I, J) = XS + I - 1; Y( I, J) = YS + J - 1; Z( I, J) = X( I, J) * @SIN( Y( I, J)) + Y( I, J ) * @SIN( X( I, J)); ); !Create the chart; @CHARTSURFACE( 'z = x * sin( y) + y * sin( x)', !Title; 'X', 'Y', 'Z', !Axis labels; '(X,Y,Z)', !Legend; X, Y, Z !Points on the surface; ); ENDCALC END

Model: CHARTSURF First off, the CHARTSURF creates a 21x21 grid of points on the (X,Y) plane centered around (0,0) and computes Z for each of these points:

606 CHAPTER 13 !First point; XS = @FLOOR( - (@SIZE( SURF1) / 2) + .5); YS = XS; !Generate remaining points and compute z; @FOR( SURF2( I, J): X( I, J) = XS + I - 1; Y( I, J) = YS + J - 1; Z( I, J) = X( I, J) * @SIN( Y( I, J)) + Y( I, J ) * @SIN( X( I, J)); );

We then use the @CHARTSURFACE function to create the surface chart: !Create the chart; @CHARTSURFACE( 'z = x * sin( y) + y * sin( x)', !Title; 'X', 'Y', 'Z', !Axis labels; '(X,Y,Z)', !Legend; X, Y, Z !Points on the surface; ); The resulting chart will appear as follows:

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Simplified Chart Function Formats All the chart functions above a simplified format where all the arguments, with the exception of the data attributes, are omitted. For instance, in the bar chart example above, we could have simply used the following: ! Bar chart of required vs. actual staffing; @CHARTBAR( REQUIRED, ONDUTY);

In which case, the resulting chart would be:

608 CHAPTER 13

Note the following differences between the charts created using the simplified chart format and the example above where we specified all the arguments:  The chart title is now the file name.  The legends are simply the names of the attributes.  The X and Y axis labels have been omitted. The simplified syntax for each of the charting functions appears below: Chart Type

Simplified Syntax

Bar

@CHARTBAR( Attribute1[, ..., Attributen]);

Bubble

@CHARTBUBBLE( X1, Y1, Size1,[, ..., Xn, Yn, Sizen]);

Contour

@CHARTCONTOUR( X, Y, Z);

Curve

@CHARTCURVE( X1, Y1[, ..., Xn, Yn]);

PROGRAMMING LINGO Histogram

@CHARTHISTO( X);

Line

@CHARTLINE( X1[, ..., Xn]);

Netarc

@CHARTNETARC( X1, Y1, X2, Y2[, ..., X1n, Y1n, X2n, Y2n]);

Netnode

@CHARTNETNODE( X1, Y1, I1, J1[, ..., Xn, Yn, In, Jn]);

Pie

@CHARTPIE( X);

Radar

@CHARTRADAR( X1[, ..., Xn]);

Scatter

@CHARTSCATTER( X1, Y1[, ..., Xn, Yn]);

Surface

@CHARTSURFACE( X, Y, Z);

609

Using Procedures to Specify Chart Points For curve, surface and contour charts you may use a procedure to specify the function to be plotted. LINGO will automatically call this procedure to generate function points for the chart, thereby allowing you to skip the step of explicitly generating the points. The syntax for these procedure based charting functions is as follows: Chart Type

Simplified Syntax

Contour

@CHARTPCONTOUR( 'Title', 'X-Axis Label', 'Y-Axis Label', 'Z-Axis Label', PROCEDURE, X, XL, XU, Y, YL, YU, 'Legend', Z);

Curve

@CHARTPCURVE( 'Title', 'X-Axis Label', 'Y-Axis Label', PROCEDURE, X, XL, XU, 'Legend1', Y1[, ...,'LegendN', YN]);

Surface

@CHARTPSURFACE( 'Title', 'X-Axis Label', 'Y-Axis Label', 'Z-Axis Label', PROCEDURE, X, XL, XU, Y, YL, YU, 'Legend', Z);

In the case of the two-dimensional curve chart, the PROCEDURE argument is the name of the procedure containing the formula for the curve, X is the independent variables, XL and XU its upper and lower bounds and Y1 is the dependent variable that gets assigned the function's value. The threedimensional contour and surface charts have two independent variables (X and Y) and the dependent variable Z. As an example, the model below uses the @CHARTPSURFACE function to generate the same surface chart presented above:

610 CHAPTER 13 MODEL: ! Creates surface chart of z = x * sin( y) + y * sin( x); ! Function points are provided by the WAVE procedure; PROCEDURE WAVE: Z = X * @SIN( Y) + Y * @SIN( X); ENDPROCEDURE CALC: B = 10; @CHARTPSURFACE( 'z = x * sin( y) + y * sin( x)', !Title; 'X','Y','Z', !Axis labels; WAVE, !Procedure name; X, -B, B, !Independent var 1 and its bounds; Y, -B, B, !Independent var 2 and its bounds; 'Z', !Legend; Z !Dependent var/function value; ); ENDCALC END

Model: CHARTPSURF Note:

Charting capabilities are only available in the interactive version of LINGO for Windows. If you are using the LINGO API and need a charting capability, then you should investigate third-party solutions or use charting tools available in your development environment.

Flow of Control In a calc section, model statements are normally executed sequentially. Flow of control statements can be used to alter the execution order of statements. In effect, this gives you a programming, or scripting, capability within the LINGO environment.

@IFC and @ELSE These statements provide you with conditional IF/THEN/ELSE branching capabilities. The syntax is as follows: @IFC( : statement_1[; …; statement_n;] [@ELSE statement_1[; …; statement_n;]] );

with the @ELSE block of statements being optional in case a pure if statement is desired. Note:

Be aware of the use of the letter ‘C’ in @IFC. This is to distinguish the flow of control if statement (@IFC) from the arithmetic if (@IF).Miscellaneous_Functions

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To illustrate, the following sample code uses if/then/else blocks as part of a binary search for a key value in an array: @IFC( KEY #EQ# X( INEW): LOC = INEW; @ELSE @IF( KEY #LT# X( INEW): IE = INEW; @ELSE IB = INEW; ); );

@FOR You’ve encountered the @FOR set looping statement previously as a way to generate constraints in the model section. @FOR is also allowed in the calc section to perform looping. The main difference is that @FOR does not generate constraints when used in the calc section. Instead, it immediately executes any assignment statements contained within its scope. The following example shows a @FOR loop extracted from a portfolio model. The loop is used to solve the portfolio model for a number of different levels of desired return. For each level of return, the model minimizes variance, and the variance is stored for later use. @FOR( POINTS( I):

);

! Compute new return level; RET_LIM = RET_MIN + (I-1)*INTERVAL; ! Re-solve the model; @SOLVE(); ! Store the return value; YRET( I) = RET_LIM; ! Store the variance too; XSTD( I) = VARIANCE^0.5;

@WHILE The @WHILE statement is used for looping over a group of statements until some termination criterion is met. The syntax is as follows: @WHILE( :

statement_1[; …; statement_n;]);

As long as the conditional expression is true, the @WHILE function will keep looping over its block of statements.

612 CHAPTER 13 As an example, the following code uses an @WHILE loop to search for a key value in an array as part of a binary search procedure: @WHILE( KEY #NE# X( LOC): INEW = @FLOOR( ( IE + IB) / 2); @IFC ( KEY #EQ# X( INEW): LOC = INEW; @ELSE @IF ( KEY #LT# X( INEW): IE = INEW - 1; @ELSE IB = INEW + 1; ); ); ); In this case, the loop executes until the current value selected from the array, X(LOC), is equal to the key value, KEY.

@BREAK The @BREAK statement is used to break out of the current loop. Execution resumes at the first statement immediately following the end of the current loop. The @BREAK statement is valid only within @FOR and @WHILE loops in calc sections and does not take any arguments. As an example, we extend the @WHILE loop in the binary search example above to include an @BREAK statement that will be executed when the key value can’t be found:

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@WHILE( KEY #NE# X( LOC): !exit loop if key can’t be found; @IFC( IE – IB #LE# 1: @PAUSE( 'Unable to find key!!!'); @BREAK; ); INEW = @FLOOR( ( IE + IB) / 2); @IFC ( KEY #EQ# X( INEW): LOC = INEW; @ELSE @IF ( KEY #LT# X( INEW): IE = INEW - 1; @ELSE IB = INEW + 1; ); ); );

@STOP( [‘MESSAGE’]) The @STOP statement terminates execution of the current model. The @STOP statement is valid only within calc sections and takes on optional text argument. When an @STOP is executed, LINGO will display error message 258: [Error Code: 258] Model execution halted.

STOP statement encountered.

If a text argument is included in the @STOP, then it will also be displayed as part of this error message. As an example, we extend the @WHILE loop in the binary search example above to include an @STOP statement that will be executed when the key value can’t be found: @WHILE( KEY #NE# X( LOC):

);

!exit if key can’t be found; @IFC( IE –IB #LE# 1: @STOP( 'Unable to find key!!!'); ); INEW = @FLOOR( ( IE + IB) / 2); @IFC ( KEY #EQ# X( INEW): LOC = INEW; @ELSE @IF ( KEY #LT# X( INEW): IE = INEW; @ELSE IB = INEW; ); );

614 CHAPTER 13

Model Generation The commands in this category are related to the model generator, i.e., the component in LINGO that translates your model’s statements into a format compatible with the solver.

@GEN( [SUBMODEL_NAME[, … SUBMODEL_NAME_N]]) The @GEN statement generates a model and displays the generated equations. @GEN converts the model into the appropriate format for the solver engine; however, it does not actually call the solver. You will primarily use @GEN for debugging your models. @GEN produces a report showing all the equations in the expanded model. This report is identical to the report created by the LINGO|Generate command. By default, the report will be sent to the terminal output window. You may use the @DIVERT statement to route the report to a file. The @GEN statement accepts an optional argument of one or more submodel names. If a submodel is specified, LINGO will only generate the model contained within the specified submodel section. If multiple submodel names are specified, LINGO will combine them all into a single, larger model. If submodel names are omitted entirely, then LINGO will generate only those model statements occurring before the @GEN statement and not contained in any submodels. As an example, below is a small staffing model that uses the @GEN statement in a calc section: MODEL: SETS: DAY / MON, TUE, WED, THU, FRI, SAT, SUN/ : NEED, START, COST; ENDSETS ! Minimize total staffing costs; [OBJ] MIN = @SUM( DAY( TODAY) : START( TODAY) * COST( TODAY)); ! Subject to meeting daily needs; @FOR( DAY( TODAY): [CONS] @SUM( DAY( COUNT)| COUNT #LE# 5: START( @WRAP(TODAY - COUNT+1, @SIZE( DAY)))) >= NEED( TODAY)); DATA: NEED = 18 15 12 16 19 14 12; COST = 400; ENDDATA CALC: @GEN(); ENDCALC END

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Running this sample model yields the following generated model report: MODEL: [OBJ] MIN= 400 * START_MON + 400 * 400 * START_WED + 400 * START_THU 400 * START_SAT + 400 * START_SUN [CONS_MON] START_MON + START_THU + START_SAT + START_SUN >= 18 ; [CONS_TUE] START_MON + START_TUE + START_SAT + START_SUN >= 15 ; [CONS_WED] START_MON + START_TUE + START_SAT + START_SUN >= 12 ; [CONS_THU] START_MON + START_TUE + START_THU + START_SUN >= 16 ; [CONS_FRI] START_MON + START_TUE + START_THU + START_FRI >= 19 ; [CONS_SAT] START_TUE + START_WED + START_FRI + START_SAT >= 14 ; [CONS_SUN] START_WED + START_THU + START_SAT + START_SUN >= 12 ; END

START_TUE + + 400 * START_FRI + ; START_FRI + START_FRI + START_WED + START_WED + START_WED + START_THU + START_FRI +

@GENDUAL( [SUBMODEL_NAME[, …, SUBMODEL_NAME_N]]) The @GENDUAL statement generates the dual formulation of a linear programming model and displays the generated equations. Every linear programming model has a corresponding, mirror-image formulation called the dual. If the original model has M constraints and N variables, then its dual will have N constraints and M variables. Some interesting properties of the dual are that any feasible solution to the dual model provides a bound on the objective to the original, primal model, while the optimal solution to the dual has the same objective value as the optimal solution to the primal problem. It's also true that the dual of the dual model is, once again, the original primal model. You may wish to refer to any good linear programming text for a further discussion of duality theory. For instance, replacing the @GEN statement with an @GENDUAL statement in the staffing model above, we’d have the following calc section: CALC: @GENDUAL(); ENDCALC

Running this sample model yields the following generated model report showing the original model's dual formulation:

616 CHAPTER 13 MODEL: MAX = 18 * CONS_MON + 15 * CONS_TUE + 12 * CONS_WED + 16 * CONS_THU + 19 * CONS_FRI + 14 * CONS_SAT + 12 * CONS_SUN; [ START_MON] CONS_MON + CONS_TUE + CONS_WED + CONS_THU + CONS_FRI <= 400; [ START_TUE] CONS_TUE + CONS_WED + CONS_THU + CONS_FRI + CONS_SAT <= 400; [ START_WED] CONS_WED + CONS_THU + CONS_FRI + CONS_SAT + CONS_SUN <= 400; [ START_THU] CONS_MON + CONS_THU + CONS_FRI + CONS_SAT + CONS_SUN <= 400; [ START_FRI] CONS_MON + CONS_TUE + CONS_FRI + CONS_SAT + CONS_SUN <= 400; [ START_SAT] CONS_MON + CONS_TUE + CONS_WED + CONS_SAT + CONS_SUN <= 400; [ START_SUN] CONS_MON + CONS_TUE + CONS_WED + CONS_THU + CONS_SUN <= 400; END

Note that the dual is the transpose of the original linear program, with variables becoming the new row names and row names becoming the new variable names.

@PIC( [SUBMODEL_NAME[, …, SUBMODEL_NAME_N]]) The @PIC statement works much like @GEN, with the exception that it displays a matrix picture rather than the generated equations. For instance, replacing the @GEN statement with an @PIC statement in the staffing model above, we’d have the following calc section: CALC: @PIC(); ENDCALC

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Running the modified model will generate the following matrix picture:

OBJ: CONS(MON): CONS(TUE): CONS(WED): CONS(THU): CONS(FRI): CONS(SAT): CONS(SUN):

S T A R T ( M O N )

S T A R T ( T U E )

S T A R T ( W E D )

S T A R T ( T H U )

C 1 1 1 1 1 ' '

C C C 1 1' ' 1 1 ' 1 1 1 1'1 1 1 1 1 1 1

S T A R T ( F R I )

S T A R T ( S A T )

S T A R T ( S U N )

C C 1 1 1'1 1

C 1 1 1 1 1' ' 1 1 ' 1 1 1

MIN > B > B > B > B > B > B > B

Refer to the PICTURE command for a discussion on how to interpret matrix picture reports.

@SMPI( ‘FILE_NAME’[, SUBMODEL_NAME[, …, SUBMODEL_NAME_N]]) The @SMPI statement generates the model and then writes it to a specified file in MPI format. MPI is a special format developed by LINDO Systems for representing all classes of mathematical programs: linear, integer, and nonlinear. This format is not intended for permanent storage of your models. LINDO API s may be interested in this format for exporting models to the LINDO API. The @SMPI statement also accepts an optional argument of one or more submodel names. If a single submodel is specified, LINGO will only generate and save the model contained within the specified submodel section. If multiple submodel names are specified, LINGO will group them all together as one larger, combined model before writing the MPI file. If no submodel name is specified, then only the model statements occurring before the @GEN statement and not contained in any submodels will be used in generating the MPI file. As an example, we could write the staffing example listed above to an MPI file by modifying the calc section to be: CALC: @SMPI( ’MYMPIFILE.MPI’); ENDCALC

@SMPS( ‘FILE_NAME’[, SUBMODEL_NAME[, …, SUBMODEL_NAME_N]]) The @SMPS statement generates the model and then writes it to a specified file in MPS format file. The MPS file format is an industry standard format developed by IBM, and is useful for ing models from one solver or platform to another. LINDO API s may be interested in this format for exporting models to the LINDO API.

618 CHAPTER 13 The @SMPS statement also accepts an optional argument of one or more submodel names. If a single submodel is specified, LINGO will only generate and save the model contained within the specified submodel section. If multiple submodel names are specified, LINGO will group them all together as one larger, combined model before writing the MPS file. If no submodel name is specified, then only the model statements occurring before the @GEN statement and not contained in any submodels will be used in generating the MPI file. As an example, we could write the staffing example listed above to an MPS file by modifying the calc section to be: CALC: @SMPS( ’MYMPSFILE.MPS'); ENDCALC

Note 1: A model must be linear or quadratic to successfully export it in MPS format.

Note 2: When exporting an MPS file, LINGO truncates all variable names to 8 characters. For instance, the two distinct LINGO names SHIP( WH1, C1) and SHIP( WH1, C2) would both be truncated to the single 8 character name SHIPWH1C under MPS. Either choose names to avoid collisions of truncated names, or enable the option for converting names to RC format when doing MPS I/O. LINGO will display an error message if potential collisions exist.

Note 3: The MPS file format is intended primarily for exporting models to other applications or platforms. The MPS format is purely scalar in nature—all set-based information is lost upon converting a LINGO model to MPS format. Thus, when saving copies of a model on your own machine, you should always use the File|Save command in order to preserve your model in its entirety.

@STATS( [SUBMODEL_NAME[, …, SUBMODEL_NAME_N]]) The @STATS statement works much like @GEN, with the exception that it displays a report containing summary statistics about the size of the generated model. Please refer to the LINGO|Model Statistics command for more information on the fields of a STATS report.

@RELEASE( VARIABLE_NAME) When a variable is assigned a value in a calc section, that variable is marked by LINGO as being permanently fixed at the value, i.e., subsequent optimization via @SOLVE will not affect the variable’s value. The @RELEASE statement is used to release such a fixed variable so that it may once again become optimizable.

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The following calc section, developed for the staff scheduling model presented above, will help to illustrate the use of @RELEASE: CALC: @SOLVE(); @FOR( DAY( D): ! Force starts to 0 for today; START( D) = 0; @SOLVE(); OBJ0( D) = OBJ; ! Allow START( D) to be optimizable again; @RELEASE( START( D)); ); @FOR( DAY( D): @WRITE( DAY( D), 10*' ', OBJ0( D), @NEWLINE( 1)); ); ENDCALC

This calc section loops through each day of the week and does the following: 1. 2. 3. 4.

Sets the number of employees starting on the day to zero: START( D) = 0 Solves for the low-cost staffing pattern given that the current day will have no starting employees: @SOLVE() Stores the objective for a later report: OBJ0( D) = OBJ Releases the number of employee starts for the current day so that it can be optimized in the next solve: RELEASE( DAY( D))

The last part of the calc section writes the following report summarizing the results: MON TUE WED THU FRI SAT SUN

9800 9000 8600 9600 9200 8800 8600

Output Statements The features in this section are for displaying output. In addition to the output functions discussed here, all the reporting functions discussed in Chapter 7, Report Functions, for data and init sections are also available for use in calc section scripts. The one exception is that the @TABLE function is only available for use in data sections.

@SOLU( [0|1[, MODEL_OBJECT[, 'REPORT_HEADER']]]) The @SOLU statement mimics the functionality of the SOLU command-line command. If all arguments are omitted, then @SOLU will display the default, LINGO solution report. All reports are sent to the screen unless an @DIVERT (see below) command is in effect, routing output to a text file.

620 CHAPTER 13 If the first argument to @SOLU is 0, then only nonzero variables and binding rows will be displayed. If the first argument is 1, then all information will be displayed. If you wish to narrow the scope of the solution report, then the optional MODEL_OBJECT argument can be either an attribute or row name. In which case, the report will be restricted to the specified object. The optional third argument, 'REPORT_HEADER', is used when you wish to place a header string on the report. You can refer to the LOOPOLE model in the Samples folder for an example of a model that uses the @SOLU command

@WRITE( ‘TEXT1’|VALUE1[, …, ‘TEXTN’|VALUEN]) The @WRITE statement is the primary tool you’ll use for displaying output in calc sections. @WRITE can display both text and variable values. Text strings are delimited with single or double-quotes. All output will be directed to the screen unless @DIVERT (discussed below) is used to route output to a file. @WRITE is useful for both debugging your calc section code and for generating custom reports. As an example, the following product mix model uses @WRITE statements in a calc section to construct a custom report: MODEL: [PROFIT] MAX = 200 * WS + 300 * NC; [CHASSIS1] WS <= 60; [CHASSIS2] NC <= 40; [DRIVES] WS + 2 * NC <= 120; CALC: @SOLVE(); @WRITE( 'Total profit = ', PROFIT, @NEWLINE( 1)); @WRITE( 'Production:', @NEWLINE( 1)); @WRITE( ' WS = ', WS, @NEWLINE( 1), ' NC = ', NC, @NEWLINE( 1), 'Total units = ', WS + NC, @NEWLINE( 2) ); @WRITE( 'Dual analysis:', @NEWLINE( 1), ' Chassis1: ', @DUAL( CHASSIS1), @NEWLINE( 1), ' Chassis2: ', @DUAL( CHASSIS2), @NEWLINE( 1), ' Drives: ', @DUAL( DRIVES), @NEWLINE( 1) ); ENDCALC END

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Running this model will yield the following report: Total profit = 21000 Production: WS = 60 NC = 30 Total units = 90 Dual analysis: Chassis1: 50 Chassis2: 0 Drives: 150

Note that in addition to the @WRITE statement, we also made use of both the @NEWLINE report function to produce line feeds and the @DUAL report function to return the dual values, or shadow prices, on the constraints. Additional information on these and other report functions may be found in section Report Functions of Chapter 7.

@PAUSE( ‘TEXT1’|VALUE1[, …, ‘TEXTN’|VALUEN]) The @PAUSE statement has the same syntax as the @WRITE statement, however, @PAUSE causes LINGO to pause execution and wait for a response. For example, under Windows, the following reference to @PAUSE: CALC: @PAUSE( 'An example of @PAUSE running on Windows.'); ENDCALC

will cause the following dialog box to be displayed:

The has the option of pressing either the Resume button to continue normal processing, or the Interrupt button to immediately terminate execution of the model. On platforms other than Windows, the output will be routed to the terminal and LINGO will wait for the to press the Enter key.

622 CHAPTER 13

@DIVERT( [‘FILE_NAME’]) By default, output generated by the @WRITE statement will be sent to the screen. However, you may wish to capture this output in a file. @DIVERT allows you to do this. As an example, we modified the product mix example from above to route its custom report to file MYREPORT.TXT as follows: CALC: @SOLVE(); @DIVERT( 'MYREPORT.TXT'); @WRITE( 'Total profit = ', PROFIT, @NEWLINE( 1)); @WRITE( 'Production:', @NEWLINE( 1)); @WRITE( ' WS = ', WS, @NEWLINE( 1), ' NC = ', NC, @NEWLINE( 1), 'Total units = ', WS + NC, @NEWLINE( 2) ); @WRITE( 'Dual analysis:', @NEWLINE( 1), ' Chassis1: ', @DUAL( CHASSIS1), @NEWLINE( 1), ' Chassis2: ', @DUAL( CHASSIS2), @NEWLINE( 1), ' Drives: ', @DUAL( DRIVES), @NEWLINE( 1) ); @DIVERT(); ENDCALC Note the two occurrences of @DIVERT. In the first instance, we specify the file we want to open, and subsequent @WRITE output is diverted to that file. The second reference to @DIVERT closes the file and reverts output back to the terminal device. @DIVERT also accepts an optional second argument of the letter ‘A’. If this argument is present, LINGO will append output to the end of the file if it already exists. If the argument is omitted, LINGO will overwrite the file if it already exists Note:

@DIVERT statements can be nested so that multiple levels of output files may be simultaneously in use.

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Setting Parameters LINGO has many optional settings that can be controlled with the LINGO|Options command. At times, you may find it necessary to alter these parameters dynamically in your model’s calc sections. For this reason, LINGO provides the @SET statement, which gives you access to the entire set of system parameters. There is also and additional function, @APISET, for setting more obscure parameters in the LINDO API (LINGO’s solver library) that aren’t available through the standard LINGO options set.

@SET( ‘PARAM_NAME’, PARAMETER_VALUE) To change a parameter’s setting, @SET requires that you it a parameter name as a text string, along with the parameter’s new value. For example, to set the integer solver’s relative optimality tolerance to .05 we’d use: @SET( ’IPTOLR’, .05); A list of all the parameter names can be found in Chapter 6’s discussion of the SET command-line command. A parameter may be returned to its default value by omitting the parameter_value argument. For instance, @SET( ’IPTOLR’); would return the IPTOLR parameter to its default value. Use: @SET( ’DEFAULT ’); to return all parameters to their default settings: Note:

Parameters whose values have been changed with @SET in a calc section will be restored to their original settings once the model run is completed.

@APISET( PARAM_INDEX, ‘INT|DOUBLE’, PARAMETER_VALUE) LINGO uses the LINDO API solver library as its underlying solver engine. The LINDO API has a wealth of parameters that may be set by the . Many of these parameters would be of use only in rare instances. Given this, LINGO does not provide direct access to all possible LINDO API parameters. However, if you need access to certain API parameters that aren’t in the standard LINGO set, you may do so with the @APISET command. To change a parameter’s setting, @APISET requires that you it a parameter index, a text string of either ‘INT’ or “DOUBLE’ indicating if the parameter is an integer or double precision quantity, along with the parameter’s new value. You can refer to the LINDO API documentation (available on LINDO Systems’ Web site) for a list of available parameters. A list of parameters and their indices is also in the lindo.h file included as part of your LINGO installation. As an example, the LINDO API adds cuts in the branch-and-bound tree every 10 nodes. If you would like to add cuts more frequently, say every 5 nodes, then you could do so with the:

624 CHAPTER 13 @APISET( 318, ‘INT’, 5); You may also force all API parameters back to their default values with: @APISET( ‘DEFAULT’);

Note:

Parameters whose values have been changed with @APISET will be restored to their original settings once the model run is completed.

Procedures PROCEDURE and ENDPROCEDURE: LINGO allows for blocks of code known as procedures. These blocks can be called from anywhere in a calc section by simply referring to the procedure's name. Procedures can be useful for portions of calc sections that must be executed multiple times. The PROCEDURE and ENDPROCEDURE statements are used to bracket a procedure's code. The PROCEDURE statement must be followed immediately by a unique name that follows the normal naming conventions. The name must then be followed by a colon. You must also place an ENDPROCEDURE statement immediately after the last statement of your procedure. Procedures form their own model section that must lie outside data, init and calc sections. Unlike a calc section, procedures are not automatically executed when solving a model. Instead, you must reference the procedure's name in a calc section to execute its code. As an example, the following procedure was extracted from the sample model, PROCEDURE.LG4: PROCEDURE PRINT_REPORT: @WRITE( ' Start: '); @WRITEFOR( DAY: @FORMAT( START, '6.0f')); @WRITE( @FORMAT( @SUM( DAY: COST*START), '8.0f')); @WRITE( @NEWLINE( 1)); @WRITE( ' On Duty: '); @WRITEFOR( DAY: @FORMAT( NEED + EXCESS, '6.0f')); @WRITE( @NEWLINE( 1)); @WRITE( ' Required: '); @WRITEFOR( DAY: @FORMAT( NEED, '6.0f')); @WRITE( @NEWLINE( 1)); @WRITE( ' Excess: '); @WRITEFOR( DAY: @FORMAT( EXCESS, '6.0f')); @WRITE( @NEWLINE( 2)); ENDPROCEDURE

This procedure prints a small report for a staffing model. The procedure is called several times from the model's main calc section to print solution reports for several permutations of the original model. Here is a code fragment from the models calc section that illustrates calling the procedure: @WRITE( 'Solution 1 - min cost:', @NEWLINE( 1)); PRINT_REPORT;

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The first line prints a header file for the report, while the second line generates the actual report by calling the PRINT_REPORT procedure above. A sample of one of the reports follows: MON TUE TOTAL Solution 1 - min cost: Start: 5 4 4600 On Duty: 19 17 Required: 19 17 Excess: 0 0

WED

THU

FRI

SAT

SUN

0

6

2

2

4

15 15 0

19 19 0

17 17 0

14 14 0

14 12 2

You may wish to open PROCECURE.LG4 in LINGO to experiment with the use of procedure calls.

Stochastic Programming LINGO 12.0 introduced a stochastic programming solver (SP) for multistage stochastic programs with recourse. Details may be found in Chapter 14, Stochastic Programming. Some additional programming features were also added for use with SP models. These additional programming features are discussed delow.

@SPLOADSCENE( SCENARIO_INDEX) This function loads the solution for scenario index SCENARIO_INDEX. The solution for this scenario then becomes the current/active solution. As an example, the following would load the solution for scenario 1: @SPLOADSCENE( 1); Suppose you have an SP model with an objective row call PROFIT_ROW. The following code will compute the expected/average profit over each scenario: I = 1; EXP_PROFIT = 0; @WHILE ( I #LE# @SPNUMSCENE()): @SPLOADSCENE( I); EXP_PROFIT = EXP_PROFIT + PROFIT_ROW; I = I + 1; ); EXP_PROFIT = EXP_PROFIT / @SPNUMSCENE(); @WRITE( 'Expected Profit = ', EXP_PROFIT);

@SPNUMSCENE() Use @SPNUMSCENE to retrieve the total number of scenarios in the model. You can refer to the code example immediately above for an example of using @SPNUMSCENE() to compute the expected/average profit over each scenario in an SP model.

@SPPRBSCENE( SCENARIO_INDEX) Use @SPPRBSCENE to retrieve the probability of a specified scenario index, SCENARIO_INDEX. In the following example, we loop to display the probabilities for each scenario in the model:

626 CHAPTER 13 CALC: ! Solve the SP; @SOLVE(); ! Loop to display scenario probabilities; I = 1; @WHILE( I #LE# @SPNUMSCENE(): @WRITE( ' Scenario: ', I, ' Probability: ', @SPPRBSCENE( I), @NEWLINE( 1)); I = I + 1; ); ENDCALC

The output from the loop is: Scenario: 1 Probability: 0.1666666666666667

Scenario: 2 Probability: 0.3333333333333333 Scenario: 3 Probability: 0.5

Utility Functions RANKING_OF_ATTRIBUTE = @RANK( ATTRIBUTE_TO_BE_RANKED) The @RANK function ranks the values of an attribute in ascending order. This can be a useful function when you need to sort the of an attribute. @RANK is available for use only in calc sections. As an example, consider the following small model: model: sets: s1: x, rankx; endsets data: x = 3.2 -1.1 5.7; enddata calc: rankx = @rank( x); endcalc end

The @RANK function will place the following values into the RANKX attribute: 2, 1, 3, indicating that X( 1) is the second from smallest value, X( 2) is the smallest, and X( 3) is the largest.

PROGRAMMING LINGO The following is another illustration of the use of @RANK. Here, we generate a list of random numbers and then sort them through the use of @RANK: model: ! Generates 40,000 random numbers, ranks them with @RANK(), and then moves them into sorted order.; sets: s1 /1..40000/: xrand, xrank, xsort; endsets calc: @set( 'terseo', 2); seed = .5; t0 = @time(); @for( s1( i): xrand( i) = @rand( seed); seed = xrand( i); ); t1 = @time(); xrank = @rank( xrand); t2 = @time(); @for( s1( i): xsort( xrank( i)) = xrand( i); ); t3 = @time(); @write( ' Time to generate: ', @format( t1 - t0, '8.2g'), @newline( 1), ' Time to rank: ', @format( t2 - t1, '8.2g'), @newline( 1), ' Time to move: ', @format( t3 - t2, '8.2g'), @newline( 2) ); endcalc end

Model: SORTRAND The following code generates the random numbers through the use of the @RAND function:

627

628 CHAPTER 13 @for( s1( i): xrand( i) = @rand( seed); seed = xrand( i); );

Then, we rank the random values via @RANK with: xrank = @rank( xrand);

The random values are then moved into sorted order with the loop: @for( s1( i): xsort( xrank( i)) = xrand( i); );

Finally, we display the time these three operations used with the @WRITE statement: @write( ' Time to generate: ', @format( t1 - t0, '8.2g'), @newline( 1), ' Time to rank: ', @format( t2 - t1, '8.2g'), @newline( 1), ' Time to move: ', @format( t3 - t2, '8.2g'), @newline( 2) );

Running the model in LINGO will yield a report similar to the following: Time to generate: Time to rank: Time to move:

0.07 0.01 0.05

Programming Example: Binary Search In this section we will illustrate some of LINGO’s programming features by developing a model to perform a binary search. Binary searches are an efficient way to look up a key in a sorted list. In the worst case, a binary search should perform no more than log2( n) comparisons to determine if a key is on a sorted list, where n is the number of items on the list. The basic idea behind a binary search is that you bracket the key in the list. You then iterate by reducing the size of the bracket by a factor of 2 each . This process continues until you either find the key, or conclude that the key is not on the list when the bracket size becomes 0. Note:

A binary search is not typically something you would do in LINGO, and we use it here merely as an example of a simple algorithm for illustrative purposes.

PROGRAMMING LINGO

The Model The following model is an example of implementing a binary search in LINGO: MODEL: ! Illustrates programming looping capabilities of LINGO by doing a binary search; SETS: S1: X; ENDSETS DATA: ! The key for which we will search; KEY = 16; ! The list (must be in sorted increasing order); X = 2 7 8 11 16 20 22 32; ENDDATA ! Do a binary search for key; CALC: ! Set begin and end points of search; IB = 1; IE = @SIZE( S1); ! Loop to find key; @WHILE( IB #LE# IE: ! Cut bracket in half; LOC = @FLOOR((IB + IE)/2); @IFC( KEY #EQ# X(LOC): @BREAK; ! Do no more loops; @ELSE @IFC( KEY #LT# X( LOC): IE = LOC-1; @ELSE IB = LOC+1; ); ); ); @IFC( IB #LE# IE) ! Display key's location; @PAUSE( 'Key is at position: ', LOC); @ELSE ! Key not in list; @STOP( ' Key not on list!!!'); @ENDIF ENDCALC END

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630 CHAPTER 13

The Details Our first step is to define a set, S1, and give it the attribute X using the following sets section: SETS: S1: X; ENDSETS

The X attribute will store the list that we will search for the key value. We populate X and input our key in the data section: DATA: ! The key for which we will search; KEY = 16; ! The list (must be in sorted increasing order); X = 2 7 8 11 16 20 22 32; ENDDATA

Note that for the purposes of this example, the list of values must be input in sorted order. An interesting exercise would be to extend this model so that it tests to determine if the data is sorted. Or, if you are really ambitious, extend the model so that it will sort a list with arbitrary ordering. Next comes the calc section, which contains our code to do the binary search. In the first part of the calc section: CALC: ! Set begin and end points of search; IB = 1; IE = @SIZE( S1);

we bracket the key by pointing to the beginning of the list, IB, and the end of the list, IE. Note that we make use of the @SIZE function to dynamically compute the size of the list. Next, we have the @WHILE loop to search for the key value: ! Loop to find key; @WHILE( IB #LE# IE: ! Cut bracket in half; LOC = @FLOOR((IB + IE)/2); @IFC( KEY #EQ# X(LOC)) @BREAK; ! Do no more loops; @ELSE @IFC( KEY #LT# X( LOC): IE = LOC-1; @ELSE IB = LOC+1; ); );

);

The @WHILE loop tests if the candidate range has become empty at the start of each iteration: @WHILE( IB #LE# IE:

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If not, we continue by dissecting the bracket in half: !cut bracket in half; INEW = @FLOOR( ( IE + IB) / 2);

We then determine if the new bracket point resides above or below the key and set IB and IE accordingly: @IFC( KEY #EQ# X(LOC): @BREAK; ! Do no more loops; @ELSE @IFC( KEY #LT# X( LOC): IE = LOC-1; @ELSE IB = LOC+1; );

);

Finally, when we fall out of the @WHILE loop we display the result of our search: @IFC( IB #LE# IE: ! Display key's location; @PAUSE( 'Key is at position: ', LOC); @ELSE ! Key not in list; @STOP( ' Key not on list!!!'); );

If the eligible range is empty, then we did not find the key. Otherwise, the key was found and report its location on the list. If you run this model, LINGO should successfully find the key and display:

632 CHAPTER 13

Programming Example: Markowitz Efficient Frontier In the March 1952 issue of Journal of Finance, Harry M. Markowitz published an article titled Portfolio Selection. In the article, he demonstrates how to reduce the risk of asset portfolios by selecting assets whose values aren't highly correlated. The concepts behind the Markowitz portfolio model were discussed in detail in the previous chapter in section Markowitz Portfolio Selection Model. The basic idea is that, given a desired level of expected return, one should select a basket of assets that minimizes risk (i.e., variance in return). Any other basket of assets would be inefficient in that it entails taking on extra risk without extra compensation. The Markowitz model allows you to evaluate tradeoffs between risk and return. By running the model for a series of different levels of return, you can see how portfolio risk must increase as desired return increases. The return/risk tradeoff may be graphed, and this graph is known as the efficient frontier. If we place risk on the vertical axis and return on the horizontal, then portfolios to the left of the efficient frontier are inefficient. This is because portfolios exist with the same amount of return but less risk. Conversely, all (return,risk) pairs that lie to the right of the curve cannot be achieved given the current available asset pool. Finally, all portfolios with (return,risk) combinations that lie on the curve are efficient—no portfolio can be found with the same level of return and lower risk. We will now demonstrate how you can use the programming capabilities in LINGO to generate points along the efficient frontier by repeatedly solving the Markowitz model for a series of portfolio return values.

The Model Here’s the model we will use to generate points along the efficient frontier: MODEL:

! Solves the generic Markowitz portfolio model in a loop to generate the points on the efficient frontier; SETS: ASSET: RATE, UB, X; COVMAT( ASSET, ASSET): V; POINTS: XRET, YVAR; ENDSETS DATA: ! Number of points on the efficient frontier graph; NPOINTS = 10; POINTS = 1..NPOINTS; ! The stocks; ASSET = GOOGLE, YAHOO, CISCO; ! Expected growth rate of each asset; RATE = 1.3 1.2 1.08;

PROGRAMMING LINGO ! Upper bound on investment in each; UB = .75 .75 .75; ! Covariance matrix; V = 3 1 -.5 1 2 -.4 -.5 -.4 1; ENDDATA ! Below are the three objectives we'll use; SUBMODEL SUB_RET_MAX: [OBJ_RET_MAX] MAX = RETURN; ENDSUBMODEL SUBMODEL SUB_RET_MIN: [OBJ_RET_MIN] MIN = RETURN; ENDSUBMODEL SUBMODEL SUB_MIN_VAR: [OBJ_MIN_VAR] MIN = @SUM( COVMAT( I, J): V( I, J) * X( I) * X( J)); ENDSUBMODEL !and the constraints; SUBMODEL SUB_CONSTRAINTS: ! Compute return; RETURN = @SUM( ASSET: RATE * X); ! Must be fully invested; @SUM( ASSET: X) = 1; ! Upper bounds on each; @FOR( ASSET: @BND( 0, X, UB)); ! Must achieve target return; RETURN >= RET_LIM; ENDSUBMODEL CALC: ! Set some parameters; ! Reset all params; @SET( 'DEFAULT'); ! Minimize output; @SET( 'TERSEO', 1); ! Suppress status window; @SET( 'STAWIN', 0); ! Capture unwanted output; @DIVERT( 'LINGO.LOG'); ! Solve to get maximum return; RET_LIM = 0; @SOLVE( SUB_RET_MAX, SUB_CONSTRAINTS); ! Save maximum return; RET_MAX = OBJ_RET_MAX;

633

634 CHAPTER 13 ! Solve to get minimum return; @SOLVE( SUB_RET_MIN, SUB_CONSTRAINTS); ! Save minimum return; RET_MIN = OBJ_RET_MIN; ! Interval between return points; INTERVAL = ( RET_MAX - RET_MIN) / ( NPOINTS-1); ! Loop over range of possible returns minimizing variance; RET_LIM = RET_MIN; @FOR( POINTS( I): @SOLVE( SUB_MIN_VAR, SUB_CONSTRAINTS); XRET( I) = RET_LIM; YVAR( I) = OBJ_MIN_VAR; RET_LIM = RET_LIM + INTERVAL; ); ! Close log file; @DIVERT(); ! Display the results; @WRITE( ' Return Variance', @NEWLINE( 1)); @FOR( POINTS: @WRITE( @FORMAT( XRET, '#12.6G'), @FORMAT( YVAR, '#12.6G'), @NEWLINE( 1)) ); ENDCALC END

Model: LOOPPORT

The Details First consider the sets section: SETS: ASSET: RATE, UB, X; COVMAT( ASSET, ASSET): V; POINTS: XRET, YVAR; ENDSETS

It defines three sets: ASSET, COVMAT and POINTS. The ASSET set will contain the set of assets available for investment. Each asset has an expected rate of return (RETURN), an upper bound on the amount of the asset we’ll allow in the portfolio (UB), and the fraction of the portfolio devoted to the asset (X). Note that the fraction of the portfolio devoted to each asset, X, constitutes our decision variables. The COVMAT set is a cross of the ASSET set on itself. We create this set for the attribute V, which will store the covariance matrix for all the assets.

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The POINTS set is used to represent the points on the efficient frontier that we will be generating. For each point, we will determine its x-coordinate, XRET, and its y-coordinate, YVAR. Note that the xcoordinate will represent risk, while the y-coordinate will represent return. What we intend to do is to solve a portfolio model once for each member of the POINTS set to get a new point on the efficient frontier. These points will be stored in XRET and YVAR. Once the loop is completed, the list of points in XRET and YVAR will give us a glimpse of what the efficient frontier looks like. Next we have the data section: DATA: ! Number of points on the efficient frontier graph; NPOINTS = 10; POINTS = 1..NPOINTS; ! The stocks; ASSET = GOOGLE, YAHOO, CISCO; ! Expected growth rate of each asset; RATE = 1.3 1.2 1.08; ! Upper bound on investment in each; UB = .75 .75 .75; ! Covariance matrix; V = 3 1 -.5 1 2 -.4 -.5 -.4 1; ENDDATA

In this data section we are setting the number of points that we will generate along the efficient frontier (NPOINTS) to 10. Once we have the number of points established, we dynamically create the POINTS set. Next, we input the set of assets, their expected rate of return, their upper bounds, and their covariance matrix. The next section of the model: ! Below are the three objectives we'll use; SUBMODEL SUB_RET_MAX: [OBJ_RET_MAX] MAX = RETURN; ENDSUBMODEL SUBMODEL SUB_RET_MIN: [OBJ_RET_MIN] MIN = RETURN; ENDSUBMODEL SUBMODEL SUB_MIN_VAR: [OBJ_MIN_VAR] MIN = @SUM( COVMAT( I, J): V( I, J) * X( I) * X( J)); ENDSUBMODEL

636 CHAPTER 13 makes use of the SUBMODEL and ENDSUBMODEL statements to establish three different objectives. The first two objectives respectively maximize and minimize portfolio return, which are used later on in the model to determine that maximum and minimum possible returns that can be generated with out basket of available stocks. The third objective, SUB_MIN_VAR, minimizes portfolio risk as measured by its variance. Following the three submodels containing our various objectives, we have another submodel that contains our three constraints: !and the constraints; SUBMODEL SUB_CONSTRAINTS: ! Compute return; RETURN = @SUM( ASSET: RATE * X); ! Must be fully invested; @SUM( ASSET: X) = 1; ! Upper bounds on each; @FOR( ASSET: @BND( 0, X, UB)); ! Must achieve target return; RETURN >= RET_LIM; ENDSUBMODEL

The first constraint of our constraint section computes portfolio return. The second constraint says that we must invest 100 percent of our capital. The third constraint puts an upper bound on the percentage of the portfolio invested in each asset. Finally, the fourth constraint forces total portfolio return to achieve some desired level. The next section, the calc section, is of primary interest. It contains the logic to solve the model multiple times to generate and store the points on the efficient frontier. First, we make use of @SET to set some parameter values: ! Set some parameters; ! Reset all params to their defaults; @SET( 'DEFAULT'); ! Minimize output; @SET( 'TERSEO', 1); ! Suppress status window; @SET( 'STAWIN', 0);

The first call to @SET restores all parameters to their default values, the second call minimizes the level of LINGO’s output to improve performance, while the third call suppresses the status window that would normally pop up when solving a model. We suppress the status window so it does not obscure the custom report we are creating at the end of the run. Next, we use @DIVERT to route LINGO’s output to a log file: ! Capture spurious output; @DIVERT( 'LINGO.LOG');

We do this to capture the output LINGO would normally generate while it’s solving the various portfolio models. We capture this output so it doesn’t distract from the custom report we will display at the end of the run.

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Our next task is to find out the possible range of portfolio returns. We want this information so that we don’t try to run our model for returns lying outside this range, which would result in an infeasible model. This requires two runs: one where we maximize return and one where we minimize return. Here is the code that performs the runs: ! Solve to get maximum return; RET_LIM = 0; @SOLVE( SUB_RET_MAX, SUB_CONSTRAINTS); ! Save maximum return; RET_MAX = OBJ_RET_MAX; ! Solve to get minimum return; @SOLVE( SUB_RET_MIN, SUB_CONSTRAINTS); ! Save minimum return; RET_MIN = OBJ_RET_MIN;

This is our first example of the the @SOLVE command. @SOLVE takes one or more submodel names as arguments. It then combines the submodels into a single model and solves them. In this case, we first solve submodel SUB_RET_MAX along with submodel SUB_CONSTRAINTS to get the maximum return possible subject to our constraint set. We then do a similar solve to find the minimal return subject to the constraints. Other items of interest are that we zero out RET_LIM, given that the constraint on return is temporarily not required, and we store the two extreme objective values in RET_MAX and RET_MIN for later use. Our end goal is to solve the model for 10 values of portfolio return, with these values being equally spaced from RET_MIN to RET_MAX. The next step in the model computes the distance, or interval, between these points: ! Interval between return points; INTERVAL = ( RET_MAX - RET_MIN) / ( NPOINTS-1);

Our next code segment is an @FOR loop that loops over each of the 10 return values and minimizes portfolio variance subject to attaining the desired level of return: ! Loop over range of possible returns minimizing variance; RET_LIM = RET_MIN; @FOR( POINTS( I): @SOLVE( SUB_MIN_VAR, SUB_CONSTRAINTS); XRET( I) = RET_LIM; YVAR( I) = OBJ_MIN_VAR; RET_LIM = RET_LIM + INTERVAL; );

We start by setting the desired level of return to RET_MIN and then increment this level of return each through the loop by the interval amount. Note that the solve command now uses the submodel containing the objective that minimizes return, SUB_MIN_VAR. Also, after each solve command, we store the coordinates of the point on the efficient frontier in XRET and YRET.

638 CHAPTER 13 Our next section of code creates a custom report: ! Close log file; @DIVERT(); ! Display the results; @WRITE( ' Return Variance', @NEWLINE( 1)); @FOR( POINTS: @WRITE( @FORMAT( XRET, '#12.6G'), @FORMAT( YVAR, '#12.6G'), @NEWLINE( 1)) );

Once again, we make use of @DIVERT, but this time we do not an argument. This results in output once again going to the report window, which will allow us to view our report on screen. After restoring terminal output with @DIVERT, we use the @WRITE command inside an @FOR loop to write out the 10 points on the efficient frontier. If you run the model, you should see the following report: Return 1.11000 1.12833 1.14667 1.16500 1.18333 1.20167 1.22000 1.23833 1.25667 1.27500

Variance 0.417375 0.417375 0.418054 0.462381 0.575957 0.758782 1.01086 1.33218 1.72275 2.18750

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In summary, returns range from 11% to a high of 27.5%, with portfolio variance ranging from .417 to 2.18. One final note, if you load this model from your Lingo samples folder you will find additional Lingo code at the end devoted to graphing the frontier. We will not go into the details of that code at this point; however, the generated graph is pictured below: Risk ^ 2.2 | * | | | | | * | | | | * | | | | * | | | * | | * 0.40 | * * * * ---------------------------------------------------------> 1.1 1.3

Model: LOOPPORT – Graph of Efficient Frontier

Programming Example: Cutting Stock An optimization problem in the paper, steel, and wood industries is the cutting-stock problem. The main feature of this problem is that finished goods of varying lengths are cut from larger raw material pieces of varying lengths. The goal is to cut the raw materials using an efficient set of patterns to minimize the total amount of raw materials required, while meeting demand for all finished products. Examples would include sheet metal and paper manufacturers that take larger rolls of product and cut them into smaller rolls for sale to customers. As an example, suppose you are a steel fabricator that takes 45 foot steel beams and cuts them into 14, 12 and 7 foot beams for sale to construction firms. Cutting one 45 foot beam into one 14 foot piece, two 12 foot pieces and one 7 foot piece would be very efficient in that it would exactly use up the 45 foot raw material piece with no trim waste. On the other hand, a pattern of one 14 foot piece, one 12 foot piece and two seven foot pieces would not be as efficient due to a 5 foot piece of trim waste.

640 CHAPTER 13 A brute force method for attacking the cutting-stock problem would be to generate all possible efficient patterns and run an optimization model to determine how may copies of each pattern to run to satisfy demand at minimal cost. The drawback here is that the number of patterns grows exponentially as the number of different finished good lengths increase. For all but the smallest problems, brute force pattern generation will yield large, intractable models. Gilmore and Gomory published a paper in 1961 titled A Linear Programming Approach to the Cutting-Stock Problem. In this paper they outline a two-stage, iterative approach for solving cuttingstock problems that dramatically reduced the number of patterns one must generate to get good solutions. The basic idea involves solving a master problem containing a limited number of patterns. The dual prices on the finished goods are then ed to a small knapsack subproblem that selects a new cutting pattern that maximizes the sum of the dual values of all the finished goods contained in the pattern subject to not exceeding the length of the raw material. This pattern is then appended to the previous master problem, which is then re-solved. This iterative process continues until no further beneficial patterns can be generated by the knapsack subproblem. In which case, we have the optimal solution to the original, linear cutting-stock problem. The remarkable feature of this algorithm is that it typically takes relatively few es to reach the optimal solution, thereby making it possible to solve very large cutting-stock models in an extremely reasonable amount of time. This approach of iteratively appending new columns to models is also referred to as column generation.

The Model For our example, we will be cutting 45 foot wide rolls of paper into smaller rolls of widths: 34, 24, 15, 10 and 18. We use Lingo’s programming capability to iteratively solve the master and subproblem until no further beneficial cutting patterns remain to be appended to the master problem. MODEL:

! Uses Lingo’s programming capability to do on-the-fly column generation for a cutting-stock problem; SETS: PATTERN: COST, X; FG: WIDTH, DEM, PRICE, Y, YIELD; FXP( FG, PATTERN): NBR; ENDSETS DATA: PATTERN = 1..20; ! Allow up to 20 patterns; RMWIDTH = 45; ! Raw material width; FG = F34 F24 F15 F10 F18;!Finished goods...; WIDTH= 34 24 15 10 18;!their widths...; DEM = 350 100 800 1001 377;!and demands; BIGM = 999; ENDDATA SUBMODEL MASTER_PROB: [MSTROBJ] MIN= @SUM( PATTERN( J)| J #LE# NPATS: COST( J)*X( J)); @FOR( FG( I):

PROGRAMMING LINGO [R_DEM] @SUM( PATTERN( J)| J #LE# NPATS: NBR( I, J) * X( J)) >= DEM( I); ); ENDSUBMODEL SUBMODEL INTEGER_REQ: @FOR( PATTERN: @GIN( X)); ENDSUBMODEL SUBMODEL PATTERN_GEN: [SUBOBJ] MAX = @SUM( FG( I): PRICE( I)* Y( I)); @SUM( FG( I): WIDTH( I)*Y( I)) <= RMWIDTH; @FOR( FG( I): @GIN(Y( I))); ENDSUBMODEL CALC: ! Send unwanted output to log file; @DIVERT( 'LINGO.LOG'); ! Set parameters; @SET( 'DEFAULT'); @SET( 'TERSEO', 1); @SET( 'STAWIN', 0); ! Max number of patterns we'll allow; MXPATS = @SIZE( PATTERN); ! Make first pattern an expensive super pattern; COST( 1) = BIGM; @FOR( FG( I): NBR( I, 1) = 1); ! Loop as long as the reduced cost is attractive and there is space; NPATS = 1; RC = -BIGM; ! Clearly attractive initially; @WHILE( RC #LT# 0 #AND# NPATS #LT# MXPATS: ! Solve for current best pattern runs; @SOLVE( MASTER_PROB); ! Copy dual prices to PATTERN_GEN submodel; @FOR( FG( I): PRICE( I) = -@DUAL( R_DEM( I))); ! Generate the current most attractive pattern; @SOLVE( PATTERN_GEN); ! Marginal value of current best pattern; RC = 1 - SUBOBJ; ! Add the pattern to the Master; NPATS = NPATS + 1; @FOR( FG( I): NBR( I, NPATS) = Y( I)); COST( NPATS) = 1; );

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642 CHAPTER 13 ! Finally solve Master as an IP; @SOLVE( MASTER_PROB, INTEGER_REQ); ! Redirect output back to terminal; @DIVERT(); ENDCALC END

Model: LOOPCUT

The Details First, we have the sets section: SETS: PATTERN: COST, X; FG: WIDTH, DEM, PRICE, Y, YIELD; FXP( FG, PATTERN): NBR; ENDSETS

The PATTERN set is used to represent the cutting patterns we will be generating. Each pattern has a cost, which, with one exception (discussed below), will be 1, i.e., each pattern uses 1 raw material piece. We also assigned an attribute called X to the patterns set. X( p) will be used to store the number of times each pattern p is to be cut and is one of the decision variables. The FG set is used to represent the set of finished goods. As input, each finished good has a width and customer demand. The PRICE attribute will be used to store the dual prices on the finished goods. These prices will be updated each time we solve the master problem. Y will be an integer, decision variable that we will use in the knapsack subproblem to represent the number of pieces of each finished good to use in the next generated pattern. YIELD will be used to store the number of pieces of each finished good that gets produced. The derived set, FXP, is derived on the finished goods and patterns sets. The attribute NBR( i, j) will store the number of finished goods I contained in pattern j. Next, we have the data section DATA: PATTERN = 1..20; ! Allow up to 20 patterns; RMWIDTH = 45; ! Raw material width; FG = F34 F24 F15 F10 F18;!Finished goods...; WIDTH= 34 24 15 10 18;!their widths...; DEM = 350 100 800 1001 377;!and demands; BIGM = 999; ENDDATA

We initialize the pattern set to have 20 . This will only allow for generation of up to 20 patterns; however, this should be more that adequate for this small example. After inputting the raw material width of 45, we input the set of five finished goods and their widths. After that, we input the demands for the finished goods. Finally, we input a parameter called BIGM, the purpose of which is discussed below.

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Next, we have our first submodel: SUBMODEL MASTER_PROB: [MSTROBJ] MIN= @SUM( PATTERN( J)| J #LE# NPATS: COST( J)*X( J)); @FOR( FG( I): [R_DEM] @SUM( PATTERN( J)| J #LE# NPATS: NBR( I, J) * X( J)) >= DEM( I); ); ENDSUBMODEL

This is the master problem we’ve been mentioning. The objective states that we wish to minimize the total cost of all the patterns used. The constraint says that we must meet, or exceed, demand for all the finished goods. The next submodel: SUBMODEL INTEGER_REQ: @FOR( PATTERN: @GIN( X)); ENDSUBMODEL

will be used in conjunction with the master problem to force the variables to take on integer values. Given that it’s not possible to cut a fractional number of a particular pattern, for our final solution we need the X variables to be integer valued. Our final submodel: SUBMODEL PATTERN_GEN: [SUBOBJ] MAX = @SUM( FG( I): PRICE( I)* Y( I)); @SUM( FG( I): WIDTH( I)*Y( I)) <= RMWIDTH; @FOR( FG( I): @GIN(Y( I))); ENDSUBMODEL

is the pattern-generation subproblem we’ve mentioned. This is a knapsack problem that finds the best pattern that can be cut given the dual prices on the finished goods. Recall that we get the dual prices on the finished goods from the demand constraints in the master problem. We then enter the calc section, which contains the programming logic to coordinate the iterative algorithm we’ve chosen. As with the previous Markowitz model, the start of the calc section is devoted to diverting output to a file and setting of parameters. You may refer to the previous Markowitz model for a discussion of why these steps are being performed. One of the features of this model most likely to change in the future would be the maximum number of patterns to generate. It’s probably not wise to assume that this number will always be fixed at 20 patterns. For this reason, we use the @SIZE function to get the current number of allowed patterns: ! Max number of patterns we'll allow; MXPATS = @SIZE( PATTERN);

Next, we construct what we refer to as a “super pattern”:

644 CHAPTER 13 ! Make first pattern an expensive super pattern; COST( 1) = BIGM; @FOR( FG( I): NBR( I, 1) = 1);

The supper pattern is a device to jumpstart our algorithm by guaranteeing that the model will be feasible from the start. We need the model to always be feasible in order to obtain a valid set of dual prices to feed into the pattern generation submodel. The super pattern is an artificial pattern that can create one piece of each finished good. In real life, such a pattern would not be possible because we can’t physically fit one of each finished good on a single raw material piece. Given that the super pattern is not physically possible, we assign it a high cost, BIGM, so it will not be used in the final solution Recall that NBR( i, j) represents the number of finished good i in pattern j. So, for our super pattern, we use a for loop over all the finished goods i, setting NBR( i, 1) to 1 for each of the finished goods. The second index of NBR is set at 1 given that the super pattern is the first pattern generated. Next, the main loop that will alternate between solving the master and subproblem until an optimal solution is found: ! Loop as long as the reduced cost is attractive and there is space; NPATS = 1; RC = -BIGM; ! Clearly attractive initially; @WHILE( RC #LT# 0 #AND# NPATS #LT# MXPATS: ! Solve for current best pattern runs; @SOLVE( MASTER_PROB); ! Copy dual prices to PATTERN_GEN submodel; @FOR( FG( I): PRICE( I) = -@DUAL( R_DEM( I))); ! Generate the current most attractive pattern; @SOLVE( PATTERN_GEN); ! Marginal value of current best pattern; RC = 1 - SUBOBJ; ! Add the pattern to the Master; NPATS = NPATS + 1; @FOR( FG( I): NBR( I, NPATS) = Y( I)); COST( NPATS) = 1; );

First, we set the pattern count, NPATS, to 1 for the one pattern already generated, i.e., the super pattern. We will increment NPATS each time we generate a new pattern in our main loop. Next, is the main @WHILE loop, which will execute until its condition evaluates false. We test two things in the condition. First, we see if the marginal impact of the current pattern, RC, is negative and will, therefore, reduce the current cost. If the current pattern will reduce our cost, then this implies that we should continue looping and generating patterns, because subsequent patterns may also be helpful. Once the marginal benefit of the current best pattern goes to zero, generating subsequent patterns will be of no benefit; if the current best pattern can’t help, subsequent patterns of less value will also be of no help. In which case, we should exit our loop.

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The second condition on the loop tests to see if we’ve generated the maximum allowed number of patterns that we’ve allocated space for. If so, we must exit the loop. Once in the loop, our first step is to invoke the @SOLVE command to solve the master problem. We then use an @FOR loop to the dual values on the finished goods to the pattern generation subproblem. We then solve the pattern generation subproblem and compute RC, the marginal rate of decrease in the main objective using the new pattern. Finally, at the end of the loop we append the new generated pattern to the master program for the next through the loop. Once we can no longer find a cutting pattern that will further reduce our costs, we exit the loop and solve the master problem one last time: ! Finally solve Master as an IP; @SOLVE( MASTER_PROB, INTEGER_REQ);

The one difference is that we now include the integer restrictions on the pattern usage array, X. As mentioned, cutting fractional quantities of a pattern is not physically possible, so we want our final solution to be integral. The version of this model in the samples folder has an additional calc section to prepare a tabular report on the cutting patterns and their usage. We will not go into the details of this table-generating code in this discussion. However, if you run the model, you should receive the following report: Total raws used:

985

Total feet yield: Total feet used:

43121 44325

Percent waste:

2.7%

Pattern: FG Demand Yield 1 2 3 4 5 6 7 8 9 ===================================================== F34 350 350 1 . . . 1 . . . . F24 100 100 1 . . . . 1 . . 1 F15 800 801 1 . 3 . . . 1 1 . F10 1001 1002 1 4 . . 1 2 1 3 . F18 377 377 1 . . 2 . . 1 . 1 ===================================================== Usage:

0

0 133

0 350

0 277 125 100

A total of 985 raw materials were used, with only 2.7% of the raw material input lost to trim waste. Note that only 9 patterns were needed to solve the model. Also note that Pattern 1, the super pattern, is not used in the final solution as per our design.

646 CHAPTER 13

Programming Example: Accessing Excel In this example, we will solve the familiar staff scheduling model once again. However, to make things interesting, we will solve several, independent models in a loop. Furthermore, the data for our models will be contained in a single Excel workbook, with each tab in the workbook storing data for an individual instance of the model. Solutions for each site will also be returned to their respective tabs in the Excel workbook. Suppose we have three hot dog stands: Pluto Dogs, Saturn Dogs and Mars Dogs. Each site has daily staffing needs that vary throughout the week. We hire employees to work 5 continuous days in a row, followed by two days off. As an example, an employee starting on Tuesday would be on duty Tuesday through Saturday and off Sunday and Monday, while someone starting on Friday would work Friday through Tuesday and be off Wednesday and Thursday. We need to know how many employees to start on each day of the week at each site so as to minimize to total number of staff hired.

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The data for this model may be found in the DOGS.XLS workbook contained in the Samples folder off the main LINGO folder. If you open this workbook, you'll see there are three tabsPluto, Saturn and Mars. Each tab contains the data for its particular site. The tab for the Pluto site appears below:

DOGS.XLS

648 CHAPTER 13 Here we see that the Pluto site requires 10 people on Mondays, 12 on Tuesdays, and so on. The staffing needs are also represented in the bar graph on the tab. Similar tabs for the remaining two sites are also included in the workbook.

The Model Here's our model loops over the three sites/tabs, solving each individual staffing model: MODEL: ! Uses a loop to solve three staff scheduling models, retrieving the data from and writing the solution to an Excel workbook; SETS: SITES / PLUTO, MARS, SATURN/; DAYS / MON TUE WED THU FRI SAT SUN/: NEEDS, START, ONDUTY; ENDSETS SUBMODEL STAFF: [OBJROW] MIN = @SUM( DAYS: START); @FOR( DAYS( D): ONDUTY( D) = @SUM( DAYS( D2) | D2 #LE# 5: START( @WRAP( D - D2 + 1, @SIZE( DAYS)))); ONDUTY( D) >= NEEDS( D); ); @FOR( DAYS: @GIN( START)); ENDSUBMODEL CALC: @SET( 'TERSEO', 2); @FOR( SITES( S): NEEDS = @OLE( '\LINGO12\SAMPLES\DOGS.XLS', SITES( S)+'NEEDS'); @SOLVE( STAFF); @SOLU( 0, ONDUTY, '

On Duty Report: ' + SITES( S));

@OLE( , SITES( S)+'START', SITES( S)+'ONDUTY', SITES( S)+'TOTAL') = START, ONDUTY, OBJROW; ); ENDCALC END

Model: LOOPOLE This model may also be found in the Samples folder of the main LINGO folder.

PROGRAMMING LINGO Note:

649

Versions of this example staffing model that interface with an Access database or that interface with text files are also available. Please see models LOOPODBC and LOOPTEXT.

The Details In the model's sets section: SETS: SITES / PLUTO, MARS, SATURN/; DAYS / MON TUE WED THU FRI SAT SUN/: NEEDS, START, ONDUTY; ENDSETS we declare two sets. The first set is our set of three sites followed by a set containing the days of the week. The days of the week set has three associated attributes: NEEDS, START, and ONDUTY, representing the staffing needs, the number of workers to start on each day of the week, and the total number of workers on duty each day.

Next, we declare a submodel that contains the staffing model that will be applied to each of the three sites: SUBMODEL STAFF: [OBJROW] MIN = @SUM( DAYS: START); @FOR( DAYS( D): ONDUTY( D) = @SUM( DAYS( D2) | D2 #LE# 5: START( @WRAP( D - D2 + 1, @SIZE( DAYS)))); ONDUTY( D) >= NEEDS( D); ); @FOR( DAYS: @GIN( START)); ENDSUBMODEL

The objective function minimizes the total number of employees hired at the site. This is then followed by an @FOR loop over the days of the week that a) computes the number of employees on duty each day, and b) requires the number on duty to equal, or exceed, the number required on each day. We also require the number starting on each day to be an integer value using the @GIN statement.

650 CHAPTER 13 The following calc section: CALC: @SET( 'TERSEO', 2); @FOR( SITES( S): NEEDS = @OLE( '\LINGO12\SAMPLES\DOGS.XLS', SITES( S)+'NEEDS'); @SOLVE( STAFF); @SOLU( 0, ONDUTY, '

On Duty Report: ' + SITES( S));

@OLE( , SITES( S)+'START', SITES( S)+'ONDUTY', SITES( S)+'TOTAL') = START, ONDUTY, OBJROW; ); ENDCALC

starts off by restricting the amount of LINGO's output by setting the TERSEO parameter to 2, which essentially eliminates all of LINGO"s output. Next, we construct an @FOR loop that loops over the three sites, allowing us to solve the submodel STAFF for each site. The first statement in the loop is an @OLE statement that reads the staffing requirements from the appropriate tab of the workbook and places the requirements into the NEEDS attribute. If you look at the workbook closely, you'll see that there are three ranges named "SaturnNeeds", "PlutoNeeds" and "MarsNeeds" that contain the staffing requirements for each of the respective sites. We are building each of these range names via the SITES(S)+'NEEDS' argument in the @OLE function call. Once the staffing requirements are loaded, we invoke the @SOLVE command to solve the staffing model for the current site. The @SOLU command prints out a small solution report to the screen showing the number of workers on duty each day for each of the sites. At the bottom of the calc section, the last @OLE call sends the solution back to the workbook. Note that for each site there are three ranges titled "<site_name>START", " ONDUTY" and "<site_name>TOTAL", where we return the daily number of workers to start, the number on duty each day, and the total number of employees required for the entire week. Note also that the workbook name was omitted in the call. When the workbook name is omitted, LINGO simply uses the current workbook in Excel. Given that we specified the workbook name in the original call to @OLE, it should now be the current workbook, thereby allowing us to omit its name in the final call.

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If you solve the model, you'll see that the tab for each of the three sites contains the number of workers to start each day, the total number on duty each day, and the total number of employees required. For the Saturn site, the tab will resemble the following:

652 CHAPTER 13

Summary There are many instances where you will need to solve a model multiple times and/or use the results of one model as input into another. The programming capabilities in LINGO allow you to automate these classes of problems. The three examples presented in this chapter introduce the programming constructs available in LINGO. In addition to the three examples presented in this chapter, several more examples of solving models in a loop may be found in the samples folder of your LINGO installation. These additional examples include: Loopdea.lg4 – A Data Envelopment Analysis model with a loop over each of the decisionmaking units to compute their individual efficiency scores. Loopts.lg4 – A traveling salesman model that loops to append subtour breaking constraints to the formulation, continuing until a complete tour of the cities is generated. Loopttt.lg4 – A looping model that implements a game of ticktacktoe. Finally, in addition to all the control structures and functions discussed in this chapter, all of the functions and operators listed in Chapter 7, LINGO's Operators and Functions, are also available for use in your calc section scripts, with the only exception being the variable domain functions. Variable domain functions are restricted for use in model section only.

14 Stochastic Programming So far, we worked with deterministic mathematical programs where all model parameters (e.g. coefficients, bounds, etc.) are known constants. A stochastic program (SP) is a mathematical program (linear, nonlinear or mixed-integer) in which some of the model parameters are not known with certainty, and the uncertainty can be expressed with known probability distributions. Applications arise in a variety of industries:           

Financial portfolio planning over multiple periods for insurance and other financial companies, in face of uncertain prices, interest rates, and exchange rates Exploration planning for petroleum companies, Fuel purchasing when facing uncertain future fuel demand, Fleet assignment: vehicle type to route assignment in face of uncertain route demand, Electricity generator unit commitment in face of uncertain demand, Hydro management and flood control in face of uncertain rainfall, Optimal time to exercise for options in face of uncertain prices, Capacity and Production planning in face of uncertain future demands and prices, Foundry metal blending in face of uncertain input scrap qualities, Product planning in face of future technology uncertainty, Revenue management in the hospitality and transport industries.

Stochastic programs fall into two major categories: a) multistage stochastic programs with recourse, and b) chance-constrained programs. With release 14, LINGO's capabilities are extended to solve models in both categories.

Multistage Decision Making Under Uncertainty Stochastic programs fall into two major categories: a) multistage stochastic programs with recourse, and b) chance-constrained programs. With version 12.0, LINGO's capabilities are extended to solve models in the first category, namely multistage stochastic recourse models. Chance-constrained models will be ed in future versions. In this chapter, the term stochastic program (SP) refers to a multistage stochastic model with recourse. The term stage is an important concept in this chapter. Usually it means the same as ‘time period’, however there are situations where a stage may consist of several time periods. The random, uncertain and stochastic are used interchangeably.

653

654 CHAPTER 14 Multistage decision making under uncertainty involves making optimal decisions for a T-stage horizon before uncertain events (random parameters) are revealed while trying to protect against unfavorable outcomes that could be observed in the future. Note:

A stage boundary is either a) the beginning of the problem, b) the end of the problem, or c) a point just after a decision event but just before a random event. A stage is the sequence of random events followed by decision events between two successive stage boundaries. Thus, decisions made in stage k are based on all information revealed up to stage k, but no more.

In its most general form, a multistage decision process with T+1 stages follows an alternating sequence of random events and decisions. Slightly more explicitly: 0.1) in stage 0, we make a decision x0, taking into that… 1.0) at the beginning of stage 1, “Nature” takes a set of random decisions ω1, leading to realizations of all random events in stage 1, and… 1.1) at the end of stage 1, having seen nature’s decision, as well as our previous decision, we make a recourse decision x1(ω1), taking into that … 2.0) at the beginning of stage 2, “Nature” takes a set of random decisions ω2, leading to realizations of all random events in stage-2, and… 2.1) at the end of stage 2, having seen nature’s decision, as well as our previous decisions, we make a recourse decision x2(ω1, ω2), taking into that … . . . T.0) At the beginning of stage T, “Nature” takes a random decision, ωT, leading to realizations of all random events in stage T, and… T.1) at the end of stage T, having seen all of nature’s T previous decisions, as well as all our previous decisions, we make the final recourse decision xT(ω1,…,ωT).

STOCHASTIC PROGRAMMING

655

This relationship between the decision variables and realizations of random data can be illustrated as follows.

Each decision, represented with a rectangle, corresponds to an uninterrupted sequence of decisions until the next random event. And each random observation corresponds to an uninterrupted sequence of random events until the next decision point.

Recourse Models The decision taken in stage 0 is called the initial decision, whereas decisions taken in succeeding stages are called recourse decisions. Recourse decisions are interpreted as corrective actions that are based on the actual values the random parameters realized so far, as well as the past decisions taken thus far. Recourse decisions provide latitude for obtaining improved overall solutions by realigning the initial decision with possible realizations of uncertainties in the best possible way.

656 CHAPTER 14 Restricting ourselves to linear multistage stochastic programs for illustration, we have the following form for a multistage stochastic program with (T+1) stages. Minimize (or maximize): c0x0 + E1[c1x1 + E2[c2x2 … + ET[cTxT ] … ]] such that: A00x0

~ b0

A(1)10x0

+ A(1)11x1

A(1,2)20x0

+ A(1,2)21x1

+ A(1,2)22x2

…

…

…

A(1,...,T)T0x0

+ A(1,...,T)T1x1 +

…

L0



 U0

L(1)1

 x1

…

x0

…

L(1,...,T)T

 xT

~ b(1)1 ~ b(1,2)2 … + A(1,...,T)TTxT

~ b(1,...,T)T

 U(1)1 …  U(1,...,T)T

where, (1,...,t) represents random outcomes from event space ( 1,...,t) up to stage-t, A(1,...,t)tp is the coefficient matrix generated by outcomes up to stage-t for all p=1…t, t=1…T, c(1,...,t)t is the objective coefficients generated by outcomes up to stage-t for all t=1…T, b(1,...,t)t is the right-hand-side values generated by outcomes up to stage-t for all t=1…T, L(1,..., t)t and U( 1,..., t)t are the lower and upper bounds generated by outcomes up to stage-t for all t=1…T, ’~’ is one of the relational operators '', ‘’, or ‘’ ; and x0 and xt  x(1,...,t)t are the decision variables (unknowns) for which optimal values are sought. The expression being optimized is called the cost due to initial-stage plus the expected cost of recourse. Note:

LINGO can solve linear, nonlinear and integer multistage stochastic programming problems.


657

Scenario Tree When the probability distributions for the random parameters (events) are discrete, there are only a finite number of outcomes in each stage. With each random parameter fixed to one of its possible outcomes, one can create a scenario representing one possible realization of the future. Enumeration of all possible combinations of outcomes allows us to represent all scenarios in a tree, with each scenario being a path from the root of the tree to one of its leaves. The nodes visited by each path correspond to values assumed by random parameters in the model. We illustrate the construction of a scenario tree with a stochastic version of the well-known Newsvendor inventory problem. In this problem, we must decide how much to order initially and then later, how much of any unsold product to return before the end of the planning horizon. There is a shortage penalty when there are lost sales and a carrying cost for left over units. The decision process takes place under uncertain demand and uncertain price per returned item: 1.

In stage 0, the order quantity has to be decided (under uncertain demand).

2.

In stage 1, at the beginning, the demand is revealed. A recourse decision, at the end of stage 1, is the number of units to be returned to the publisher (for an uncertain refund price)

3.

In stage 2 at the beginning, the refund price is announced by the publisher. The price per returned item can be either:

4.



Positive (i.e. publisher accepts them at a high price which covers the cost of shipping and handling) or



Negative (i.e. publisher accepts them at a low price which doesn’t cover the cost of shipping and handling).

The objective is to maximize the total expected profit at the end of planning horizon (stage 2).

658 CHAPTER 14

In the scenario tree above, x0 represents the initial decision, or order size to be determined before seeing any of the random outcomes. x1 represents the quantity to return to the publisher of any portion of the unsold units. Profit2 represents the total profit collected at the end of planning horizon. The notation 1 represents the event space for the unknown demand, for which there are three different possible outcomes 1 = {Low, Medium, and High} with probabilities {0.4, 0.3, 0.3}, respectively. Once we observe the demand 1  1, we make a recourse decision x1 based upon which 1 nature chose and our previous decision x0. The notation 2 represents the event space for refund price per unsold newspapers if returned to the publisher in stage 2. This event has two different outcomes 2 = {Positive, Negative} with probabilities {0.7, 0.3}. Once the refund price 2  2 in stage 2 is observed, the total profit would be computed by the model as the final decision Profit2. It should be clear from the scenario tree that,   

There are as many distinct scenarios in the SP as there are leaf-nodes. Each root-leaf path defines a scenario, induced by a full observation of all random events. There is a one-to-one correspondence between the scenarios and the leaf-nodes in the tree.

STOCHASTIC PROGRAMMING   

659

The unconditional probability of a node is computed by multiplying the conditional probabilities of the nodes positioned on the path, which starts from the root and terminates at that node. The unconditional probability of each leaf-node corresponds to the probability of the associated scenario. Each node in the tree corresponds to a vector of random parameter with a particular history up to that node in some scenario.

The branches out of each node enumerate all possible outcomes associated with random parameters associated with it in order to construct the history of random parameters that belong to the next stage.

Monte Carlo Sampling In stochastic programming (SP), when one or more random variables have a continuous (or discrete but infinite) event space, it is obviously computationally impossible to explicitly for the infinite number of possible outcomes. For such cases, Monte Carlo sampling (also called presampling) can be used to approximate the problem using a finite scenario tree. As illustrated in the figure below, if the model has a single stochastic parameter with a continuous distribution such as the normal distribution, one can discretize the event space simply by generating N sample points and thereby construct a finite and tractable scenario tree. This is also the case for discrete distributions with infinite event spaces, such as the Poisson distribution.

Note:

Since the point probability of each scenario in the original model is zero, it is customary to set the probabilities of sampled scenarios to 1/N.

660 CHAPTER 14 Given the parametric distribution of each stochastic parameter, LINGO's sampling routines can be used to efficiently generate univariate samples from these distributions. When sampling, LINGO uses Latin-hyper-square techniques to reduce the sample variance. One may also specify a correlation matrix for the continuous random variable, which the sampling routines will use to induce correlation in the sample points. These sample points are then used to define discrete distributions, which approximate the original distributions. Repeating this for all continuous stochastic parameters, LINGO recasts the infinite event space model into an approximate model with a finite event space.

Setting up SP Models There are four steps to setting up an SP model. Below is a list of these steps and the primary LINGO functions that you will call to perform each step. Each of these steps will be illustrated in detail in the example SP model later in this chapter. Step

Task

Description

How

1

Defining core model

The core model is built just like any other deterministic LINGO model. The random variables are used directly in the core model's expressions as if they were determinate.

Entered like any other deterministic LINGO model.

2

Identifying the random variables

Each random variable must be identified along with the stage of the model where it becomes known.

@SPSTGRNDV

3

Identifying the initial decision and recourse variables

Each initial decision variable and each recourse variable must be identified along with the stage where it must be decided. The initial decision variables are assigned stage 0, while the subsequent recourse variables are assigned to stages 1,2,... .

@SPSTGVAR

4

Declaring distributions

The probability distributions of the random variables must be declared. The techniques for doing so will depend on whether the variable of interest has a distribution defined by either a) a discrete outcome table, or b) a parametric distribution.

Outcome Tables:

An outcome table is a finite set of all possible outcomes, their probabilities and values for the random variable(s). Outcome tables may be constructed using either scalar values or attribute vectors. A parametric distribution, on the other hand, is defined by a particular type of probability density function, e.g., the normal distribution. Parametric distributions may have a finite or infinite number of possible outcomes, depending on the distribution type.

Scalar-based: @SPTABLESHAPE @SPTABLEOUTC @SPTABLEINST @SPTABLERNDV Vector-based: @SPDISTTABLE Parametic Distributions: @SPDIST @SPSAMPSIZE @SPCORR<METHOD>


661

Language Features for SP Models The primary features in the LINGO language used for building a stochastic programming model can be partitioned into the following three categories, corresponding to steps 2-4 of the Setting up SP Models section above: 1. Identifying random variables 2. Identifying initial decision and recourse variables 3. Declaring distributions Each of the categories are discussed in the following five sections, which are primarily intended for reference use. The reader may wish to jump directly to the first SP modeling example, Gas Buying Example, in order to see LINGO's SP modeling features used in the context of an actual model. There are additional SP language constructs available for use in calc sections for model's that exploit LINGO's programming capability. These features are discussed in Chapter 13, Programming LINGO.

Identifying Random Variables The fundamental component of any SP model is its set of random variables. The random variables are the stochastic component of the model. Their values are not known with certainty beforehand, however, their uncertainty can be expressed with known probability distributions. LINGO requires each random variable to be explicitly identified. When we identify each random variable, we must also provide the stage in the scenario tree where each variable's value is revealed. The @SPSTGRNDV function is used for this purpose: @SPSTGRNDV( STAGE, VARIABLE_NAME) This function takes two arguments: the random variable's name and the integer valued stage at which the random variable's value is revealed. Note that this function merely flags a variable as being random, and does not communicate any information regarding the actual distribution of the variable. A distribution is associated with the random variable in the Binding random variables to distribution instances, discussed below. Some examples follow: Example 1: @SPSTGRNDV( 1, DEMAND); Flags variable DEMAND as being a random variable in stage 1. Example 2:

@FOR( TIME_ASSET( T, A) | T #GT# 1: @SPSTGRNDV( T - 1, RETURN( T, A))); Flags RETURN( T, A) as being a random variable in stage T-1. Note that by using @SPSTGRNDV in an @FOR loop, we are able to identify multiple variables as being random.

662 CHAPTER 14

Identifying Initial Decision and Recourse Variables After identifying the random variables, all the remaining variables in the model are either initial decision variables or recourse variables. These are the variables that are in our control, as opposed to being stochastic and in the control of Mother Nature. The initial decision variables are those variables whose values must be determined at time 0, before nature has revealed values for any of the random variables. The recourse variables are the variables whose values are decided at a later period in response to outcomes for the random variables. Both initial decision and recourse variables are identified with the @SPSTGVAR function, with initial decision variables being distinguished by a stage of 0 and recourse variables by a stage >0. Row stages can be established using @SPSTGROW. @SPSTGVAR( STAGE, VARIABLE_NAME) This function takes two arguments: an integer value equal to the stage at which the variable's value must be decided and the variable's name. Some examples follow: Example 1: @SPSTGVAR( 0, PURCHASE); Flags variable PURCHASE as being an initial decision variable. Example 2: @FOR( TIME_ASSET( T, A): @SPSTGVAR( T - 1, INVEST( T, A))); Flags INVEST( T, A) as being recourse variable in stage T-1 (or an initial decision variable when T-1=0). Note that by using @SPSTGVAR in an @FOR loop, we are able to identify multiple recourse variables. Before an SP model can be ed to the SP solver engine, all initial and recourse variables must be assigned stage values. Having to explicitly assign stage values to all these variables can be a tedious and error-prone process. Fortunately, LINGO can deduce the stages for many of the variables based on the inputting a minimal set of stages for the variables. We refer to this process a stage deduction and it is discussed more in section The Solution below of the Gas Buying Example. Guidelines for determining the variables that require explicit stage identification are listed below. @SPSTGROW( VARIABLE_NAME, STAGE) This function takes two arguments: an integer-valued stage at which the row's value becomes known and the row's name. This function should generally not be required, in that LINGO can automatically deduce all row stages. If you do use this function, you must name the rows whose stages you wish to set. The syntax of asg names to rows may be found in the Constraint Names section of Chapter 1. Example: @SPSTGROW( 4, CAPACITY); Assigns row CAPACITY to stage 4.


663

Note: Guidelines for determining the variables and rows that must be explicitly staged are:  All random variables must be assigned to a stage using @SPSTGRNDV.  Any initial decision or recourse variable that belongs to stage N that does not depend either directly, or indirectly, on another variable (random or otherwise) declared to be in stage N must be explicitly assigned to stage N using @SPSTGVAR.  If you are uncertain whether a particular variable must be explicitly assigned to a stage, or not, then it doesn't hurt to assign it anyway using @SPSTGVAR.  In general, stage assignment for rows should be left to LINGO. Each row will be assigned to the stage equal to the maximum stage of the variables appearing in that row. However, if you wish to explicitly specify a row’s stage, you may always do so using @SPSTGROW.

Declaring Distributions After identifying the random variables, you will need to assign them to appropriate probability distributions. There are two classes of distributions: parametric distributions and outcome table distributions, with each requiring different declarations. Declaring parametric distributions involves declaring a particular type of probability density function (e.g., normal) and its parameters (e.g., mean and standard deviation). Outcome table distributions may be declared using scalar values or, assuming your model makes use of sets and attributes, they may also be declared very conveniently using attribute vectors and matrices. In either case, you will need to identify all possible outcomes and their respective probabilities.

Declaring Parametric Distributions LINGO allows both independent and t parametric probability distributions, as well as continuous and discrete distributions. The functions used to declare these distributions are of the form @SPDIST , where represents the type of distribution being declared. In addition, there are the @SPSAMPSIZE and @SPCORR<METHOD> functions for, respectively, setting sample sizes and inducing correlations, where <METHOD> refers to the particular method used to induce correlation.

664 CHAPTER 14 @SPDIST ( PARAM_1[, ... , PARAM_N], RANDOM_VAR) This function is used to apply both continuous and discrete parametric probability distributions to random variables. At least one distribution parameter argument will always be required. Additional parameters may be required depending on the particular distribution. Parameter values are always numeric. The random variable must be declared beforehand via @SPSTGRNDV. Below is a list of ed continuous distributions. Distribution

Syntax

Parameters

Beta

@SPDISTBETA( A, B, RNDVAR)

Cauchy Chi-Square

@SPDISTCACY( LOC, SCALE, RNDVAR) @SPDISTCHIS( DF, RNDVAR)

Exponential F

@SPDISTEXPO( LAMDA, RNDVAR) @SPDISTFDST( DF1, DF2, RNDVAR)

Gamma

Student's t

@SPDISTGAMM(SCALE, SHAPE, RNDVAR) @SPDISTGMBL( LOC, SCALE, RNDVAR) @SPDISTLAPL( LOC, SCALE, RNDVAR) @SPDISTLGST( LOC, SCALE, RNDVAR) @SPDISTLOGN( MU, SIGMA, RNDVAR) @SPDISTNORM( MU, SIGMA, RNDVAR) @SPDISTPRTO( SCALE, SHAPE, RNDVAR) @SPDISTSTUT( DF, RNDVAR)

A = alpha > 0 B = beta > 0 LOC = location SCALE = scale > 0 DF = degrees of freedom = a positive integer LAMDA = rate parameter > 0 DF1 = degrees of freedom 1 = a positive integer DF2 = degrees of freedom 2 = a positive integer SCALE = scale parameter > 0 SHAPE = shape parameter > 0 LOC = location SCALE = scale > 0 LOC = location SCALE = scale > 0 LOC = location SCALE = scale > 0 MU = mu parameter SIGMA = sigma parameter > 0 MU = mean SIGMA = standard deviation > 0 SCALE = scale parameter > 0 SHAPE = shape parameter > 0 DF = degrees of freedom = a positive integer

Symetric Stable

@SPDISTSMST( A, RNDVAR)

Triangular

@SPDISTTRIA( L, U, M, RNDVAR)

Uniform

@SPDISTUNIF( L, U, RNDVAR)

Weibull

@SPDISTWEIB(SCALE, SHAPE, RNDVAR)

Gumbel Laplace Logistic Lognormal Normal Pareto

A = alpha  [.02,2.0] L = lowest point H = high point M = middle point L = lower point U = upper point SCALE = scale parameter > 0 SHAPE = shape parameter > 0


665

Example: @SPDISTNORM( 82.3, 18.1, RND_TEMP); In this example, we apply a normal distribution to the random variable RND_TEMP, with a mean of 82.3 inches and a standard deviation of 18.1. Here's the list of discrete distributions currently ed: Distribution

Syntax

Parameters

Beta Binomial

@SPDISTBTBN( N, A, B, RNDVAR)

Binomial

@SPDISTBINO( N, P, RNDVAR)

Geometric Hypergeometric

@SPDISTGEOM( P, RNDVAR) @SPDISTHYPG( N, D, K, RNDVAR)

Logarithmic Negative Binomial Poisson

@SPDISTLOGR( P, RNDVAR) @SPDISTNEGB( R, P, RNDVAR)

N = trials = a positive integer  0 A = alpha > 0 B = beta > 0 N = trials = a positive integer  1 P = probability of success, 0  P  1 P = probability of success, 0 < P  1 N = population = a positive integer  1 D = number defective  {0,1,...,N} K = sample size {1,2,...,N} P = P-factor, 0 < P < 1 R = R-factor > 0 P = probability of success, 0 < P < 1 LAMDA = arrival rate > 0

Example:

@SPDISTPOIS( LAMDA, RNDVAR) @SPDISTPOIS( 33, CUSTOMERS);

In this example, we apply a Poisson distribution with an arrival rate of 33 to the random variable CUSTOMERS.

666 CHAPTER 14 @SPSAMPSIZE( STAGE, SAMPLE_SIZE) This function is used to set the sample size for probability distributions by stage. Most instances of the parametric distributions listed in this section have a large, if not an infinite, number of outcomes. Trying to incorporate all possible outcomes into the model would be impossible, therefore, LINGO must sample from the parametric distributions. If a specific sample size is not set using @SPSAMPSIZE, then LINGO defaults to a sample size equal to the Default Sample Size/Stage parameter, which is typically set to 2. However, in some models you may wish to have different sample sizes across stages, typically having larger samples in the more critical early stages, and less sampling in the latter stages. @SPSAMPSIZE can enforce this type of sampling strategy. Example 1: @SPSAMPSIZE( 1, 8); Here we set the sample size to 8 outcomes in stage 1. Example 2:

@FOR( PERIODS( P) | P #GT# @INDEX( P0): @SPSAMPSIZE( P - 1, SAMP_SZ( P)) ); In this example, we set the sample size equal to SAMP_SZ( P) in periods 1 through N-1. Note that by adding the conditional expression P #GT# @INDEX( P0) to the @FOR loop, the loop starts with P0=2, and we avoid setting the sample size in stage 0. We do this because stage 0 never contains random variables. Random variables first occur in stage 1, with stage 0 being reserved for the initial decision variables. In this example, we have also not assigned a sample size to the final stage, because we either want the final stage to use the default sample size, or the final stage does not have random variables.

@SPCORR<METHOD>( RANDOM_VAR_1, RANDOM_VAR_2, RHO) This function causes the sampling code to induce correlations between two random variables. The ordering of the two variables is not significant. The correlation coefficient, RHO, must lie in the interval [-1,1]. You must also choose between three methods available for inducing correlations: Pearson, Kendall or Spearman. Only one correlation method is allowed per model. Some examples follow: Example 1: @SPCORPEARSON( X, Y, .9); Here we set the correlation between random variables X and Y to be .9 using the Pearson method. Example 2:

@FOR( CORR_RNDM_PLANTS( P1, P2): @SPCORRKENDALL( CAPACITY( P1), CAPACITY( P2), RHO( P1, P2)); ); In this example, we have a derived, two-dimensional set. CORR_RNDM_PLANTS, that contains the pairs of electrical plants whose output capacities are correlated. We loop over these pairs, setting each pair's correlation to RHO( P1, P2) using the Kendall method.


667

Matrix Declarations of Outcome Table Distributions LINGO allows probability density functions to be input as discrete outcome tables. You will need to input the values for the random variables under each outcome, as well as the probability of each outcome. Matrix-based declarations are considerably more concise than scalar-based declarations, in that one simple function call can declare the outcome table, it's probabilities, as well as assign multiple random variables to the distribution. The function used for vector-based outcome table declaration is @SPDISTTABLE: @SPDISTTABLE( OUTCOME_TABLE, RANDOM_VARIABLE_ATTR[, PROBABILITY_ATTR]) This function identifies an outcome table, the random variables to be assigned to the outcome table, and, optionally, probabilities for each of the outcomes. The first argument, or outcome table, must be either a one or two-dimensional attribute containing the outcome values for the random variables. If it is one-dimensional, then the random variable argument must be a scalar. If the outcome table is twodimensional, say M x N, then it must contain M outcomes, with each row being a single outcome for N random variables. Also, in the two-dimensional case the random variable argument must be a onedimensional attribute of length N. If the probability argument is included, it must be a onedimensional attribute of length equal to the number of outcomes. If the probability argument is omitted, then each outcome is assumed to be equally probable. Some examples follow: Example 1: @SPDISTTABLE( DIST_RETURNS, RETURNS); Declares an outcome table with the two-dimensional attribute DIST_RETURNS containing one outcome per row for the random variables in the one-dimensional RETURNS attribute. The number of columns in DIST_RETURNS must equal the number of elements in RETURNS. The optional probability argument was omitted in this example, meaning that all outcomes are equally likely. Example 2: @SPDISTTABLE( DIST_SALES, SALES, PROB); Declares an outcome table with the one-dimensional attribute DIST_SALES containing the possible outcomes for the scalar random variable SALES. The PROB attribute is a onedimensional attribute of the same length as DIST_SALES that contains the probabilities of all the outcomes. Sample SP models installed with LINGO that utilize @SPDISTTABLE include: SPPUTOPT1, SPNEWSDISC and SPHYDRO.

Scalar Declarations of Outcome Table Distributions In the previous section, we discussed how to declare outcome tables using matrix-based declarations. In general, the matrix form of declaration is the simplest and easiest to use. However, there may be models that are not set-based that would be unable to use matrix declarations. Also, there may be instances when you only wish to declare portions of an attribute to be random. In these cases, you will need to use scalar-based outcome table declarations. The scalar-based functions used to declare outcome tables are: @SPTABLESHAPE, @SPTABLEOUTC., @SPTABLEINST and @SPTABLERNDV.

668 CHAPTER 14 @SPTABLESHAPE( 'TABLE_NAME', NUMBER_OF_OUTCOMES, NUMBER_OF_RANDOM_VARS) This function initializes an outcome table and requires three arguments: a unique name for the distribution table (entered as text), the number of possible outcomes, and the number of random variables in the distribution. Some examples follow: Example 1: @SPTABLESHAPE( 'WEATHER', 3, 1); Declares an outcome table called WEATHER with 3 outcomes and 1 independent variable. Example 2: @SPTABLESHAPE( 'RETURN', @SIZE( OUTCOMES), @SIZE( ASSETS)); Declares an outcome table called RETURN with the number of outcomes equal to the size of set OUTCOMES and the number of tly distributed variables equal to the size of set ASSETS. @SPTABLEOUTC( 'TABLE_NAME', OUTC_PROB, OUTC_VAL1[ ,..., OUTC_VALN]) Once an outcome table has been declared with @SPTABLESHAPE, the next step is to load the table with its set of outcome values and their probabilities. This is accomplished through calls to @SPTABLEOUTC. @SPTABLEOUTC accepts a multiple number of arguments; however, there must always be at least two arguments, with argument 1 always being the distribution name and the remaining arguments being numeric values. The first numeric value ed will be taken as the probability of outcome 1. Assuming the distribution has N variables, the next N arguments will be taken as the values for variables 1 through N for outcome 1. This process repeats for each outcome until a total of M * (N + 1) numeric values have been ed, where M is the number of outcomes. These numeric values may be ed in as many calls as is convenient, but, again, the first argument of each call must be the name of the relevant distribution. The following examples should help to illustrate: Example 1: @SPTABLEOUTC( 'WEATHER', .3, 56, .6, 77, .1, 92); In this example, we have a 3-outcome, single-variable, outcome table named 'WEATHER'. We make a single call to @SPTABLEOUTC to load all three outcomes. Outcome 1 has a probability of .3 and a value of 56, outcome 2 has a .6 probability and a value of 77, while outcome 3 has a .1 probability and a value of 92. Example 2:

@FOR( OUTCOMES( O): @SPTABLEOUTC( 'D1', 1 / @SIZE( OUTCOMES)); @FOR( ASSETS( A): @SPTABLEOUTC( 'D1', O_RETURN( O, A))); ); Here we are making two calls to @SPTABLEOUTC for each member of the OUTCOMES set. The first call es the probability of the outcome, while the second call es the outcomes for each member of the ASSETS set. This example was taken from the SPCOLLEGEDISC.LG4 example, which you may wish to review for additional clarification.


669

@SPTABLEINST( 'PARENT_TABLE_NAME', 'TABLE_INSTANCE_NAME') In order to actually use an outcome table distribution in a model we must also declare an instance of the relevant distribution. This allows us to use a particular distribution a multiple number of times in a model, saving us from having to reenter the distribution's parameters each time. @SPTABLEINST is the function for declaring distribution instances. This function takes two arguments: the parent outcome table's name as established in the call to @SPTABLESHAPE, and a unique name to apply to the new instance of the distribution. A particular outcome table may be used many times in a single model, however, each particular instance requires a unique name. Some examples follow: Example 1: @SPTABLEINST( 'WEATHER', 'WEATHER_1'); Declares an instance of the parent distribution 'WEATHER', giving it a name 'WEATHER_1'. Example 2:

@FOR( TIME( T): @SPTABLEINST( 'D1', 'D1_' + TIME( T))); Declares multiple instances of the parent distribution 'D1', asg each one the name 'D1_'+TIME( T). In this case, TIME( T) is a primitive set member whose name gets appended to the 'D1_' string to form the instance name. As an example, if set TIME has values T1..T12, then the instance name for the first distribution would be 'D1_T1'.

@SPTABLERNDV( 'DIST_INSTANCE_NAME', RANDOM_VAR_1[, ... , RANDOM_VAR_N]) The final step in setting up an outcome table distribution is binding the random variables to the distribution. This step associates specific random variables with the outcome table instance that was defined as described above. The @SPTABLERNDV function is used for this purpose. This function takes two, or more, arguments. The first argument is the distribution instance name, entered as text. The second argument is the name of the random variable to bind to the distribution instance. If the distribution has multiple variables, then you may include the additional variable names as arguments, or you may call @SPTABLERNDV multiple times, until all relevant random variable names have been assigned to the distribution instance. Some examples follow: Example 1: @SPTABLERNDV( 'WEATHER_1', RV_WEATHER_1); Binds random variable RV_WEATHER_1 to the 'WEATHER_1' distribution instance. Example 2:

@FOR( TIME( T): @FOR( ASSETS( A): @SPTABLERNDV( 'D1_' + TIME( T), RETURN( T, A)))); Assuming the TIME set consists of T1..T12, this example assigns one random for each member of the ASSETS set to distribution instance 'D1_Ti', for i =1,...,12. Assuming the ASSETS set has two , then two random variables are assigned to each the 12 distribution instances.

Sample SP models installed with LINGO that utilize the above techniques include: SPGAS1 and SPCOLLEGEDISC.

670 CHAPTER 14

Gas Buying Example Our first example of how to formulate and solve an SP is a gas buying example. This example was taken from the Committee on Stochastic Programming's (COSP) website: http://stoprog.org/index.html?spintroduction.html. The complete LINGO model can be found in the LINGO samples folder under the name SPGAS1.LG4. A gas company must come up with a plan for its gas purchases under uncertain weather conditions. The demand for the current period (period 1) is known and is 100 units. The demand for the period is not known with certainty and will depend on how cold the weather is. There are three possible outcomes for the weather: normal, cold and very cold. Each of these outcomes are equally likely. The following table lists the costs and demands under the three outcomes: Outcome

Probability

Gas Cost/Unit ($)

Units Demand

Normal

1/3

5.0

100

Cold

1/3

6.0

150

Very Cold

1/3

7.5

180

Gas for the current period is bought now and delivered directly to the customers at a cost of $5 per unit. Gas in the period can either be bought now and held in storage for period 2 use, or it can be purchased in period 2 at a price that will depend on the weather as per the table above. Storing gas bought in period 1 for use in period 2 costs the company $1 per unit. The question the company is faced with is: How much gas should be bought in periods 1 and 2 to meet total customer demand at minimal expected cost?

The Core Model Step 1 - Defining the Core Model: The core model is the same optimization model we would construct if all the random variables were known with certainty. There is nothing in the core model that addresses the stochastic nature of the model. The stochastic properties of the model will be introduced in a subsequent model section. For our current example, the core model is formulated as follows:


671

! Minimize Total Cost = Purchases + Holding; [R_OBJ] MIN= PURCHASE_COST + HOLD_COST; ! Compute purchase cost; [R_PC] PURCHASE_COST = 5 * PURCHASE_1 + COST_2 * PURCHASE_2; ! Compute holding cost; [R_HC] HOLD_COST = INVENTORY_1 + INVENTORY_2; ! Compute inventory levels; [R_I1] INVENTORY_1 = PURCHASE_1 - 100; [R_I2] INVENTORY_2 = INVENTORY_1 + PURCHASE_2 - DEMAND_2;

Core Model for Gas Buying Example In the first expression: ! Minimize Total Cost = Purchases + Holding; [R_OBJ] MIN= PURCHASE_COST + HOLD_COST;

we are saying that we want to minimize total cost, which consists of two components: the purchase cost of buying gas in periods 1 and 2, as well as holding costs of buying gas in a current period for use in a future period. The following expression: ! Compute purchase cost; [R_PC] PURCHASE_COST = 5 * PURCHASE_1 + COST_2 * PURCHASE_2;

calculates the total purchase cost, which is $5 per unit of gas purchased in period 1 plus the cost of gas in period 2 (COST_2) multiplied by the number of units of gas purchased in period 2. Note that COST_2 is stochastic and will depend on how the weather plays out. We will convey the probabilistic nature of period 2 cost in the next section. We next compute total holding costs, which, given the $1/unit holding cost, is simply the sum of units held over in periods 1 and 2: ! Compute holding cost; [R_HC] HOLD_COST = INVENTORY_1 + INVENTORY_2;

Finally, we compute the inventory levels for the two periods: ! Compute inventory levels; [R_I1] INVENTORY_1 = PURCHASE_1 - 100 ; [R_I2] INVENTORY_2 = INVENTORY_1 + PURCHASE_2 - DEMAND_2;

Period 1's inventory is the amount purchased in period 1 (PURCHASE_1) less the known demand in period 1. While inventory in period 2 (INVENTORY_2) if equal to any inventory carried over from period 2 plus the purchase in period 2 (PURCHASE_2) minus the demand in period 2 (DEMAND_2). Once again, DEMAND_2 is not known with certainty. The uncertain nature of DEMAND_2 will be established in the next section of the model - the stochastic declarations section.

672 CHAPTER 14

Stochastic Declarations After the core model, the remainder of the model is the stochastic declarations section: ! *** STEP 2 *** - Define Random Variables; !The random variables are period 2's demand and cost.; @SPSTGRNDV( 1, COST_2); @SPSTGRNDV( 1, DEMAND_2); ! *** STEP 3 *** - Define initial decision and recourse variables; !The initial decision is how much to purchase in period 1; @SPSTGVAR( 0, PURCHASE_1); !Period 2 purchases are a recourse variable after the weather reveals itself; @SPSTGVAR( 1, PURCHASE_2); ! *** STEP 4 *** - Assign distributions to the random variables; !Declare a discrete distribution called 'DST_DMD' with three outcomes and two tly distributed variables (i.e., Demand and Cost); @SPTABLESHAPE( 'DST_DMD', 3, 2); !Load the three equally likely outcomes into 'DST_DMD'; !Dist Name Probability Cost Demand; @SPTABLEOUTC( 'DST_DMD', 1/3, 5.0, 100); @SPTABLEOUTC( 'DST_DMD', 1/3, 6.0, 150); @SPTABLEOUTC( 'DST_DMD', 1/3, 7.5, 180); !Declare a specific instance of the 'DST_DMD' distribution, naming the instance 'DST_DMD_1'; @SPTABLEINST( 'DST_DMD', 'DST_DMD_1'); !Bind Period 2 Cost and Demand to the distribution instance; @SPTABLERNDV( 'DST_DMD_1', COST_2, DEMAND_2);

Stochastic Declarations for Gas Buying Example We use this section to identify all the random variables and their distributions. A detailed breakdown of the stochastic declarations follows. Step 2 - Identifying the Random Variables: The next step in building our sample SP model is to identify the random variables. The random variables are the variables that are stochastic by nature and whose values are not known before we must make our initial decisions. The @SPSTGRNDV function is used to identify random variables. @SPSTGRNDV accepts two arguments: the random variable's name and its stage. In this example, there are two random variables, the second period cost and demand, and we identify them via @SPSTGRNDV in the following to statements:


673

!The random variables are period 2's demand and cost.; @SPSTGRNDV( 1, COST_2); @SPSTGRNDV( 1, DEMAND_2);

Note that we have indicated that the two variables are in stage 1. This may seem somewhat odd, in that they are relevant to period 2. However, you will recall that the first set of random variables to become known after our initial decision belong to stage 1. If this was a multiperiod model, period 3's random variables would belong to stage 2, and so on. Step 3 - Identifying the Initial Decision and Recourse Variables: The next step is to identify the the initial decision variables and the recourse variables. Unlike the random variables, which are under Mother Nature's control, the initial decision and recourse variables are under our control. The initial decision variables occur at the very outset, before any of the random variables become known, and are always assigned to stage 0. The recourse variables are the subsequent decisions we make after learning of the outcomes of the random variables. Recourse variables that are decided after the stage N random variables become known are assigned to stage N as well. The @SPSTGVAR function is used to identify initial decision and recourse variables. @SPSTGVAR accepts two arguments: the variable's stage and its name. In our example, there is one initial decision, which is PURCHASE_1, the amount of gas to purchase in period 1. The weather then reveals itself and our recourse variable is PURCHASE_2, the amount to purchase in period 2. We identify this information with the two statements: !The initial decision is how much to purchase in period 1; @SPSTGVAR( 0, PURCHASE_1); !Period 2 purchases are a recourse variable after the weather reveals itself; @SPSTGVAR( 1, PURCHASE_2);

It turns out that before an SP model can be ed to the SP solver engine, all variables and rows must be assigned stage values. You will note that we have not done this. Specifically, we have not assigned stages to any of the rows nor to the variables: PURCHASE_COST, HOLD_COST, PURCHASE_2, INVENTORY_1 and INVENTORY_2. The reason we have not made these stage assignments is that LINGO can deduce the stages of the remaining rows and variables from the minimal stage information we've already supplied. We will examine this stage deduction feature in more detail in the next section. In the meantime, the following guidelines should suffice:

674 CHAPTER 14 Note: Guidelines for determining the variables and rows that must be explicitly staged are:  All random variables must be assigned to a stage using @SPSTGRNDV.  Any initial decision or recourse variable that belongs to stage N that does not depend either directly, or indirectly, on another variable (random or otherwise) declared to be in stage N must be explicitly assigned to stage N using @SPSTGVAR.  If you are uncertain whether a particular variable must be explicitly assigned to a stage, or not, then it doesn't hurt to assign it anyway using @SPSTGVAR.  In general, stage assignment for rows should be left to LINGO. Each row will be assigned to the stage equal to the maximum stage of the variables appearing in that row. However, if you wish to explicitly specify a row’s stage, you may always do so using @SPSTGROW. Step 4 - Declare Distributions Next, we need to declare the t probability distribution for the random variables COST_2 and DEMAMD_2. In this case, we will be using an outcome table distribution, and in order to declare our distribution we will make use of the scalar-based functions: @SPTABLESHAPE and @SPTABLEOUTC, @SPTABLEINST and @SPTABLERNDV. @SPTABLESHAPE initializes the distribution with a name, number of outcomes and number of tly distributed variables, while @SPTABLEOUTC is called once for each outcome to load information relevant to each outcome: !Declare a discrete distribution called 'DST_DMD' with three outcomes and two tly distributed variables (i.e., Demand and Cost); @SPTABLESHAPE( 'DST_DMD', 3, 2); !Load the three equally likely outcomes into 'DST_DMD'; ! Dist Name Probability Cost Demand; @SPTABLEOUTC( 'DST_DMD', 1/3, 5.0, 100); @SPTABLEOUTC( 'DST_DMD', 1/3, 6.0, 150); @SPTABLEOUTC( 'DST_DMD', 1/3, 7.5, 180);

@SPTABLESHAPE accepts three arguments: a name for the distribution, the number of outcomes and the number of tly distributed variables in the distribution. In this case, we've titled the distribution 'DST_DMD', and it has three outcomes along with two tly distributed variables. Now, to be able to actually apply the distribution to random variables we need to declare an instance of the distribution. By doing things this way, it's possible to reuse the same outcome table on more than one set of random variables. We declare an instance of a particular distribution using the @SPTABLEINST function, which accepts two arguments - the name of the parent distribution and a new name for the instance of the distribution. In the case of our example, we do the following:


675

!Declare a specific instance of the 'DST_DMD' distribution, naming the instance 'DST_DMD_1'; @SPTABLEINST( 'DST_DMD', 'DST_DMD_1'); Or, in words, we create an instance of the DST_DMD distribution, which we name DST_DMD_1.

Our last step is to associate, or bind, the random variables to the instance of the distribution. Specifically, we wish to bind the cost and demand random variables from period 2 to the DST_DMD_1 distribution instance. We do this with the following expression: !Bind Period 2 Cost and Demand to the distribution instance; @SPTABLERNDV( 'DST_DMD_1', COST_2, DEMAND_2);

Our SP model has now been fully set up and we are now ready to have LINGO solve the model.

The Solution SP models are solved just like all other models by using the LINGO|Solve command. After solving our sample gas buying model, SPGAS_SCALAR.LG4, you will see a solution report that begins with the following section: Global optimal solution found. Objective value: Infeasibilities: Total solver iterations:

1400.000 0.000000 2

The first line tells us that the solver found a globally optimal solution. This solution status information is critical. If the status is anything other than globally optimal, locally optimal, or feasible then the solution will not be valid. The objective value is the expected objective value of the objective over all the scenarios. Thus, in this example, the gas companies expected cost over the three weather scenarios is $1400. The infeasibilities figure measure the amount by which all constraints were violated in the deterministic equivalent (DE) model. The amount of infeasibilities should always be a very small number and should generally not exceed 1.e-7. Total solver iterations lists the number of iterations, or es, the solver required to solve the DE model. The next section of the solution report lists a number of expected values: Expected value of: Objective (EV): Wait-and-see model's objective (WS): Perfect information (EVPI = |EV - WS|): Policy based on mean outcome (EM): Modeling uncertainty (EVMU = |EM - EV|):

1400.000 1326.667 73.33333 1479.444 79.44444

These values are a guide as to how the stochastic nature of the model is impacting the objective value. The following is a brief description of these expected values:

676 CHAPTER 14 Expected Value of Objective (EV) - is the expected value for the model's objective over all the scenarios, and is the same as the reported objective value for the model. Expected Value of Wait-and-See Model's Objective (WS) - reports the expected value of the objective if we could wait and see the outcomes of all the random variables before making our decisions. Such a policy would allow us to always make the best decision regardless of the outcomes for the random variables, and, of course, is not possible in practice. For a minimization, it's true that WS <= EV, with the converse holding for a maximization. Technically speaking, WS is a relaxation of the true SP model, obtained by dropping the nonanticipativity constraints. Expected Value of Perfect Information (EVPI) - is the absolute value of the difference between EV and WS. This corresponds to the expected improvement to the objective were we to obtain perfect information about the random outcomes. As such, this is a expected measure of how much we should be willing to pay to obtain perfect information regarding the outcomes of the random variables. Expected Value of Policy Based On Mean Outcome (EM) - is the expected true objective value if we (mistakenly) assume that all random variables will always take on exactly their mean values. EM is computed using a two-step process. First, the values of all random variables are fixed at their means, and the resulting deterministic model is solved to yield the optimal values for the stage 0 decision variables. Next, a) the stage 0 variables are fixed at their optimal values from the previous step, b) the random variables are freed up, c) the nonanticipativity constraints are dropped, and d) this wait-and-see model is solved. EM is the objective value from this WS model. Expected Value of Modeling Uncertainty (EVMU) - is the absolute value of the difference EV - EM. It is a measure of what we can expect to gain by taking into uncertainty in our modeling analysis, as opposed to mistakenly assuming that random variables always take on their mean outcomes. Note:


STOCHASTIC PROGRAMMING Note:

677


The next component of the solution lists various statistics regarding the class and size of the model: Model Class:

LP

Total scenarios/leaf nodes: Total random variables: Total stages: Total variables: Nonlinear variables: Integer variables: Total constraints: Nonlinear constraints: Total nonzeros: Nonlinear nonzeros:

3 2 1 Core 6 0 0

Deteq 18 0 0

5 0

17 0

12 0

30 0

Model class lists the DE's model class. In the case of our gas buying sample we have an LP, or linear program. Possible model classes are detailed in the Solver Status Window section of Chapter 1. Total scenarios/leaf nodes lists the total number of scenarios in the model. In this example, there are three scenarios: Normal, Cold and Very Cold. Note:

The total number of scenarios will grow exponentially in the number of stages. For example, suppose we have just one random variable with two possible outcomes in each of N stages, then the number of scenarios will equal 2N. Thus, it doesn't take many stages before an SP can become quite large.

Total random variables lists the number of random variables in the SP. In this example, we have just two: COST_2 and PURCHASE_2. The Total stages quantity is the model's stage count. In this case there is only one round of random variable realizations and recourse decisions, giving a stage count of 1. Next, LINGO displays statistics regarding the model's size. Statistics for both the core model and DE are given. The more relevant statistics are the ones for the DE, which is the actual model that must be solved. Definitions for these statistics may be found in the Solver Status Window section of Chapter 1. The next section of the solution reports lists the stage-0 solution:

678 CHAPTER 14 Stage 0 Solution ---------------Variable PURCHASE_1 INVENTORY_1 Row R_I1

Value 200.0000 100.0000 Slack or Surplus 0.000000

The stage-0 solution lists the values for all variables and rows that are part of the initial decision. These values are of pressing importance, in that they must be implemented currently. For this reason, they are displayed in their own separate section near the top of the report. In the case of our gas company, the optimal initial decision to minimize expected cost is to purchase 200 units of gas in period 1, storing 100 units in inventory. If period 2 is normal the company can fulfill demand entirely from inventory, otherwise it must make up the difference through additional purchases in period 2.


679

Next, we have the Staging Report: Staging Report -------------Random Variable COST_2 DEMAND_2

Stage 1 1

Variable PURCHASE_COST HOLD_COST PURCHASE_1 PURCHASE_2 INVENTORY_1 INVENTORY_2

Stage 1* 1* 0 1 0* 1*

Row R_OBJ R_PC R_HC R_I1 R_I2

Stage 1* 1* 1* 0* 1*

Distribution DST_DMD(DST_DMD_1,1) DST_DMD(DST_DMD_1,2)

(*) Stage was inferred

The staging report lists all variables and rows along with their stages. When an SP model is solved, all variables and rows need to be assigned stage values before the model is ed to the SP solver engine. We could have done this explicitly through the use of the @SPSTGVAR, @SPSTGROW and @SPSTGRNDV functions; however, this can be a tedious process prone to errors - particularly for large models. On the one hand, all random variables must be flagged using @SPSTGRNDV, but only a minimal set of the recourse variables and the initial decision variables need to have their stages explicitly assigned. The reason we only need to specify a minimal set is that LINGO can deduce the stages of the remaining variables by tracing their dependency chains. We refer to this process as stage deduction, and any variable or row whose stage has been deduced will be flagged with an asterisk in the staging report. As an example, from the staging report we see that variable INVENTORY_1 was deduced to be in stage 0. LINGO determined this from the following row: [R_I1] INVENTORY_1 = PURCHASE_1 - 100;

Given that we declared PURCHASE_1 to be in stage 0, we can also compute INVENTORY_1's value in stage 0. In which case, INVENTORY_1 can be deduced to be in stage 0. In turn, we can then use this staging information for INVENTORY_1 in the following expression: [R_I2] INVENTORY_2 = INVENTORY_1 + PURCHASE_2 - DEMAND_2;

and deduce that INVENTORY_2 is in stage 1, by the fact that the maximum of the stages of all the right-hand side variables is 1. LINGO continues in this fashion, iteratively tracing the dependencies in the model and asg stages to each remaining row and variable. Explicit stage assignments made by the are never overridden.

680 CHAPTER 14 Guidelines for determining the variables and rows that require explicit staging were presented in section, Identifying Initial Decision and Recourse Variables. The next report is the Random Variable Distribution Report, which lists information on the distribution assigned to each of the random variables, as well as their observed sample means and standard deviations: Random Variable Distribution Report ----------------------------------Random Variable Distribution COST_2 DST_DMD,DST_DMD_1,1 DEMAND_2 DST_DMD,DST_DMD_1,2

Sample Mean

Sample StdDev

6.166667

1.027402

143.3333

32.99832

From this, we can see that the random variable COST_2 is assigned to the first column of the DST_DMD_1 instance of the tabular distribution DST_DMD. Furthermore, COST_2 had a sample mean of 6.166667 and a sample standard deviation of 32.99832. The remainder of the solution report contains sub-reports for each of the scenarios. Information regarding the each scenario's probability, objective value and variable values are displayed. The values for the random variables for the scenario are contained in a separate section of the report. Here we reproduce the report for scenario 1: Scenario: 1 Probability: 0.3333333 Objective: 1100.000 ---------------------------------------------------------Random Variable COST_2 DEMAND_2

Value 5.000000 100.0000

Variable PURCHASE_COST HOLD_COST PURCHASE_1 PURCHASE_2 INVENTORY_1 INVENTORY_2

Value 1000.000 100.0000 200.0000 0.000000 100.0000 0.000000

Row R_PC R_HC R_I1 R_I2

Value 0.000000 0.000000 0.000000 0.000000

STOCHASTIC PROGRAMMING Note:

681

Moderate to large SP models can have a huge number of scenarios. In which case, the standard solution report, with its detailed reporting of each scenario, can become unwieldy. In which case, you may wish to reduce the amount of output by placing LINGO into Terse output mode. You may then request more specific solution information using the LINGO|Solution command or add output statements to your model (see SPCOLLEGE2.LG4 for an example of adding output statements to an SP model).

If you examine the reports for each of the three scenarios in this example, you will find that the costs are as follows: $1,100 for Normal, $1,400 for Cold and $1,700 for Very Cold. This results in an expected cost of $1,400 = (1/3) * $1100 + (1/3) * $1400 + (1/3) * $1700. The company buys 200 units of gas in period 1, applying 100 units immediately to period 1 demand, while storing 100 units for period 2. If period 2''s weather is normal, then period 2 demand is satisfied entirely from inventory. If the period 2 turns out to be cold, an additional 50 units must be purchased in period 2. If the very cold scenario occurs, then an additional 80 units must be purchased.

Stock Option Example Here we will walk through a stochastic programming example of a six-period American put-option model. The holder of an American put option has the right to sell a specified (underlying) stock at any time (the feature of American options) between now and a specified expiration date at a specified strike price. The holder makes a profit in the period of exercise if the strike price exceeds the market price of the stock at the time of sale, otherwise the profit is 0. The objective will be to maximize the expected present value of the option in period 0, which also corresponds to the maximum amount we should be willing to pay for the option. In this example, the underlying stock's initial price is $100, with the option's strike price equal to $99. This initially places the option "out of the money" and make its value $0. However, should the stock's price decline over subsequent periods to the point where it becomes less than the strike price, then the option will have a positive value. We will also use a risk free interest rate of 3% for discounting revenue back to the initial period. The price behavior of the underlying stock is the stochastic element in this example. In any given period, we have determined that there are four equally likely outcomes for the stock's price: Return (%) +9.0 +3.0 -1.0 -8.0

Probability 1/4 1/4 1/4 1/4

Using the following reasoning, a casual observer might analyze the above data and declare the option as being worthless: The stock is presently out of the money by $1. Before it will move into the money, the stock must decline by at least $1. However, this seems highly unlikely given that the expected return in any period is positive +0.75% = .25 * (9.0+3.0-1.0-8.0)%. On average, the stock will just go up in value,

682 CHAPTER 14 and the option will never go into the money. Given this, we should not be willing to pay anything for this option. We will see below if our casual observer in on the right track or not. The complete LINGO model can be found in the LINGO samples folder under the name SPPUTOPT1.LG4.

The Core Model Step 1 - Defining the Core Model: First off, we have the model's sets section. Unlike the previous gas-buying model, which was entirely scalar, we will make use of LINGO's set-based modeling capabilities in this example. Any set-based model will have at least one sets section, and for the core model for this example we have the following: SETS: PERIODS: !six periods (1 initial and 5 recourse); RETURN, !return of underlying stock, a r.v.; PRICE, !current price of underlying stock; SELL, ! = 1 if option exercised in period; WLTH, !wealth in period from exercise of option; SAMP_SZ; !sample size; ENDSETS

Sets Section As mentioned, this is a six-period (one initial and five recourse) model. Across the six periods, we have the following attributes:     

RETURN - the stock's return (fractional price change), a random variable, PRICE - the current price of the stock (a function of the RETURN), SELL - a recourse variable indicating the fraction of the option position sold, WLTH - wealth in the current period consisting of the wealth from the previous period multiplied by the risk free rate plus any proceeds from exercising all, or part, of the option, and SAMP_SZ - the sample sizes used for sampling the distribution for RETURN.

Next we have the data section containing the parameters we mentioned above, consisting of the stock's initial price, the option's strike price and the risk free rate for discounting: DATA: PERIODS = P0..P5; INIT_PRICE = 100; STRIKE_PRICE = 99; RISK_FREE_RATE = .03; ENDDATA

Data Section

!names for the periods; !initial underlying stock price; !option strike price; !risk free rate for discounting;


683

Next, we have the core model. You will recall that the core model is the model we would construct if the random variables were know with certainty. ! Core Model -------------------------------------------+; NP = @SIZE( PERIODS); ! Number of periods; ! Maximize the present value of the option in period 0; [R_OBJ] MAX = WLTH( NP)/( 1 + RISK_FREE_RATE) ^ ( NP - 1); ! Compute stock's price in each period; [R_PRICE_INIT] PRICE( 1) = INIT_PRICE; @FOR( PERIODS( P) | P #GT# 1: [R_PRICE] PRICE( P) = PRICE( P - 1) * ( 1 + RETURN( P)); ); ! Calculate our compounded wealth in each period; [R_PV_1] WLTH( 1) = SELL( 1) * ( STRIKE_PRICE - PRICE( 1)); @FOR( PERIODS( P) | P #GT# 1: [R_PV] WLTH( P) = SELL( P) * ( STRIKE_PRICE - PRICE( P)) + WLTH( P - 1) * ( 1 + RISK_FREE_RATE); ); ! Can't sell more than we have; [R_SELL_EQ_1] @SUM( PERIODS( P): SELL( P)) <= 1; ! Random variables not allowed in initial period; [R_RETURN_P0] RETURN( 1) = 0;

Core Model for Stock Option Example As mentioned, our objective is to maximize the present value of the option in the initial period, which will tell us the maximum we should be willing to pay for the option. To compute the present value in period 0, we discount wealth in the final period back to the initial period using the risk-free rate: ! Maximize the present value of the option in period 0; [R_OBJ] MAX = WLTH( NP)/( 1 + RISK_FREE_RATE) ^ ( NP - 1);

where NP is the number of periods, calculated using the @SIZE function: NP = @SIZE( PERIODS); ! Number of periods;

Technically, NP is actually equal to the number of periods plus 1, due to P0 being part of the PERIODS set. So, when we discount wealth, we discount by NP - 1 periods as opposed to NP periods. In the next section of the core model we compute wealth in each period:

684 CHAPTER 14 ! Calculate our compounded wealth in each period; [R_PV_1] WLTH( 1) = SELL( 1) * ( STRIKE_PRICE - PRICE( 1)); @FOR( PERIODS( P) | P #GT# 1: [R_PV] WLTH( P) = SELL( P) * ( STRIKE_PRICE - PRICE( P)) + WLTH( P - 1) * ( 1 + RISK_FREE_RATE); );

Wealth can potentially come from two sources: sale of the option and/or carryover wealth from the previous period. In the first period, carryover wealth is not possible, so period 1 is handled as a special case with sale of the option being the only source of wealth. Subsequent periods' wealth are then computed in an @FOR loop and include both sales and carryover sources of wealth. Revenue from sales is computed as: SELLp * ( STRIKE_PRICE - PRICEp). Note, that the SELL variable may be fractional if we choose to sell less than the full position on a period. Carryover wealth is computed as: WLTHp-1 * ( 1 + RISK_FREE_RATE), where we multiply by the risk free rate to for interest payments on wealth. Clearly, we can't sell more of full option position, so the sum of the SELL attribute across all periods cannot exceed 1: ! Can't sell more than we have; [R_SELL_EQ_1] @SUM( PERIODS( P): SELL( P)) <= 1;

Finally, there can be no return in period 0, so we set RETURN to 0 in the initial period: ! Random variables not allowed in initial period; [R_RETURN_P0] RETURN( @INDEX( P0)) = 0;

In the next section, we will cover the final part of the model listing the stochastic properties.


Stochastic Declarations The remainder of the model after the core model is the stochastic declarations section: ! SP Related Declarations -----------------------------+; ! Returns are the random variables; @FOR( PERIODS( P) | P #GT# @INDEX( P0): @SPSTGRNDV( P - 1, RETURN( P)); ); ! Sell(P0) is the init decision, Sell(P1..P5) are the recourse variables; @FOR( PERIODS( P): @SPSTGVAR( P - 1, SELL( P)); ); SETS: ! Declare the outcome table distribution for returns; OUTCOMES: O_RETURN; !An outcome set with a single; ! attribute of stock's return; ENDSETS DATA: ! Probability distribution and sampling data; SAMP_SZ = 0 4 4 4 4 4; !sample size/period; O_RETURN = .09 .03 -.01 -.08; !4 outcomes for return; ENDDATA ! Declare outcome table distribution for return; @FOR( PERIODS( P) | P #GT# 1: @SPDISTTABLE( O_RETURN, RETURN( P)); ); ! Set the sample sizes for the stages; @FOR( PERIODS( P) | P #GT# 1: @SPSAMPSIZE( P - 1, SAMP_SZ( P)); ); END

Stochastic Declarations for Stock Option Example

685

686 CHAPTER 14 Step 2 - Identifying the Random Variables: As mentioned, the random variables in this example are the stock price returns, RETURN, for periods 1 through 5. We indicate this with the expression: ! Returns are the random variables; @FOR( PERIODS( P) | P #GT# @INDEX( P0): @SPSTGRNDV( P - 1, RETURN( P)); );

Note that return in the initial period, P0, is not valid in this model and we zeroed it out above. Step 3 - Identifying the Initial Decision and Recourse Variables: The initial decision is deciding whether or not to sell some, or all, of the option position in the initial period. The recourse decisions are whether or not to sell some of our position in periods 1 through 5. We set this up with the following: ! Sell(P0) is the initial decision, Sell(P1..P5) are the recourse variables; @FOR( PERIODS( P): @SPSTGVAR( P - 1, SELL( P)); );

Step 4 - Declare Distributions We have a single discrete distribution for the returns on the underlying stock. As with the previous gas-buying example, this distribution is represented by an outcome table. However, unlike the previous example, we will make use of the matrix-based @SPDISTTABLE function for declaring our outcome table and its random variables. The code in the model devoted to declaring this distribution is:


687

SETS: ! Declare the outcome table distribution for returns; OUTCOMES: O_RETURN; !An outcome set with a single; ! attribute of stock's return; ENDSETS DATA: ! Probability distribution and sampling data; SAMP_SZ = 0 4 4 4 4 4; !sample size/period; O_RETURN = .09 .03 -.01 -.08; !4 outcomes for return; ENDDATA ! Declare outcome table distribution for return; @FOR( PERIODS( P) | P #GT# 1: @SPDISTTABLE( O_RETURN, RETURN( P)); ); ! Set the sample sizes for the stages; @FOR( PERIODS( P) | P #GT# 1: @SPSAMPSIZE( P - 1, SAMP_SZ( P)); );

END

First, we define a set OUTCOMES with an attribute O_RETURN. O_RETURN will be used to hold the four possible outcomes from the stock's return in a period. Next, we call @SPDISTTABLE to declare the outcome table for each RETURN random variable in periods 1 through 5. We then set the samples size for each of the periods to 4 in the data section: SAMP_SZ = 0 4 4 4 4 4;

!sample size/period;

and then them to LINGO's SP solver in the @FOR loop: ! Set the sample sizes for the stages; @FOR( PERIODS( P) | P #GT# @INDEX( P0): @SPSAMPSIZE( P - 1, SAMP_SZ( P)); );

Note that a sample size is not relevant for the initial period, P0, given that no random variables are permitted in the initial period. So, we have explicitly avoided ing a sample size for period P0. In the previous gas-buying example, we had only one stage and three outcomes, resulting in only three possible scenarios. This current model has a much larger set of scenarios. In this example, there are 5 stages with 4 possible outcomes for return in each stage. This results in 4 5=1024 scenarios. In order to keep the number of scenarios down to a more manageable size in larger models, you may need to reduce the sample sizes in each stage. As an example you might want to try running this same model with the following sample sizes:

688 CHAPTER 14 SAMP_SZ = 0 4 4 2 2 2;

!sample size/period;

This cuts the number of scenarios in the underlying SP model to 4*4*2*2*2=128 scenarios. This should help to dramatically cut the runtime, but at the expense of potentially less accurate results due to sampling error.

The Solution After solving the model, you'll find that the expected value of the option is $3.37: Local optimal solution found. Objective value: Infeasibilities: Total solver iterations:

3.451804 0.000000 19608

Thus, it turns out that our casual observation of the option being worthless was incorrect. Even though the option is initially out of the money, and, on average, is expected to go even further out of the money as time es, the option still has a positive expected value of $3.45. This is due to the volatility in the stock price, which can temporarily send the stock price down, even though the long run expectation is for the price to climb. Based on our sample size of 4 in each of teh 5 stages, there was a total of 1024 scenarios. In fully 598 of these scenarios, the option's value did prove to be worthless. However, in the remaining 426 scenarios the option had a positive value of as high as $13.54. Here's a histogram of the full range of outcomes:


689

The key initial decision was whether or not to sell the option. From the stage 0 (initial decision) solution we see that the optimal decision is to not sell: Stage 0 Solution ---------------Variable SELL( P0)

Value 0.000000

Of course, we didn't really need LINGO to tell us not to sell in the first period  we could have deduced this ourselves. The worst outcome we could experience is to gain $0. By selling in the initial period, we lock in the $0 return, never allowing for the possibility of the stock to go down in price and the option to come into the money. We will rely on LINGO's solution, however, to tell us the best time to sell in subsequent periods. Here is a fragment of the solution report for scenario 65, which is one of the scenarios with the maximum option value of $13.54:

690 CHAPTER 14 Scenario: 65 13.53568

Probability: 0.9765625E-03

Objective:

--------------------------------------------------------Random Variable RETURN( P1) RETURN( P2) RETURN( P3) RETURN( P4) RETURN( P5) Variable NP RETURN( P0) PRICE( P0) SELL( P0) WLTH( P0) PRICE( P1) SELL( P1) WLTH( P1) PRICE( P2) SELL( P2) WLTH( P2) PRICE( P3) SELL( P3) WLTH( P3) PRICE( P4) SELL( P4) WLTH( P4) PRICE( P5) SELL( P5) WLTH( P5)

Value -0.8000000E-01 -0.8000000E-01 -0.1000000E-01 -0.1000000E-01 0.3000000E-01 Value 6.000000 0.000000 100.0000 0.000000 0.000000 92.00000 0.000000 0.000000 84.64000 1.000000 14.36000 83.79360 0.000000 14.79080 82.95566 0.000000 15.23452 85.44433 0.000000 15.69156

Note that the stock goes down in the first four periods, but that we sold in period 2 for a gain of $14.36, which discounted back to the initial period is equal to a gain of $13.54. Had we been omniscient, we could have held on to the option for two more periods for additional gains. But, of course, we can't see the future, and we must base our decisions on maximizing the option's expected value over all scenarios.


691

Investing Under Uncertainty Example In the previous two examples, the random variable distributions were expressed as discrete outcome tables. In this example, we will look at a model with distributions that are not discreet. In particular, we will be using a normal distribution for our random variables. Given the infinite number of outcomes in a normal distribution, we will also need to use @SPSAMPSIZE to declare the samples sizes to be used by the SP solver. In addition, we will use @SPCORRPEARSON to input a correlation coefficient for some of the random variables. In this example, we are investing to fund a college education (Birge and Louveaux, 1997). We may invest in either stocks or bonds, both of which are assumed to have returns that are normally distributed as per the following table: Asset Bonds Stocks

Mean Return 12% 16%

Standard Deviation 1% 10%

It also turns out that the returns on stocks and bonds are correlated, with a correlation coefficient of .5. Our initial wealth is $55,000, while our target at the end of four periods is $80,000. Our goal is to maximize wealth at the end of the three periods, however, we will be very disappointed if we don't meet our goal. For this reason, we penalize ourselves by a factor of four for each dollar that we are short of our goal. We also have the option of readjusting our portfolio at the start of each period. The complete LINGO model can be found in the LINGO samples folder under the name SPCOLLEGENORM.LG4.

692 CHAPTER 14

The Core Model The core model for our investing example follows: ! Core Model -------------------------------------------+; SETS: TIME; ASSETS: MU, SIGMA; AXT( ASSETS, TIME): INVEST, RETURN; AXA( ASSETS, ASSETS) | &2 #GT# &1: RHO; ENDSETS DATA: INITIAL = 55; GOAL = 80; PENALTY = 4; TIME = T0..T3; ASSETS, MU, SIGMA = BONDS 1.12 .01 STOCKS 1.16 .10; RHO = .5; ENDDATA MIN = PENALTY * UNDER - OVER; @FOR( TIME( T) | T #GT# 1: @SUM( ASSETS( A): RETURN( A, T) * INVEST( A, T - 1)) = @SUM( ASSETS( A): INVEST( A, T)) ); FINAL = @SUM( ASSETS( A): INVEST( A, @SIZE( TIME))); FINAL = GOAL + OVER - UNDER; @SUM( ASSETS( A): INVEST( A, @INDEX( TIME, T0))) = INITIAL; @FOR( ASSETS( A): RETURN( A, @INDEX( TIME, T0)) = 0; );

Core Model for College Investing Example In the sets section, we have two primitive sets: TIME and ASSETS. The TIME set will be used to represent our time periods and will have four : T0 will represent the initial decision period, while T1, T2 and T3 will represent the three time periods. The set ASSETS will contain two : BONDS and STOCKS. We also define two derived sets in the sets section: AXT and AXA. The AXT set is a dense, 2dimensional set consisting of all (ASSETS,TIME) pairs. We define the two attributes INVEST and RETURN on AXT. The INVESTa,t attribute is our decision variable of how much to invest in asset a in period t, while RETURNa,t is a random variable representing the return of asset a in period t.


693

Next, in the data section we input the following pieces of data:      

INITIAL - the initial wealth, GOAL - the final goal for wealth PENALTY - the penalty factor to apply to goal shortfalls TIME - the four of the TIME set ASSETS,MU,SIGMA - the normal distribution data for returns on bonds and stocks, and RHO - the correlation coefficient between bond and stock returns

Getting into the model section, the first expression: MIN = PENALTY * UNDER - OVER; is the objective function, which can be viewed as a penalty function that we will be minimizing. For each dollar under our goal (UNDER) we get penalized 4 units, while for each dollar over our goal (OVER) we reduce the penalty by 1 unit. This will force the solver to more heavily weight solutions that meet our goal, perhaps at the expense of maximizing total wealth. In the next expression: @FOR( TIME( T) | T #GT# 1: @SUM( ASSETS( A): RETURN( A, T) * INVEST( A, T - 1)) = @SUM( ASSETS( A): INVEST( A, T)) );

we are setting total investments in period t equal to total investments plus returns in period t-1. Note that by formulating the investment flows in this manner, we are free to reallocate investments each period. The following two expressions are used to calculate the amount that we are either over (OVER) or under (UNDER) goal: FINAL = @SUM( ASSETS( A): INVEST( A, @SIZE( TIME))); FINAL = GOAL + OVER - UNDER;

Next, we add a constraint to limit the initial investments to our initial wealth level: @SUM( ASSETS( A): INVEST( A, @INDEX( TIME, T0))) = INITIAL;

Without this expression, the model would allow infinite investment, leading to an unbounded solution. Finally, there are no investment returns in the initial decision period, so we zero out them out: @FOR( ASSETS( A): RETURN( A, @INDEX( TIME, T0)) = 0; );

694 CHAPTER 14

Stochastic Declarations After the core model, we enter the stochastic declarations identifying the stochastic features of the model: ! SP Related Declarations -----------------------------+; ! The initial decision and recourse variables; @FOR( AXT( A, T): @SPSTGVAR( T - 1, INVEST( A, T)); ); @FOR( AXT( A, T) | T #GT# 1: ! Return is a random variable...; @SPSTGRNDV( T - 1, RETURN( A, T)); ! ...and it's normally distributed; @SPDISTNORM( MU( A), SIGMA( A), RETURN( A, T)) ); @FOR( TIME( T)| T #GT# @INDEX( TIME, T0): !Set a sample size for each period; @SPSAMPSIZE( T-1, 8); !Load the stock and bond correlation for the period; @FOR( AXA( I, J): @SPCORRPEARSON( RETURN( I, T), RETURN( J, T), RHO( I, J)) ); );

Stochastic Declarations for College Investing Example The initial decision variables in the model are how much to allocate to bonds and how much to allocate to stocks in the initial time period, or variables INVEST( BONDS, T0) and INVEST( STOCKS, T0). The recourse variables are how much to allocate to both assets in subsequent periods as the returns for each period are observed, or INVEST( BONDS, T1), INVEST( STOCKS, T1), ..., INVEST( STOCKS, T3). We identify these variables and assign them to their respective stages as follows: ! The initial decision recourse variables; @FOR( AXT( A, T): @SPSTGVAR( T - 1, INVEST( A, T)); );

In the following loop:


695

@FOR( AXT( A, T) | T #GT# 1: ! Return is a random variable...; @SPSTGRNDV( T - 1, RETURN( A, T)); ! ...and it's normally distributed; @SPDISTNORM( MU( A), SIGMA( A), RETURN( A, T)) );

we declare the asset return variables as random variables and declare that they have a normal distribution using the @SPDISTNORM function. Note that @SPDISTNORM requires the parameters of the distribution, which, in the case of a normal distribution, are the mean and standard deviation. Finally, we set the sample size to 8 samples in each stage/period and load the correlation coefficient between the two assets in each period in the loop: @FOR( TIME( T)| T #GT# @INDEX( TIME, T0): !Set a sample size for each period; @SPSAMPSIZE( T-1, 8); !Load the stock and bond correlation for the period; @FOR( AXA( I, J): @SPCORRPEARSON( RETURN( I, T), RETURN( J, T), RHO( I, J)) ); );

In this particular instance, we are using the Pearson method for inducing correlations in the sample values. Other alternatives are the Kendall and Spearman methods.

The Solution After solving the model, you'll find that the expected value of the penalty function is approximately 2.78: Global optimal solution found. Objective value: Infeasibilities: Total solver iterations:

2.775631 0.000000 1092

You will recall that the penalty function used in the model's objective is a weighted combination of 4 times the dollars under target minus the number of dollars over target. So, does the positive expected objective value mean that we miss our target on average? Perhaps not, given that we weight the under target dollars 4 times more than the over target dollars. To get a more meaningful result, a calc section was added to the model:

696 CHAPTER 14 CALC: @SET( 'TERSEO', 1); @SOLVE(); I = 1; NOVER = 0; NUNDER = 0; @WRITE( ' Surplus '); @FOR( TIME( T) | T #GT# @INDEX( T0): @WRITE( ' ', TIME( T))); @WRITE( @NEWLINE( 1)); @WRITE( ' Scenario Return Prob'); @FOR( TIME( T) | T #GT# @INDEX( T0): @WRITE( ' Bond Stock')); @WRITE( @NEWLINE( 1)); X_SURPLUS = 0; @WHILE( I #LE# @SPNUMSCENE(): @SPLOADSCENE( I); @WRITE( @FORMAT( I, '10.0f'), @FORMAT( OVER - UNDER, '15.3f'), @FORMAT( @SPPRBSCENE( I), '10.3f') ); @FOR( TIME( T) | T #GT# @INDEX( T0): @FOR( ASSETS( A): @WRITE( ' ', @FORMAT( 100*(RETURN( A, T) - 1), '4.1f'), '%'))); @WRITE( @NEWLINE( 1)); X_SURPLUS = X_SURPLUS + @SPPRBSCENE( I) * ( OVER - UNDER); I = I + 1; @IFC( UNDER #LE# 1.E-8: NOVER = NOVER + 1; @ELSE NUNDER = NUNDER + 1; ); ); @WRITE( @NEWLINE( 1)); @WRITE( ' Expected surplus: ', @FORMAT( X_SURPLUS, '15.3f')); @WRITE( @NEWLINE( 1)); @WRITE( ' Scenarios over goal: ', @FORMAT( NOVER, '12g')); @WRITE( @NEWLINE( 1)); @WRITE( ' Scenarios under goal: ', @FORMAT( NUNDER, '11g')); @WRITE( @NEWLINE( 1)); @WRITE( ' Success ratio: ', @FORMAT( NOVER/(NOVER+NUNDER), '18.3f')); ENDCALC


697

This calc section uses the scripting capabilities in LINGO to generate a custom report that displays each of the 512 scenarios in the model, their return values and their expected surpluses of funds over the goal. (More details on LINGO scripting capabilities may be found in Chapter 13, Programming LINGO.) The report also lists summary information on the expected surplus dollars, along with the number of scenarios over target and the number below target. Portions of this report follow:

Scenario 1 2 3 4 5

Surplus Return 0.528 14.604 12.203 27.423 18.706

Prob 0.002 0.002 0.002 0.002 0.002

T1 Bond 11.0% 11.0% 11.0% 11.0% 11.0%

T2 Stock 35.4% 35.4% 35.4% 35.4% 35.4%

T3

Bond 13.3% 13.3% 13.3% 13.3% 13.3%

Stock 7.7% 7.7% 7.7% 7.7% 7.7%

Bond 9.5% 11.9% 11.0% 11.6% 12.9%

Stock 0.3% 17.9% 14.9% 33.9% 23.0%

11.4% 11.4% 11.4% 11.4% 11.4% 12.8% 12.8% 12.8% 12.8% 12.8% 12.8% 12.8% 12.8%

0.5% 0.5% 0.5% 0.5% 0.5% 27.7% 27.7% 27.7% 27.7% 27.7% 27.7% 27.7% 27.7%

11.6% 12.9% 12.3% 12.3% 13.2% 9.5% 11.9% 11.0% 11.6% 12.9% 12.3% 12.3% 13.2%

33.9% 23.0% 5.3% 10.8% 20.9% 0.3% 17.9% 14.9% 33.9% 23.0% 5.3% 10.8% 20.9%

<...> 500 501 502 503 504 505 506 507 508 509 510 511 512

9.946 3.249 -7.809 -4.413 1.961 6.928 22.123 19.531 35.960 26.551 11.241 15.953 24.741

Expected surplus: Scenarios over goal: Scenarios under goal: Success ratio:

0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

12.7% 12.7% 12.7% 12.7% 12.7% 12.7% 12.7% 12.7% 12.7% 12.7% 12.7% 12.7% 12.7%

23.3% 23.3% 23.3% 23.3% 23.3% 23.3% 23.3% 23.3% 23.3% 23.3% 23.3% 23.3% 23.3%

4.413 344 168 0.672

Based on this report, we see that the expected surplus is $4,413, meaning that on average we can expect to meet our goal by that amount. Furthermore, 344 of the 512 scenarios ended over target for a success ratio of 67.2%.

698 CHAPTER 14

Chance-Constrained Programs (Cs) The second major class of models in stochastic programming is chance-constrained programs (C). A C model is a) similar to general stochastic programs in that the model contains random quantities with known distributions, but b) simpler in that the model has just a single decision stage and a single random outcome stage. The goal in C is to make an optimal decision prior to realization of random data, while controlling the chance that constraints are violated. Consider an LP with random matrix Ξ and random right-handside ω: Min c x Ξxi ≥ ωi

i=1...m

If we required all m realizations of Ξ x ≥ ω to be satisfied, then we would get a very conservative/expensive solution x, or no feasible solution at all. The distinctive feature of C is that we require that Ξ x ≥ ω be satisfied with some prespecified probability, 0 < p < 1, as opposed to it being satisfied for all possible realizations of (Ξ,ω). An example of a C model would be a blending model where the quality level of the raw materials is not known with certainty, but have known probability distributions. Given this variability in raw material quality, it may not always be practical to satisfy quality requirements in the final blend 100 percent of the time. Instead, we seek to find a blend of the raw materials that will satisfy quality constraints to a specified precentage, say 90 percent, of the time. We present this example later in the chapter in section A C Fuel Blending Model.


699

Setting up C Models Setting up a C model requires four steps. Three of the steps are identical to those used in setting up an SP. The one step found in SPs that's not found in Cs is identifying initial decision and recourse variables, and this is because in a C all decision variables are considered to be in stage 0, i.e., they are all initial decision variables. This contrasts with an SP, which may have any number of recourse stages in addition to the initial decision stage. The fourth (and new) step in setting up Cs is identifying the chance-constraint sets. These four steps are summarized in the table below and are illustrated in detail in the example C model that follows. Step

Task

Description

How

1

Defining core model The core model is built just like any other Entered like any other deterministic LINGO model. The deterministic LINGO random variables are used directly in the model. core model's expressions as if they were determinate.

2

Identifying the random variables

Each random variable must be identified @SPSTGRNDV along with the stage of the model where it becomes known. In a C, random variables must always belong to stage 1.

3

Declaring distributions

The probability distributions of the Outcome Tables: random variables must be declared. The Scalar-based: techniques for doing so will depend on @SPTABLESHAPE whether the variable of interest has a @SPTABLEOUTC distribution defined by either a) a discrete @SPTABLEINST outcome table, or b) a parametric @SPTABLERNDV distribution. Vector-based: An outcome table is a finite set of all @SPDISTTABLE possible outcomes, their probabilities and values for the random variable(s). Outcome tables may be constructed using either scalar values or attribute vectors. A parametric distribution, on the other hand, is defined by a particular type of probability density function, e.g., the normal distribution. Parametric distributions may have a finite or infinite number of possible outcomes, depending on the distribution type.

4

Identify chanceconstraint sets.

One or more sets of constraints are @SPCHANCE classified as being chance-constrained. An individual constraint may appear in no more than one chance-constraint set.

700 CHAPTER 14

Language Features for C Models The primary features in the LINGO language used for building a stochastic programming model can be partitioned into the following three categories, corresponding to steps 2-4 of the Setting Up C Models section above: 1. Identifying random variables 2. Declaring distributions 3. Identifying chance-constraint sets Each of the categories are discussed in the following three sections, which are primarily intended for reference use. The reader may wish to jump directly to the C modeling example, A C Fuel Blending Example, in order to see LINGO's C modeling features used in the context of an actual model. There are additional C language constructs available for use in calc sections for model's that exploit LINGO's programming capability. These features are discussed in Chapter 13, Programming LINGO.

Identifying Random Variables A fundamental component of any C model is its set of random variables. The random variables are the stochastic component of the model. Their values are not known with certainty beforehand, however, their uncertainty can be expressed with known probability distributions. LINGO requires each random variable to be explicitly identified. When we identify each random variable, we must also provide the stage in the scenario tree where each variable's value is revealed. Currently, unlike SPs, Cs are limited to single stages, given this, the stage value must always be 1. The @SPSTGRNDV function is used for this for identifying the random variables in a C model: @SPSTGRNDV( STAGE, VARIABLE_NAME) This function takes two arguments: the random variable's name and the integer valued stage at which the random variable's value is revealed. Note that this function merely flags a variable as being random, and does not communicate any information regarding the actual distribution of the variable. A distribution is associated with the random variable in the Binding random variables to distribution instances, discussed below. Some examples follow: Example 1: @SPSTGRNDV( 1, DEMAND); Flags variable DEMAND as being a random variable in stage 1. Example 2: @FOR( CUSTOMERS( I): @SPSTGRNDV( 1, DEMAND( I)))); Flags DEMAND( I) as being a random variable in stage 1. Note that by using @SPSTGRNDV in an @FOR loop, we are able to identify multiple variables as being random.

Declaring Distributions In a C, the distributions for the random variables are declared exactly as they are in SPs. You can refer to the section Declaring Distributions for SPs above to see how this is done.


701

Identifying Chance-Constraint Sets A C model will have one or more sets of chance constraints. Each set is assigned a) a name, b) a direction and c) a probability. You make these declarations using the @SPCHANCE function using the syntax: @SPCHANCE( 'Set_Name', '>='|'<=', Probability) The set name must be unique, enclosed in quotes, and must follow normal LINGO naming conventions. The direction must be either '>=' or '<=', while the probability is a real number in the range of [0,1]. The direction argument indicates whether the probability that the set is satisfied is either greater-than-or-equal-to or less-than-or-equal-to the specified probability. An example follows: Example 1: @SPCHANCE( 'C_DEMAND', '>=', .95); Here we declare a C constraint set with name C_DEMAND, a direction of greater-than-or-equalto, and a set probability of .95. This says that the constraints assigned to this set must all be satisfied in at least 95% of the scenarios. If just one constraint is unsatisfied in a particular scenario, then the whole set is considered unsatisfied for that scenario. Once you've declared a C set, there is still an additional step of asg actual constraints to the set. This is again done with the @SPCHANCE function, however, the syntax is slightly different: @SPCHANCE( 'Set_Name',

Constraint_Ref)

The same set name is used as was used in the set declaration, however, you now must enter a constraint reference, consisting of the name of a constraint in the core model. Here's an example: Example 2: @FOR( CUST( I): @SPCHANCE( 'C_DEMAND', R_DEMAND( I))); Here we use an @FOR loop to assign multiple constraints with name R_DEMAND to the C constraint set C_DEMAND. When asg constraints to a C constraint set there are a few rules to :  A constraint may be assigned to no more than one C set.  Each constraint belonging to a C set must reference at least one random variable.  The objective function may not be assigned to a C set. In the next section, we will walk through building a complete C model for fuel blending.

A C Fuel Blending Example  The Model As production manager at a fuel processing plant, you would like to maximize profit from blending the raw materials butane, catalytic reformate, and naphtha into the two finished products Regular and . These final products must satisfy certain ranges of quality requirements on octane, vapor pressure, and volatility. You also know the minimum required for each finished good, the maximum you will be able to sell, and the profit contribution per unit.

702 CHAPTER 14 Each raw material has a limited availability and cost per unit which are known, however, the quality level of the three raw materials is not known with certainty. We do know that the quality levels are normally distributed with known means and standard deviations. We'd like to chose blending ratios so that the finished products don't fall below their lower limits on requirements at least 90 percent of the time and don't exceed their upper limits at least 70 percent of the time. We will use a C to solve this problem, with the quality constraints forming the chance-constrained sets. The model we will use is available in the LINGO Samples folder under the name BLENDC.LG4. Our model's sets section is: SETS: !Each raw material has an availability and cost/unit; RAWMAT: AVAIL, COST; !Each finished good has a min required, max sellable, selling price, and batch size to be determined; FINGOOD: MINREQ, MAXSELL, PRICE, BATCH; !The quality measures set; QUALMES; !For each combo of raw material and quality measure there is an average quality level, its standard deviation, and the actual quality level received; RXQ( RAWMAT, QUALMES): QLEVMU, QLEVSD, QLEV; !For each combination of quality measure and finished good there are upper and lower limits on quality; QXF( QUALMES, FINGOOD): QUP, QLOW; !For each combination of raw material and finished good there is an amount of raw material used to be solved for; RXF( RAWMAT, FINGOOD): USED; ENDSETS

A feature to note in the sets section is the attribute QLEV, which will store the actual quality levels of the raw materials. The actual quality levels are not known with certainty, but they are normally distributed with means QLEVMU and standard deviations QLEVSD. Also, at the bottom of the sets section, we've also declared the attribute USED, which constitutes our decision variables and will tell us the amount of each raw material to use in each finished product


703

Next is the model's data section: DATA: !Raw materials; RAWMAT = BUTANE, CATREF, NAPTHA; !Raw material availability; AVAIL = 1000, 4000, 5000; !Raw material costs; COST = 7.3, 18.2,

12.5;

!Finished goods; FINGOOD = REGU, PREM; !Limits on finished goods; MINREQ = 4000, 2000; MAXSELL = 8000, 6000; !Finished goods prices; PRICE = 18.4, 22; !Quality measures (octane, vapor, volatility); QUALMES = OCT, VAP, VOL; !Average quality parameters of raw materials; QLEVMU = 120, 60, 105, 100, 2.6, 3, 74, 4.1, 12; QLEVSD =

5, 2, 3,

3, .11, .2,

6, .13, .4;

!Upper and lower limits on quality for each finished good; QUP = 100, 120, 11, 11, 25, 25; QLOW = 90, 95, 8, 8, 17, 17; ENDDATA

At the bottom of the data section, we initialize the averages and standard deviations (QLEVMU and QLEVSD) for the raw material quality levels. As an example, the average vapor level for catalytic reformate is 60 with a standard deviation of 3. We also input the upper and lower quality limits for each quality and each finished product (QUP and QLOW). As an example here, the lower and upper limits on octane for are 95 and 120. As mentioned, the quality constraints on the finished products will be chance constrained, so we will be able to violate these limits at times.

704 CHAPTER 14 Next comes the core model: !Core model ++++++++++++++++++++++++++++++++++++; !We want to maximize the profit contribution; [R_OBJ] MAX = @SUM( FINGOOD( F): PRICE( F) * BATCH( F)) @SUM( RAWMAT( R): COST( R) * @SUM( FINGOOD( F): USED( R, F))); !Subject to raw material availability; @FOR( RAWMAT( R): [R_RMLIM] @SUM( FINGOOD( F): USED( R, F)) <= AVAIL( R); ); @FOR( FINGOOD( F): !Batch size limits; @BND( MINREQ( F), BATCH( F), MAXSELL( F)); !Batch size computation; [R_BATCOMP] BATCH( F) = @SUM( RAWMAT( R): USED( R, F)); ); @FOR( QXF( Q, F): !Quality restrictions for each quality measure; [R_QUP] @SUM( RAWMAT( R): QLEV( R, Q) * USED( R, F)) <= QUP( Q, F) * BATCH( F); [R_QDN] @SUM( RAWMAT( R): QLEV( R, Q) * USED( R, F)) >= QLOW( Q, F) * BATCH( F); );

Once again, the core model contains the deterministic version of the model without the introduction of any stochastic elements. Basically, we want to maximize profit, while not using more raw materials than are available, not exceeding batch size limits and strictly meeting the quality restrictions on the finished goods. As we transition the model to a C, we will relax this final condition on the finished goods' quality levels.


705

Finally, at the end of our model are the declarations that convert the core model into a C: !C declarations ++++++++++++++++++++++++++++++; DATA: PROB_UP = .7; PROB_DN = .9; NSAMP = 20; ENDDATA !Declare QLEV(Q,F) as normally distributed; @FOR( RXQ( R, Q): @SPSTGRNDV( 1, QLEV( R, Q)); @SPDISTNORM( QLEVMU( R, Q), QLEVSD( R, Q), QLEV( R, Q)); ); !Establish a minimal @SPCHANCE( @SPCHANCE(

two empty C row sets with satisfaction probability level; 'C_QUAL_UP', '>=', PROB_UP); 'C_QUAL_DN', '>=', PROB_DN);

!Add all the quality constraints to the C set; @FOR( RXF( R, F): @SPCHANCE( 'C_QUAL_UP', R_QUP( R, F)); @SPCHANCE( 'C_QUAL_DN', R_QDN( R, F)); ); !Set sample size; @SPSAMPSIZE( 1, NSAMP);

In the following section: !Declare QLEV(Q,F) as normally distributed; @FOR( RXQ( R, Q): @SPSTGRNDV( 1, QLEV( R, Q)); @SPDISTNORM( QLEVMU( R, Q), QLEVSD( R, Q), QLEV( R, Q)); );

we declare the raw material quality levels as random variables in stage 1 via the @SPSTGRNDV function. We then declare them as being normally distributed using @SPDISTNORM, along with ing their means and standard deviations. Next, we need to declare two C sets: one for the upper limits on quality, and one for the lower limits on quality. As mentioned above, we use the @SPCHANCE function to do this as follows: !Establish a minimal @SPCHANCE( @SPCHANCE(

two empty C row sets with satisfaction probability level; 'C_QUAL_UP', '>=', PROB_UP); 'C_QUAL_DN', '>=', PROB_DN);

706 CHAPTER 14 Here, we assign the names C_QUAL_UP and C_QUAL_DN for, respectively, the upper and lower limit constraint sets. The direction for both sets is greater-than-or-equal-to ('>='), given that we want the probability that the sets are satisfied to equal, or exceed, a specified probability. The desired probability hurdles, PROB_UP and PROB_DN, were initialized in an earlier data section to be .7 and .9. Once we've declared our C sets we need to assign constraints to them. We do this again with @SPCHANCE: !Add all the quality constraints to the C set; @FOR( RXF( R, F): @SPCHANCE( 'C_QUAL_UP', R_QUP( R, F)); @SPCHANCE( 'C_QUAL_DN', R_QDN( R, F)); );

Here, we loop over every (raw material, finished good) pair, loading constraint references (i.e., constraint names) into their respective sets. You will note that the constraint names were assigned to the quality constraints earlier in the core model: @FOR( QXF( Q, F): !Quality restrictions for each quality measure; [R_QUP] @SUM( RAWMAT( R): QLEV( R, Q) * USED( R, F)) <= QUP( Q, F) * BATCH( F); [R_QDN] @SUM( RAWMAT( R): QLEV( R, Q) * USED( R, F)) >= QLOW( Q, F) * BATCH( F); );

You may refer to section Constraint Names for more information on how to establish constraint names. Lastly, we set the sample size for the model to be 20 with the statement: !Set sample size; @SPSAMPSIZE( 1, NSAMP);

This is the number of times each random variable will be sampled, and, in the case of Cs will also be the total number of scenarios. Note that currently for Cs there is only one stage, so the sample size is always set for stage 1. Note that NSAMP was initialized earlier in the model to be 20. In the next section, we will discuss the solution to our fuel blending example.


707

A C Fuel Blending Example  The Solution Solving the C model from the previous section, we find the following summary at the top of the solution report: Global optimal solution found. Objective value: Infeasibilities: Extended solver steps: Total solver iterations: Stochastic Model Class: Deteq Model Class:

33396.85 0.000000 621 28081 CHANCE-CONSTRAINED MILP

Total scenarios/leaf nodes: Total random variables:

20 9 Core 8 0 0

Deteq 48 0 40


18 0

248 0


34 0

1262 0


The solver found a globally optimal solution of approximately $33,397 in 621 branches and 28,801 iterations. LINGO correctly identifies the model as CHANCE-CONSTRAINED. One interesting result is that the deterministic equivalent (deteq) generated by LINGO for solving the stochastic model was a mixed integer linear program (MILP). This is of note because the core model is a pure linear program with no integer variables. You can this further down on the line listing the integer variable counts of 0 in the core model and 40 in the deterministic equivalent. Binary variables were added to the deteq to work as indicator variables indicating whether or not a particular C set is satisfied in a particular scenario. Here, we have 2 sets and 20 scenarios resulting in 2 x 20 = 40 binary variables. The remaining section of the summary report lists the sizes of the various components of the model. The main thing to note here is the size of the deteq relative to the core model, which turns out to be considerably larger. It is the dimensions of the deteq that are most relevant with respect to runtimes and any size limitations (e.g., variable limits) of your installation. Note that a deteq will grow as you increase the sample size; on the one hand, a large samples size is nice for reducing sampling error, but, on the other hand, you will experience longer runtimes when increasing sample size. Those interested in viewing the actual deteq can run the LINGO|Generate|Explicit Deteq command.

708 CHAPTER 14 Next up we have the C Sets Report: C Sets Report --------------Set Name/

Unsatisfied Scenarios

C_QUAL_UP R_QUP( OCT, REGU) R_QUP( OCT, PREM) R_QUP( VAP, REGU) R_QUP( VAP, PREM) R_QUP( VOL, REGU) R_QUP( VOL, PREM)

6

C_QUAL_DN R_QDN( OCT, REGU) R_QDN( OCT, PREM) R_QDN( VAP, REGU) R_QDN( VAP, PREM) R_QDN( VOL, REGU) R_QDN( VOL, PREM)

2

Actual Probability 0.7000000

0.9000000

Target Dir Probability >=

0.7000000

>=

0.9000000

The C Sets Report list the following information: 1. Set Name/ - Each C set is listed by name immediately followed by the names of its member constraints. 2. Unsatisfied Scenarios - This reports on the number of scenarios where the C set was violated. 3. Actual Probability - This is the observed probability that a constraint set was satisfied in a particular scenario, and is computed for C set i as (TotalScenarios - UnsatisfiedScenariosForSeti) / TotalScenarios. 4. Dir - This is the direction of the C set, indicating whether the observed probability should be greater-than-or-equal-to or less-than-orequal to the target probability. 5. Target - This column lists the target probability for the C set. For this particular example, we see that the upper limits on quality were unsatisfied in a total of 6 out of 20 scenarios for an actual probability of .7, exactly matching the target probability. The lower limits set was violated only twice for an actual probability of .9, again exactly matching the target.


709

After the C Sets Report LINGO generates the Stage 0 Solution: Stage 0 Solution ---------------USED( USED( USED( USED( USED( USED(

Variable BUTANE, REGU) BUTANE, PREM) CATREF, REGU) CATREF, PREM) NAPTHA, REGU) NAPTHA, PREM)

Row R_RMLIM( BUTANE) R_RMLIM( CATREF) R_RMLIM( NAPTHA)

Value 536.1166 401.7255 2053.043 1946.957 1410.840 569.2198 Slack or Surplus 62.15789 0.000000 3019.940

The Stage 0 Solution lists the values for all the non-random variables and constraints. These are all part of stage 0 and will not vary from one scenario the next, hence, their values are reported only once in the Stage 0 Solution. Here we've reproduced a a fragment of the report showing the items of most interest: the amount of each raw material used in each finished product (USED) and the slack values on the raw material supply constraints (R_RMLIM). After the Stage 0 Solution LINGO generates the Random Variable Distribution Report: Random Variable Distribution Report ----------------------------------Random Variable QLEV( BUTANE, OCT) QLEV( BUTANE, VAP) QLEV( BUTANE, VOL) QLEV( CATREF, OCT) QLEV( CATREF, VAP) QLEV( CATREF, VOL) QLEV( NAPTHA, OCT) QLEV( NAPTHA, VAP) QLEV( NAPTHA, VOL)

Sample Mean 119.7697 59.94561 104.9087 99.94025 2.600531 2.999666 73.99625 4.102662 12.00060

Sample StdDev 4.981368 2.969037 6.133397 2.124726 0.1093538 0.1257882 3.077703 0.1975493 0.3970604

Distribution NORMAL,120,5 NORMAL,60,3 NORMAL,105,6 NORMAL,100,2 NORMAL,2.6,0.11 NORMAL,3,0.13 NORMAL,74,3 NORMAL,4.1,0.2 NORMAL,12,0.4

This report lists each of the random variables in the model, their sample mean and standard deviation, and their distribution type.

710 CHAPTER 14 Finally, at the end of the report LINGO generates a individual reports for each of the scenarios. For brevity, we reproduce the report just for the 12th scenario: Scenario: 12 Probability: 0.5000000E-01 ----------------------------------------Random Variable QLEV( BUTANE, OCT) QLEV( BUTANE, VAP) QLEV( BUTANE, VOL) QLEV( CATREF, OCT) QLEV( CATREF, VAP) QLEV( CATREF, VOL) QLEV( NAPTHA, OCT) QLEV( NAPTHA, VAP) QLEV( NAPTHA, VOL)

Value 126.4475 65.45897 109.6616 100.5477 2.504830 3.098372 73.84414 4.081760 11.41373

C Set C_QUAL_UP C_QUAL_DN

Status UNSATISFIED SATISFIED

Row REGU) REGU) PREM) PREM) REGU) REGU) PREM) PREM) REGU) REGU) PREM) PREM)

Value 21598.34 18401.66 61555.49 11392.05 -1994.877 13994.88 -1399.827 10153.53 18744.58 13255.42 16364.39 6978.823

R_QUP( R_QDN( R_QUP( R_QDN( R_QUP( R_QDN( R_QUP( R_QDN( R_QUP( R_QDN( R_QUP( R_QDN(

OCT, OCT, OCT, OCT, VAP, VAP, VAP, VAP, VOL, VOL, VOL, VOL,

The scenario reports list the values for the random variables for the scenarios as well as the slack values of the C constraints. An interesting feature of the scenario reports for a C is the middle section, where each C set is listed along with whether or not the set was satisfied in the scenario. In this case, the upper limit quality constraints are listed as being unsatisfied, which is verified by the negative slacks on the row R_QUP( VAP, REGU) and R_QUP( VAP_PREM). This then makes scenario 12 one of the 6 scenarios where the upper limit quality constraints were allowed to be violated.

15 On Mathematical Modeling When developing a model in LINGO, it helps to understand how the model is processed internally by the LINGO solver. The relationships in your model influence the computation time, the solution methods used by LINGO, and the type of answer returned. Here we’ll explain some of the different types of relationships in a LINGO model and how each type affects the solution search. An understanding of these topics is not required to use LINGO, but it can help you use the software more effectively.

Solvers Used Internally by LINGO LINGO has four solvers it uses to solve different types of models. These solvers are:    

a direct solver, a linear solver, a nonlinear solver, and a branch-and-bound manager.

The LINGO solvers, unlike solvers sold with other modeling languages, are all part of the same program. In other words, they are linked directly to the modeling language. This allows LINGO to data to its solvers directly through memory, rather than through intermediate files. Direct links to LINGO’s solvers also minimize compatibility problems between the modeling language component and the solver components. When you solve a model, the direct solver first computes the values for as many variables as possible. If the direct solver finds an equality constraint with only one unknown variable, it determines a value for the variable that satisfies the constraint. The direct solver stops when it runs out of unknown variables or there are no longer any equality constraints with a single remaining unknown variable. Once the direct solver is finished, if all variables have been computed, LINGO displays the solution report. If unknown variables remain, LINGO determines what solvers to use on a model by examining its structure and mathematical content. For a continuous linear model, LINGO calls the linear solver. If the model contains one or more nonlinear constraints, LINGO calls the nonlinear solver. When the model contains any integer restrictions, the branch-and-bound manager is invoked to enforce them. The branch-and-bound manager will, in turn, call either the linear or nonlinear solver depending upon the nature of the model.

711

712 CHAPTER 15 The linear solver in LINGO uses the revised simplex method with product form inverse. A barrier solver may also be obtained, as an option, for solving linear models. LINGO’s nonlinear solver employs both successive linear programming (SLP) and generalized reduced gradient (GRG) algorithms. Integer models are solved using the branch-and-bound method. On linear integer models, LINGO does considerable preprocessing, adding constraint “cuts” to restrict the noninteger feasible region. These cuts will greatly improve solution times for most integer programming models.

Type of Constraints Through the use of the direct solver, LINGO substitutes out all the fixed variables and constraints from the model. The remaining reduced set of constraints and variables are then classified as being either linear or nonlinear. LINGO’s solver status window, which by default opens every time you solve a model, gives a count of the linear and nonlinear variables and constraints in a model. If any nonlinear variables or constraints are found in the model, the entire model is considered nonlinear and the relatively slower nonlinear solver must be invoked in place of the linear solver.

Linear Constraints If all the of a constraint are of the first order, the constraint is said to be linear. This means the constraint doesn’t contain a variable squared, cubed, or raised to any power other than one, a term divided by a variable, or variables multiplied by each other. Also, proportionality must exist. In other words, for every unit increase or decrease in a variable, the value of the constraint increases or decreases by a fixed amount. Linear formulas are “straight line” relationships. The basic form of a linear formula is: Y=mX+b where m and b are constants. For example, suppose you’re buying tomatoes for $1.50 per pound. The expression or function used to calculate the cost (C) in of the amount of tomatoes purchased (T) is: C = 1.5 * T.

ON MATHEMATICAL MODELING

713

As you might expect, a graph of this function for cost is a straight line:

$60 $50 $40 Cost $30 $20 $10 $0 0

10 20 30 Tomatoes Purchased (lbs.)

40

Linear expressions can have multiple variables. For example, if you added potatoes (P) at $0.75 per pound and apples (A) at $1.25 per pound, your cost function would become: C = 1.5 * T + 0.75 * P + 1.25 * A This new cost expression is also linear. You could think of it as the sum of three simpler linear expressions. Because linear models can be solved much faster and with more accuracy than nonlinear models, it’s preferable to formulate your models using linear expressions whenever possible. When the LINGO|Solve command is issued, LINGO analyzes the relationships in the model. If all the expressions are linear, LINGO will recognize and take advantage of this fact. Relative to other types of models, problems expressed using exclusively linear relationships can be solved quickly. If allowed to run to completion, LINGO will return the answer that yields the highest value for a maximization objective, or the lowest value for a minimization objective. One way to learn whether or not all expressions in your model are linear is to note the classification statistics displayed during the solution process in the solver status window. The Nonlinear categories of the Variables and Constraints boxes display the number of nonlinear relationships in the model. If zeros appear in both these categories, the model is linear. If LINGO displays a number greater than zero for the nonlinear relationships, you may want to investigate whether the constraints and variables in your model could be reformulated in a linear manner. For example, consider the following constraint: X / Y = 10; As written, this constraint is nonlinear because we are dividing by Y. By simply multiplying both sides of the equation through by Y, we can convert it to the equivalent linear constraint: X = 10 * Y;

714 CHAPTER 15

Nonlinear Constraints By definition, all constraints that are not linear are nonlinear. Nonlinear expressions include relationships with variables that are squared, cubed, taken to powers other than one, or multiplied or divided by each other. Models with nonlinear expressions are much more difficult to solve than linear models. Unlike linear models, nonlinear models may prevent LINGO from finding a solution, though one exists. Or LINGO may find a solution to a nonlinear model that appears to be the “best”, even though a better one may exist. These results are obviously undesirable. For more on what you can do to help minimize the occurrence of these undesirable results, see the Guidelines for Nonlinear Modeling section below.

Local Optima vs. Global Optima When LINGO finds a solution to a linear optimization model, it is the definitive best solution—we say it is the global optimum. Multiple optima may exist. However, a globally optimal solution is a feasible solution with an objective value that is as good as or better than all other feasible solutions to the model. The ability to obtain a globally optimal solution is attributable to certain properties of linear models. This is not the case for nonlinear optimization. Nonlinear optimization models may have several solutions that are locally optimal. All gradient based nonlinear solvers converge to a locally optimal point (i.e., a solution for which no better feasible solutions can be found in the immediate neighborhood of the given solution). Additional local optimal points may exist some distance away from the current solution. These additional locally optimal points may have objective values substantially better than the solver's current local optimum. Thus, when a nonlinear model is solved, we say the solution is merely a local optimum. The should be aware that other local optimums may, or may not, exist with better objective values. Conditions may exist where you may be assured that a local optimum is in fact a global optimum. See the Convexity section below for more information. Consider the following small nonlinear model involving the highly nonlinear cosine function: MIN = X * @COS( 3.1416 * X); X < 6; The following graph shows a plot of the objective function for values of X between 0 and 6. If you're searching for a minimum, there are local optimal points at X values of 0, 1.09, 3.03, and 5.02 in the "valleys." The global optimum for this problem is at X = 5.02, because it is the lowest feasible valley.


715

Graph of X * @COS( 3.1416 * X) Imagine the graph as a series of hills. You're searching in the dark for the minimum or lowest elevation. If you are at X = 2.5, every step towards 2 takes you uphill and every step towards 3 takes you downhill. Therefore, you move towards 3 in your search for the lowest point. You'll continue to move in that direction as long as it leads to lower ground. When you reach X=3.03, you'll notice a small flat area (slope is equal to zero). Continuing begins to lead uphill and retreating leads up the hill you just came down. You're in a valley, the lowest point in the immediate neighborhooda local optimum. However, is it the lowest possible point? In the dark, you are unable to answer this question with certainty. LINGO lets you enter initial values for variables (i.e., the point from which LINGO begins its search) using an INIT section. A different local optimum may be reached when a model is solved with different initial values for X. In this example, one might imagine that starting at a value of X=6 would lead to the global optimum at X=5.02. Unfortunately, this is not guaranteed, because LINGO approximates the true underlying nonlinear functions using linear and/or quadratic functions. In the early stages of solving the model, these approximations can be somewhat rough in nature. Therefore, the solver can be sent off to distant points, effectively ing over nearby local optimums to the true, underlying model. When "good" initial values for the variables are known, you should input these values in an INIT section. Additionally, you may want to use the @BND function to bound the variables within a reasonable neighborhood of the starting point. When you have no idea what the optimal solution to your model is, you may find that observing the results of solving the model several times with different initial values can help you find the best solution. LINGO has several optional strategies to help overcome the problem of stalling at local optimal points. The global solver employs branch-and-bound methods to break a model down into many convex subregions. LINGO also has a multistart feature that restarts the nonlinear solver from a number of intelligently generated points. This allows the solver to find a number of locally optimal points and

716 CHAPTER 15 report the best one found. Finally, LINGO can automatically linearize a number of nonlinear relationships through the addition of constraints and integer variables so that the transformed linear model is mathematically equivalent to the original nonlinear model. Keep in mind, however, that each of these strategies will require additional computation time. Thus, whenever possible, you are always better off formulating models to be convex so that they contain a single extreme point.

Convexity The characteristic of an expression called convexity along with a closely related feature called concavity are the most useful features of a model for guaranteeing that a local optimum is actually a global optimum. Let’s consider the convexity property of a minimization problem. The mathematical definition of a convex model is as follows: f(y)  a f(x) + (1-a) f(z), where y=a*x + (1-a)*z In words, a function is convex if, for any two points on the function, a straight line connecting the two points lies entirely on or above the function. Determining the convexity of a multiple variable problem is no easy task. Mathematicians call a function convex if the matrix of the second derivatives is positive definite or has positive Eigen values. However, if you can identify your LINGO model as being convex for a minimization problem, you can ensure that any solution you reach is a global optimum (including nonlinear models).

Strictly Convex A strictly convex function with no constraints has a single global minimum. Therefore, minimizing a strictly convex function will yield the unique global optimal solution regardless of the initial value of the variables. The graph below shows a convex function of a single variable:

A Strictly Convex Function: .4*(x-3)2+.5


717

In this strictly convex function, the unique global minimum can be defined at the point on the function where the variable x is equal to 3. Changing the value of x to be more or less than 3 will increase the result of the function.

Loosely Convex A loosely convex function with no constraints has multiple local minima, which are also global minima. Therefore, minimizing a loosely convex function may yield different solutions for different initial values of the variables. However, if you know the function is loosely convex, you will know the solution is a global minimum. For example, note the flat section (i.e., slope is zero) from approximately –200 to 150 in the following function that makes it loosely convex:

A Loosely Convex Function In this loosely convex function, the global minimum of 3 can be found between about –200 through just over 150 on the x-axis. Although variable values may vary, the result of the function will be the same for all local minima.

Concavity While convexity applies to minimization problems, concavity ensures the corresponding attribute of global optimality in maximization problems. Concavity can be defined as the negative of convexity (see above). In other words, a function is concave if, for any two points on the function, a straight line connecting the two points lies entirely on or below the function.

718 CHAPTER 15

300

250

200

150

100

50

-50

-100

-150

-200

-250

0

0

-300

The following function is strictly concave:

-10000 -20000 -30000 -40000 -50000 -60000 -70000 -80000 -90000 A Strictly Concave Function: Graph of -(x2) In this strictly concave function, the unique global maximum can be defined at the point on the function where the variable x is equal to zero. Changing the value of the variable to be any more or less than 0 will decrease the result of the function. A loosely concave function might look similar to the negative of the loosely convex function shown in the previous section. The only functions that are both convex and concave are straight lines (i.e., hyperplanes). Therefore LINGO classifies the solutions to all linear optimization problems as globally optimal. Due to the difficulty in determining convexity and concavity, LINGO classifies all nonlinear optimization models as locally optimal. However, you can ensure you have a global optimum if you can determine that your nonlinear optimization model is convex or concave, or you can reformulate it in such a way to make it convex or concave. On the other hand, if you can determine your nonlinear optimization model is definitely not convex or concave, you know that it is a mixed function and multiple optima exist. It might be a good idea to try solving with different starting values for the variables. See the section above Local Optima vs. Global Optima for an example of a mixed function and more information on the strategies LINGO offers to help overcome the problems with solving mixed functions.


719

Smooth vs. Nonsmooth Functions Smooth functions have a unique defined first derivative (slope or gradient) at every point. Graphically, a smooth function of a single variable can be plotted as a single continuous line with no abrupt bends or breaks. All the examples you’ve seen so far in this chapter have been smooth. Nonsmooth functions include nondifferentiable and discontinuous functions. Functions with first derivatives with undefined regions are called nondifferentiable. Graphs of nondifferentiable functions may have abrupt bends. The absolute value of a variable, @ABS(X), is an example of a nondifferentiable expression, as illustrated in the following graph: 100 90 80 70 60 50 40 30 20 0

-100

-50

0

50

100

Graph of ABS(X) Here, there is an abrupt bend in the graph at zero. This can dramatically increase solution times as well as affect the accuracy of the solution. Additional nonsmooth functions are @MAX, @MIN, @SMAX, @SMIN, and any of the probability functions that use linear interpolation to return results for nonintegral arguments. Perhaps even more confounding than functions with sharp bends are discontinuous functions that have actual breaks in their graphs. Discontinuous functions in LINGO include @SIGN and @FLOOR. In simplified , LINGO searches along the path of a function to find a maximum or minimum point representing an optimal solution. As LINGO is searching along a function, it uses the function’s derivatives to help guide the search. When derivatives vanish, as they do at sharp corners, LINGO is “blinded” and is unable to “see” around the sharp corner. Thus, dealing with functions containing breaks and sharp bends is considerably more difficult than dealing with smooth, continuous functions. Where possible, nonsmooth functions should be replaced with linear constraints or some combination of linear constraints and integer variables. As an example, refer to the assembly line balancing example (ASLBAL.LNG) on page 723 to see how we constructed a model to avoid the use of the @MAX function in order to maintain linearity.

720 CHAPTER 15

Guidelines for Nonlinear Modeling As shown in the previous sections, nonlinear models can be extremely complex to solve. Spending a little extra time to make sure the model is formulated in a way that is most efficient to solve can pay off in of solution speed and reliability. This section gives some general guidelines to consider when building and solving nonlinear models.

Supplying Bounds for Variables Intelligent use of upper and lower bounds on variables can help make LINGO’s solution search as efficient as possible. Supplying good bounds can keep LINGO from wasting time searching regions unlikely to yield good solutions. For example, suppose you know that, even though the feasible range for a particular variable is between 0 and 100, it is highly improbable the optimal value is outside the range of 50 to 75. In this case, using the @BND function to specify a lower bound of 50 and an upper bound of 75 could significantly reduce solution times. Bounding can also help keep the solution search clear of mathematically troublesome areas like undefined regions. For example, if you have a constraint with the term 1/X, it may be helpful to add a lower bound on X, so it does not get too close to zero.

Supplying Initial Values for Variables The initial values you provide for the variables in a model can affect the “path” LINGO takes to the solution. Starting with values close to the optimal solution can noticeably reduce the solution time. In many situations, you may not know “good” initial values. However, when you do know reasonable ones, it may be to your benefit to use them in the INIT section. Consider changing your supplied initial values and re-solving the model if you suspect: 1) there is an answer better than the answer returned by LINGO, or 2) a feasible solution exists even though LINGO returns the message “No feasible solution found”.

Scale the Model to a Reasonable Range of Units Try to model your problem such that the units involved are of similar orders of magnitude. If the largest number in the model is greater than 1000 times the smallest number in the model, LINGO may encounter problems when solving the model. This may also affect the accuracy of the solution by introducing rounding problems. For example, consider a financial problem with equations expressing an interest rate of 8.5% (.085) and budget constraints of $12,850,000. The difference in magnitude between these numbers is on the order of 10^9 (1/100th compared to 10,000,000). A difference of 10^4 or less between the largest and smallest units would be preferable. In this case, the budget could be expressed in units of millions of dollars. That is, $12.85 would be used to represent $12,850,000. This lowers the difference in magnitude of the units of these numbers to 10^4.


721

Simplify Relationships When practical, use linear rather than nonlinear relationships. Some nonlinear expressions can be reformulated in a linear manner. A simple example is a constraint on the ratio of two variables. Using the example from earlier in the chapter, consider the following constraint: X / Y < 10; This constraint is nonlinear because we are dividing by Y. To linearize the constraint, you can multiply both sides by Y. The equivalent, linear constraint becomes: X < 10 * Y; Avoid nonsmooth relationships when possible. Models with nonsmooth constraints are generally much more difficult to solve. When possible, approximate the nonsmooth relationship with smooth expressions and, perhaps, integer variables.

Reduce Integer Restrictions Minimizing the use of integer restrictions can drastically reduce the solution time. In instances involving larger variable values, you may find solving the model without integer restrictions and then rounding yields acceptable answers in a fraction of the time required by the integer model. Be forewarned, however, that rounding a solution will not necessarily yield a feasible or optimal solution.

Appendix A: Additional Examples of LINGO Modeling In this Appendix, we list, in alphabetical order by model name, many of the models contained in the SAMPLES directory off your main LINGO directory. Most of the models contain a brief background discussion. Many have appeared in more detailed descriptions earlier in this manual and are reproduced here for reference.

Assembly Line Balancing

Model: ASLBAL

The following model illustrates how to balance an assembly line in order to minimize its total cycle time. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: ! Assembly line balancing model; ! This model involves asg tasks to stations in an assembly line so bottlenecks are avoided. Ideally, each station would be assigned an equal amount of work.; SETS: ! The set of tasks to be assigned are A through K, and each task has a time to complete, T; TASK/ A B C D E F G H I J K/: T; ! Some predecessor,successor pairings must be observed(e.g. A must be done before B, B before C, etc.); PRED(TASK, TASK)/ A,B B,C C,F C,G F,J G,J J,K D,E E,H E,I H,J I,J /; ! There are 4 workstations; STATION/1..4/; TXS(TASK, STATION): X; ! X is the attribute from the derived set TXS that represents the assignment. X(I,K) = 1 if task I is assigned to station K; ENDSETS

723

724 APPENDIX A DATA: ! Data taken from Chase and Aquilano, POM; ! Each task has an estimated time required: A B C D E F G H I J K; T = 45 11 9 50 15 12 12 12 12 8 9; ENDDATA ! The model; ! *Warning* may be slow for more than 15 tasks; ! For each task, there must be one assigned station; @FOR(TASK(I): @SUM(STATION(K): X(I, K)) = 1); ! Precedence constraints; ! For each precedence pair, the predecessor task I cannot be assigned to a later station than its successor task J; @FOR(PRED(I, J): @SUM(STATION(K): K * X(J, K) - K * X(I, K)) >= 0); ! For each station, the total time for the assigned tasks must be less than the maximum cycle time, CYCTIME; @FOR(STATION(K): @SUM(TXS(I, K): T(I) * X(I, K)) <= CYCTIME); ! Minimize the maximum cycle time; MIN = CYCTIME; ! The X(I,J) assignment variables are binary integers; @FOR(TXS: @BIN(X)); END

Model: ASLBAL

ADDITIONAL EXAMPLES

Bayes Rule; Conditional Probabilities

Model: Bayes

MODEL: SETS: ! Computing probabilities using Bayes rule; ACTUAL/1..3/:MPA;!Marginal probability of actual; FCAST/1..3/:MPF;!Marginal probability of forecast; FXA(FCAST, ACTUAL): CAGF, !Conditional prob of actual given forecast; CFGA, !Conditional prob of forecast given actual; JP; ! t probability of both; ENDSETS DATA: !Conditional probability of forecast, given actual; CFGA = .80 .15 .20 .10 .70 .20 .10 .15 .60; ! Marginal probabilities of actual; MPA = .5 .3 .2; ENDDATA ! The calculations; ! Marginal probabilities are the sum of t probabilities; @FOR(ACTUAL(J): MPA(J) = @SUM(FCAST(I): JP(I, J)) ); @FOR(FCAST(I): MPF(I) = @SUM(ACTUAL(J): JP(I, J)) ); ! Bayes rule relating t to conditional probabilities; @FOR(FXA(I, J): JP(I, J) = MPF(I) * CAGF(I, J); JP(I, J) = MPA(J) * CFGA(I, J) ); END

Model: BAYES

725

726 APPENDIX A

Blending of Ingredients I

Model: BLEND

In blending problems, two or more raw materials are to be blended into one or more finished goods, satisfying one or more quality requirements on the finished goods. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: TITLE BLEND; SETS: !Each raw material has an availability and cost/unit; RAWMAT/ BUTANE, CATREF, NAPHTHA/: AVAIL, COST; !Each finished good has a min required, max sellable, selling price, and batch size to be determined; FINGOOD/ REGULAR, /: MINREQ, MAXSELL, PRICE, BATCH; !Here is the set of quality measures; QUALMES/ OCTANE, VAPOR, VOLATILITY/; !For each combo of raw material and quality measure there is a quality level; RXQ(RAWMAT, QUALMES): QLEVEL; !For each combination of quality measure and finished good there are upper and lower limits on quality, and a slack on upper quality to be determined; QXF(QUALMES, FINGOOD): QUP, QLOW, QSLACK; !For each combination of raw material and finished good there is an amount of raw material used to be solved for; RXF(RAWMAT, FINGOOD): USED; ENDSETS DATA: !Raw material availability; AVAIL = 1000, 4000, 5000; !Raw material costs; COST = 7.3, 18.2, 12.5; !Quality parameters of raw materials; QLEVEL = 120, 60, 105, 100, 2.6, 3, 74, 4.1, 12;

ADDITIONAL EXAMPLES !Limits on finished goods; MINREQ = 4000, 2000; MAXSELL = 8000, 6000; !Finished goods prices; PRICE = 18.4, 22; !Upper and lower limits on quality for each finished good; QUP = 110, 110, 11, 11, 25, 25; QLOW = 90, 95, 8, 8, 17, 17; ENDDATA !Subject to raw material availability; @FOR(RAWMAT(R): [RMLIM] @SUM(FINGOOD(F): USED(R, F)) <= AVAIL(R); ); @FOR(FINGOOD(F): !Batch size computation; [BATCOMP] BATCH(F) = @SUM(RAWMAT(R): USED(R, F)); !Batch size limits; @BND(MINREQ, BATCH, MAXSELL); !Quality restrictions for each quality measure; @FOR(QUALMES(Q): [QRESUP] @SUM(RAWMAT(R): QLEVEL(R, Q) * USED(R, F)) + QSLACK(Q, F) = QUP(Q, F) * BATCH(F); [QRESDN] QSLACK(Q, F) <= (QUP(Q, F) - QLOW(Q, F)) * BATCH(F); ); ); ! We want to maximize the profit contribution; [OBJECTIVE] MAX = @SUM(FINGOOD: PRICE * BATCH) @SUM(RAWMAT(R): COST(R) *

727

728 APPENDIX A @SUM(FINGOOD(F): USED(R, F))); END

Model: BLEND

ADDITIONAL EXAMPLES

Plant Location

Model: CAPLOC

In this example, we build a model to help decide what plants to open and how much product to ship from each plant to each customer. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: ! Capacitated Plant Location Problem; SETS: PLANTS: FCOST, CAP, OPEN; CUSTOMERS: DEM; ARCS( PLANTS, CUSTOMERS) : COST, VOL; ENDSETS DATA: ! The plant, their fixed costs and capacity; PLANTS, FCOST, CAP = P1 91 39 P2 70 35 P3 24 31; ! Customers and their demands; CUSTOMERS, DEM = C1 15 C2 17 C3 22 C4 12; ! The plant to cust cost/unit shipment matrix; COST = 6 2 6 7 4 9 5 3 8 8 1 5; ENDDATA ! The objective; [TTL_COST] MIN = @SUM( ARCS: COST * VOL) + @SUM( PLANTS: FCOST * OPEN); ! The demand constraints; @FOR( CUSTOMERS( J): [DEMAND] @SUM( PLANTS( I): VOL( I, J)) >= DEM( J) ); ! The supply constraints; @FOR( PLANTS( I): [SUPPLY] @SUM( CUSTOMERS( J): VOL( I, J)) <= CAP( I) * OPEN( I) );

729

730 APPENDIX A ! Make OPEN binary(0/1); @FOR( PLANTS: @BIN( OPEN)); END

Model: CAPLOC

ADDITIONAL EXAMPLES

Blending of Ingredients II

731

Model: CHESS

In blending problems, two or more raw materials are to be blended into one or more finished goods, satisfying one or more quality requirements on the finished goods. In this example, we blend mixed nuts into four different brands with a goal of maximizing revenue. A detailed discussion of this model may be found in Chapter 2, Using Sets. MODEL: SETS: NUTS / PEANUTS, CASHEWS/: SUPPLY; BRANDS / PAWN, KNIGHT, BISHOP, KING/: PRICE, PRODUCE; NCROSSB(NUTS, BRANDS): FORMULA; ENDSETS DATA: SUPPLY = 750 250; PRICE = 2 3 4 5; FORMULA = 15 10 6 2 1 6 10 14; ENDDATA MAX = @SUM(BRANDS(I): PRICE(I) * PRODUCE(I)); @FOR(NUTS(I): @SUM(BRANDS(J): FORMULA(I, J) * PRODUCE(J) / 16) <= SUPPLY(I) ); END

Model: CHESS

732 APPENDIX A

Chemical Equilibrium

Model: CHMBL1

In a chemical equilibrium problem of a closed system (such as a sealed container), there are two or more opposing processes, such as evaporation and condensation. The question is: at a given temperature and pressure, what portion of the material will be found in each possible state—water and vapor, for instance—at equilibrium. The typical equilibrium conditions are that the ratios of fractions must equal known temperature and/or pressure-dependent constants. Note that, in many cases, the numbers you will be dealing with in chemical equilibrium models will be infinitesimal. Such tiny numbers will make it next to impossible to solve your model. Transforming values by taking their logarithms is a useful strategy that is used here. This tends to result in a transformed model with more manageable data. MODEL: !Chemical equilibrium problem of Peters, Hayes & !Hieftje. Calculate concentrations of various !components of phosphoric acid(H3PO4) with pH of 8 !and total phosphate concentration of .10. The !equilibrium equations in obvious form look like: ! ! H2P * H/ H3P = .0075; ! !However, for scale reasons it is better to take !logs thus; LH2P + LH - LH3P = @LOG(.0075); ! Ditto for other equilibrium equations; LHP + LH - LH2P = @LOG(6.2 * 10^-8); LH + LP - LHP = @LOG(4.8 * 10^-13); LH = @LOG(10 ^-8); ! Convert back to original variables; H = @EXP(LH); P = @EXP(LP); HP = @EXP(LHP); H2P = @EXP(LH2P); H3P = @EXP(LH3P); H3P + H2P + HP + P = .1; ! Must unconstrain log variables; @FREE(LH2P); @FREE(LH); @FREE(LH3P); @FREE(LHP); @FREE(LP); ! Solution should be: LH2P= -4.2767, LH= -18.4207, LH3P= -17.8045, LHP= -2.4522, LP= -12.3965; END

Model: CHMBL1

ADDITIONAL EXAMPLES

Cont Analysis

733

Model: CONJNT

When deg a product, it’s useful to know how much customers value various attributes of that product. This allows us to design the product most preferred by consumers within a limited budget. For instance, if we determine consumers place a very high value on a long product warranty, then we might be more successful in offering a long warranty with fewer color options. The basic idea behind cont analysis is, while it may be difficult to get consumers to accurately reveal their relative utilities for product attributes, it’s easy to get them to state whether they prefer one product configuration to another. Given these rank preferences, we can use cont analysis to work backwards and determine the implied utility functions for the product attributes. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: ! Cont analysis model to decide how much weight to give to the two product attributes of warranty length and price; SETS: ! The three possible warranty lengths; WARRANTY /LONG, MEDIUM, SHORT/ : WWT; ! where WWT(i) = utility assigned to warranty i; !The three possible price levels(high,medium,low); PRICE /HIGH, MEDIUM, LOW/ : PWT; ! where PWT(j) = utility assigned to price j; ! We have a customer preference ranking for each combination; WP(WARRANTY, PRICE) : RANK; ENDSETS DATA: ! Here is the customer preference rankings running from a least preferred score of 1 to the most preferred of 9. Note that long warranty and low price are most preferred with a score of 9, while short warranty and high price are least preferred with a score of 1; RANK = 7 8 9 3 4 6 1 2 5; ENDDATA SETS: ! The next set generates all unique pairs of product configurations such that the second configuration is preferred to the first; WPWP(WP, WP) | RANK(&1, &2) #LT# RANK(&3, &4): ERROR;

734 APPENDIX A ! The attribute ERROR computes the error of our estimated preference from the preferences given us by the customer; ENDSETS ! For every pair of rankings, compute the amount by which our computed ranking violates the true ranking. Our computed ranking for the (i,j) combination is given by the sum WWT(i) + PWT(j). (NOTE: This makes the bold assumption that utilities are additive!); @FOR(WPWP(i, j, k, l): ERROR(i, j, k, l) >= 1 + (WWT(i) + PWT(j)) (WWT(k) + PWT(l)) ); ! The 1 is required on the right-hand-side of the above equation to force ERROR to be nonzero in the case where our weighting scheme incorrectly predicts that the combination (i,j) is equally preferred to the (k,l) combination. Since variables in LINGO have a default lower bound of 0, ERROR will be driven to zero when we correctly predict that (k,l) is preferred to (i,j). Next, we minimize the sum of all errors in order to make our computed utilities as accurate as possible; MIN = @SUM(WPWP: ERROR); END

Model: CONJNT

ADDITIONAL EXAMPLES

Data Envelopment Analysis

735

Model: DEAMOD

Data Envelopment Analysis (DEA) was developed to help compare the relative performance of decision-making units. DEA generates an efficiency score between 0 and 1 for each unit, indicating how effectively they are managing their resources. A compelling feature of DEA is it develops a unique rating system for each unit designed to make them look their best. This should facilitate acceptance of DEA within an organization. For more information on DEA, see Schrage (2006). MODEL: ! Data Envelope Analysis of Decision Maker Efficiency; SETS: DMU/BL HW NT OP YK EL/: ! Six schools; SCORE; ! Each decision making unit has a; ! score to be computed; FACTOR/COST RICH WRIT SCIN/; ! There is a set of factors, input & output; DXF(DMU, FACTOR): F; ! F(I, J) = Jth factor of DMU I; ENDSETS DATA: ! Inputs are spending/pupil, % not low income; ! Outputs are Writing score and Science score; NINPUTS = 2; ! The first NINPUTS factors are inputs; ! The inputs, the outputs; F = 8939 64.3 25.2 223 8625 99 28.2 287 10813 99.6 29.4 317 10638 96 26.4 291 6240 96.2 27.2 295 4719 79.9 25.5 222; ENDDATA SETS: ! Weights used to compute DMU I's score; DXFXD(DMU,FACTOR) : W; ENDSETS ! Try to make everyone's score as high as possible; MAX = @SUM(DMU: SCORE); ! The LP for each DMU to get its score; @FOR(DMU(I): SCORE(I) = @SUM(FACTOR(J)|J #GT# NINPUTS: F(I, J)* W(I, J)); ! Sum of inputs(denominator) = 1; @SUM(FACTOR(J)| J #LE# NINPUTS: F(I, J)* W(I, J)) = 1; ! Using DMU I's weights, no DMU can score better than 1; @FOR(DMU(K): @SUM(FACTOR(J)| J #GT# NINPUTS: F(K, J) * W(I, J)) <= @SUM(FACTOR(J)| J #LE# NINPUTS: F(K, J) * W(I, J))

736 APPENDIX A ) ); ! The weights must be greater than zero; @FOR(DXFXD(I, J): @BND(.00001, X, 100000)); END

Model: DEAMOD

ADDITIONAL EXAMPLES

Generating Random Numbers

737

Model: DEMRND

In cases where you are modeling under uncertainty, it is useful to have a random number generator to help simulate a situation. LINGO s the @RAND function, which can be used to generate sequences of pseudo random numbers with uniform distribution between 0 and 1. By using the same seed value, you can recreate a series of numbers. In this example, we generate 15 random numbers, and then use them with the @PSN and @PTD functions to generate observations from both the unit Normal and T distributions. MODEL: ! Generate a series of Normal and T distributed random variables ; SETS: SERIES/1..15/: U, ZNORM, ZT; ENDSETS ! First uniform is arbitrary; U(1) = @RAND(.1234); ! Generate the rest recursively; @FOR(SERIES(I)| I #GT# 1: U(I) = @RAND(U(I - 1)) ); ! Generate some...; @FOR(SERIES(I): ! Normal deviates...; @PSN(ZNORM(I)) = U(I); ! and t deviates(2 degrees of freedom); @PTD(2, ZT(I)) = U(I); ! ZNORM and ZT may take on negative values; @FREE(ZNORM(I)); @FREE(ZT(I)); ); END

Model: DEMRND

738 APPENDIX A

Scenario-based Portfolio Model

Model: DNRISK

In this model, we are attempting to come up with an optimal portfolio that meets a certain level of return while minimizing downside risk. Downside risk is a measure of the risk of falling below our target return. An additional feature of this model is it is scenario-based. More specifically, we have seven scenarios that will each occur with a given probability. The model incorporates this distribution of predicted outcomes in deriving the optimal portfolio. MODEL: ! (dnrisk.lng); ! Downside risk portfolio model; SETS: ASSET: INVEST; ! Amount to invest in each asset; SCENARIO: TRETRN, ! Return under this scenario; DRISK; ! Downside risk under this scenario; TABLE( SCENARIO, ASSET): ARETRN; ! Return in scenario I of asset J; ENDSETS DATA: ! Number of scenarios; SCENARIO = 1..7; ! The available assets in which to invest; ASSET = ATT GMC USX; ARETRN = -.071 .144 .169 .056 .107 -.035 .038 .321 .133 .089 .305 .732 .090 .195 .021 .083 .390 .131 .035 -.072 .006; ! Desired return; DRETURN = .13; ! Threshold, below which we are unhappy; THRESH = .11; ! Power to use for risk(1 or 2); ! When NPOW = 1, it is a linear program; ! When NPOW = 2 and threshold = desired return; ! it is the semi-variance; NPOW = 2; ENDDATA NSCEN = @SIZE( SCENARIO); ! Minimize average downside risk; MIN = @SUM( SCENARIO: DRISK ^ NPOW)/ NSCEN; ! Compute return for each scenario;

ADDITIONAL EXAMPLES @FOR( SCENARIO( I): TRETRN( I) = @SUM( ASSET( J): ARETRN( I, J) * INVEST( J)); ! .. and how much we fall short of threshold ; DRISK( I) >= THRESH - TRETRN( I); ! Return in a period could be negative; @FREE( TRETRN( I)); ); ! Our budget constraint(divided by a billion); [BUDGET] @SUM( ASSET: INVEST ) = 1; ! Our desired return; [PRICER] @SUM( SCENARIO( I): TRETRN( I))/ NSCEN >= DRETURN; END

Model: DNRISK

739

740 APPENDIX A

Staff Scheduling

Model: STAFFDEM

This covering model assigns start days to employees to cover staffing needs, minimizing the total number of staff, subject to the requirements that each staff member must have two consecutive days off each week. A detailed discussion of this model may be found in Chapter 2, Using Sets. MODEL: SETS: DAYS / MON TUE WED THU FRI SAT SUN/: REQUIRED, START; ENDSETS DATA: REQUIRED = 20 16 13 16 19 14 12; ENDDATA MIN = @SUM(DAYS(I): START(I)); @FOR(DAYS(J): @SUM(DAYS(I) | I #LE# 5: START(@WRAP(J - I + 1, 7))) >= REQUIRED(J) ); END

Model: STAFFDEM

ADDITIONAL EXAMPLES

Dynamic Programming

741

Model: DYNAMB

Dynamic programming (DP) is a creative approach to problem solving that involves breaking a large, difficult problem into a series of smaller, easy to solve problems. By solving this series of smaller problems, we are able to assemble the optimal solution to the initial large problem. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: SETS: ! Dynamic programming illustration (see Anderson, Sweeney & Williams, An Intro to Mgt Science, 6th Ed.). We have a network of 10 cities. We want to find the length of the shortest route from city 1 to city 10.; ! Here is our primitive set of ten cities, where F(i) represents the shortest path distance from city i to the last city; CITIES /1..10/: F; ! The derived set ROADS lists the roads that exist between the cities (note: not all city pairs are directly linked by a road, and roads are assumed to be one way.); ROADS(CITIES, CITIES)/ 1,2 1,3 1,4 2,5 2,6 2,7 3,5 3,6 3,7 4,5 4,6 5,8 5,9 6,8 6,9 7,8 7,9 8,10 9,10/: D; ! D(i, j) is the distance from city i to j; ENDSETS DATA: ! The distances corresponding to the links; D = 1 5 2 13 12 11 6 10 4 12 14 3 9 6 5 8 10 5 2; ENDDATA ! If you are already in City 10, then the cost to travel to City 10 is 0; F(@SIZE(CITIES)) = 0;

742 APPENDIX A ! The following is the classic dynamic programming recursion. In words, the shortest distance from City i to City 10 is the minimum over all cities j reachable from i of the sum of the distance from i to j plus the minimal distance from j to City 10; @FOR(CITIES(i)| i #LT# @SIZE(CITIES): F(i) = @MIN(ROADS(i, j): D(i, j) + F(j)) ); END

ADDITIONAL EXAMPLES

Capacitated EOQ

743

Model: EOQCAP

In this model, we have three products with known demand, production rates, setup costs, and holding costs. The question is how much of each product should we produce to minimize total setup and holding costs without exceeding the capacity of our production facility. MODEL: ! Production capacity constrained EOQ; ! Demand rates for three products; d1 = 400; d2 = 300; d3 = 300; ! Production rates; p1 = 1300; p2 = 1100; p3 = 900; ! Setup costs for producing individual products; sc1 = 10000; sc2 = 12000; sc3 = 13000; ! Per unit holding costs; hc1 = 1; hc2 = 1.1; hc3 = 1.4; ! !

The model; Single machine capacity constraint; [CAP] d1 / p1 + d2 / p2 + d3 / p3 + 1.5 * (d1 / q1 + d2 / q2 + d3 / q3) <= 1;

!

Minimize setup + holding costs; min = setup + holding;

!

Total setup costs; setup = (sc1 * d1 / q1) + (sc2 * d2 / q2) + (sc3 * d3 / q3);

!

Total holding costs; holding = (hc1 * q1 * (1 - d1 / p1) + hc2 * q2 * (1 - d2 / p2) + hc3 * q3 * (1 - d3 / p3)) / 2;

!; @BND(.01, Q1, 99999); @BND(.01, Q2, 99999); @BND(.01, Q3, 99999); @FREE(SETUP); @FREE(HOLDING); END

Model: EOQCAP

744 APPENDIX A

Machine Repair Problem

Model: EZMREPAR

This model analyzes a queuing system with a fixed population of that periodically require servicing. Although we model a computer timesharing system in this example, this model is typically referred to as the machine repair problem, because one can think of it as applying to a group of machines periodically requiring repair by a number of service personnel. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: ! Model of a computer timesharing system; ! The mean think time for each (more generally, Mean Time Between Failures in a repair system); MTBF = 40; ! The mean time to process each compute request (more generally, Mean Time To Repair in seconds); MTTR = 2; ! The number of s; N = 32; ! The number of servers/repairmen; NREPR = 1; ! The mean number of s waiting or in service (more generally, the mean number of machines down); NDOWN = @PFS(MTTR * N/ MTBF, NREPR, N); ! The overall request for service rate (more generally, overall failure rate), FR, must satisfy; FR = (N - NDOWN)/ MTBF; ! The mean time waiting for or in service (more generally, the mean time down), MTD, must satisfy; NDOWN = FR * MTD; END

Model: EZMREPAR

ADDITIONAL EXAMPLES

Newsboy Problem

745

Model: EZNEWS

A common inventory management problem occurs when the product in question has limited shelf life (e.g., newspapers, produce, and computer hardware). There is a cost of over ordering, because the product will shortly become obsolete and worthless. There is also an opportunity cost of under ordering associated with forgone sales. Under such a situation, the question of how much of a product to order to maximize expected profit is classically referred to as the newsboy problem. In this example, we assume demand has a Poisson distribution. However, this is not mandatory. Refer to any operations research textbook for a derivation of the formulas involved. MODEL: DATA: ! The average demand; MU = 144; ! Opportunity cost of each unit of lost demand; P = 11; ! Cost/unit of excess inventory; H = 5; ENDDATA ! Calculate the order-up-to point, S, using the newsboy problem equation; @PPS(MU, S) = P / (P + H); ! PS is the expected profit of being at S; PS=P * MU - H * (S -MU) - (P + H) * @PPL(MU, S); END

Model: EZNEWS

746 APPENDIX A

Simple Queuing Example

Model: EZQUEUE

Given a queue with a certain arriving load and number of servers, the @PEL function determines the fraction of customers lost due to all servers being busy. In this example, we use @PEL to solve for the number of servers that limit customer loss to 5%. MODEL: ! Arrival rate of customers/ hour; AR = 25; ! Service time per customer in minutes; STM = 6; ! Service time per customer in hours; STH = STM/ 60; ! Fraction customers finding all servers busy; FB = .05; !The PEL function finds number of servers needed, NS; FB = @PEL(AR * STH, NS); END

Model: EZQUEUE

ADDITIONAL EXAMPLES

General Equilibrium of an Economy

Model: GENEQ1

MODEL: ! General Equilibrium Model of an economy; ! Data based on Kehoe, Math Prog, Study 23(1985); ! Find clearing prices for commodities/goods and equilibrium production levels for processes in an economy; SETS: GOOD/1..4/: PRICE, H; SECTOR/1..4/; GXS(GOOD, SECTOR): ALPHA, W; PROCESS/1..2/: LEVEL; GXP(GOOD, PROCESS): MAKE; ENDSETS DATA: ! Demand curve parameter for each good and SECTOR; ALPHA = .5200 .8600 .5000 .0600 .4000 .1 .2 .25 .04 .02 .2975 .0025 .04 .02 .0025 .6875; ! Initial wealth of Good I by Market J; W = 50 0 0 0 0 50 0 0 0 0 400 0 0 0 0 400; ! Amount produced of good I by process J; MAKE = 6 -1 -1 3 -4 -1 -1 -1; ! Weights for price normalization constraint; H = .25 .25 .25 .25; ENDDATA !————————————————————————————; ! Model based on Stone, Tech. Rep. Stanford OR(1988); ! Minimize the artificial variable; MIN = V; ! Supply is >= demand; @FOR(GOOD(G): @SUM(SECTOR(M): W(G, M)) + @SUM(PROCESS(P): MAKE(G, P) * LEVEL(P))

747

748 APPENDIX A - @SUM(SECTOR(S): ALPHA(G, S) * @SUM(GOOD(I): PRICE(I) * W(I, S))/ PRICE(G)) + H(G) * V >= 0; ); ! Each process at best breaks even; @FOR(PROCESS(P): @SUM(GOOD(G): - MAKE(G, P) * PRICE(G)) >= 0; ); ! Prices scale to 1; @SUM(GOOD(G): - H(G) * PRICE(G)) = -1; END

Model: GENEQ1

ADDITIONAL EXAMPLES

Markowitz Portfolio Example

749

Model: GENPRT

In the March, 1952 issue of Journal of Finance, Harry M. Markowitz published an article titled Portfolio Selection. In the article, he demonstrates how to reduce the risk of asset portfolios by selecting assets whose values aren’t highly correlated. The following model implements these ideas in constructing a simple portfolio with three assets. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: ! GENPRT: Generic Markowitz portfolio; SETS: ASSET/1..3/: RATE, UB, X; COVMAT(ASSET, ASSET): V; ENDSETS DATA: ! The data; ! Expected growth rate of each asset; RATE = 1.3 1.2 1.08; !

Upper bound on investment in each; UB = .75 .75 .75;

!

Covariance matrix; V = 3 1 -.5 1 2 -.4 -.5 -.4 1;

!

Desired growth rate of portfolio; GROWTH = 1.12; ENDDATA ! The model; ! Min the variance; [VAR] MIN = @SUM(COVMAT(I, J): V(I, J) * X(I) * X(J)); ! Must be fully invested; [FULL] @SUM(ASSET: X) = 1; ! Upper bounds on each; @FOR(ASSET: @BND(0, X, UB)); ! Desired value or return after 1 period; [RET] @SUM(ASSET: RATE * X) >= GROWTH; END

Model: GENPRT

750 APPENDIX A

Job Shop Scheduling

Model: JOBSLT

In this model, there are six jobs that can be done on one machine. The machine can only work on one job at a time. Each job has a due date. If we can’t complete a job by its due date, we do not take the job. Our objective is to maximize the total value of the jobs selected. Although we have based this model on a job shop scenario, the basic principles should be applicable in any area where time is the limiting factor in deciding what projects to undertake. MODEL: ! One machine job selection; SETS: !There are six jobs each of which has a Due Date, Processing Time, Value, and a flag variable Y indicating if the job has been selected.; JOB/1..6/: ! Each job has a...; DD, ! Due date; PT, ! Processing time; VAL, ! Value if job is selected; Y; ! = 1 if job is selected, else 0; ENDSETS DATA: VAL = 9 DD = 9 PT = 5 ENDDATA

2 3 2

4 6 4

2 5 3

4 7 1

6; 2; 2;

! Maximize the total value of the jobs taken; MAX = TVAL; TVAL = @SUM(JOB: VAL * Y); ! For the jobs we do, we do in due date order; @FOR(JOB(J): ! Only jobs with earlier due dates can precede job J, and jobs must be completed by their due dates; @SUM(JOB(I)| DD(I) #LT# DD(J) #OR# (DD(I) #EQ# DD(J) #AND# I #LE# J): PT(I) * Y(I)) <= DD(J); ! Make the Y's binary; @BIN(Y); ); END

Model: JOBSLT

ADDITIONAL EXAMPLES

Knapsack Model

751

Model: KNAPSACK

In the knapsack model, one wants to select items to place into a knapsack to maximize a measure of utility without exceeding the capacity of the knapsack. This model can be generalized to many other areas such as truck loading, bin packing, choosing science experiments for the Space Shuttle, and so on. An in-depth description of this model can be found in Chapter 3, Using Variable Domain Functions. SETS: ITEMS / ANT_REPEL, BEER, BLANKET, BRATWURST, BROWNIES, FRISBEE, SALAD, WATERMELON/: INCLUDE, WEIGHT, RATING; ENDSETS DATA: WEIGHT RATING = 1 2 3 9 4 3 3 8 3 10 1 6 5 4 10 10; KNAPSACK_CAPACITY = 15; ENDDATA MAX = @SUM(ITEMS: RATING * INCLUDE); @SUM(ITEMS: WEIGHT * INCLUDE) <= KNAPSACK_CAPACITY; @FOR(ITEMS: @BIN(INCLUDE));

Model: KNAPSACK

752 APPENDIX A

Learning Curve

Model: LEARNC

The cost, labor, and/or time it takes to perform a task will often decrease the more times it is performed. A manufacturer may need to estimate the cost to produce 1,000 units of a product after producing only 100. The average unit cost of the first 100 is likely to be considerably higher than the average unit cost of the last 100. Learning curve theory assumes each time the quantity produced doubles, the cost per unit decreases at a constant rate. In our example, we wish to estimate the cost (in hours) to produce paper based on the cumulative number of tons produced so far. The data is fitted to a curve of the form: COST(i) = a * VOLUME(i) b where COST is the dependent variable and VOLUME is the independent variable. By taking logarithms, we can linearize the model: ln[ COST(i)] = ln(a) + b * ln[ VOLUME(i)] We can then use the theory of linear regression to find estimates of ln(a) and b that minimize the sum of the squared prediction errors. Note that since the regression involves only a single independent variable, the formulas for computing the parameters are straightforward. Refer to any theoretical statistics text for a derivation of these formulas. MODEL: ! Learning curve model; ! Assuming that each time the number produced doubles, the cost per unit decreases by a constant rate, predict COST per unit with the equation: COST(i) = A * VOLUME(i) ^ B; SETS: ! The OBS set contains the data for COST and VOLUME; OBS/1..4/: COST, ! The dependent variable; VOLUME; ! The independent variable; ! The OUT set contains the outputs of the model. Note: R will contain the output results.; OUT/ A, B, RATE, RSQRU, RSQRA/: R; ENDSETS ! Data on hours per ton, cumulative tons for a papermill based on Balof, J. Ind. Eng., Jan. 1966; DATA: COST = .1666, .1428, .1250, .1111; VOLUME = 8, 60, 100 190; ENDDATA ! The model; SETS: ! The derived set OBSN contains the set of

ADDITIONAL EXAMPLES logarithms of our dependent and independent variables as well the mean shifted values; OBSN(OBS): LX, LY, XS, YS; ENDSETS NK = @SIZE(OBS); ! Take the logs; @FOR(OBSN(I): LX(I) = @LOG(VOLUME(I)); LY(I) = @LOG(COST(I)); ); ! Compute means; XBAR = @SUM(OBSN: LX)/ NK; YBAR = @SUM(OBSN: LY)/ NK; ! Shift the observations by their means; @FOR(OBSN: XS = LX - XBAR; YS = LY - YBAR); ! Compute various XYBAR = @SUM(OBSN: XXBAR = @SUM(OBSN: YYBAR = @SUM(OBSN:

sums XS * XS * YS *

of squares; YS); XS); YS);

! Finally, the regression equation; SLOPE = XYBAR/ XXBAR; CONS = YBAR - SLOPE * XBAR; RESID = @SUM(OBSN: (YS - SLOPE * XS)^2); ! The unadjusted/adjusted fraction of variance explained; [X1]R(@INDEX(RSQRU)) = 1 - RESID/ YYBAR; [X2]R(@INDEX(RSQRA)) = 1 - (RESID/ YYBAR) * (NK - 1)/(NK - 2); [X3]R(@INDEX(A)) = @EXP(CONS); [X4]R(@INDEX(B)) = - SLOPE; [X5]R(@INDEX(RATE)) = 2 ^ SLOPE; ! Some variables must be unconstrained in sign; @FOR(OBSN: @FREE(LY); @FREE(XS); @FREE(YS)); @FREE(YBAR); @FREE(XBAR); @FREE(SLOPE); @FREE(XYBAR); @FREE(CONS); END

Model: LEARNC

753

754 APPENDIX A

Markov Chain Model

Model: MARKOV

A standard approach used in modeling random variables over time is the Markov chain approach. The basic idea is to think of the system as being in one of a discrete number of states at each point in time. The behavior of the system is described by a transition probability matrix, which gives the probability the system will move to a specified other state from some given state. Some example situations are: System Consumer brand switching Inventory System

States Brand of product most recently purchased by consumer Amount of inventory on hand

Cause of Transition Consumer changes mind, advertising Orders for new material, demands

In the following model, we use Markov chain analysis to determine the long-term, steady state probabilities of the system. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: ! Markov chain model; SETS: ! There are four states in our model and over time the model will arrive at a steady state equilibrium. SPROB(J) = steady state probability; STATE/ A B C D/: SPROB; ! For each state, there's a probability of moving to each other state. TPROB(I, J) = transition probability; SXS(STATE, STATE): TPROB; ENDSETS DATA: ! The transition probabilities. These are probabilities of moving from one state to the next in each time period. Our model has four states, for each time period there's a probability of moving to each of the four states. The sum of probabilities across each of the rows is 1, since the system either moves to a new state or remains in the current one.; TPROB = .75 .1 .05 .1 .4 .2 .1 .3 .1 .2 .4 .3 .2 .2 .3 .3; ENDDATA ! The model; ! Steady state equations; ! Only need N equations, so drop last; @FOR(STATE(J)| J #LT# @SIZE(STATE): SPROB(J) = @SUM(SXS(I, J): SPROB(I) *

ADDITIONAL EXAMPLES TPROB(I, J)) ); ! The steady state probabilities must sum to 1; @SUM(STATE: SPROB) = 1; ! Check the input data, warn the if the sum of probabilities in a row does not equal 1.; @FOR(STATE(I): @WARN('Probabilities in a row must sum to 1.', @ABS(1 - @SUM(SXS(I, K): TPROB(I, K))) #GT# .000001); ); END

Model: MARKOV

755

756 APPENDIX A

Matching Model

Model: MATCHD

Pair-matching problems require a number of objects be grouped into pairs subject to some criteria. The objective may be to minimize cost, to group like objects, etc. As an example, the following matching model pairs workers into offices to minimize total incompatibilities between paired individuals. A detailed discussion of this model may be found in Chapter 2, Using Sets. MODEL: SETS: ANALYSTS / 1..8/; PAIRS(ANALYSTS, ANALYSTS) | &2 #GT# &1: RATING, MATCH; ENDSETS DATA: RATING = 9 3 4 1 7 4

2 3 4 1

1 5 2 5 8

5 2 9 5 7 2

6 1 2 2 6 3 4;

ENDDATA MIN = @SUM(PAIRS(I, J): RATING(I, J) * MATCH(I, J)); @FOR(ANALYSTS(I): @SUM(PAIRS(J, K) | J #EQ# I #OR# K #EQ# I: MATCH(J, K)) = 1 ); @FOR(PAIRS(I, J): @BIN(MATCH(I, J))); END

Model: MATCHD

ADDITIONAL EXAMPLES

Computing Demand Backlog

757

Model: METRIC

The METRIC model estimates the average number of backorders in a two level inventory system composed of a single depot or distribution center and any number of outlets served by the depot. Inventory policy at each location is described by a single number S. Whenever the sum of the amount on hand and on order drops below S, an order is placed to bring inventory back up to S. This is an evaluation model, rather than an optimization model. The is prompted for the S values (SDEPOT and SOUTLET). MODEL: ! The two level METRIC inventory model. ; SETS: OUTLET/1..2/: ! Each outlet has a...; ROUTLET, ! Resupply time from depot to outlet; DEM, ! Demand rate at outlet; SOUTLET, ! Stock level to use at outlet; ERT, ! Effective resupply time to outlet; AL; ! Average level of backlogged demand; ENDSETS DATA: ROUTLET = 2 DEM = .1 RDEPOT = 14;

2; .1;

!

Get the stock levels from the ; SDEPOT = ?; SOUTLET = ?, ?; ENDDATA

! Compute total demand; DEM0 = @SUM(OUTLET: DEM); ! Effective expected wait at depot; EWT0 = @PPL(DEM0 * RDEPOT, SDEPOT)/ DEM0; @FOR(OUTLET(I): ! Estimate the resupply time including depot delay; ERT(I) = ROUTLET(I) + EWT0; ! Expected demand on backorder; AL(I) = @PPL(DEM(I) * ERT(I), SOUTLET(I)); ); ! Total expected demand on backorder; TBACK = @SUM(OUTLET: AL); END

Model: METRIC

758 APPENDIX A

The Mexican Steel Problem

Model: MEXICO

The following model is a production-planning model adapted from the GAMS documentation. MODEL: ! The Mexican Steel problem; SETS: ! Steel plants; PLANT /AHMSA FUNDIDA SICARTSA HYLSA HYLSAP/: RD2, MUE; ! Markets; MARKET / MEXICO MONTE GUADA /: DD, RD3, MUV; ! Final products; CF /STEEL/: PV, PE, EB; ! Intermediate products; CI / SPONGE PIGIRON/; ! Raw materials; CR /PELLETS COKE NATGAS ELECTRIC SCRAP/ : PD; ! Processes; PR /PIGIRON1 SPONGE1 STEELOH STEELEL STEELBOF/; ! Productive units; UNIT / BLASTFUR OPENHEAR BOF DIRECT ELECARC/; ! ; CRXP(CR, PR) : A1; CIXP(CI, PR) : A2; CFXP(CF, PR) : A3; UXP(UNIT, PR) : B; UXI(UNIT, PLANT) : K; PXM(PLANT, MARKET) : RD1, MUF; PXI(PR, PLANT) : Z; CFXPXM(CF, PLANT, MARKET) : X; CRXI(CR, PLANT) : U; CFXI(CF, PLANT) : E; CFXM(CF, MARKET) : D, V; ENDSETS ! Demand equations; @FOR(CFXM(C, J): D(C,J) = 5.209 * (1 + 40/ 100) * DD(J)/100); ! Transport rate equations; @FOR(PXM(I, J)| RD1(I, J) #GT# 0: MUF(I, J) = 2.48 + .0084 * RD1(I, J)); @FOR(PXM(I, J)| RD1(I, J) #LE# 0: MUF(I, J) = 0); @FOR(PLANT(I)| RD2(I) #GT# 0: MUE(I) = 2.48 + .0084 * RD2(I)); @FOR(PLANT(I)| RD2(I) #LE# 0: MUE(I) = 0);

ADDITIONAL EXAMPLES @FOR(MARKET(J)| RD3(J) #GT# 0: MUV(J) = 2.48 + .0084 * RD3(J)); @FOR(MARKET(J)| RD3(J) #LE# 0: MUV(J) = 0); ! For each plant I1; @FOR(PLANT(I1): ! Sources >= uses, final products; @FOR(CF(CF1): [MBF] @SUM(PR(P1): A3(CF1, P1) * Z(P1, I1)) >= @SUM(MARKET(J1): X(CF1, I1, J1)) + E(CF1, I1)); ! Intermediate products; @FOR(CI(CI1): [MBI] @SUM(PR(P1): A2(CI1, P1) * Z(P1, I1)) >= 0); ! Raw materials; @FOR(CR(CR1): [MBR] @SUM(PR(P1): A1(CR1, P1) * Z(P1, I1)) + U(CR1, I1) >= 0); ! Capacity of each productive unit M1; @FOR(UNIT(M1): [CC] @SUM(PR(P1): B(M1, P1) * Z(P1, I1)) <= K(M1, I1)); ); ! For each final product CF1; @FOR(CF(CF1): ! Demand requirements for each market J1; @FOR(MARKET(J1): [MR] @SUM(PLANT(I1): X(CF1, I1, J1)) + V(CF1, J1) >= D(CF1, J1)); ! Upper limit on exports ; [ME] @SUM(PLANT(I1): E(CF1, I1)) <= EB(CF1); ); ! Components of objective; PHIPSI = @SUM(CR(CR1): @SUM(PLANT(I1): PD(CR1) * U(CR1, I1))); PHILAM = @SUM(CF(CF1): @SUM(PLANT(I1): @SUM(MARKET(J1):MUF(I1,J1)* X(CF1, I1, J1))) + @SUM(MARKET(J1): MUV(J1) * V(CF1, J1)) + @SUM(PLANT(I1): MUE(I1) * E(CF1, I1))); PHIPI = @SUM(CFXM(CF1, I1): PV(CF1) * V(CF1, I1)); PHIEPS = @SUM(CFXP(CF1, I1): PE(CF1) * E(CF1, I1)); [OBJROW] MIN = PHIPSI + PHILAM + PHIPI - PHIEPS;

759

760 APPENDIX A

ADDITIONAL EXAMPLES DATA: A1= -1.58, -1.38, 0, 0, 0, -0.63, 0, 0, 0, 0, 0, -0.57, 0, 0, 0, 0, 0, 0, -0.58, 0, 0, 0, -0.33, 0, -0.12; A2= 1, 0, -0.77, 0, -0.95, 0, 1, 0, -1.09, 0; A3= 0, 0, 1, 1, 1; B = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0; K = 3.25, 1.40, 1.10, 0, 0, 1.50, 0.85, 0, 0, 0, 2.07, 1.50, 1.30, 0, 0, 0, 0, 0, 0.98, 1, 0, 0, 0, 1.13, 0.56; RD1= 1204 218 1125 1017 0 1030 819 1305 704 1017 0 1030 185 1085 760; RD2 = 739, 521, 0, 521, 315; RD3 = 428, 521, 300; PD = 18.7, 52.17, 14, 24, 105; PV = 150; PE = 140; EB = 1; DD = 55, 30, 15; ENDDATA END

Model: MEXICO

761

762 APPENDIX A

Multiprod. Capac. Lot Sizing

Model: MPSCHD

The following model is a production-planning model where multiple products compete for scarce capacity. Each product has a setup time, setup cost, production time, production cost, and holding cost. The question is how much of each product should we produce in each period to minimize total production costs. MODEL: ! Multiproduct Capacitated lot sizing; SETS: PROD/1..2/: ! Each product has a ...; ST, ! Setup time; VT, ! Production time per unit; SC, ! Setup cost; VC, ! Production cost per unit; HC; ! Holding cost per unit per period; TIME/1..6/:; PXT(PROD, TIME): ! Each product in each period has...; DEM, ! Demand; MAKE, ! Amount to produce; Y; ! = 1 if anything is produced; ENDSETS DATA: CAP = 200; ! Capacity per period; ST = 0 0; ! Setup times; VT = 1 1; ! Production time per unit; SC = 150 45; ! Setup costs; VC = 7 4; ! Cost per unit to produce; HC = 2 1; ! Holding cost per unit; DEM = 40 60 130 0 100 200 ! Demands; 0 45 50 35 20 35; ENDDATA !——————————————————; ! The Eppen/Martin model; SETS: PXTXT(PROD, TIME, TIME)| &3 #GE# &2: PCOF, CCOF, X; ENDSETS ! Compute cost and production amounts for various production runs; @FOR(PROD(I): @FOR(TIME(S): PCOF(I, S, S) = DEM(I, S); CCOF(I, S, S) = VC(I) * DEM(I, S);

ADDITIONAL EXAMPLES @FOR(TIME(T)| T #GT# S: PCOF(I, S, T) = PCOF(I, S, T - 1) + DEM(I, T); CCOF(I, S, T) = CCOF(I, S, T - 1) + (VC(I) + HC(I) * (T - S)) * DEM(I, T); ) ) ); ! The objective; MIN = TCOST; TCOST = @SUM(PXTXT: CCOF * X) + @SUM(PXT(I, S): SC(I) * Y(I, S)); @FOR(PROD(I): ! In period 1, some production run must be started; ! Note, watch out for periods without demand; @SUM(PXTXT(I, S, T) | S #EQ# 1: X(I, S, T)) = 1; @FOR(TIME(K)| K #GT# 1: !

If we ended a run in period K - 1...; @SUM(PXTXT(I, S, T)| T #EQ# K - 1: X(I, S, K - 1))

!

then we must start a run in period k; = @SUM(PXTXT(I, K, T): X(I, K, T)); );

! Setup forcing; @FOR(TIME(S): Y(I, S) = @SUM(PXTXT(I, S, T) : (PCOF(I, S, T) #GT# 0) * X(I, S, T)); ! Calc amount made each period; MAKE(I, S) = @SUM(PXTXT(I, S, T): PCOF(I, S, T) * X(I, S, T)) ) ); ! The capacity constraints; @FOR(TIME(S): @SUM(PROD(I): ST(I) * Y(I, S)) + @SUM(PXTXT(I, S, T): VT(I) * PCOF(I, S, T) * X(I, S, T)) <= CAP ); ! Make the Y's integer; @FOR(PXT: @GIN(Y)); END

Model: MPSCHD

763

764 APPENDIX A

Machine Repair Model

Model: MREPAR

This model illustrates the tradeoff between the cost of service people and the costs of machine downtime. In our model, we have ten machines that have a tendency to break down randomly. It costs $350/hour in lost production whenever a machine is broken. The question we need to answer is how many service people should we hire to minimize the total cost of down time and salaries for service people. MODEL: ! Machine repair model; SETS: NREP/1..5/: !Consider 5 possible repair persons; NDOWN, !Expected no. of down machines; ERHR,Expected cost/hour of down machines; TCOST; !Total expected cost/hour; ENDSETS ! The input data; NMACH = 10;! No. machines subject to breakdown; RTIME = 1; ! Average repair time; UPTIME = 5;! Mean time between failures; CR = 30; ! Hourly cost of a repair person; CM = 350; ! Hourly cost of a down machine ! The machine repairman queuing model; ! For each case of 1 - 5 service people calculate expected number of machines down, cost per hour of down machines, and total cost per hour of operations. @PFS calculates the Probability in a Finite Source, in this case expected number of machines under repair. ; @FOR(NREP(I): NDOWN(I) = @PFS(NMACH * RTIME / UPTIME, I, NMACH); ERHR(I) = CM * NDOWN(I); TCOST(I) = ERHR(I) + CR * I ); END

Model: MREPAR

ADDITIONAL EXAMPLES

Material Requirements Planning

765

Model: MRP

Material Requirements Planning, or MRP, is used to generate production schedules for the manufacture of complex products. MRP takes the demand schedule for a finished product and the lead times to produce the finished product and all the various subcomponents that go into the finished product, and then works backwards to come up with a detailed, just-in-time production schedule that meets the demand schedule. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: ! Data for this model is read from MRP.LDT; SETS: ! The set of parts; PART: LT; ! LT(i) = Lead time to produce part i; ! The set of time periods; TIME; ! A relationship called USES between pairs of parts; USES( PART, PART): NEEDS; ! Parent part i needs NEEDS(i, j) units of child part j; ! For each part and time period we're interested in; PXT( PART, TIME): ED, TD; ! ED(i, j) = External demand for part i at time j; ! TD(i, j) = Total demand for part i at time j; ENDSETS DATA: ! Load the data from an external file; ! Parts list; PART = @FILE( 'MRP.LDT'); ! Time periods; TIME = @FILE( 'MRP.LDT'); ! Get the parent child relations and the number of parts required; USES, NEEDS = @FILE( 'MRP.LDT'); ! Get the lead times from the file; LT = @FILE( 'MRP.LDT'); ! Get the external demands over time for each part;

766 APPENDIX A ED = @FILE( 'MRP.LDT'); ENDDATA ! Set NP = no. of time periods in the problem; NP = @SIZE( TIME); ! For each part P and period T, the total demand = external demand + demand generated by parents one lead time in the future; @FOR( PXT( P, T) | T + LT( P) #LE# NP : TD( P, T) = ED( P, T + LT( P)) + @SUM( USES( P2, P): TD( P2, T + LT( P)) * NEEDS( P2, P)); ); DATA: ! Display a table showing the production schedule; @TEXT() = ' The production schedule:'; @TEXT() = @TABLE( TD); ENDDATA END

Model: MRP

ADDITIONAL EXAMPLES

Minimal Spanning Tree

767

Model: MSPAN

In the minimal spanning tree, we need to find a set of links (a tree) in a network that connects all cities. Furthermore, the sum of the distances over all the links in the tree should be minimized. Among other things, this application is useful in constructing communications networks at minimal cost. It turns out that this becomes a very difficult problem to solve using optimization as the number of nodes grows. For large versions of this problem, the optimization techniques provided by LINGO are not the appropriate tool. One would be wise to pursue alternatives such as heuristics or dynamic programming. MODEL: !Given the number of nodes and the distance between them, finding the shortest total distance of links on the network to connect all the nodes is the classic problem called minimal spanning tree (MST). This model finds the (MST) connecting Atlanta, Chicago, Cincinnati, Houston, LA, and Montreal so that messages can be sent from Atlanta (base) to other cities through the network at minimum cost; SETS: CITY / 1.. 6/: U; ! U(I) = level of city I; ! U(1) = 0; LINK(CITY, CITY): DIST, ! The distance matrix; X; ! X(I, J) = 1 if we use link I, J; ENDSETS DATA:

! Distance matrix need not be symmetric; ! However, city 1 is base of the tree; !to: Atl Chi Cin Hou LA Mon ; DIST = 0 702 454 842 2396 1196 !from Atl; 702 0 324 1093 2136 764 !from Chi; 454 324 0 1137 2180 798 !from Cin; 842 1093 1137 0 1616 1857 !from Hou; 2396 2136 2180 1616 0 2900 !from LA; 1196 764 798 1857 2900 0; !from Mon; ENDDATA

! The model size: Warning, may be slow for N >= 8; N = @SIZE(CITY); ! Minimize total distance of the links; MIN = @SUM(LINK: DIST * X); ! For city K, except the base, ... ; @FOR(CITY(K)| K #GT# 1: ! It must be entered; @SUM(CITY(I)| I #NE# K: X(I, K)) = 1;

768 APPENDIX A ! If there are 2 dist tours from 1 city to another, we can remove a link without breaking connections. Note: These are not very powerful for large problems; @FOR(CITY(J)| J #GT# 1 #AND# J #NE# K: U(J) >= U(K) + X (K, J) (N - 2) * (1 - X(K, J)) + (N - 3) * X(J, K); ); ); ! There must be an arc out of city 1; @SUM(CITY(J)| J #GT# 1: X(1, J)) >= 1; ! Make the X's 0/1; @FOR(LINK: @BIN(X); ); ! The level of a city except the base is at least 1 but no more than N-1, and is 1 if it links to the base; @FOR(CITY(K)| K #GT# 1: @BND(1, U(K), 999999); U(K) <= N - 1 - (N - 2) * X(1, K); ); END

Model: MSPAN

ADDITIONAL EXAMPLES

Multilevel Distribution

769

Model: MULLDC

In this model, we minimize shipping costs over a three tiered distribution system consisting of plants, distribution centers, and customers. Plants produce multiple products that are shipped to distribution centers. If a distribution center is used, it incurs a fixed cost. Customers are supplied by a single distribution center. MODEL: ! MULLDC; ! Multilevel DC location model, based on Geoffrion/Graves, Man. Sci., Jan., 1974; ! Original LINGO model by Kamaryn Tanner; SETS: ! Two products; PRODUCT/ A, B/; ! Three plants; PLANT/ P1, P2, P3/; ! Each DC has an associated fixed cost, F, and an "open" indicator, Z.; DISTCTR/ DC1, DC2, DC3, DC4/: F, Z; ! Five customers; CUSTOMER/ C1, C2, C3, C4, C5/; ! D = Demand for a product by a customer.; DEMLINK(PRODUCT, CUSTOMER): D; ! S = Capacity for a product at a plant.; SUPLINK(PRODUCT, PLANT): S; ! Each customer is served by one DC, indicated by Y.; YLINK(DISTCTR, CUSTOMER): Y; ! C= Cost/ton of a product from a plant to a DC, X= tons shipped.; CLINK(PRODUCT, PLANT, DISTCTR): C, X; ! G= Cost/ton of a product from a DC to a customer.; GLINK(PRODUCT, DISTCTR, CUSTOMER): G; ENDSETS DATA: ! Plant Capacities; S = 80, 40, 75, 20, 60, 75; ! Shipping costs, plant to DC; C = 1, 3, 3, 5, ! Product A; 4, 4.5, 1.5, 3.8, 2, 3.3, 2.2, 3.2, 1, 2, 2, 5, ! Product B;

770 APPENDIX A 4, 4.6, 1.3, 3.5, 1.8, 3, 2, 3.5; ! DC fixed costs; F = 100, 150, 160, 139; ! Shipping costs, DC to customer; G = 5, 5, 3, 2, 4, ! Product A; 5.1, 4.9, 3.3, 2.5, 2.7, 3.5, 2, 1.9, 4, 4.3, 1, 1.8, 4.9, 4.8, 2, 5, 4.9, 3.3, 2.5, 4.1, ! Product B; 5, 4.8, 3, 2.2, 2.5, 3.2, 2, 1.7, 3.5, 4, 1.5, 2, 5, 5, 2.3; ! Customer Demands; D = 25, 30, 50, 25, 8, 0, ENDDATA

15, 30,

35, 30;

!—————————————————————————; ! Objective function minimizes costs.; [OBJ] MIN = SHIPDC + SHIPCUST + FXCOST; SHIPDC = @SUM(CLINK: C * X); SHIPCUST = @SUM(GLINK(I, K, L): G(I, K, L) * D(I, L) * Y(K, L)); FXCOST = @SUM(DISTCTR: F * Z); ! Supply Constraints; @FOR(PRODUCT(I): @FOR(PLANT(J): @SUM(DISTCTR(K): X(I, J, K)) <= S(I, J)) ); ! DC balance constraints; @FOR(PRODUCT(I): @FOR(DISTCTR(K): @SUM(PLANT(J): X(I, J, K)) = @SUM(CUSTOMER(L): D(I, L)* Y(K, L))) ); ! Demand; @FOR(CUSTOMER(L): @SUM(DISTCTR(K): Y(K, L)) = 1 );

Model: MULLDC

ADDITIONAL EXAMPLES

Network Equilibrium

771

Model: NETEQ1

In this example, we have a network of pipelines capable of transporting either fluid or gas. There are a total of eight nodes and 11 arcs between them. Two of the nodes are source nodes, while the remaining nodes are net demanders of product. The pressures at the source nodes are given. The model determines the flow of product down each arc and the pressures at each node. MODEL: ! Network equilibrium NETEQ1:based on Hansen et.al., Math. Prog. vol. 52, no.1; SETS: NODE/A, B, C, D, E, F, G, H/: P; ! Pressure at this node; ARC(NODE, NODE)/ B A, C A, C B, D C, E D, F D, G D, F E, H E, G F, H F/ : R, ! Resistance on this arc; FLO; ! Flow on this arc; SRC(NODE)/ G, H/: PFIXED; ! Fixed pressure at source nodes; DEST(NODE) | #NOT# @IN(SRC, &1): DEMAND; ! Given demand at destination nodes; ENDSETS DATA: PFIXED = 240, 240; DEMAND = 1, 2, 4, 6, 8, 7; R = 1, 25, 1, 3, 18, 45, 1, 12, 1, 30, 1; ! For incompressible fluids and electricity: PPAM = 1, for gases: PPAM = 2; PPAM = 1; ! For optimization networks: FPAM = 0 (for arcs withflow >= 0) electrical networks: FPAM = 1 other fluids: 1.8 <= FPAM <= 2; FPAM = 1.852; ENDDATA !Set the pressures for the source/reservoir nodes; @FOR(SRC(I): P(I) = PFIXED(I)); ! Conservation of flow at non-source nodes; @FOR(DEST(J): @SUM(ARC(I, J): FLO(I, J)) = DEMAND(J) + @SUM(ARC(J, K): FLO(J, K))); ! Relate pressures at 2 ends of each arc; @FOR(ARC(I, J): P(I)^ PPAM - P(J)^ PPAM = R(I, J) * FLO(I, J) ^ FPAM;); END

Model: NETEQ1

772 APPENDIX A

Minimize Traffic Congestion

Model: NLTRAZ

In this model, we have a network with seven nodes. Three of the nodes are source nodes and four of the nodes are destination nodes. Each source node can ship to any of the destination nodes. The problem is the more of a product you send down an arc, the longer it takes for it to arrive. This might be the case if the underlying network were a railroad, for instance. We assume that shipping times obey the following relationship: Time = Rate / (1 - Flow / Limit) where, Time time to ship one unit of product down a route, Rate time required to transport one unit down a route with no congestion along the route, Flow amount of product shipped down a route, and Limit maximum limit of product flow down a route. Based on this relationship, we see shipping times go to infinity as the capacity of an arc is approached. The goal of the model is to determine how much of a product to ship from each source to each destination so as to minimize total shipping times. MODEL: ! Traffic congestion transportation problem. Cost/unit increases to infinity as traffic on link approaches its link capacity. Truncated variation of an AMPL example; SETS: ORIG/ CHIC CINC ERIE/: SUPPLY; DEST / HAM AKR COL DAY/ : DEMAND; OXD(ORIG, DEST): RATE, LIMIT, TRAF; ENDSETS DATA: SUPPLY = 1200 800 1400; DEMAND = 1000 1200 700 500 ; RATE = 39 14 11 14 27 9 12 9 24 14 17 13 ; LIMIT = 500 1000 1000 1000 500 800 800 800 800 600 600 600 ; ENDDATA [TOTCOST] MIN = @SUM(OXD: RATE * TRAF/(1 - TRAF/ LIMIT)); @FOR(ORIG(I): @SUM(OXD(I, J): TRAF(I, J)) = SUPPLY(I)); @FOR(DEST(J): @SUM(OXD(I, J): TRAF(I, J)) = DEMAND(J)); @FOR(OXD: @BND(0, TRAF, LIMIT);); END Model: NLTRAZ

ADDITIONAL EXAMPLES

Newsboy with Fixed Order Charge

773

Model: NUSBOY

In the simple newsboy model (EZNEWS), presented earlier in this chapter, we did not have a fixed ordering charge to deal with. We add this minor complication to the model below. Assuming you decide to order, the fixed charge is a sunk cost and you should therefore order up to the same quantity, S, as in the standard newsboy model. However, there may be cases where preexisting inventory is of a level close enough to S that the expected gains of a minimal increase in inventory are not outweighed by the fixed order charge. The problem now is to not only determine S (or “big S”), but also to determine the additional parameter, s (or “little s”), where when inventory exceeds s the optimal decision is to not incur a fixed charge by foregoing an order for additional stock. Inventory strategies such as this are referred to as “little s-big S”, or (s,S), policies. One additional feature to note in this model is we now assume demand to be normally distributed. This is acceptable because the newsboy model does not demand any particular form of demand distribution. As with the EZNEWS model, we could have also used a Poisson distribution if we felt it was more appropriate. MODEL: ! Newsboy inventory model; ! This model calculates the optimal stock levels for a product with normally distributed demand and a fixed ordering cost; ATA: P = 11; H = 5; MU = 144; SIGMA = 25; K = 15; ENDDATA

! ! ! ! !

Penalty/unit for not having enough; Holding cost/unit for excess; Mean demand; Standard deviation in demand; Fixed cost of placing an order;

! Compute reorder point, SLIL, and order up to point, SBIG; ! Calculate the order up to point, SBIG, using standard newsboy formula; @PSN(ZBIG ) = P /(P + H); ZBIG = (SBIG - MU)/ SIGMA; ! and the expected cost of being there, CSBIG; CSBIG = SIGMA * @PSL(ZBIG) * (P+H) + H * (SBIG-MU); ! The expected cost at the reorder point should differ from the expected cost at SBIG by the fixed order cost, K; CSLIL = K + CSBIG; ! Solve for SLIL; CSLIL=SIGMA*@PSL(ZLIL)*(P+H)*H*(ZLIL*SIGMA); ZLIL = (SLIL - MU)/ SIGMA; END

Model: NUSBOY

774 APPENDIX A

Optimal Airline Overbooking I

Model: OBOOKO

Closely related to the newsboy problem (in a mathematical sense) is the airline-overbooking problem. Given that a certain percentage of fliers with reservations will not show up for a flight, airlines that don’t overbook will be sending most planes up with empty seats. Assuming the penalty cost for overbooking is not too high, an airline that hopes to maximize revenue should overbook its flights. The following model determines the optimal number of reservations to allow on a flight, and assumes the number of no-shows on a flight has a binomial distribution. MODEL: ! This overbooking model determines the number of reservations, M, to allow on a flight if the no-show distribution is binomial; ! Some available data ; N = 16; ! total seats available; V = 225; !Revenue from a sold seat; P = 100; !Penalty for a turned down customer; Q = .04; !Probability a customer is a no-show; ! The probability to turn down customers is @PBN(Q, M, M - N), therefore the corresponding expected loss due to imperfect information is: (V + P) * @PBN(Q, M, M - N), and we want the loss to equal the revenue V on the margin. So, the break-even equation is:; (V + P) * @PBN(Q, M, M - N) = V; ! Note, you should round up if M is fractional; END

Model: OBOOKO

ADDITIONAL EXAMPLES

Optimal Airline Overbooking II

775

Model: OBOOKT

For those of you uncomfortable with the previous overbooking example, we use a “brute force” method here to compute the expected profits from overbooking 1 to 6 seats. Solving this model, you will find the results agree with the previous—the overbooking level that maximizes expected revenue is 1 enger. MODEL: ! A strategy for airlines to minimize loss from no-shows is to overbook flights. Too little overbooking results in lost revenue. Too much overbooking results in excessive penalties. This model computes expected profits for various levels of overbooking.; SETS: SEAT/1..16/; EXTRA/1..6/: EPROFIT;

! seats available ; ! expected profits from overbooking 1-6 seats;

ENDSETS ! Available data; V = 225; ! Revenue from a sold seat; P = 100; ! Penalty for a turned down customer; Q = .04; ! Probability customer is a no-show; ! No. of seats available; N = @SIZE(SEAT); ! Expected profit with no overbooking; EPROFIT0 = V * @SUM(SEAT(I): (1 - @PBN(1- Q, N, I - 1))); ! Expected profit if we overbook by 1 is: EPROFIT0 + Prob(he shows) * (V - (V + P) * Prob(we have no room)); EPROFIT(1) = EPROFIT0 + (1 - Q) * (V - (V + P) * @PBN(Q, N, 0)); ! In general; @FOR(EXTRA(I)| I #GT# 1: EPROFIT(I) = EPROFIT(I - 1) + (1 - Q) * (V - (V + P) * @PBN(Q, N + I - 1, I - 1)); ); END

Model: OBOOKT

776 APPENDIX A

Black & Scholes Options Pricing

Model: OPTION

A call option is a financial instrument that gives the holder the right to buy one share of a stock at a given price (the exercise price) on or before some specified expiration date. A frequent question is, “How much should one be willing to pay for such an option?” An exact answer to this question eluded researchers for many years until Fischer Black and Myron Scholes derived an option pricing formula in 1973. A Nobel Prize was subsequently awarded for their work in 1997. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: ! Computing the value of an option using the Black Scholes formula (see "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, May-June, 1973); SETS: ! We have 27 weeks of prices P(t), LOGP(t) is log of prices; WEEK/1..27/: P, LOGP; ENDSETS DATA: ! Weekly prices of National Semiconductor; P = 26.375, 27.125, 28.875, 29.625, 32.250, 35.000, 36.000, 38.625, 38.250, 40.250, 36.250, 41.500, 38.250, 41.125, 42.250, 41.500, 39.250, 37.500, 37.750, 42.000, 44.000, 49.750, 42.750, 42.000, 38.625, 41.000, 40.750; ! The current share price; S = 40.75; ! Time until expiration of the option, expressed in years; T = .3644; ! The exercise price at expiration; K = 40; ! The yearly interest rate; I = .163; ENDDATA SETS: ! We will have one less week of differences; WEEK1(WEEK)| &1 #LT# @SIZE(WEEK): LDIF; ENDSETS ! Take log of each week's price; @FOR(WEEK: LOGP = @LOG(P)); ! and the differences in the logs; @FOR(WEEK1(J): LDIF(J) = LOGP(J + 1) - LOGP(J));

ADDITIONAL EXAMPLES ! Compute the mean of the differences; MEAN = @SUM(WEEK1: LDIF)/ @SIZE(WEEK1); ! and the variance; WVAR = @SUM(WEEK1: (LDIF - MEAN)^2)/ (@SIZE(WEEK1) - 1); ! Get the yearly variance and standard deviation; YVAR = 52 * WVAR; YSD = YVAR^.5; ! Here is the Black-Scholes option pricing formula; Z = ((I + YVAR/2) * T + @LOG(S/ K))/(YSD * T^.5); ! where VALUE is the expected value of the option; VALUE = S *@PSN(Z) - K *@EXP(- I * T) * @PSN(Z - YSD *T^.5); ! LDIF may take on negative values; @FOR(WEEK1: @FREE(LDIF)); ! The price quoted in the Wall Street Journal for this option when there were 133 days left was $6.625; END

Model: OPTION

777

778 APPENDIX A

Binomial Options Pricing

Model: OPTIONB

Compared to the Black & Scholes example above, we take a slightly different approach to options pricing in this example. We now assume a stock’s return has a binomial distribution, and use dynamic programming to compute the option’s value. MODEL: SETS: ! Binomial option pricing model: We assume that a stock can either go up in value from one period to the next with probability PUP, or down with probability (1 - PUP). Under this assumption, a stock's return will be binomially distributed. In addition, the symmetric probabilities allow us to build a dynamic programming recursion to determine the option's value; ! No. of periods, e.g., weeks; PERIOD /1..20/:; ENDSETS DATA: ! Current price of the stock; PNOW = 40.75; ! Exercise price at option expiration; STRIKE = 40; ! Yearly interest rate; IRATE = .163; ! Weekly variance in log of price; WVAR = .005216191 ; ENDDATA SETS: ! Generate our state matrix for the DP. STATE(S, T) may be entered from STATE(S, T - 1) if stock lost value, or it may be entered from STATE(S - 1, T - 1) if stock gained; STATE(PERIOD, PERIOD)| &1 #LE# &2: PRICE, ! There is a stock price, and...; VAL; ! a value of the option; ENDSETS ! Compute number of periods; LASTP = @SIZE(PERIOD); ! Get the weekly interest rate; (1 + WRATE) ^ 52 = (1 + IRATE); ! The weekly discount factor; DISF = 1/(1 + WRATE); ! Use the fact that if LOG(P) is normal with mean LOGM and variance WVAR, then P has

ADDITIONAL EXAMPLES mean EXP(LOGM + WVAR/2), solving for LOGM...; LOGM = @LOG(1 + WRATE) - WVAR/ 2; ! Get the log of the up factor; LUPF = (LOGM * LOGM + WVAR) ^ .5; ! The actual up move factor; UPF = @EXP(LUPF); ! and the down move factor; DNF = 1/ UPF; ! Probability of an up move; PUP = .5 * (1 + LOGM/ LUPF); ! Initialize the price table; PRICE(1, 1) = PNOW; ! First the states where it goes down every period; @FOR(PERIOD(T) | T #GT# 1: PRICE(1, T) = PRICE(1, T - 1) * DNF); ! Now compute for all other states S, period T; @FOR(STATE(S, T)| T #GT# 1 #AND# S #GT# 1: PRICE(S, T) = PRICE(S - 1, T - 1) * UPF); ! Set values in the final period; @FOR(PERIOD(S): VAL(S, LASTP) = @SMAX(PRICE(S, LASTP) - STRIKE, 0)); ! Do the dynamic programing; @FOR(STATE(S, T) | T #LT# LASTP: VAL(S, T) = DISF * (PUP * VAL(S + 1, T + 1) + (1 - PUP) * VAL(S, T + 1))); ! Finally, the value of the option now; VALUE = VAL(1, 1); END

Model: OPTION

779

780 APPENDIX A

Bond Portfolio Optimization

Model: PBOND

In certain situations, a business or individual may be faced with financial obligations over a future number of periods. In order to defease (i.e., eliminate) this future debt, the debtor can determine a minimal cost mix of current assets (e.g., cash and bonds) that can be used to cover the future stream of payments. This problem is sometimes referred to as the cash flow matching problem, or the debt defeasance problem. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: !Bond portfolio/cash matching problem. Given cash needs in a series of future periods, what collection of bonds should we buy to meet needs?; SETS: BOND/A B/ : MATAT, ! Maturity period; PRICE, ! Price; CAMNT, ! Coupon; BUY; ! Amount to buy; PERIOD/1..15/: NEED, ! Cash needed each period; SINVEST; !Short term investment each period; ENDSETS DATA: STRTE = .04; ! Short term interest rate; MATAT = 6, 13; ! Years to maturity; PRICE = .980, .965; ! Bond purchase prices; CAMNT = .060, .065; ! Bond coupon amounts; NEED = 10, 11, 12, 14, 15, 17, 19, 20, 22, 24, 26, 29, 31, 33, 36; ! Cash needs; ENDDATA ! Minimize the total investment required to generate the stream of future cash needs; MIN = LUMP; ! First period is slightly special; LUMP = NEED(1) + SINVEST(1) + @SUM(BOND: PRICE * BUY); ! For subsequent periods; @FOR(PERIOD(I)| I #GT# 1: @SUM(BOND(J)| MATAT(J) #GE# I: CAMNT(J) * BUY(J)) + @SUM(BOND(J)| MATAT(J) #EQ# I: BUY(J)) + (1 + STRTE) * SINVEST(I - 1) = NEED(I) + SINVEST(I); ); ! Can only buy integer bonds; @FOR(BOND(J): @GIN(BUY(J)));

Model: PBOND

ADDITIONAL EXAMPLES

Simple Product-Mix

781

Model: PC

In this example, we illustrate a simple product-mix model for deciding how many of two types of computers to produce. Note, LINGO allows you to use scalar variables and forgo the use of sets, thus allowing straightforward entry of simpler models. MODEL: ! Total profit for the week; MAX = 200 * WS + 300 * NC; ! The total number of Wordsmiths produced is limited by the supply of graphics chips; WS <= 60; ! The total number of Numbercrunchers produced is limited by the supply of math coprocessors; NC <= 40; ! The total amount of memory used in all machines manufactured for the week can't exceed 120 Mb; WS + 2 * NC <= 120; END

Model: PC

782 APPENDIX A

Project Management

Model: PERT

In this example, we will set up a PERT model to determine the critical path of tasks in a project involving the roll out of a new product. For those not familiar, PERT stands for Project Evaluation and Review Technique. PERT is a simple, but powerful, technique developed in the 1950s to assist managers in tracking the progress of large projects. A detailed discussion of this model may be found in Chapter 2, Using Sets. MODEL: SETS: TASKS / DESIGN, FORECAST, SURVEY, PRICE, SCHEDULE, COSTOUT, TRAIN/: TIME, ES, LS, SLACK; PRED( TASKS, TASKS) / DESIGN,FORECAST, DESIGN,SURVEY, FORECAST,PRICE, FORECAST,SCHEDULE, SURVEY,PRICE, SCHEDULE,COSTOUT, PRICE,TRAIN, COSTOUT,TRAIN /; ENDSETS DATA: TIME = 10, 14, 3, 3, 7, 4, 10; ENDDATA @FOR( TASKS( J)| J #GT# 1: ES( J) = @MAX( PRED( I, J): ES( I) + TIME( I)) ); @FOR( TASKS( I)| I #LT# LTASK: LS( I) = @MIN( PRED( I, J): ES( J) - TIME( I)); ); @FOR( TASKS( I): SLACK( I) = LS( I) - ES( I)); ES( 1) = 0; LTASK = @SIZE( TASKS); LS( LTASK) = ES( LTASK); DATA: !Use @TABLE() to display the precedence relations set, PRED; @TEXT() = @TABLE( PRED); END

Model: PERT

ADDITIONAL EXAMPLES

Proj. Management with Crashing

783

Model: PERTC

In the previous example, PERT, it was assumed each task takes a fixed amount of time to complete. However, if we allocated additional resources to a task, it is reasonable to assume that we could complete that task in a shorter time period. This option of speeding up a task by spending more on it is referred to as crashing. In this next model, we incorporate crashing as an option. The goal is to meet the project’s due date, while minimizing total crashing costs. MODEL: ! A PERT/M model with crashing; ! The precedence diagram is: ! /FCAST\—SCHED——COSTOUT\ ! / \ \ ! FIRST \ \ ! \ \ \ ! \SURVEY-PRICE—————————FINAL; SETS: TASK/ FIRST, FCAST, SURVEY, PRICE, SCHED, COSTOUT, FINAL/: TIME, ! Normal time for task; TMIN, ! Min time at max crash; CCOST, ! Crash cost/unit time; EF, ! Earliest finish; CRASH; ! Amount of crashing; ! Here are the precedence relations; PRED(TASK, TASK)/ FIRST,FCAST FIRST,SURVEY, FCAST,PRICE FCAST,SCHED SURVEY,PRICE, SCHED,COSTOUT PRICE,FINAL COSTOUT,FINAL/; ENDSETS DATA: TIME = 0 14 3 3 7 4 10; ! Normal times; TMIN = 0 8 2 1 6 3 8; ! Crash times; CCOST = 0 4 1 2 4 5 3; ! Cost/unit to crash; DUEDATE = 31; ! Project due date; ENDDATA ! The crashing LP model; ! Define earliest finish, each predecessor of a task constrains when the earliest time the task can be completed. The earliest the preceding task can be finished plus the time required for the task minus any time that could be reduced by crashing this task.; @FOR(PRED(I, J): EF(J) >= EF(I) + TIME(J) - CRASH(J) );

784 APPENDIX A ! For each task, the most it can be crashed is the regular time of that task minus minimum time for that task; @FOR(TASK(J): CRASH(J) <= TIME(J) - TMIN(J) ); ! Meet the due date; ! This assumes that there is a single last task; EF(@SIZE(TASK)) <= DUEDATE; ! Minimize the sum of crash costs; MIN = @SUM(TASK: CCOST * CRASH); END

Model: PERTC

ADDITIONAL EXAMPLES

Product-Mix with Setup Costs

785

Model: PRODMIX

In a product-mix model, the decision is how much of a number of different products should be produced to maximize total revenue. Each product competes for a number of scarce resources. In this example, we produce six different flying machines from six different raw materials. This model also has the feature that, should we produce a given product, we incur a fixed setup cost. MODEL: SETS: PLANES/ ROCKET, METEOR, STREAK, COMET, JET, BIPLANE /: PROFIT, SETUP, QUANTITY, BUILD; RESOURCES /STEEL, COPPER, PLASTIC, RUBBER, GLASS, PAINT/: AVAILABLE; RXP(RESOURCES, PLANES): USAGE; ENDSETS DATA: PROFIT SETUP = 30 35 45 20 24 60 26 70 24 75 30 30; AVAILABLE = 800 1160 1780 1050 1360 1240; USAGE = 1 4 0 4 2 0 4 5 3 0 1 0 0 3 8 0 1 0 2 0 1 2 1 5 2 4 2 2 2 4 1 4 1 4 3 4; ENDDATA MAX = @SUM(PLANES: PROFIT*QUANTITY - SETUP*BUILD); @FOR(RESOURCES(I): @SUM(PLANES(J): USAGE(I,J) * QUANTITY(J)) <= AVAILABLE(I) ); @FOR(PLANES: QUANTITY <= 400 * BUILD; @BIN(BUILD) ); @FOR(PLANES: @GIN(QUANTITY) ); END

Model: PRODMIX

786 APPENDIX A

Scenario Portfolio Selection

Model: PRTSCEN

Scenarios here refer to outcomes of events with an influence on the return of a portfolio. Examples might include an increase in interest rates, war in the Middle East, etc. In the scenario-based approach to portfolio selection, the modeler comes up with a set of scenarios, each with a certain probability of occurring over the next period. Given this set of scenarios and their probabilities, the goal is to select a portfolio that minimizes some measure of risk, while meeting a target return level. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: ! Scenario portfolio model; SETS: SCENE/1..12/: PRB, R, DVU, DVL; STOCKS/ ATT, GMT, USX/: X; STXSC(SCENE, STOCKS): VE; ENDSETS DATA: TARGET = 1.15; ! Data based VE = 1.300 1.103 1.216 0.954 0.929 1.056 1.038 1.089 1.090 1.083 1.035 1.176

on original Markowitz example; 1.225 1.290 1.216 0.728 1.144 1.107 1.321 1.305 1.195 1.390 0.928 1.715

1.149 1.260 1.419 0.922 1.169 0.965 1.133 1.732 1.021 1.131 1.006 1.908;

! All scenarios happen to be equally likely; PRB= .08333; ENDDATA ! Compute expected value of ending position; AVG = @SUM(SCENE: PRB * R); ! Target ending value; AVG >= TARGET; @FOR(SCENE(S): ! Compute value under each scenario; R(S) = @SUM(STOCKS(J): VE(S, J) * X(J)); ! Measure deviations from average; DVU(S) - DVL(S) = R(S) - AVG ); ! Budget; @SUM(STOCKS: X) = 1;

ADDITIONAL EXAMPLES ! Our three measures of risk; [VARI] VAR = @SUM(SCENE: PRB * (DVU + DVL)^2); [SEMI] SEMIVAR = @SUM(SCENE: PRB * (DVL) ^2); [DOWN] DNRISK = @SUM(SCENE: PRB * DVL); ! Set objective to VAR, SEMIVAR, or DNRISK; [OBJ] MIN = VAR; END

Model: PRTSCEN

787

788 APPENDIX A

Quadratic Assignment

Model: QASGN

In this example, we need to assign airline flights to gates at a hub to minimize the distance traveled from gate to gate by engers transferring between flights. This model is called the quadratic assignment model because we are asg planes to gates, and a straightforward formulation would involve the use of quadratic in the objective. By complicating things slightly through the introduction of an additional variable (Y in this case), we are able to replace each quadratic objective term with one of the new variables. The result of this substitution is a linear model, allowing us to tackle much larger models. MODEL: ! A quadratic assignment problem: Given transfers between flights and distance between gates, assign flights to gates to minimize total transfer distance; SETS: FLIGHT/1..3/; ! There are three flights; GATE/1..4/; ! There are five gates; FXG(FLIGHT, GATE): X; !Flight-gate assignment; GXG(GATE, GATE): T; !Distance between gates; FXF(FLIGHT, FLIGHT): N; !Transfers btwn flights; ENDSETS DATA: N = 0 30 5 ! No. transfers between flights; 20 0 0 30 40 0 ; T = 0 5 10 14 ! distance between gates; 5 0 5 10 10 4 0 6 15 10 5 0 ; ENDDATA SETS: ! Transfer between 2 flights must be required and related to 2 different gates. Warning: this set gets big fast.; TGTG(FLIGHT, GATE, FLIGHT, GATE)| &1 #LT# &3 #AND# ((N(&1, &3) #NE# 0) #AND# (T(&2, &4) #NE# 0) #OR# (N(&3, &1) #NE# 0) #AND# (T(&4, &2) #NE# 0)): Y; ENDSETS ! Each flight, B, must be assigned to a gate; @FOR(FLIGHT(B): @SUM(GATE(J): X(B, J)) = 1); ! Each gate, J, can receive at most one flight; @FOR(GATE(J): @SUM(FLIGHT(B): X(B, J)) <= 1);

ADDITIONAL EXAMPLES ! Force Y(B,J,C,K)=1 if B assigned to J and C assigned to K; ! Assumes the T and N matrices are nonnegative; @FOR(TGTG(B, J, C, K): Y(B, J, C, K) >= X(B, J) + X(C, K) - 1); ! Min the sum of transfers * distance; MIN = @SUM(TGTG(B, J, C, K): Y(B, J, C, K) * (N(B, C) * T(J, K) + N(C, B) * T(K, J))); ! Make the X's 0/1 (Y's will naturally be 0/1); @FOR(FXG: @BIN(X)); END

Model: QASGN

789

790 APPENDIX A

Economic Order Quantity

Model: QDISCX

The classic EOQ (Economic Order Quantity) inventory model detailed in every introductory operations research text tells us the optimal order quantity of an item given its demand rate, unit holding cost, and fixed order cost. A common situation not handled by the standard EOQ model is quantity discounts. In this model, we extend the EOQ analysis by allowing for quantity discounts. MODEL: ! Economic order quantity with quantity discounts; ! This model determines the optimal order quantity for a product that has quantity discounts; SETS: ! Each order size range has; RANGE/1..4/: B, ! An upper breakpoint; P, ! A price/unit over this range; H, ! A holding cost/unit over this range; EOQ, ! An EOQ using this ranges H and K; Q, ! An optimal order qty within this range; AC; ! Average cost/year using this range's Q; ENDSETS DATA: D = 40000; ! The yearly demand; K = 90; ! The fixed cost of an order; IRATE = .2; ! Yearly interest rate; !The upper break points, B, and price per unit, P: Range: 1 2 3 4; B = 10000, 20000, 40000, 60000; P = .35225, .34525, .34175, .33825; ENDDATA ! The model; ! Calculate holding cost, H, and EOQ for each range; @FOR(RANGE: H = IRATE * P; EOQ = (2 * K * D/ H) ^.5; ); ! For the first range, the optimal order quantity is equal to the EOQ ...; Q(1) = EOQ(1) ! but, if the EOQ is over the first breakpoint, lower it; - (EOQ(1) - B(1) + 1) * (EOQ(1) #GE# B(1)); @FOR(RANGE(J)| J #GT# 1: ! Similarly, for the rest of the ranges, Q = EOQ; Q(J) = EOQ(J) +

ADDITIONAL EXAMPLES

791

! but, if EOQ is below the lower breakpoint, raise it up; (B(J-1) - EOQ(J)) * (EOQ(J) #LT# B(J - 1)) ! or if EOQ is above the upper breakpoint, lower it down; - (EOQ(J) - B(J) + 1) * (EOQ(J) #GE# B(J)); ); ! Calculate average cost per year, AC, for each stage; @FOR(RANGE: AC = P * D + H * Q/ 2 + K * D/ Q); ! Find the lowest average cost, ACMIN.; ACMIN = @MIN(RANGE: AC); ! Select the Q that gives the lowest AC per year; ! Note: TRUE = 1, FALSE = 0; QUSE = @SUM(RANGE: Q * (AC #EQ# ACMIN)); END

Model: QDISCX An interesting feature to note in this model is the use of logical expressions as in the following: QUSE = @SUM(RANGE: Q * (AC #EQ# ACMIN));

In this formula, we have the logical expression: AC #EQ# ACMIN

Logical expressions will return the value 1 if they evaluate to TRUE, or 0 if they evaluate to FALSE. As you know, expressions of this nature are discontinuous, and will make it very difficult for the solver to find reliable answers to an optimization model. It turns out in this model, however, that all the variables and formulas are fixed. When a formula is fixed in value, logical expressions contained in the formula do not cause a problem.

792 APPENDIX A

Simple Queuing System

Model: QMMC

In this simple queuing model, we use LINGO’s runtime prompt feature to prompt the for the arrival rate of service customers, the time required by a typical service call, and the number of available servers. Once you enter these values, LINGO computes various statistics about the system including using the Erlang busy function (@PEB) to compute the probability that a customer must wait for service. MODEL: ! Compute statistics for a multi-server system with Poisson arrivals, exponential service time distribution. ! We prompt the for he system parameters; DATA: ARV_RATE = ?; SRV_TIME = ?; NO_SRVRS = ?; ENDDATA ! The model; ! Average no. of busy servers; LOAD = ARV_RATE * SRV_TIME; ! Probability a given call must wait; PWAIT = @PEB(LOAD, NO_SRVRS); !Conditional expected wait, i.e., given must wait; WAITCND = SRV_TIME/(NO_SRVRS - LOAD); ! Unconditional expected wait; WAITUNC = PWAIT * WAITCND; END

Model: QMMC

ADDITIONAL EXAMPLES

Minimal Cost Queuing

793

Model: QUEUEL

The objective of this model is to choose the number of servers in a queuing system that minimizes total cost. If all servers are busy when a customer arrives, then the customer is lost. Total cost is composed of the cost of hiring our servers plus the expected cost of lost customers. The @PEL function is used to get the fraction of customers lost due to all servers being busy when they arrive. MODEL: ! Model of a queuing system with N servers, each of which costs $17/hour. Arrivals occur at a rate of 70 per hour in a Poisson stream. Arrivals finding all servers busy are lost. A lost customer costs $35. The average time to process a customer is 5 minutes; ! Minimize total cost = service costs + lost customer cost; [COST] MIN = SCOST + LCOST ; ! Cost of servers; SCOST = 17 * N ; ! Cost of lost customers; LCOST = 35 * 70 * FLOST ; ! The fraction of customers lost; FLOST = @PEL(70 * 5 / 60 , N); END

Model: QUEUEL

794 APPENDIX A

Steady State Queuing Model

Model: QUEUEM

A useful approach for tackling general queuing models is to use the Rate In = Rate Out Principle (RIRO) to derive a set of steady state equations for a queuing system. RIRO assumes a system can reach a state of equilibrium. In equilibrium, the tendency to move out of a certain state must equal the tendency to move towards that state. Given the steady state equations derived from this assumption, we can solve for the probability that a system is in a given state at any particular moment. A detailed discussion of this model may be found in Chapter 12, Developing More Advanced Models. MODEL: ! Model of a queue with arrivals in batches. In this particular example, arrivals may show up in batches of 1, 2, 3, or 4 units; SETS: ! Look at enough states so that P(i) for large i is effectively zero, where P(i) is the steady state probability of i customers in the system; STATE/ 1..41/: P; ! Potential batch sizes are 1, 2, 3 or 4 customers, and A(i) = the probability that an arriving batch contains i customers; BSIZE/ 1..4/: A; ENDSETS DATA: ! Batch size distribution; A = .1, .2, .3, .4; ! Number of batches arriving per day; LMDA = 1.5; ! Number of servers; S = 7; ! Number of customers a server can process per day; MU = 2; ENDDATA ! LAST = number of STATES; LAST = @SIZE(STATE); ! Balance equations for states where the number of customers in the system is less than or equal to the number of servers; @FOR(STATE(N)| N #LE# S: P(N) * ((N - 1)* MU + LMDA) = P(N + 1) * MU * N + LMDA * @SUM(BSIZE(I)| I #LT# N: A(I) * P(N - I)) ); ! Balance equations for states where number in system is

ADDITIONAL EXAMPLES greater than the number of servers, but less than the limit; @FOR(STATE(N)| N #GT# S #AND# N #LT# LAST: P(N) * (S * MU + LMDA) = P(N + 1) * MU * S + LMDA * @SUM(BSIZE(I)| I #LT# N: A(I) * P(N - I)) ); ! Probabilities must sum to 1; @SUM(STATE: P) = 1; END

Model: QUEUEM

795

796 APPENDIX A

Deg a Computer Cabinet

Model: BOX

In this example, we create a nonlinear optimization model to design the case for a computer. MODEL: ! Design a box at minimum cost that meets area, volume, marketing and aesthetic requirements; [COST] min = 2*(.05*(d*w + d*h) +.1*w*h); [SURFACE] 2*(h*d + h*w + d*w) >= 888; [VOLUME] h*d*w >= 1512; ! These two enforce aesthetics; [NOTNARRO] h/w <= .718; [NOTHIGH] h/w >= .518; ! Marketing requires a small footprint; [FOOTPRNT] d*w <= 252; END

Model: BOX

ADDITIONAL EXAMPLES

Linear Regression

797

Model: REGRES

Linear Regression is a forecasting technique used to predict the value of one variable (called the dependent variable) based upon the value of one or more other variables (the independent variables). Our example is a simple linear regression model with one independent variable. The data is fit to a linear equation of the form: Y(i) = CONS + SLOPE * X(i) where Y is the dependent variable, X is the independent variable, CONS is the value of Y when X = 0, and SLOPE is the rate of change in Y with a unit change in X. For our example, the dependent variable, Y, is the number of annual road casualties and the independent variable, X, is the number of licensed vehicles. We have 11 years of data. MODEL: ! Linear Regression with one independent variable; ! Linear regression is a forecasting method that models the relationship between a dependent variable to one or more independent variable. For this model we wish to predict Y with the equation: Y(i) = CONS + SLOPE * X(i); SETS: ! The OBS set contains the data points for X and Y; OBS/1..11/: Y, ! The dependent variable (annual road casualties); X; ! The independent or explanatory variable (annual licensed vehicles; ! The OUT set contains model output.; OUT/ CONS, SLOPE, RSQRU, RSQRA/: R; ENDSETS ! Our data on yearly road casualties vs. licensed vehicles, was taken from Johnston, Econometric Methods; DATA: Y = 166 153 177 201 216 208 227 238 268 268 274; X = 352 373 411 441 462 490 529 577 641 692 743; ENDDATA SETS: ! The derived set OBS contains the mean shifted values of the independent and dependent variables; OBSN(OBS): XS, YS; ENDSETS

798 APPENDIX A ! Number of observations; NK = @SIZE(OBS); ! Compute means; XBAR = @SUM(OBS: X)/ NK; YBAR = @SUM(OBS: Y)/ NK; ! Shift the observations by their means; @FOR(OBS(I): XS(I) = X(I) - XBAR; YS(I) = Y(I) - YBAR); ! Compute various sums XYBAR = @SUM(OBSN: XS XXBAR = @SUM(OBSN: XS YYBAR = @SUM(OBSN: YS

of squares; * YS); * XS); * YS);

! Finally, the regression equation; R(@INDEX(SLOPE)) = XYBAR/ XXBAR; R(@INDEX(CONS)) = YBAR - R(@INDEX(SLOPE)) * XBAR; RESID= @SUM(OBSN:(YS - R(@INDEX(SLOPE)) * XS)^2); ! A measure of how well X can be used to predict Y - the unadjusted (RSQRU) and adjusted (RSQRA) fractions of variance explained; R(@INDEX(RSQRU)) = 1 - RESID/ YYBAR; R(@INDEX(RSQRA)) = 1 - (RESID/ YYBAR) * (NK - 1)/(NK - 2); ! XS and YS may take on negative values; @FOR(OBSN: @FREE(XS); @FREE(YS)); END

Model: REGRES

ADDITIONAL EXAMPLES

Acceptance Sampling I

799

Model: SAMPLE

In this example, we have a lot of 400 items. We take a sample of 100 items from the lot. We accept the entire lot as being good if the sample has 2 or less defective items. We use the hypergeometric distribution (@PHG) to determine the exact producer risk (probability of rejecting a good lot), and the exact consumer risk (probability of accepting a bad lot). In the days before computers were widely available, statisticians had to rely on published tables of the probability distributions to compute probabilities such as these. Because the hypergeometric distribution is specified by four parameters, it would have been unrealistic to carry around hypergeometric tables that covered all possible scenarios. Instead, statisticians routinely used distributions of fewer parameters to approximate the hypergeometric. So, in deference to the good old days, we make use of the binomial, Poisson, and normal approximations to the hypergeometric to compute these same risk probabilities. The interested reader can compare the accuracy of the various approximations. MODEL: ! Acceptance sampling: taking one or more samples at random from a lot, inspecting each of the items in the sample(s), and deciding on the basis of inspection results whether to accept or reject the entire lot. This Acceptance Sampling model illustrates the effect choice of distribution.; ! From a lot of 400 items; LOTSIZE = 400; ! We take a sample of size 100; SAMPSIZE = 100; ! Producer considers the lot good if the lot fraction defective is .0075 or less; FGOOD = .0075; ! Consumer considers the lot bad if the lot fraction defective is .025 or more; FBAD = .025; ! We accept the lot if sample contains 2 or less; ACCEPTAT = 2; ! The model; ! What is producer risk of rejecting a good lot; !Using the (exact) hypergeometric distribution; PGOODH = 1 - @PHG(LOTSIZE, LOTSIZE * FGOOD, SAMPSIZE, ACCEPTAT); ! Using binomial approx. to the hypergeometric; PGOODB = 1 - @PBN(FGOOD, SAMPSIZE, ACCEPTAT); ! Using the Poisson approx. to the binomial; PGOODP = 1 - @PPS(FGOOD * SAMPSIZE, ACCEPTAT); ! Using Normal approximation; PGOODN = 1 - @PSN((ACCEPTAT + .5 - MUG) / SIGMAG);

800 APPENDIX A ! where; MUG = SAMPSIZE * FGOOD; SIGMAG = (MUG * (1 - FGOOD)) ^ .5; !What is the consumer risk of accepting a bad lot; ! Using the hypergeometric; PBADH = @PHG(LOTSIZE, LOTSIZE * FBAD, SAMPSIZE, ACCEPTAT); ! Binomial; PBADB = @PBN(FBAD, SAMPSIZE, ACCEPTAT); ! Poisson; PBADP = @PPS(FBAD * SAMPSIZE, ACCEPTAT); ! Using Normal approximation; PBADN = @PSN((ACCEPTAT + .5 - MUB) / SIGMAB); ! where; MUB = SAMPSIZE * FBAD; SIGMAB = (MUB * (1 - FBAD)) ^ .5; END

Model: SAMPLE

ADDITIONAL EXAMPLES

Stratified Sampling Design

801

Model: SAMPLE2

In this model, we want to come up with a sampling strategy that yields a variance within a specified target at minimal cost. We have four strata of a population that we will be querying on two topics. There is a maximum variance limit on the two questions. We know the variance in responses for each stratum on each question. How many respondents must you select from each stratum to meet your maximal variance requirements at minimal cost? MODEL: ! Stratified sampling plan design, taken from Bracken and McCormick. Minimize the cost of sampling from 4 strata, subject to constraints on the variances of the sample based estimates of two categories; SETS: STRATUM/1..4/: SIZE, POP, COST, WEIGHT; CATEGORY/1..2/: VARMAX, K2; SXC(STRATUM, CATEGORY): VAR, K1; ENDSETS !

POP = population of each stratum. COST = cost of sampling in each. VARMAX = variance limits. VAR = variance for each category in each stratum. CFIX = a fixed cost;

DATA: POP = 400000, 300000, 200000, 100000; COST = 1, 1, 1, 1; VARMAX = .043, .014; VAR = 25 1 25 4 25 16 25 64; CFIX = 1; ENDDATA [OBJ] MIN = CFIX + @SUM(STRATUM: SIZE * COST); ! Compute some parameters; TOTP = @SUM(STRATUM(I): POP(I)); @FOR(STRATUM(I): ! Weight given each stratum; WEIGHT(I) = POP(I)/TOTP; @GIN(SIZE(I)); ); @FOR(CATEGORY(J): K2(J) = @SUM(STRATUM(I): VAR(I, J)^2 * WEIGHT(I)/ POP(I)); ); @FOR(SXC(I, J):

802 APPENDIX A K1(I, J) = );

VAR(I, J)^2* WEIGHT(I)^2;

@FOR(CATEGORY(J): @SUM(STRATUM(I): K1(I, J) / SIZE(I)) - K2(J) <= VARMAX(J) ); @FOR(STRATUM(I): @BND(0.0001, SIZE(I), POP(I) -1); ); END

Model: SAMPLE2

ADDITIONAL EXAMPLES

Acceptance Sampling II

803

Model: SAMSIZ

We are sampling items from a large lot. If the number of defectives in the lot is 3% or less, the lot is considered “good”. If the defects exceed 8%, the lot is considered “bad”. We want a producer risk (probability of rejecting a good lot) below 9% and a consumer risk (probability of accepting a bad lot) below 5%. We need to determine N and C, where N is the minimal sample size, and C is the critical level of defects such that, if defects observed in the sample are less-than-or-equal-to C, we accept the lot. MODEL: ! Acceptance sampling design. From a large lot, take a sample of size N, accept if C or less are defective; ! Poisson approximation to number defective is used; DATA: AQL = .03; LTFD = .08; PRDRISK = .09; CONRISK = .05; MINSMP = 125; ENDDATA

! ! ! ! !

"Good" lot fraction defective; "Bad" lot fraction defective; Tolerance for rejecting good lot; Tolerance for accepting bad lot; Lower bound on sample size to help solver;

[OBJ] MIN = N; ! Tolerance for rejecting a good lot; 1 - @PPS(N * AQL, C) <= PRDRISK; ! Tolerance for accepting a bad lot; @PPS(N * LTFD, C) <= CONRISK; ! Give solver some help in getting into range; N >= MINSMP; C>1; ! Make variables general integer; @GIN(N); @GIN(C); END

Model: SAMSIZ

804 APPENDIX A

Seasonal Sales Forecasting

Model: SHADES

We have quarterly observations of sales for the last two years. We would like to estimate a base, trend, and seasonal factors to form a sales forecasting function that minimizes the sum of squared prediction errors when applied to the historical sales. A detailed discussion of this model may be found in Chapter 3, Using Variable Domain Functions. MODEL: SETS: PERIODS /1..8/: OBSERVED, PREDICT, ERROR; QUARTERS /1..4/: SEASFAC; ENDSETS DATA: OBSERVED = 10 14 12 19 14 21 19 26; ENDDATA MIN = @SUM(PERIODS: ERROR ^ 2); @FOR(PERIODS: ERROR = PREDICT - OBSERVED); @FOR(PERIODS(P): PREDICT(P) = SEASFAC(@WRAP(P, 4)) * (BASE + P * TREND)); @SUM(QUARTERS: SEASFAC) = 4; @FOR(PERIODS: @FREE(ERROR)); END

Model: SHADES

ADDITIONAL EXAMPLES

Exponential Smoothing

805

Model: SIMXPO

Exponential smoothing is a technique that is relatively easy to implement, and yet has proven to be an effective tool at forecasting sales. In its simplest form, this technique assumes, on average, sales are constant, but include a random error term about the average. It is also assumed the underlying average can drift from period to period. These assumptions lead to the following smoothing equation: St = Xt + (1 - ) St-1 where, St = predicted sales, or signal, in period t, Xt = observed sales in period t, and  = a constant term between 0 and 1. From this equation, we can see the closer  is to 1, the more the current observation affects our signal and, subsequently, the less “memory” our equation has. Frequently, a value for is chosen in the range of .01 to .3. In this example, we will solve for an  that minimizes the sum of the squared prediction errors. For more information on exponential smoothing, see Winston (1995). MODEL: SETS: PERIODS /1..8/: OBSERVED, ERROR, PREDICT; ENDSETS DATA: ! The degree of the objective. N may be changed to 1 to minimize absolute deviation; N = 2; ! The observed values of the time series; OBSERVED = 10 14 12 19 14 21 19 26; ENDDATA ! Force Period 1 prediction to 10; PREDICT(1) = 10; ! The objective function; [OBJ] MIN= @SUM(PERIODS: @ABS(ERROR) ^ N); ! Calculate the forecasts; @FOR(PERIODS(T) | T #GT# 1: PREDICT(T) = ALPHA * OBSERVED(T - 1) + (1 - ALPHA) * PREDICT(T - 1)); ! Calculate forecast errors; @FOR(PERIODS: ERROR = PREDICT - OBSERVED); ! Error may be negative as well as positive; @FOR(PERIODS: @FREE(ERROR));

806 APPENDIX A ! Exclude meaningless Alphas of zero or one; @BND(.01, ALPHA,.9999); END

Model: SIMXPO

ADDITIONAL EXAMPLES

Placing Songs on a Cassette Tape

807

Model: SONGS

In this model, we have seven songs, each with a different length, that must be placed on a cassette tape. The goal is to maximize the number of songs on one side of the tape without exceeding half of the total time of the music on the other side. MODEL: SETS: SONG/1..7/: LENGTH, Y; ENDSETS ! Maximize number of songs on short side; MAX = @SUM(SONG: Y); ! It must contain at most half the music; @SUM(SONG: LENGTH * Y) <= HALF; ! Compute half the length; HALF = @SUM(SONG: LENGTH)/ 2; ! We want the Y's to be 0/1; @FOR(SONG: @BIN(Y)); DATA: LENGTH = 7, 5, 2, 2, 2, 2, 2; ENDDATA END

Model: SONGS

808 APPENDIX A

Computing Sort Order

Model: SORTIN

This simple model sorts cities according to their distance from the equator. MODEL: ! Compute sort order ; SETS: CITY: ! Some cities; LAT, ! Their latitudes; RANKLT, ! Compute rank in distance from equator; RDRLIST; ! Store in this ordered list; ENDSETS DATA: CITY = BEIJING LONDON PARIS NYC LA MOSCOW TOKYO; LAT = 39.6 51.3 48.5 40.4 34.1 55.5 35.4; ENDDATA CALC: ! Minimize output; @SET( 'TERSEO', 2); ! Compute rank of each city; RANKLT = @RANK( LAT); ! Put the original indices in order in a list; @FOR( CITY(i): RDRLIST( RANKLT(i)) = i; ); !Display them; @WRITE(' The cities from closest to farthest from equator:', @NEWLINE(1)); @WRITE(' Latitude City', @NEWLINE(1)); @FOR( CITY(i): @WRITE(' ', LAT(RDRLIST(i)),' ', CITY(RDRLIST(i)), @NEWLINE(1)); ); ENDCALC END

Model: SORTIN

ADDITIONAL EXAMPLES

Traveling Salesman Problem

809

Model: TSP

In the traveling salesman problem (TSP), we have a network of cities connected by roads. We need to find a tour that visits each city exactly once, minimizing the total distance traveled. As it turns, large TSP models are difficult to solve using optimization, and are best approached using some form of heuristic (see Lin and Kernighan, 1973). The problem lies in the fact that solutions to large models tend to contain subtours. A subtour is a tour of a subset of cities unconnected to the main tour. One can add constraints to break the subtours, but the number of constraints required grows dramatically as the number of cities increase. MODEL: ! Traveling Salesman Problem for the cities of Atlanta, Chicago, Cincinnati, Houston, LA, Montreal; SETS: CITY / 1.. 6/: U; ! U(I) = sequence no. of city; LINK(CITY, CITY): DIST, ! The distance matrix; X; ! X(I, J) = 1 if we use link I, J; ENDSETS DATA: !Distance matrix, it need not be symmetric; DIST = 0 702 454 842 2396 1196 702 0 324 1093 2136 764 454 324 0 1137 2180 798 842 1093 1137 0 1616 1857 2396 2136 2180 1616 0 2900 1196 764 798 1857 2900 0; ENDDATA !The model:Ref. Desrochers & Laporte, OR Letters, Feb. 91; N = @SIZE(CITY); MIN = @SUM(LINK: DIST * X); @FOR(CITY(K): ! It must be entered; @SUM(CITY(I)| I #NE# K: X(I, K)) = 1; ! It must be departed; @SUM(CITY(J)| J #NE# K: X(K, J)) = 1; !Weak form of the subtour breaking constraints; !These are not very powerful for large problems; @FOR(CITY(J)| J #GT# 1 #AND# J #NE# K: U(J) >= U(K) + X (K, J) (N - 2) * (1 - X(K, J)) + (N - 3) * X(J, K) ); );

810 APPENDIX A ! Make the X's 0/1; @FOR(LINK: @BIN(X)); ! For the first and last stop we know...; @FOR(CITY(K)| K #GT# 1: U(K) <= N - 1 - (N - 2) * X(1, K); U(K) >= 1 + (N - 2) * X(K, 1) ); END

Model: TSP

ADDITIONAL EXAMPLES

The Log Gamma Function

811

Model: EZCOUNT

The factorial function is used in many probability computations. Unfortunately, the factorial function can generate some very large numbers that can exceed the fixed word size of most computers. A common way around this is to use the Log Gamma function (@LGM), which returns the logarithm of the factorial function. In the following model, we use @LGM to compute the number of possible poker hands. MODEL: ! This model computes the number of ways of selecting 5 objects from a set of 52 objects; ! This is expressed by 52! / (5! * 47!). The actual computation uses the log-gamma function; WAYS = @EXP(@LGM(53) - @LGM(6) - @LGM(48)); ! Note that the arguments of the @LGM functions are one greater than the corresponding arguments of the factorial functions, due to the definition of the Gamma function; END

Model: EZCOUNT

812 APPENDIX A

Vehicle Routing Problem

Model: VROUTE

The vehicle routing problem occurs in many service systems such as delivery, customer pick-up, repair and maintenance. A fleet of vehicles, each with fixed capacity, starts at a common depot and returns to the depot after visiting locations where service is demanded. The objective is to minimize the total distance of all the routes. In general, it takes much longer to find the best routes as the number of locations grow. Large versions of this model may have to be tackled using some form of heuristics. In this particular example, we are delivering one product to seven cities with the depot at city 1. MODEL: ! The Vehicle Routing Problem (VRP) occurs in many service systems such as delivery, customer pick-up, repair and maintenance. A fleet of vehicles, each with fixed capacity, starts at a common depot and returns to the depot after visiting locations where service is demanded. The objective is to minimize total routes. In general it takes much longer time to find the best routes when the number of locations become larger. This model involves delivering the required amount of goods to seven cities with depot at city 1 ; SETS: ! Q(I) is the amount required at city I U(I) is the accumulated delivers at city I ; CITY/1..8/: Q, U; ! DIST(I,J) is the distance from city I to city J X(I,J) is 0-1 variable: It is 1 if some vehicle travels from city I to J, 0 if none ; CXC(CITY, CITY): DIST, X; ENDSETS DATA: ! city 1 represent the common depot, i.e. Q(1) = 0; Q = 0 6 3 7 7 18 4 5; ! distance from city I to city J is same from city J to city I distance from city I to the depot is 0, since the vehicle has to return to the depot ; DIST = ! To City; ! Chi Den Frsn Hous KC LA Oakl Anah From; 0 996 2162 1067 499 2054 2134 2050! Chicago; 0 0 1167 1019 596 1059 1227 1055! Denver; 0 1167 0 1747 1723 214 168 250! Fresno; 0 1019 1747 0 710 1538 1904 1528! Houston; 0 596 1723 710 0 1589 1827 1579! K. City; 0 1059 214 1538 1589 0 371 36! L. A.; 0 1227 168 1904 1827 371 0 407! Oakland; 0 1055 250 1528 1579 36 407 0;! Anaheim;

ADDITIONAL EXAMPLES ! VCAP is the capacity of a vehicle ; VCAP = 18; ENDDATA ! The objective is to minimize total travel distance; MIN = @SUM(CXC: DIST * X); ! for each city, except depot....; @FOR(CITY(K)| K #GT# 1: ! a vehicle does not traval inside itself,...; X(K, K) = 0; ! a vehicle must enter it,... ; @SUM(CITY(I)| I #NE# K #AND# (I #EQ# 1 #OR# Q(I) + Q(K) #LE# VCAP): X(I, K)) = 1; ! a vehicle must leave it after service ; @SUM(CITY(J)| J #NE# K #AND# (J #EQ# 1 #OR# Q(J) + Q(K) #LE# VCAP): X(K, J)) = 1; ! U(K) is at least amount needed at K but can't exceed capacity; @BND(Q(K), U(K), VCAP); ! If K follows I, then can bound U(K) - U(I); @FOR(CITY(I)| I #NE# K #AND# I #NE# 1: U(K) >= U(I) + Q(K) - VCAP + VCAP*(X(K, I) + X(I, K)) - (Q(K) + Q(I)) * X(K, I); ) ; ! If K is 1st stop, then U(K) = Q(K); U(K) <= VCAP - (VCAP - Q(K)) * X(1, K); ! If K is not 1st stop...; U(K)>= Q(K)+ @SUM(CITY(I)| I #GT# 1: Q(I) * X(I, K)); ); ! Make the X's binary; @FOR(CXC(I, J): @BIN(X(I, J)) ; ); ! Minimum no. vehicles required, fractional and rounded; VEHCLF = @SUM(CITY(I)| I #GT# 1: Q(I))/ VCAP; VEHCLR = VEHCLF + 1.999 - @WRAP(VEHCLF - .001, 1); ! Must send enough vehicles out of depot; @SUM(CITY(J)| J #GT# 1: X(1, J)) >= VEHCLR; END

Model: VROUTE

813

814 APPENDIX A

Home Mortgage Calculation

Model: WHATIF

This example models a home mortgage. The is prompted for the monthly payment, the length of the mortgage, and the interest rate. The model then solves for the value of the home that can be purchased with the mortgage. MODEL: ! A model of a home mortgage(WHATIF.LNG); ! The is prompted for values for the payment, years, and interest rate. The face value of the mortgage (LUMP) is solved for.; DATA: ! is prompted PAYMENT = ?; ! YEARS = ?; ! YRATE = ?; ! ENDDATA

for these: Monthly payment; No. of years; Interest rate;

! Relate no. of months to no. of years; MONTHS = YEARS * 12; ! Relate monthly interest rate to yearly rate; (1 + MRATE) ^ 12 = 1 + YRATE; ! Relate lump sum to monthly payment, monthly interest rate, and no. of months; LUMP = PAYMENT * @FPA(MRATE, MONTHS); END

Model: WHATIF

ADDITIONAL EXAMPLES

Transportation Problem

815

Model: WIDGETS

In this example, we want to ship a product from warehouses to vendors at minimal cost. An in-depth description of this model can be found in Chapter 1, Getting Started with LINGO. MODEL: ! A 6 Warehouse 8 Vendor Transportation Problem; SETS: WAREHOUSES / WH1 WH2 WH3 WH4 WH5 WH6/: CAPACITY; VENDORS / V1 V2 V3 V4 V5 V6 V7 V8/ : DEMAND; LINKS(WAREHOUSES, VENDORS): COST, VOLUME; ENDSETS ! The objective; MIN = @SUM(LINKS(I, J): COST(I, J) * VOLUME(I, J)); ! The demand constraints; @FOR(VENDORS(J): @SUM(WAREHOUSES(I): VOLUME(I, J)) = DEMAND(J)); ! The capacity constraints; @FOR(WAREHOUSES(I): @SUM(VENDORS(J): VOLUME(I, J)) <= CAPACITY(I)); ! Here is the data; DATA: CAPACITY = 60 55 51 43 41 52; DEMAND = 35 37 22 32 41 32 43 38; COST = 6 2 6 7 4 2 5 9 4 9 5 3 8 5 8 2 5 2 1 9 7 4 3 3 7 6 7 3 9 2 7 1 2 3 9 5 7 2 6 5 5 5 2 2 8 1 4 3; ENDDATA END

Model: WIDGETS

Appendix B: Error Messages Listed below by code number are the error messages you may encounter when using LINGO. Suggestions for overcoming the errors are also included. 0.

THE MODEL GENERATOR RAN OUT OF MEMORY. LINGO’s model generator ran out of working memory. The model generator converts the text of your LINGO model to an expanded form suitable for an appropriate solver engine. On most platforms, you can increase the amount of working memory allocated to the model generator. Note that memory set aside for LINGO’s model generator will not be available to LINGO’s various solver engines. Given this, you should not allocate an overly excessive amount of memory to the generator. In Windows versions, select the LINGO|Options command, then the General tab, and in the Generator Memory Limit box increase the amount of working memory. Press the Save button and then restart LINGO. You can the new memory allotment by issuing the Help|About LINGO command. On other platforms, use the following commands: SET MAXMEMB n FREEZE

where n is the new memory allotment in megabytes. Exit LINGO and restart. Once LINGO restarts, you can the new memory allotment with the MEM command. 1.

TOO MANY LINES OF TEXT IN THE MODEL. There are too many lines in the model’s text. For all practical purposes, the limit on total lines of text is large enough that this error message should never be encountered.

2.

TOO MANY CHARACTERS IN THE MODEL TEXT. There are too many characters in the model’s text. For all practical purposes, the limit on the number of characters is large enough that this message should never be encountered.

3.

OVERLENGTH LINE, CHARACTERS MAY HAVE BEEN LOST OFF END. USE CARRIAGE RETURN TO BREAK UP OVER SEVERAL INPUT LINES. Input lines are limited to 200 characters. You will need to break up long input lines into shorter ones.

4.

VALID LINES ARE 1 TO N. TYPE ‘ALL’ TO REFERENCE ALL LINES. The LOOK command expects a range of row numbers. If the range is invalid, you will get this message. Enter a new range with valid numbers.

817

818 APPENDIX B 5.

THERE IS NO CURRENT MODEL. Any command that makes sense only if there is a current model in memory will print this message if invoked without the presence of a model. You need to load a model with the File|Open command in Windows or the TAKE command on other platforms, or enter a new model with the File|New command in Windows or the MODEL command on other platforms.

6.

TOO MANY NESTED TAKE COMMANDS. You have exceeded LINGO’s limit of ten nested TAKE commands within a command script. If possible, try combining some commands into a single file.

7.

UNABLE TO OPEN FILE: FILENAME. The file you tried to read doesn’t exist, or you misspelled its name. Try opening the file again.

8.

TOO MANY CONSECUTIVE COMMAND ERRORS. REVERT TO TERMINAL INPUT. LINGO prints this message after having encountered a number of consecutive errors in a command script. LINGO assumes that something has gone seriously awry, closes the script file, and returns you to command level.

9.

NOT USED.

10.

NOT USED.

11.

INVALID INPUT. A SYNTAX ERROR HAS OCCURRED. This is the generic error issued by the LINGO compiler when it detects a syntax error. In Windows, when you close the error box, the cursor will be on the line where the error occurred. Other versions of LINGO will try print out the general area where the error has occurred, but LINGO cannot always pinpoint the exact line. Examine this area for any obvious syntax errors. If you are unable to find any obvious errors, a useful technique is to comment out small sections of the model until the error goes away. This should give you a good idea of exactly where the error is occurring. Syntax errors may also occur if you are not invoking the correct compiler in LINGO. Most s will choose to build models using the native LINGO syntax, however, some s may prefer building their models using LINDO syntax. LINGO can compile models written in either native LINGO syntax or LINDO syntax. LINGO chooses the compiler based on a model's file extension. LINGO models must have an extension of lg4 (the default) or lng. LINDO models must have an ltx extension. The default model extension may be set by clicking on: LINGO | Options | Interface | File Format. Each model window's title bar displays whether it is a LINGO or LINDO model.

12.

MISSING RIGHT PARENTHESIS. A LINGO expression is missing at least one closing right parenthesis. LINGO will point to the end of the expression where the error occurred. Count the number of parentheses in this expression to if you have input the correct number, or, in Window use the Edit|Match Parenthesis command to find the unmatched parenthesis.

ERROR MESSAGES

819

13.

A SPECIFIED SPREADSHEET RANGE WAS NOT FOUND. RANGE_NAME. You specified a spreadsheet range titled RANGE_NAME that LINGO was unable to find. Check the spelling of the range name and be sure that the range is defined in the spreadsheet.

14.

NOT ENOUGH TEMPORARY OPERATOR STACK SPACE. LINGO uses a stack to temporarily hold operators and prefix functions during compilation of an expression. It is possible, though unlikely, that this stack will overflow. If so, try breaking up lengthy expressions, or adding parentheses to offending expressions.

15.

NO RELATIONAL OPERATOR FOUND. Each LINGO expression (with the exception of variable domain expressions) must contain one relational operator (e.g., =, <, >). Check to be sure that all expressions contain relational operators.

16.

ALL MODEL OBJECTS MUST HAVE THE SAME PARENT SET FOR THIS OPERATION. You have attempted to use an import or export function in the data section that involves two or more attributes from different parent sets. Break the function into multiple calls, so each instance refers to attributes belonging to the same parent set.

17.

NOT ENOUGH INDEX STACK SPACE. LINGO uses a stack to keep track of indices referenced by set operators. It is possible, though unlikely, that this stack will overflow. The only way to remedy this situation is to rewrite the given expression, so it uses fewer indices.

18.

OUT OF SET STACK SPACE. LINGO uses a stack to keep track of sets referenced by set operators. This stack may overflow, though it’s unlikely. The only way to remedy this situation is to rewrite the given expression, so the maximum number of pending set operators is reduced.

19.

INVALID USE OF @INDEX FUNCTION. The @INDEX function expects an optional set name followed by a mandatory set number. Check to see that your arguments comply with these restrictions.

20.

IMPROPER USE OF SET NAME. A set name has been used in an improper manner. For instance, you may have attempted to set the name equal to a quantity. Check your model’s syntax.

21.

IMPROPER USE OF ATTRIBUTE NAME. This message is printed if an attribute name is used incorrectly. For instance, you may have attempted to use it as a scalar (single value) variable. Check your model’s syntax.

22.

TOO MANY INEQUALITY OR EQUALITY RELATIONS. A constraint may have only one relational operator (i.e., =, <, or >). A two-sided constraint such as 2 < X < 4 is not permitted. Instead, write it as two one sided constraints: X > 2; and X < 4;.

820 APPENDIX B 23.

IMPROPER NUMBER OF ARGUMENTS. LINGO’s predefined functions generally expect a specific number of arguments. You’ll get this message if you are ing an incorrect number of arguments. Check the syntax of the function in question.

24.

INVALID SET NAME. If LINGO was expecting a set name and didn’t find it, you will get this message. Note that all sets must be defined in a sets section before they are referenced in a model expression.

25.

NOT USED.

26.

IMPROPER NUMBER OF INDEX VARIABLES. When using indices in conjunction with a set in a set operator function, LINGO checks to be sure that you have specified the correct number of indices for the set. If not, it prints this message.

27.

THE FOLLOWING SPREADSHEET RANGE IS DISCONTINUOUS: RANGE_NAME. At present, LINGO only s continuous ranges. Continuous ranges are simple, rectangular ranges. Discontinuous ranges are unions of two, or more, continuous ranges. You will need to break your discontinuous range up into a set of equivalent, continuous ranges.

28.

INVALID USE OF A ROW NAME. A row name may be input within brackets at the start of a constraint and may contain up to 32 characters. This error message indicates some improper use. Check your model’s syntax.

29.

INVALID NUMBER OF INITIALIZATION VALUES IN A DATA/INIT/CALC SECTION. THE REQUIRED NUMBER OF VALUES IS: N When asg values to an array, you must assign a value to every element in data, init and calc statements. LINGO keeps track of the number of values you specified and checks this against the length of each array being assigned. If the two numbers don’t agree, LINGO prints this message along with the number of values that are required.

30.

A GENERAL FAILURE OCCURRED WHILE ATTEMPTING A LINK TO EXCEL. LINGO attempted to open an OLE link to Excel, but was unsuccessful. Be sure that Excel is installed on your machine (version 5, or later). Also, if your machine is busy, this error message can occur because it is taking an excessive amount of time to load Excel. In which case, simply retry the action when the machine is less busy.

31.

INVALID ARGUMENT LIST FOR AN @TEXT() FUNCTION CALL. An instance of the @TEXT function has an invalid argument list. Check the documentation on this function for the correct syntax.

32.

ATTEMPT TO IMPORT A BLANK SET MEMBER NAME. You have attempted to import a set member name that is entirely blank. All set member names must conform to standard LINGO naming conventions and may not be blank. Assign a nonblank name to the set member and retry.

ERROR MESSAGES

821

33.

INVALID ARGUMENT LIST FOR AN @OLE() FUNCTION CALL. Check the documentation on the @OLE function to determine the correct syntax.

34.

RANGE LENGTHS MUST BE IDENTICAL WHEN IMPORTING DERIVED SETS. You are importing a derived set from multiple ranges of varying lengths. When using multiple ranges, each range must contain the same number of cells. Adjust the sizes of the ranges so they agree and retry.

35.

UNRECOGNIZED NAME IN AN OUTPUT FUNCTION. You have attempted to use an output function (e.g., @OLE or @ODBC) to export the value of a variable that does not exist. Check your output functions to see if all the variables exist and that they are spelled correctly.

36.

ATTEMPT TO IMPORT INVALID PRIMITIVE SET ELEMENT NAME: NAME. A primitive set name imported from an external source is invalid. Make sure the name conforms to normal LINGO naming standards.

37.

NAME ALREADY IN USE: NAME. CHOOSE A DIFFERENT NAME. A model can’t use duplicate names. Select a new, unique name.

38.

THE ODBC SERVER RETURNED THE FOLLOWING ERROR MESSAGE: MESSAGE-TEXT. The ODBC server encountered an error. The text of the message will be displayed. In many cases, the text of the message should help clarify the problem. Unfortunately, some ODBC servers will not return an explanatory message for all error conditions, in which case, you may need to experiment with the format of your data to determine the cause of the problem.

39.

THE FOLLOWING SET ELEMENT WAS NOT DERIVED FROM ITS PARENT PRIMITIVE SET: SET-ELEMENT. When defining a sparse derived set, LINGO checks each set element against the parent set. If the element is not found in the parent set, you will get this message. Please check the spelling and ordering of the set elements in the sparse derived set you are defining.

40.

INVALID NUMBER OF DERIVED SET INDEX ELEMENTS. If a sparse derived set is formed from, say, three primitive sets, then there must be a multiple of three primitive set element names in the explicit definition of the derived set. LINGO checks this, and issues this message if there is not a match.

41.

OUT OF ATTRIBUTE INDEX SPACE. LINGO imposes a limit on the total number of primitive sets used in defining derived sets. The current limit is quite large and should not pose a problem.

42.

EXPLICIT VARIABLE IN A SET CONDITION FOR SET: SET_NAME. When using a conditional expression to define the of a derived set, you cannot reference a variable that has not been fixed in a previous data statement. LINGO must be able to completely evaluate these conditional expressions during compilation so it knows the size of the set.

822 APPENDIX B 43.

EXECUTOR ERROR IN SET CONDITION FOR SET: SET_NAME. LINGO prints this message if an arithmetic error occurred when it was trying to evaluate a conditional expression used to define the of a derived set. Check all arithmetic operations in the set definition in question.

44.

UNTERMINATED CONDITION. Each conditional expression placed on a set operator must be terminated with a colon (:). LINGO prints this message if you fail to add one.

45.

INVALID ARGUMENT LIST FOR AN @ODBC FUNCTION CALL. You have a syntax error in an argument list of an @ODBC function call. Check the documentation on the @ODBC function to determine the proper syntax.

46.

INISSIBLE FILE NAME: FILENAME. The file name is either too long or it contains characters not permitted on this platform. Use a different file name and try again.

47.

TOO MANY FILES OPEN: NAME_OF_LAST_FILE. LINGO imposes a limit on the total number of files that can be opened simultaneously through use of the @FILE command. You can try placing all the data in fewer files. Also, avoid using the LINGO end-of-record mark (~) at the end of files. This allows LINGO to “see” the end of the file, forcing it to close the file down, thus allowing for an additional open file.

48.

UNABLE TO OPEN FILE: FILENAME. LINGO prints this message when it is unable to open a file. Check the spelling of the filename. Be sure a copy of the file exists on your disk and that the file is not open in another application.

49.

ERROR READING FILE: FILENAME. LINGO prints this message in case an error occurred while reading a file with the @FILE function. Check the file to be sure it is not damaged. Another possibility is that you do not have read access to the disk or directory where the file resides.

50.

IMPROPER USE OF @FOR() FUNCTION. LINGO prints this message if you’ve attempted to nest an @FOR function inside some other set operator. You can nest @FOR functions within other @FORs, and other set operators within @FORs, but nesting an @FOR inside any function other than another @FOR is not permitted.

51.

RAN OUT OF GENERATOR MEMORY COMPILING MODEL. LINGO exhausted available generator memory compiling a model and was forced to halt. See error message 0 for suggestions on increasing the model generator’s allotment of working memory.

52.

IMPROPER USE OF @IN() FUNCTION. You ed incorrect arguments to the @IN function. Check the documentation on the @IN function.

ERROR MESSAGES

823

53.

UNABLE TO LOCATE RANGE NAME: RANGE_NAME. You specified a range name in a spreadsheet interface function, which is either inissible or was not found in the specified worksheet. Please check the worksheet file to be sure the range name exists as spelled.

54.

ERROR (N) READING FROM FILE. LINGO was unable to read from the file FILE_NAME. Possible causes include the file being locked by another or the file being corrupted. Please check to see that the file is valid and available for reading.

55.

UNABLE TO OPEN @TEXT DATA FILE: FILENAME. You have specified a file as part of the @TEXT function that could not be opened. Check to see that the file exists and that you have spelled the name correctly.

56.

ERROR READING FROM @TEXT DATA FILE: FILENAME. A read error was encountered when LINGO attempted to read data from a file specified in an @TEXT function. Check to be sure that the file is not corrupted.

57.

INVALID INPUT ENCOUNTERED IN @TEXT DATA FILE: TEXT_STRING. The @TEXT function may be used in the data section to read and write numeric values. This error message results when nonnumeric data is encountered in a file being read with @TEXT. If you need to import nonnumeric data from a text file, use the @FILE function.

58.

NOT ENOUGH VALUES FOUND IN @TEXT DATA FILE: FILENAME. N VALUES NEEDED. M VALUES FOUND. The @TEXT function may be used in the data section to read and write numeric values. This error message results when an insufficient number of data points were found in a file being read with @TEXT. Add enough data to the file to fully initialize the attributes of interest.

59.

TOO MANY VALUES ENCOUNTERED IN @TEXT DATA FILE: FILENAME. The @TEXT function may be used in the data section to read and write numeric values. This error message results when too many data points were found in a file being read with @TEXT. Remove data from the file until you have the exact number required to initialize the attributes of interest.

60.

FILE NAME REQUIRED IN AN @TEXT() INPUT OPERATION. The @TEXT function may be used in the data section to read and write numeric values. This error message results when an input file was not specified when attempting to read data points from a file using @TEXT.

61.

COMMAND DISREGARDED. LINGO prints this message when it was unable to interpret a command you typed at the colon prompt (:). Check the spelling of the command and that you are using correct syntax for the command.

62.

RAN OUT OF WORKSPACE IN MODEL GENERATION (N). LINGO ran out of working memory generating your model. Refer to error message 0 for strategies to increase the model generator’s working memory allotment.

824 APPENDIX B 63.

MODEL IS ILL DEFINED. CHECK FOR UNDEFINED INDICES AND/OR CONDITIONS IN EXPRESSION: EXPRESSION. LINGO will print this message for one of two reasons: 1) a conditional expression used to qualify a set operator function cannot be evaluated, or 2) a subscript expression cannot be evaluated. When we say an expression cannot be evaluated, we mean that one or more variables in the expression are not fixed. LINGO will report the expression number, or row name (if used), where the fault occurred. Please be sure to add constraints or data statements to fix all the variables in the conditional or subscript expression in question. Also, if you are using primitive set member names in the model’s equations, you must use the @INDEX function to get the index of the primitive set member. If you don’t use the @INDEX function, LINGO will treat the primitive set member as if it is a new scalar variable. You can have LINGO check for primitive set names that are in use in both the data and the model equations by checking the Check for duplicate names in data and model box on the General Solver tab of the LINGO|Options command dialog box (or the SET CHKDUP 1 command on other platforms). Finally, if you would like to use primitive set names directly in your model’s equations (a practice we don’t recommend), you can force LINGO to allow this by checking the Allow unrestricted use of primitive set member names checkbox on the General Solver tab (or the SET USEPNM 1 command on other platforms).

64.

TOO MANY NESTED @FOR OPERATORS. LINGO maintains a stack to keep track of pending nested @FOR functions. Too many nested @FORs could cause this stack to overflow, although it would be an unusual model that triggers this error. You can avoid nesting some @FORs by forming derived sets, and looping over a single derived set as opposed to many primitive sets.

65.

IMPROPER USE OF @WARN FUNCTION. You have used the @WARN function incorrectly. Check the documentation on @WARN to determine where you went wrong.

66.

WARNING: TOTAL FIXED ROWS WITH NONUNIQUE ROOTS: N When LINGO generates a model, it determines if a variable can be solved for directly and substituted out of the model. These variables are referred to as being fixed, and the row in the model used to solve for a fixed variable is referred to as a fixed row. When LINGO solves for the value of a fixed variable in its fixed row, it will look around a neighborhood of the variable’s value to determine if multiple roots exist, and display this message if any are found. An example of an equation with such multiple roots would be: @SIGN(X ) = 1. All non-negative values of X would satisfy this relation. LINGO will display the names of up to three variables and their corresponding fixed rows with this symptom. LINGO will continue to process the model, however. You should examine the model closely when this error occurs, because it would be unusual to find a well formulated model displaying this characteristic. Keep in mind that there might be a better solution involving a different value for the fixed variable than was chosen by LINGO.

67.

UNED STRING ARITHMETIC OPERATION. You've attempted to perform an arithmetic operation on a text object.

ERROR MESSAGES

825

68.

MULTIPLE OBJECTIVE FUNCTIONS IN MODEL. The model contains more than one objective function. Possible solutions are to delete some objectives, convert some objectives to constraints, or combine the multiple objectives into a single objective using a weighting scheme.

69.

UNDEFINED ARITHMETIC OPERATION. LINGO ran into an undefined arithmetic operation during execution (e.g., 1/0). Check the constraints to be sure all operations are defined.

70.

SUBSCRIPT OUT OF RANGE ON ATTRIBUTE: ATTRIB. While executing your model, LINGO found a subscript that was out of range. For example, if you defined the attribute STAFF with the set /1..5/, then referencing STAFF(6) would result in this error message. Nonintegral subscripts will also generate this message. Please be sure your sets are defined across the intended range and that any subscript computations are correctly specified.

71.

IMPROPER USE OF A VARIABLE DOMAIN FUNCTION (E.G. , @GIN, @BIN, @FREE, @BND). This error results when the syntax of a variable domain function has been violated. Check the model’s use of these functions.

72.

UNABLE TO SOLVE FOR FIXED VARIABLE VAR_NAME IN CONSTRAINT CONSTRAINT_NAME. LINGO has determined that it should be able to solve for a particular variable in a given row. The root finder was not able to converge on a solution. Be sure a solution exists for the row (e.g., the expression: -1=x^.5, would have no real solution). If possible, rewrite the equation in the form x = f(.), where x appears only once and is on the left-hand side of the expression. LINGO is always able to solve expressions in this form, as long as the function is defined and evaluates to a real number (e.g., x = @LOG(-3) evaluates to an imaginary number; and x/0 is undefined). If this is not possible, then you might try solving the expression by some other method, and enter the variable’s value as a constant in LINGO.

73.

A INTERRUPT OCCURRED. LINGO was interrupted before the solver completed its run.

74.

MAGNITUDE OF BOUND EXCEEDS: 1.E+21 ON VARIABLE: VARIABLE_NAME. You have entered a bound outside LINGO’s allowable limit. The magnitude of bounds input using @BND may not exceed 1019.

75.

CONFLICTING BOUNDS ON VARIABLE: VAR_NAME. LINGO has detected a bound on the named variable that is outside of the range of another bound. For instance, @BND(-6, X, 6), followed by @BND(-5, X, 5) would not yield this error. However, following it with @BND(7, X, 9) would.

76.

ERROR GENERATING MODEL. LINGO was unable to the model to the optimizer. If this error occurs, LINDO Systems technical .

826 APPENDIX B 77.

ARITHMETIC ERROR GENERATING MODEL. LINGO was unable to generate the model due to an undefined arithmetic operation (e.g., division by zero). Remove all undefined arithmetic operations from your model.

78.

SET SECTIONS NOT ALLOWED IN SUBMODELS. Set sections can not be placed inside submodels. You can move the set section before the relevant submodel to correct the problem.

79.

NOT USED.

80.

NOT ENOUGH WORKSPACE GENERATING MODEL. LINGO ran out of working memory while trying to generate the model. Please refer to message 0 for strategies to increase the model generator’s working memory.

81.

NO FEASIBLE SOLUTION FOUND. LINGO was unable to find a solution that simultaneously satisfies all the constraints. Check the model’s consistency. Try dropping constraints until the problem goes away to get an idea of where the trouble may lie. Also, check the solution report. Constraints contributing to the infeasibility will have a nonzero dual price.

82.

UNBOUNDED SOLUTION. LINGO was able to increase the objective function without bound. Be sure that you have added all constraints to your model and that they have been input correctly.

83.

ATTEMPT TO USE A FUNCTION THAT IS NOT VALID IN A MODEL SECTION. Not all functions may be used in a model's constraints. Some functions, for example @WRITE, only make sense in and may only be used in calc and data sections. You will receive this error message if you attempt to use such functions in a model's constraints or objective function.

84.

@ODBC AND @POINTER MAY NOT BE USED WITH @WRITE/@WRITEFOR IN CALC SECTIONS. Calc sections do not presently the use of the @ODBC and @POINTER output operators in conjunction with @WRITE/@WRITEFOR. In order to use @ODBC and @POINTER, you must complete attributes as per the following example: CALC: @POINTER( 1) = X; @POINTER( 'MYDATA','MYTABLE','MYCOLUMN') = X; ENDCALC

85.

STACK OVERFLOW. EXPRESSION TOO COMPLEX. LINGO uses a stack to store temporary values while executing the expressions in a model. The default stack size is quite large. Thus, this message should not occur. Should you receive this message, LINDO Systems technical .

ERROR MESSAGES

827

86.

ARITHMETIC ERROR IN CONSTRAINT: CONSTRAINT. INSTRUCTION POINTER: N An undefined arithmetic operation (e.g., 1/0 or @LOG(-1)) occurred while LINGO was generating the model. If you have specified a row name for the constraint, LINGO will print the name of the constraint. If you haven’t specified row names in your model, you may want to add them to assist in tracking down this error. Check the referenced constraint for any undefined operations.

87.

IMPROPER USE OF @IN FUNCTION. You have specified improper arguments for the @IN function. Refer to the documentation on this function for more details.

88.

SOLUTION IS CURRENTLY UNDEFINED. LINGO was not able to solve the model to completion for some reason. In which case, any attempt to print out a solution will result in this message. Try re-solving the model and determining the reason LINGO was unable to solve the model.

89.

RUNTIME ERROR IN SOLVER ROUTINES. LINDO SYSTEMS. An unexpected runtime error has occurred in LINGO's solver routines. Please LINDO Systems for assistance.

90.

THE SOLVER TIME LIMIT WAS EXCEEDED. The solver time limit was reached and the solution process was halted. If you believe that there should be no time limit, then you should check to see what the current time limit is. In Windows, check to see what the current setting is using the LINGO|Options command to check the Time box on the General Solver tab. On other platforms, run the HELP SET command to see what the setting is for the TIMLIM parameter.

91.

INVALID RUNTIME PARAMETER VALUE. At runtime, LINGO will prompt for any variable set to equal a question mark in the data section. If LINGO receives an invalid value, you will get this error message. Correct your data and try again.

92.

WARNING: THE CURRENT SOLUTION MAY BE NONOPTIMAL/INFEASIBLE FOR THE CURRENT MODEL. If you’ve solved a model and brought another model window to the front, LINGO prints this message to remind you that the solution you’ve asked for may not belong to the frontmost model window. LINGO also prints this message whenever you attempt to examine a nonoptimal solution. For instance, when you’ve interrupted the solver, or when LINGO couldn’t find a feasible answer. In the latter case, correct any errors in the model and re-solve.

93.

INVALID SWITCH IN COMMAND LINE. Some LINGO commands accept switches, or modifiers. If there was an error in one of these command-line modifiers, LINGO will print this message. Refer to the documentation on the specific command to learn the available modifiers.

828 APPENDIX B 94.

NOT USED.

95.

NOT USED.

96.

NOT USED.

97.

NOT USED.

98.

NOT USED.

99.

NOT USED.

100.

A SUBMODEL REFERENCE WAS EXPECTED. A reference was made to an undefined submodel. Submodels are "models within models" that may be solved independently or combined with other submodels. You will need to either define the referenced submodel, or move it to an earlier part of the main model before any references to the submodel occur. Refer to the SUBMODEL statement for more information.

101.

S DON'T MATCH -- MODEL NOT EXTENDED. You've attempted to extend a protected model using a model fragment with a that differs from the main model's . The s must be identical in order to add the new model fragment, so you will need to re-save either the main model or the model fragment with a matching .

102.

UNRECOGNIZED MODEL OBJECT NAME: OBJECT_NAME. The model object name you have specified does not exist. Please check your spelling.

103.

TEXT LITERALS NOT ALLOWED IN VECTOR-BASED, NON-TEXT OUTPUT OPERATIONS. When outputting solutions in vector format (i.e., ing an entire attribute at a time), text literals may not be included in the list of output objects. For example, suppose you had the following output statement in your model: DATA: @OLE( 'MYBOOK.XLS', 'HEADER', 'VOLUME') = 'SHIP=', VOLUME; ENDDATA

In this case, VOLUME is an attribute containing multiple values, while 'SHIP=' is a single text literal. This is a vector-based output statement in that we aren't embedding output objects in @WRITE or @WRITEFOR statements. The use of text literals is not permitted in vector based output. You will need to break the statement up into two statements, one using @WRITE to output the text literal: DATA: @OLE( 'MYBOOK.XLS', 'HEADER') = @WRITE( 'SHIP='); @OLE( 'MYBOOK.XLS', 'VOLUME') = VOLUME; ENDDATA

104.

MODEL NOT SOLVED YET, OR THE MODEL HAS BEEN SOLVED AND FOUND TO CONTAIN NO CONSTRAINTS OR VARIABLES. If the model has not already been solved then issue the SolveLINGO_Solve command. If the model has been solved then it was found to be vacuous.

ERROR MESSAGES

829

105.

THE FOLLOWING TABLE IS TOO SMALL TO RECEIVE ALL REQUESTED OUTPUT. An ODBC output statement to a database has more rows than is contained in the output statement. You will need to either increase the size of the table in the database or reduce the number of rows in the output table.

106.

MINIMUM AND MAXIMUM VALUES FOR WIDTH ARE: 68 800. When changing terminal width using the WIDTH command, you must enter an integer between 68 and 800.

107.

INVALID @POINTER INDEX VALUE Your model contains a reference to the @POINTER function with an invalid argument list. The @POINTER function requires one argument that must be integer valued and greaterthan-or-equal-to 1.

108.

THE MODEL’S DIMENSIONS EXCEED THE CAPACITY OF THIS VERSION. Your model is too large for your version of LINGO. Some versions of LINGO limit one or more of the following model properties: total variables, integer variables, nonlinear variables, and constraints. This error message displays the dimensions of the model and the limits of your version. Refer to this information to determine the specific limit that is being exceeded. You can also view the limits of your version of LINGO by issuing the Help|About command. Nonlinear variables are allowed only if you have purchased the nonlinear option for your LINGO system. If you aren't sure which parts of yout model, you can run the Generate | Display nonlinear rows command to display the rows in the model containing nonlinearities. It's possible that some or all of the nonlinear rows can be rewritten so that they become linear. In general, you must either make your model smaller by simplifying it or a larger version of LINGO. If you are exceeding the limit on constraints and if you have simple bounds entered as constraints, you should enter them with the @BND function. Constraints entered using @BND don’t count against the constraint limit. The limits for the various versions of LINGO are: Version Demo/Web Solver Suite Super Hyper Industrial Extended

Total Variables 300 500 2,000 8,000 32,000 Unlimited

Integer Variables 30 50 200 800 3,200 Unlimited

Nonlinear Variables 30 50 200 800 3,200 Unlimited

Constraints 150 250 1,000 4,000 16,000 Unlimited

If your version of LINGO doesn't include the nonlinear option, then your models can't contain any nonlinear variables.

830 APPENDIX B 109.

THE SOLVER ITERATION LIMIT WAS EXCEEDED. The solver iteration limit was reached and the solution process was halted. If you believe that there should be no iteration limit, then you should check to see what the current iteration limit is. In Windows, check to see what the current setting is using the LINGO|Options command to check the Iterations box on the General Solver tab. On other platforms, run the HELP SET command to see what the setting is for the ITRLIM parameter.

110.

S ARE LIMITED TO 56 CHARACTERS. The associated with your HIDE command is too long. It must be 56 characters, or less.

111.

COMMAND NOT AVAILABLE WHEN MODEL IS HIDDEN. Any command that reveals the content of a model is not permitted when the model is hidden. If you know the to unhide the model, enter it with the HIDE command and try again.

112.

INCORRECT ... MODEL REMAINS HIDDEN. You have entered an incorrect . The model may not be viewed without the correct .

113.

LOOK/SAVE FAILED... THE FOLLOWING LINE IS TOO LONG: N. INCREASE THE TERMINAL WIDTH. An attempt to view a given line, all lines, or to save the model, has failed because a line is too long. Lines can't exceed 200 characters in length. You will need to break each long line into two or more shorter lines.

114.

VERIFICATION FAILED... MODEL WILL NOT BE HIDDEN. You must enter the same twice to it.

115.

MODEL MUST BE AT LEAST 64 CHARACTERS LONG FOR ENCRYPTION. In order to encrypt a model with the HIDE command it must be at least 64 characters long. If need be, you can extend the model's length using comments.

116.

THE MAXIMUM NUMBER OF CHARACTERS IN A FILE NAME IS: N. The maximum number of characters in a file name, including path, has been exceeded.

117.

INVALID COMMAND. TYPE ‘COM’ TO SEE VALID COMMANDS. You have entered a command that is not implemented in LINGO. Check the list of available commands with the COMMANDS command to see whether you’ve misspelled a command.

118.

AMBIGUOUS COMMAND. TYPE ‘COM’ TO SEE VALID COMMANDS. You have probably typed an abbreviated command that can be interpreted in more than one way. Spell out the command name completely.

119.

TOO MANY CONSECUTIVE ERRORS IN BATCH MODE. LINGO WILL STOP. When running a command script, LINGO will stop after the sixth consecutive error.

ERROR MESSAGES

831

120.

UNABLE TO WRITE CNF FILE TO STARTUP AND WORKING DIRECTORIES. LINGO was unable to write its configuration file. You must have write permission for either the working directory or the startup directory. If you’re connected to a network, check with your network to determine your disk privileges.

121.

RANGE ANALYSIS NOT ALLOWED ON INTEGER PROGRAMMING MODELS. Range reports are not possible on IP models, given that they have no meaning in this context.

122.

RANGE REPORTS NOT POSSIBLE WHEN RANGE ANALYSIS IS DISABLED. Range computations are currently disabled. To enable range computations in Windows versions of LINGO, run the LINGO|Options command, click the General Solver tab, and select the Prices and Ranges option from the Dual Computations list box. To enable range computations in command-line versions of LINGO, use the command: SET DUALCO 2. Be aware that range computations will increase solution times.

123.

MODELS MUST BE EITHER INFEASIBLE OR UNBOUNDED IN ORDER TO BE DEBUGGED. Debugging is permitted only on models that are either infeasible or unbounded.

124.

ATTEMPT TO INITIALIZE AN ATTRIBUTE OF A NULL SET. You are attempting to assign values to an attribute of a null set. Given that the set is null, each of its attributes also have no , and, therefore, may not be assigned values. Check the initialization of the attribute’s parent set to be sure it is being assigned at least one member.

125.

NOT USED. Error code not currently in use.

126.

THE FOLLOWING VARIABLE NAME WAS NOT RECOGNIZED: VAR_NAME. You have attempted to use an ODBC link to export an unknown variable to a database. Check the spelling of the variable’s name and try again.

127.

THE FOLLOWING VARIABLE IS OF A DIFFERENT DIMENSION: VAR_NAME. You have attempted to use an ODBC link to export multiple variables of different dimensions to a database. All variables being simultaneously exported must be of the same dimension.

128.

THE PARAMETER INDEX MUST BE BETWEEN 1 AND N. You have input an invalid parameter index as part of the SET command. Use the HELP SET command to determine the valid indices.

129.

THE PARAMETER VALUE IS NOT VALID. You have input an invalid parameter value as part of the SET command. LINGO will print the valid range of values. You should adjust the value of the parameter, so it falls within the valid range, and reenter the command.

130.

THE FOLLOWING PARAMETER NAME WAS NOT RECOGNIZED: PARAMETER_NAME. You have input an invalid parameter name as part of the SET command. Use the HELP SET command to determine the valid parameter names.

832 APPENDIX B 131.

UNABLE TO WRITE PARAMETERS TO CONFIGURATION FILE. You have attempted to save LINGO’s current configuration using the FREEZE command. LINGO was unable to write the parameter values to its configuration file (LINGO.CNF). Perhaps you don’t have write access to the drive or the disk is full.

132.

ONE OR MORE ERRORS OCCURRED WHILE READING THE CONFIGURATION FILE. LINGO was unable to successfully read its configuration file (LINGO.CNF). Perhaps the file has become corrupted. Beware that some of the parameters you have set may revert back to their default values.

133.

UNABLE TO INITIALIZE WORKBOOK FOR OLE TRANSFERS. You have attempted to link to a spreadsheet that LINGO was unable to load into Excel. Try reading the file into Excel yourself, and check to be sure that some other isn’t accessing the file.

134.

UNABLE TO COMPLETE ALL OUTPUT OPERATIONS. You have attempted to use interface functions in the data section to export parts of the solution. One or more of these operations failed. You will receive additional messages from LINGO detailing the problem.

135.

THE FOLLOWING FIXED VARIABLE VIOLATES ITS SEMICONTINUOUS BOUNDS: VARIABLE. You have specified semicontinuous bounds for the specified variable using the @SEMIC function. The variable was determined to be fixed to a particular value that violates its semicontinuous property. For example, if X were specified to be semicontinuous using the statement @SEMIC( 5, X, 100), then X must be either 0 or lie within the range [5,100]. If the model generator determines that X is fixed at some value other than this, you will receive this error message. The most likely cause of this error is incorrect semicontinuous bounds, i.e., in this example, the bounds of 5 and 100

136.

ERROR PERFORMING @TEXT() OUTPUT OPERATION: N. You have attempted to use the @TEXT interface function to export data to a text file. For some reason, this operation has failed. Perhaps the disk is full or you don’t have write access to the drive.

137.

ENCRYPTED FILE NOT IN PROPER FORMAT. An encrypted model (see the HIDE command) was found to not be in the proper format. Please check the file name to be sure you are referring to the correct file.

138.

TOO MANY NESTED DIVERT FILES. LINGO allows you to nest DIVERT commands up to 10 levels deep. If you attempt to go beyond 10 levels, you will get this error message.

139.

DIVERT COMMAND FAILED. UNABLE TO OPEN FILE. You have attempted to open a file for output using the DIVERT command. LINGO was unable to open the files. Perhaps the disk is full or you don’t have write access to the drive.

ERROR MESSAGES

833

140.

DUAL VALUES WERE REQUESTED BUT DUAL COMPUTATIONS ARE DISABLED. You have attempted to export dual values when the solver is in primals only mode. In primals only mode, dual values are not computed and, therefore, can't be exported. In Windows versions of LINGO, to enable dual computations, run the LINGO|Options command, click the General Solver Tab, and select the Prices option from the Dual Computations list box. To enable range computations, as well as, dual computations, select the Prices & Ranges option from the Dual Computations list box. In command-line versions of LINGO, enable dual computations by using the command: SET DUALCO 1. To enable range computations, as well as, dual computations, use the command: SET DUALCO 2.

141.

RANGE VALUES WERE REQUESTED ON A ROW THAT WAS SUBSTITUTED FROM THE MODEL. You have attempted to export range values on a row that is fixed in value. LINGO substitutes these rows from the model. Therefore, range values are not computed on these rows.

142.

AN UNEXPECTED ERROR OCCURRED. PLEASE LINDO SYSTEMS TECHNICAL . In general, this error message should never occur. Should you receive this message, please a technical representative.

143.

OUTPUT OPERATION FAILED. MODEL OBJECTS NOT OF SAME LENGTH. You have attempted to use an interface function to export two or more model objects simultaneously. This interface function requires all objects to be of the same dimension. Break the output operation up into individual operations that all contain objects of the same dimension.

144.

INVALID ARGUMENT LIST FOR @POINTER FUNCTION. The @POINTER function only accepts a single positive integer as an argument. Review the documentation on the use of the @POINTER function.

145.

ERROR N PERFORMING @POINTER OUTPUT OPERATION. A @POINTER output function operation failed. Some of the values for N and their interpretations are: No. 2 3 4 5 6

Interpretation attempt to export an invalid variable ran out of working memory requested duals in primals only mode range values were requested on fixed rows unexpected error, call LINDO Systems Technical

834 APPENDIX B 146.

THE FOLLOWING NAMES APPEARED IN THE MODEL AND THE DATA: NAME1 NAME2 NAME3. If you go to the Model Generator tab on the LINGO|Options command’s dialog box, you will see a Check for duplicates names in data and model checkbox. When this option is enabled, LINGO will compare primal set member names to all the variable names used in the model’s equations. If any duplicates are found, LINGO will print the first three and print this error message. To enable this option in command-line versions, use the SET CHKDUP 1 command.

147.

UNABLE TO EVALUATE ALL @WARN FUNCTIONS. Conditional expressions contained in @WARN functions must contain fixed variables only. When this is not the case, LINGO can’t evaluate the @WARN functions and you will receive this error message.

148.

@OLE FUNCTION NOT ED ON THIS PLATFORM. At present, the @OLE function is ed only in Windows versions of LINGO. If you don’t have a Windows version of LINGO, you can export the data from your spreadsheet to a file and use the @FILE function in your LINGO model to import the data.

149.

TARGET RANGE NAMES AND ODBC COLUMN NAMES CAN'T TAKE DEFAULT VALUES FOR INDEX VARIABLE OUTPUT. If you are outputting an index variable's value to a workbook or database, you must explicitly provide the name of the workbook range or the database column. LINGO cannot provide default names in this instance.

150.

ODBC INTERFACE NOT ED ON THIS PLATFORM. LINGO’s ODBC link to databases is ed only in Windows versions. If you don’t have a Windows version of LINGO, you can use text files to move data in and out of LINGO. See Chapter 8, Interfacing with External Files, for more details.

151.

@POINTER NOT ED ON THIS PLATFORM. LINGO’s @POINTER function for interfacing with calling applications is ed only in Windows versions. If you don’t have a Windows version of LINGO, you can use text files to move data in and out of LINGO. See Chapter 8, Interfacing with External Files, for more details.

152.

COMMAND NOT ED ON THIS PLATFORM. You have selected a command that is not ed on your platform.

153.

SET DEFINITIONS NOT ALLOWED IN INIT SECTIONS. Sets can’t be initialized in an init section. You must change the model, so the set is initialized in either a sets section or in a data section.

154.

ATTEMPT TO REDEFINE A PREVIOUSLY DEFINED SET. You have attempted to define a set twice in the same model. A set name can only be used once. Choose a different name for the two sets and try again.

ERROR MESSAGES

835

155.

SET MEMBER NAMES MAY NOT BE OMITTED IN DATA STATEMENTS. When initializing a set in a model’s data section, you must explicitly list each member. You may not skip elements as you can with set attributes.

156.

INCORRECT NUMBER OF ARGUMENTS IN A DATA SECTION. ARGUMENT MUST BE A MULTIPLE OF: N NUMBER OF ARGUMENTS FOUND: M You have a data, init or calc statement in your model that doesn’t have the correct number of values to initialize a specified list of attributes and/or sets. LINGO will let you know how many values it found and what the number of arguments should be a multiple of. Add or subtract values in the data statement and try again.

157.

ATTEMPT TO USE @INDEX ON AN UNDEFINED SET. LINGO can’t compile an instance of the @INDEX function without knowing of the existence of the set that @INDEX is referencing. Move the expression with the @INDEX function after the set definition to correct this problem.

158.

A SET MEMBER NAME WAS EXPECTED. You have used a function that was expecting a set member name as an argument. Correct the arguments to the function and try again.

159.

THE FOLLOWING DERIVED SET MEMBER IS NOT CONTAINED IN ITS PARENT SET. You have specified a set member in a derived set that is not contained in the parent set. Each member in a derived set must be derived from a member of its parent set(s). You have probably misspelled the name of the derived set member. Check the spelling and try again.

160.

ONLY ONE SET MAY BE DEFINED IN EACH DATA STATEMENT. You have attempted to define more than one set in an individual data statement. Break the data statement into multiple statements with no more that one set per statement.

161.

INDEX VARIABLES MAY NOT SHARE NAMES WITH OTHER VARIABLES. The index variables used in set looping functions may not use the same names as those used by the structural variables in the model. For example, in the following model: MODEL: SETS: S1/1..5/: X; ENDSETS MAX=I; @SUM( S1( I): X( I)) <= 100; END

the variable name i is used in the objective row for a structural variable. In the next to the last statement in the model, the name i is also being used as an index variable, which will trigger the error. In the case, you would need to change the name of either the variable in the objective or the name of the index variable in the @SUM loop.

836 APPENDIX B 162.

ATTEMPT TO INITIALIZE MIXED DATA TYPES (TEXT AND NUMERIC) FROM A SINGLE WORKBOOK RANGE. When specifying a single range for multiple model objects, all the objects must be of the same data type—either text (set ) or numeric (set attributes). LINGO’s spreadsheet interface can’t handle ranges with mixed data types. Break the data statement up into two— one containing text model objects and the other containing numeric objects.

163.

INVALID NUMBER OF DATA VALUES FOR OBJECT: OBJECT_NAME. You have attempted to initialize a model object of known length with an incorrect number of values. Check the initialization statement to be sure that the values are specified correctly.

164.

INVALID LINGO NAME: NAME. You have used a symbol name that doesn’t conform to LINGO’s naming conventions. Please correct the name and try again.

165.

SET EXPORTS NOT ED WITH THIS FUNCTION. Not all of LINGO’s export functions can handle set . Switch to a different export function and try again.

166.

ATTEMPT TO OUTPUT OBJECTS OF VARYING LENGTHS TO A TABULAR DEVICE. LINGO requires output to be in tabular form for certain output devices (databases and text files). You have constructed an output statement with two or more model objects of varying length. In which case, it is not obvious how to transform the data into tabular form. Break the output function call up into two or more function calls such that each call contains model objects of identical length.

167.

INCORRECT NUMBER OF RANGES SPECIFIED: N. NUMBER OF RANGES REQUIRED: M. You didn’t specify the correct number of ranges in an import/export function. In general, you will need one range for each model object.

168.

OUTPUT MODIFIERS NOT ALLOWED ON TEXT DATA. When exporting set , use of the @DUAL, @RANGEU, and @RANGED modifier functions are not allowed. Remove the modifier function and try again.

169.

RUNTIME INPUT OF SET NOT ALLOWED. When initializing attributes in a data section, you can initialize all or part of an attribute with a question mark. In which case, LINGO will prompt you for the values each time the model is run. This is not the case with sets—all set must be explicitly listed when initializing a set in the data section.

170.

LICENSE KEY WAS NOT FOUND OR IS INVALID. LINGO did not recognize your license key. Please check to make sure you have entered it correctly. If you received it as part of an email, then you may cut-and-paste it from the email into LINGO's license dialog box.

ERROR MESSAGES

837

171.

LICENSE KEY IS INVALID. Your license key is not recognized by LINGO. LINGO will continue to operate, but only in demonstration mode.

172.

INTERNAL SOLVER ERROR. LINDO SYSTEMS. LINGO’s solver encountered an unexpected error condition and was unable to continue. Please LINDO Systems for assistance.

173.

NUMERIC ERROR IN THE SOLVER. LINGO’s solver experienced numerical problems and was unable to continue. Scaling the model’s coefficients so that they don’t cover as large a range may be helpful in eliminating this error. Also, check for potentially undefined arithmetic operations in the model. If these remedies fail, please LINDO Systems for assistance.

174.

OUT OF MEMORY IN PREPROCESSOR. LINGO’s solver ran out of memory during the preprocessor phase. Refer to error message 175 for possible remedies.

175.

NOT ENOUGH VIRTUAL SYSTEM MEMORY. LINGO’s solver ran out of virtual system memory. LINGO’s solver accesses memory from the system heap. Adding more memory to your machine may help. Note, also, that the memory allocated to LINGO’s model generator is not available to the solver. Thus, if too much memory is allocated to the generator there may not be enough left over for the solver. If you are using a Windows version of LINGO, see the LINGO|Options command for information on adjusting the generator memory level, otherwise, refer to the SET command.

176.

OUT OF MEMORY DURING SOLVER POSTPROCESSING. LINGO’s solver ran out of memory during the post-processing phase. Refer to error message 175 for possible remedies.

177.

OUT OF MEMORY ALLOCATING SET: SET_NAME. LINGO failed to allocate the internal data structures for a set due to lack of generator memory. First, check to see that the set's dimensions are not unreasonably large, perhaps indicating a typo in the model. If all appears well, then you will need to boost the available generator memory, restart LINGO and try again.

178.

OUT OF STACK SPACE ALLOCATING SET: SET_NAME. When LINGO allocates the data structures for a set, it may find that the set is derived from another set. If this parent set is also unallocated, LINGO must back track and allocate it, too. In some rare instances, LINGO might have to backtrack many generations. As LINGO back tracks, it keeps information on a stack. This error message is triggered when this stack runs out of space. However, it would be a highly unusual model that causes this error.

838 APPENDIX B 179.

THE MODEL TRANSLATOR HAD TO PATCH NAMES TO MAKE THEM COMPATIBLE VAR NAMES PATCHED: N ROW NAMES PATCHED: M This message occurs when LINGO imports a non-native file converting it to a LINGO model. Non-native file formats (e.g., MPS, MPI and/or LP) allow for variable and row names that aren’t entirely compatible with LINGO’s syntax (e.g., spaces in names). In order to overcome this, LINGO will patch these names by replacing inissible characters in the names with underscores and truncating names longer than 64 characters. This can create problems because two or more unique MPS names may get mapped into one LINGO name . A technique for avoiding this problem is to use R/C naming conventions. Refer to the Use R/C format names for MPS I/O option for more information.

180.

UNABLE TO CREATE MODEL DATA STRUCTURE. LINGO was unable to create the internal data structures required for transferring an MPS file. The most likely cause of the problem is insufficient system memory. Try cutting back on LINGO’s allotment of generator memory or using a machine with more memory.

181.

ERROR EXTRACTING DATA FROM MODEL STRUCTURE. LINGO was unable to extract model information from a data structure. The most likely cause of the problem is insufficient system memory. Try cutting back on LINGO’s allotment of generator memory, or try using a machine with more memory.

182.

ERROR IN MPS OR MPI FILE ON LINE NUMBER: N TEXT OF INVALID LINE: TEXT LINGO encountered an error reading an MPS or MPI format file. Go to the line number in the file, correct the error, and try again.

183.

INVALID USE OF SP TABLE DECLARATION FUNCTION @SPDISTTABLE. The @SPDISTTABLE function is a matrix oriented function that is uses in stochastic programming models to declare outcome tables and assign to assign random variables to those tables. Correct usage is: @SPDISTTABLE( OUTCOME_TABLE_ATTR, RANDOM_VAR_VECTOR[, PROBABILITY_VECTOR]); The outcome table argument must be either one or two dimensions, with each row of the table corresponding to an outcome of the random variable(s) in the random variable vector. The number of elements in the random variable vector must equal the number of columns in the outcome table argument. The third argument is optional, is used to the probabilities of the outcomes, and must have one element for each row of the outcome table.

ERROR MESSAGES

839

184.

RANGE VALUES CAN'T BE REPORTED WHEN RANGE ANALYSIS IS DISABLED. You have attempted to reference range values while the solver is not set to compute ranges. To enable range computations in Windows versions of LINGO, run the LINGO|Options command, click the General Solver Tab, and select the Prices and Ranges option from the Dual Computations list box. To enable dual and range computations in command-line versions of LINGO use the command: SET DUALCO 2.

185.

BARRIER SOLVER REQUESTED WITHOUT A LICENSE. LINGO’s barrier solver capability is an additional option. To enable the option, LINDO Systems for licensing information and fees.

186.

SETS MAY NOT BE INITIALIZED WITH @QRAND. The @QRAND function may be used for initializing attributes only. Set can not be initialized with this function.

187.

ONLY ONE LHS ATTRIBUTE MAY BE INITIALIZED IN A @QRAND STATEMENT. You can initialize only one attribute at a time with the @QRAND function. For more information, refer to the Probability Functions section in Chapter 7, Operators & Functions.

188.

ATTRIBUTES INITIALIZED WITH @QRAND MUST BELONG TO DENSE SETS. You can initialize only one attribute of dense sets with the @QRAND function. For more information, refer to the Probability Functions section in Chapter 7, Operators & Functions.

189.

INVALID SEED. You have attempted to use an invalid seed value for a random number generating function. Seed values must have non-negative, integer values.

190.

INVALID IMPLICIT SET DEFINITION. LINGO allows you to express sets implicitly. An example would be Jan..Dec, which would give you a set of 12 elements consisting of the months of the year. There are a number of syntax rules required of implicit set definitions. For information on the specific rules, refer to Chapter 2, Using Sets.

191.

THE LINDO API RETURNED THE FOLLOWING ERROR CODE: N ERROR_TEXT. The LINDO API is the core solver engine used by LINGO. On rare occasions, the LINDO API may raise an unexpected error condition. LINGO will display a text message from the API regarding the error. In most cases, this message should clarify the situation. If not, please LINDO Systems technical .

192.

@WKX NO LONGER ED…USE @OLE INSTEAD. The @WKX function is no longer ed. You must now use the @OLE function to interface with spreadsheets.

193.

A SOLUTION IS NOT AVAILABLE FOR THIS MODEL. There is no solution associated with the model. Either an error occurred while attempting to solve the model, or the model has not been solved yet.

840 APPENDIX B 194.

UNABLE TO CREATE ENVIRONMENT DATA STRUCTURE. LINGO was not able to allocate some internal data structures. This is most likely due to insufficient memory in your system’s dynamic memory pool. You can try running on a machine with more memory, increasing Windows allotment of virtual memory, and/or reducing the amount of generator memory allocated to LINGO’s model generator.

195.

AN ERROR OCCURRED WHILE ATTEMPTING TO WRITE TO A FILE. LINGO experienced problems writing to a file. Be sure your disk is not full and that you have write access to the target file.

196.

A DUAL SOLUTION DOES NOT EXIST FOR THIS MODEL. The solver was unable to successfully compute the dual solution for the model. Given this, the solution report will only display primal values. If you don't require dual information, you can avoid the error by disabling Dual Computations on the General Solver Tab.

197.

THE MODEL CONTAINS ONE OR MORE VACUOUS @MAX OR @MIN FUNCTIONS. The model contains an @MAX or @MIN function that has no arguments. An example of this situation is: MODEL: SETS: S1/1..5/: X; ENDSETS THEMAX = @MAX( S1( I) | I #LT# 0: X( I)); END

Note that the index variable I will take on values 1, 2, 3, 4 and 5. In no case will I ever be less than 0. This means that the condition (I #LT# 0) on the @MAX function will never be satisfied, meaning that there are no explicit arguments to the @MAX function. Please check that all the @MAX and @MIN functions in the model are correctly specified. 198.

QUADRATIC MODEL IS NOT CONVEX. Quadratic models must be convex in order for them to be solved by the quadratic solver. You must disable the quadratic solver by turning off Quadratic Recognition and then re-solve. You can disable Quadratic Recognition with the command: LINGO | Options | Nonlinear Solver | Quadratic Recognition With command-line versions of LINGO, use the command: SET USEQPR 0 This will cause the general purpose, nonlinear solver to be invoked. Please see the section titled Convexity for more information.

199.

A BARRIER SOLVER LICENSE IS REQUIRED FOR THE QUADRATIC SOLVER. You will need a license for the barrier option to run the quadratic solver. You can proceed by disabling quadratic recognition and re-solving. This will cause the general purpose, nonlinear solver to be invoked.

ERROR MESSAGES

841

200.

UNABLE TO COMPUTE DUAL SOLUTION. LINGO was unable to compute the dual values. You can proceed by turning off dual calculations and re-solving.

201.

THE MODEL IS LOCALLY INFEASIBLE. The solver was unable to find a feasible solution within a local region. However, a feasible solution may exist elsewhere. The global or multistart solvers may have more success.

202.

THE NUMBER OF NONLINEAR VARIABLES IN THE MODEL: N1 EXCEEDS THE GLOBAL SOLVER LIMIT IN THIS VERSION: N2 Some versions of LINGO impose a limit on the total number of nonlinear variables when running the global solver. You will either need to reduce the number of nonlinear variables, turn off the global solver, or a larger version of LINGO.

203.

THE GLOBAL SOLVER OPTION WAS REQUESTED WITHOUT A LICENSE. LINGO WILL REVERT TO USING THE DEFAULT NONLINEAR SOLVER. Your installation of LINGO does not have the global solver option enabled. The global solver is an add-on option to LINGO. LINGO will use the standard nonlinear solver in upgrading your license in order to enable this option.

204.

THE MULTISTART OPTION WAS REQUESTED WITHOUT A LICENSE. LINGO WILL REVERT TO USING THE DEFAULT NONLINEAR SOLVER. Your installation of LINGO does not have the global solver option enabled. The multistart solver is a component of the global solver add-on option. LINGO will use the standard nonlinear solver in place of the multistart solver. You can LINDO Systems for information on upgrading your license in order to enable this option.

205.

THE MODEL IS POORLY SCALED AND MAY YIELD ERRATIC RESULTS. THE UNITS OF THE ROWS AND VARIABLES SHOULD BE RESCALED SO THE COEFFICIENTS COVER A MUCH SMALLER RANGE. After LINGO generates a model, it checks all the nonzero coefficients in the model and computes the ratio of the largest to smallest coefficients. This ratio is an indicator of how well the model is scaled. When the ratio gets to be too high, scaling is considered to be poor, and numerical difficulties may result during the solution phase. If the scaling ratio exceeds the value of the SCALEW parameter, LINGO will display this error message. The default value for SCALEW is 1e9. Instead of simply increasing the SCALEW setting to eliminate error 205, we strongly suggest that you attempt to rescale the units of your model so as to reduce the largest-to-smallest coefficient ratio. In some instances, changing the units of measure can be an easy way to improve a model’s scaling. For instance, suppose we have a model with the following budget constraint in dollar units: 1000000 * X + 2200000 * Y + 2900000 * Z <= 5000000; This constraint introduces several large coefficients into the model. If we rewrote the constraint so that it is in units of millions of dollars, then we would have: X + 2.2 * Y + 2.9 * Z <= 5;

842 APPENDIX B The coefficients in this new constraint are much less likely to present a problem. As part of this error message, LINGO reports the values of the largest and smallest coefficients, as well as where they appear in the model. This information should help in tracking down the problem. You may also run the LINGO|Generate command to track down other extreme coefficients. This Generate command displays the full, generated model and specifically lists all the coefficients. 206.

A MODEL MAY NOT BE SOLVED WITH LINEARIZATIONAND THE GLOBAL SOLVER SIMULTANEOUSLY ENABLED. LINEARIZATION WILL BE TEMPORARILY DISABLED. The linearization and global solver options may not be simultaneously selected when solving a model. LINGO will default to using the global solver. You can set the linearization option on the General tab of the LINGO|Options command, while the global solver option is controlled on the Global Solver tab.

207.

MISSING LEFT PARENTHESIS. A unmatched right parenthesis was encountered in the model. Use the Edit|Match Parenthesis command to help pair up parentheses.

208.

@WRITEFOR() MAY ONLY APPEAR IN A DATA AND CALC SECTIONS. The @WRITEFOR function is permitted only in the data and calc section of a model. You will need to move the expression to a data or calc section.

209.

RELATIONAL OPERATORS NOT ALLOWED IN @WRITEFOR() STATEMENTS. The @WRITEFOR function is used to display output. Constraint relational operators may not be used inside the @WRITEFOR function.

210.

INVALID USAGE OF @WRITEFOR() FUNCTION. The @WRITEFOR function is being used incorrectly. @WRITEFOR may only be used in the data and calc sections for the purpose of creating reports. Refer to the documentation for the correct usage of @WRITEFOR.

211.

ARITHMETIC ERROR IN OUTPUT OPERATION. The @WRITEFOR function is being used incorrectly. @WRITEFOR may only be used in the data section for creating reports. Refer to the documentation for the correct usage of @WRITEFOR.

212.

SUBSCRIPT OUT OF RANGE ON SET NAME: SET_NAME A subscript was found to be out of range while attempting to output a set member, . Check all output operations that refer to the set SET_NAME for correctness.

213.

TEXT OPERAND NOT PERMITTED HERE. A text argument to a function was encountered where something other than text was expected. Please check all function arguments to be sure they are correct.

214.

DUPLICATE INITIALIZATION OF A VARIABLE. A variable has been initialized more than one time in the model's data sections. Eliminate all duplicate variable initializations.

ERROR MESSAGES

843

215.

OUTPUT MODIFIERS NOT ALLOWED HERE (E.G., @DUAL) Output modifiers (e.g., @DUAL, @RANGEU, @RANGED) are not allowed here. Remove them to continue.

216.

PREFIX FUNCITON EXPECTED A TEXT ARGUMENT. Output modifiers (e.g., @DUAL, @RANGEU, @RANGED) are not allowed here. Remove them to continue.

217.

PREFIX FUNCTION EXPECTED A NUMERIC ARGUMENT. A function was expecting a numeric argument, but found something other than a numeric value. Check all function arguments for correctness.

218.

PREFIX FUNCITON EXPECTED A ROW OR VARIABLE INDEX ARGUMENT. A function was expecting a variable or row name as an arguments but found something different. Check all function arguments for correctness.

219.

UNABLE TO FIND A SPECIFIED INSTANCE OF A ROW NAME. You've requested the value of a row that does not exist in the model.

220.

UNED OPERATION ON TEXT OPERAND. You've attempted to perform an undefined operation on a text operand.

221.

ARGUMENT OVERFLOW IN @WRITE() OR @WRITEFOR(). This error occurs when there are too many text objects being output within a single @WRITE or @WRITEFOR function call. Reduce the size of any large arguments lists to these functions.

222.

A VARIABLE OR ROW REFERENCE WAS EXPECTED. A variable or row reference was expected here.

223.

A DYNAMIC RANGE EXCEEDED BOUNDARY LIMITS WRITING TO RANGE: RANGE_NAME You attempted to write more values to a spreadsheet than it can actually hold. In general, this means the dynamic range created by LINGO to receive all the values has overflowed the row limit of the spreadsheet. An option is to specify the range yourself so that it stays within the limits of the workbook.

224.

THE FOLLOWING RANGE IS TOO SMALL TO RECEIVE ALL REQUESTED OUTPUT: RANGE_NAME You've specified a workbook range that is too small to receive all the exported values. You'll need to increase the size of the range.

225.

INVALID FORMAT FOR A CALC EXPRESSION. A calc section is only for performing computations. Various functions are not permitted in the calc section (e.g., @GIN and @FREE). Remove any disallowed functions to continue.

844 APPENDIX B 226.

DEFAULT WORKBOOK NOT OPEN. If all arguments are omitted to the @OLE spreadsheet interface function, then Excel must be open with a workbook in memory (@OLE will default to using this open workbook). Open Excel and then load the workbook containing the data for your model.

227.

THE FOLLOWING RANGE: RANGE_NAME MUST HAVE A COLUMN COUNT EQUAL TO THE NUMBER OF MODEL OBJECTS BEING IMPORTED/EXPORTED. When importing values from or exporting values to a workbook you need to consider the number of model objects involved. When multiple objects are being either sent to or received from a single range in a workbook, the number of columns in the range must coincide with the number of model objects. Keep in mind that a derived set counts as more than one object depending upon its dimension. You must either adjust the size of the workbook range to the correct number of columns or edit the set of model objects involved in the interface statement.

228.

INVALID API PARAMETER INDEX. The parameter index you specified as part of the APISET command is not valid. Refer to the LINDO API documentation for a list of valid parameter indices.

229.

INVALID API PARAMETER TYPE. The parameter type you specified as part of the APISET command is not valid. The parameter type must be specified as either “int” for integer or “double” for double precision floating point. Refer to the LINDO API documentation for a list of valid parameter indices and their types.

230.

INVALID API PARAMETER VALUE. The parameter value you specified as part of the APISET command is not valid. Refer to the LINDO API documentation for a list of valid parameter indices and their permissible values.

231.

UNABLE TO SET LINDO API PARAMETER. The parameter value you specified as part of the APISET command is not valid. Refer to the LINDO API documentation for a list of valid parameter indices and their permissible values. To clear all parameter values set with APISET enter the command: APISET DEFAULT.

232.

RANGE ANALYSIS NOT AVAILABLE ON QUADRATIC PROGRAMS. LINGO cannot currently perform range analysis on quadratic programs. You’ll need to disable range analysis by running the LINGO|Options command, selecting the General Solver tab, and set the Dual Computations option to disable range analysis. On platforms other than Windows, you’ll need to set the DUALCO option to disable range analysis.

233.

THE DEBUGGING PROCESS WAS INTERRUPTED. You interrupted the model debugger, before it could complete its analysis. Given this, the resulting report will be incomplete or empty.

ERROR MESSAGES

845

234.

N LARGE NUMBER(S) WERE TRUNCATED TO A VALUE OF: <MACHINE_INFINITY>. You attempted to input one or more numbers that are too large to be handled on your hardware. LINGO reduced the numbers to your machine’s infinity. In general, you should scale your models so that very large numbers are not required.

235.

THE MODEL IS NONLINEAR AND MAY NOT BE GENERATED AS A LINEAR PROGRAM You have attempted to generate a nonlinear model with the Assume model is linear option enabled. You will need to run the LINGO|Options command, select the Model Generator tab, and disable this option. On platforms other than Windows, you will need to issue the SET LINEAR 0 command.

236.

INCORRECT @FORMAT USAGE IN AN OUTPUT STATEMENT. The arguments to an @FORMAT() reference are not valid. @FORMAT() requires one numeric argument (the numeric value to be formatted) and one text argument (the format template). For example, @FORMAT( X,

’14.5g’)

will cause X to be displayed using 5 significant digits in a field of 14 characters. 237.

RANGE ANALYSIS NOT CURRENTLY AVAILABLE ON THIS MODEL. LINGO does not have range analysis data available for the model. You’ll need to disable range analysis by running the LINGO|Options command, selecting the General Solver tab, and set the Dual Computations option to enable range analysis. You should then re-solve the model.

238.

OBJECTS DISPLAYED BY @TABLE() ARE LIMITED TO A DIMENSION OF: 16. The @TABLE() function can be used to display sets and attributes in tabular format. At the moment, these objects are limited to a maximum dimension of 16, which should be more than adequate for most models.

239.

A REPORT IS TOO WIDE TO DISPLAY. INCREASE THE PAGE WIDTH LIMIT AND TRY AGAIN. You have requested an report that is too wide for the current page width setting. You will need to increase this setting by running the LINGO|Options command, selecting the Interface tab and then increase the Width parameter in the Page Size Limits box. On platforms other than Windows, you will need to adjust the line width parameter with the SET LINLEN command.

240.

INVALID USE OF @TABLE() OUTPUT FUNCTION. The model contains an invalid reference to the @TABLE() function. The valid forms of arguments lists for @TABLE() are:    

@TABLE( AttrName|SetName) @TABLE( AttrName|SetName, HzPrimSet) @TABLE( AttrName|SetName, HzPrimSet1,…, HzPrimSetI) @TABLE( AttrName|SetName, HzPrimSet1,…, HzPrimSetK)

846 APPENDIX B 

@TABLE( AttrName|SetName, HzPrimSet1,…, HzPrimSetN, NHz)

Currently, @TABLE() output can only be routed to etiher the standard output device or text files via the @TEXT() interface function. 241.

@IFC STRUCTURE IS TOO DEEPLY NESTED. Calc sections can use @IFC/@ELSE statements to implement conditional branching. Presently, you may nest @IF statements up to 64 levels deep. You will receive this error message if your model exceeds this limit.

242.

INVALID USE OF @IFC CONSTRUCT. Calc sections can use @IFC/@ELSE statements to implement conditional branching. You will receive this error message if an @IFC is note being used correctly. An example would be an @ELSE statement appearing before a corresponding @IFC. Refer to the documentation on the use of @IFC for more information.

243.

MULTIPLE @ELSE BLOCKS WERE FOUND. Calc sections can use the @IFC/@ELSE statements to implement conditional branching. You will receive this error message if a compound @IFC statement contained more than one @ELSE branch.

244.

@IFC CONTROL STRUCTURE ONLY ALLOWED IN CALC SECTIONS. Calc sections can use the @IFC/@ELSE statements to implement conditional branching. You have attempted to use these statements outside of a Calc section. You will need to either move them to a Calc section or delete them.

245.

AN UNTERMINATED @IFC STRUCTURE WAS FOUND. Calc sections can use the @IFC/@ELSE statements to implement conditional branching. An @IFC/@ELSE compound statement is terminated with a right parenthesis, which, in this instance, is missing. You will need to add the terminating right parenthesis at the appropriate place.

246.

THE @SOLVE() FUNCTION IS ONLY VALID IN CALC SECTIONS. A number of command functions are allowed only in Calc sections, one of which is the @SOLVE() function for solving a model. You will need to either delete the function reference or move it to a Calc section.

247.

INVALID USE OF A SUBMODEL NAME. LINGO allows you to define submodels within a main model. These submodels can be solved using the @SOLVE() function in Calc sections. Your model has an invalid reference to a submodel. You must remove this reference in order to proceed.

248.

SUBMODEL NAMES MUST BE UNIQUE. LINGO allows you to define submodels within a main model using the SUBMODEL statement. These submodels can be solved using the @SOLVE() function in Calc sections. You have attempted to use the same submodel name more that once. You must change the names of the submodels so that all the names are unique.

ERROR MESSAGES

847

249.

SUBMODEL NOT VALID HERE. LINGO allows you to define submodels within a main model using the SUBMODEL statement. These submodels can be solved using the @SOLVE() function in Calc sections. You have attempted to locate a submodel in an invalid section of the model. Please move the submodel to the main part of the model, outside of all Data, Calc and Init sections.

250.

SUBMODEL NOT DEFINED YET. LINGO allows you to define submodels within a main model using the SUBMODEL statement. These submodels can be solved using the @SOLVE() function in Calc sections. You have referenced a submodel that has not been defined. Submodels must be defined in the model before any occurrences of references to them. You may be able to solve this error by moving the submodel in question to an earlier position in the model text.

251.

MODEL CONTAINS AN UNTERMINATED SUBMODEL STATEMENT. LINGO allows you to define submodels within a main model using the SUBMODEL statement. These submodels can be solved using the @SOLVE() function in Calc sections. Your model contains a submodel without a terminating ENDSUBMODEL statement.

252.

A VARIABLE OR ROW REFERENCE WAS EXPECTED. LINGO was expecting a reference to a row or variable name. You will need to correct the problem to continue.

253.

OPERATION ALLOWED ONLY IN CALC SECTIONS. A number of operators are allowed only in Calc sections. An example would be the @SOLVE() function. You will need to either delete the function reference or move it to a Calc section.

254.

INVALID PARAMETER ARGUMENT FOR @SET( PARAM, VALUE). The @SET() function can be used in a Calc section to set any of LINGO’s parameters . You must provide the parameter’s name as text, and a permissible value for the parameter. For instance, to turn set the iteration limit to one million, you would use @SET( ‘ITRLIM’, 1.e6). You can refer to documentation on the SET command to learn all the parameters and their names.

255.

IMPROPER USE OF A CALC COMMAND FUNCTION. You may include a Calc section in your model that contains a series of commands for LINGO to execute. Examples would include the @SOLVE() and @WRITE() command functions. This error is triggered when one of these command functions is used incorrectly. You should refer to the documentation to learn the correct usage and syntax for the function.

256.

@WHILE() ALLOWED ONLY IN CALC SECTION. At present, the @WHILE() statement is allowed only in calc sections and may not appear in the model section. You may be able to use the @FOR() statement instead.

257.

INVALID ARGUMENT FOR @SET( ‘PARAM’, VALUE). In order to set a parameter via the @SET command you must specify a correct parameter name in quotes and a correct value. Please be sure you have spelled the parameter name

848 APPENDIX B correctly and that you have placed it in quotes. Also, check that the specified value is permissible for the parameter. 258.

MODEL EXECUTION HALTED. STOP STATEMENT ENCOUNTERED. LINGO encountered an @STOP() statement while processing a calc section. This terminates execution of the current model. If a text string was specified in the reference to @STOP(), then it will also be displayed as part of this message. @STOP is typically used in response to some unexpected condition.

259.

FILE NAME LENGTH EXCEEDS: N You’ve specified a file name whose length exceeds the maximum length allowed of N characters. You will need to shorten the length of the file name to continue.

260.

@BREAK() MAY ONLY APPEAR IN @WHILE() AND @FOR() LOOPS INSIDE CALC SECTIONS The @BREAK() statement is used for unconditional breaks out of @FOR() and @WHILE() loops. It may also only be used inside calc sections, and is not valid in the model section. You will need to delete the reference to @BREAK() to continue.

261.

INVALID ARGUMENT FOR @RELEASE(). ARGUMENTS MUST BE A SINGLE VARIABLE REFERENCE, E.G., @RELEASE( X). The @RELEASE() statement is used in calc sections to release a variable that was previously fixed so it may once again become optimizable. To release an entire attribute, place the @RELEASE() reference in an @FOR() loop, e.g., @FOR( SET( I): @RELEASE( X( I))).

262.

INVALID ARGUMENT FOR @APISET( PARAM-ID, ‘INT|DOUBLE’, VALUE). The @APISET() statement is used in calc sections to set options in the LINDO API (LINGO’s solver engine) that aren’t available through the standard option set in LINGO. The argument list consists of a parameter-id (an integer value), a string specifying if the parameter is an integer or a double precision value, and the parameter value. More information on the parameters available in the LINDO API may be found in the LINDO API documentation and the Lindo.h header file included with your LINGO installation.

263.

MODEL WAS FOUND TO BE FEASIBLE AND CANNOT BE DEBUGGED. You have run the LINGO|Debug to find the infeasibilities in a model that was found to be feasible. In which case the model debugger will stop its search for an infeasible subset of constraints.

264.

LINEARIZED MODEL TOO LARGE FOR THIS VERSION. You have enabled LINGO's linearization option, which attempts to rewrite nonlinear models in an equivalent, but linear, manner. This requires the addition of variables and constraints, whose increased counts have exceeded the limits of your version. You will need to either disable linearization or a larger capacity version of LINGO.

265.

DUPLICATE VARIABLES WERE FOUND IN THE SEMI-CONTINUOUS SET. LINGO s semi-continuous (SC) variables. SC variables are restricted to being either 0, or to lie within some non-negative range. Semi-continuous variables are declared using the @SEMIC statement. A variable may only be declared semi-continuous once in a model.

ERROR MESSAGES

849

You will receive this error message if one or more variables are declared semi-continuous more than one time. 266.

BOUNDS OUT OF RANGE ON A SEMI-CONTINUOUS VARIABLE. LINGO s semi-continuous (SC) variables. SC variables are restricted to being either 0, or to lie within some non-negative range. Semi-continuous variables are declared using the @SEMIC statement. The syntax for @SEMIC is: @SEMIC( lower_bound, variable, upper_bound) You will receive this error whenever the variable bounds fall outside the range permitted by LINGO.

267.

@SPCHANCE, @SOS OR @CARD SET LABEL EXCEEDS LIMIT OF: LINGO s chance-constrained constraint sets, special ordered sets (SOS) of variables as well as cardinality sets of variables. Each set is denoted by a unique name/label, which is limited to characters in length. You will need to choose a shorter label name for the variable set.

268.

VARIABLE SET LABEL USED TO REPRESENT MORE THAN ONE SET TYPE. LINGO s chance-constrained constraint sets, special ordered sets (SOS) of variables as well as cardinality sets of variables. Each set is denoted by a unique label/name. You have attempted to use the same set label for different set types (e.g., SOS1 and SOS2). Check to be sure you've specified the correct set name and/or set type.

269.

THE FIRST ARGUMENT TO @SOS/@CARD MUST BE A STRING. LINGO s special ordered sets (SOS) of variables as well as cardinality sets of variables. Each set is denoted by a unique label/name. The first argument to the @SOS and @CARD functions must be a string representing the label name for the particular variable set. Furthermore, the label name must be contained in single or double quotation marks.

270.

THE SOS/CARD SET: <SET_NAME> CONTAINS THE FIXED VARIABLE: LINGO s special ordered sets (SOS) of variables as well as cardinality sets of variables. All variables belonging to these sets must not be fixed, i.e., they must be optimizable decision variables. You will need to remove the fixed variable from the designatged set.

271.

NO CARDINALITY WAS SPECIFIED FOR @CARD SET: <SET_NAME> LINGO s cardinality sets of variables. All variables belonging to a particular cardinality set must sum to a specified integer value. To form a cardinbality set for variable X, Y and Z that sums to 2, you would specify the following in your model: @CARD( @CARD( @CARD( @CARD(

'MYSET', 'MYSET', 'MYSET', 'MYSET',

X); Y); Z); 2);

850 APPENDIX B You will receive this error message when the cardinality value, @CARD( 'MYSET', 2) in this case, is omitted. You will need to add a cardinality value reference for the set. 272.

DUAL FORMULATIONS MAY ONLY BE GENERATED FOR LINEAR PROGRAMS. LINGO can generate the dual formulation for linear programming models. You have attempted to generate the dual for a model that is not a linear program, i.e., the model is either nonlinear or it contains integer variables. If it is possible to rewrite the model so that it becomes linear without any integer variables, then you should be able to generate the dual formulation. Otherwise, it will not be possible to generate the dual formulation.

273.

THE LICENSE MANAGER RETURNED THE FOLLOWING ERROR: LINGO's license manager was unable to find a valid license file. The license file is normally located in your main LINGO subdirectory under the name: lndlng .lic, where is the version number of your LINGO installation. You may want to examine this file for any obvious errors or problems. The license manager will also return a brief string with more specific information (e.g., "No dongle found"), which will be displayed as part of this message. This additional information should help you narrow down the problem. If you are still unable to resolve this problem, please LINDO Systems Technical for assistance.

274.

THE MODEL CONTAINS FUNCTIONS NOT ED BY THE GLOBAL SOLVER. The model contains functions not ed by the global solver. You will need to either remove these functions or disable the global solver options. The global solver s most of the functions of the LINGO language. However, there are few that aren’t currently, or can’t be ed. You may refer the Use Global Solver section of Chapter 5 for a listing of functions not ed by the global solver.

275.

A BARRIER LICENSE IS REQUIRED TO RUN BARRIER IN A CORE. LINGO can run multiple linear programming solvers in parallel on multi-core machines. You have requested to run the barrier solver in parallel, but your installation does not have a license for the barrier solver. You will need to run the LINGO|Options command, go to the Linear Solver tab, and disable the request to run the barrier solver in parallel.

276.

PRIMAL2 MAY NOT BE RUN WITHOUT BARRIER RUNNING IN PARALLEL. LINGO can run multiple linear programming solvers in parallel on multi-core machines. You have requested to run the Primal2 solver in parallel without also requesting the barrier solver. You will need to run the LINGO|Options command, go to the Linear Solver tab, and either enable the barrier solver to run parallel or disable Primal2 from running in parallel.

277.

INCONSISTENT BOUNDS ON VARIABLE: IN THE SOS/CARD SET: <SET> LINGO s special ordered sets (SOS) of variables as well as cardinality sets of variables. You have specified bounds on variables in the designated set that are inconsistent

ERROR MESSAGES

851

with variables belonging to such a set. Check to be sure you've specified the bounds correctly. 278.

ERROR IN DYNAMIC SET DEFINITION FOR SET: SET_NAME. In some instances, you may not need to explicitly define a set. Instead, you can have LINGO infer the dimensions of the set from the number data points assigned to an attribute of the set in the model's Data section. This is referred to as a dynamic set definition. It's not always possible for LINGO to accurately deduce a set's dimension from a data statement, in which case, you will receive this error message. A valid solution is to explicitly define the set in question in the model's Sets section.

279.

AN UNALLOCATED SET WAS REFERENCED: <SET_NAME> You've referenced of a set that has not been allocated. You must allocate sets in a model before they can be referenced. Many times, all that is required is to move the set's allocation to a point in the model that precedes any references to it. Set allocation refers to the declaration of the set's . Allocation is generally done in a sets section or in a data section.

280.

INVALID SYNTAX FOR A CALC SECTION I/O OPERATION. You have incorrectly specified an I/O function (@TEXT, @OLE, @ODBC, @POINTER) in a calc section. Please refer to the documentation for the correct usage of these functions.

281.

I/O OPERATIONS MAY ONLY APPEAR IN CALC, DATA AND INIT SECTIONS. You have attempted to us an I/O function (@TEXT, @OLE, @ODBC, @POINTER) in the model section. These fucntions are available only in calc, data and init sections of a model. Please refer to the documentation for the correct usage

282.

SETS MAY ONLY BE INITIALIZED IN SETS AND DATA SECTIONS. You have attempted to initialize a set outside of a set and data section. You will need to move the initiation statement to the correct model section.

283.

TEXT AND NUMERIC OBJECTS CAN'T BE COMBINED IN A SINGLE @POINTER OPERATION. You have attempted to reference both text and numeric objects in a single @POINTER statement. A given reference to the @POINTER function can only import/export items of one type, i.e., text or numeric. You will need to split the statement up into two or more statements, such that each new statement only references either text or only references numeric objects.

284.

INVALID @POINTER SET I/O OPERATION. @POINTER CAN ONLY REFERENCE ONE SET AT A TIME. You have attempted to reference both more than one set in a single @POINTER statement. A given reference to the @POINTER function can only import/export one set at a time. You will need to split the statement up into two or more statements, such that each new statement only references no more than one set.

285.

INCORRECT USAGE OF @RANK. You used the @RANK sorting function incorrectly. Proper usage is:

852 APPENDIX B ATTR_RANK = @RANK( ATTR1[,...,ATTRN); where ATTR_RANK receives the ranking of attribute ATTR1, with ties being broken by the optionally supplied attributes ATTR2 through ATTRN. In addition, @RANK is only valid in calc sections. Please refer to the documentation on @RANK for more details. 286.

INCORRECT USAGE OF @SOLU. You used the @SOLU sorting solution report function incorrectly. This function displays the standard LINGO solution report and may be called in calc sections. Proper usage is: @SOLU( [0|1[, OBJECT_NAME[, ''HEADER_TEXT'']]]); where the first argument is set to 1 if only nonzero variable values are desired (and binding rows), OBJECT_NAME is the optional name of an object to report on (the entire report will be generated otherwise), and the optionalHEADER_TEXT is a header line to display in the report. Please refer to the documentation on @SOLU for more details.

287.

INCORRECT USAGE OF @SPSTGVAR/@SPSTGRNDV/@SPSTGROW. You used either the @SPSTGVAR, @SPSTGRNDV or @SPSTGROW function incorrectly. Proper usage is: @SPSTG[VAR | RNDV | ROW]( STAGE, VARIABLE_OR_ROW_NAME); where the first argument is the stage of the variable or row and the second argument the name of the variable or row.

288.

INVALID STAGE VALUE. An invalid stage value was specified. Stage values must be non-negative, integer values. In addition, random variables must have stages of 1, or higher. Please correct the indicated stage value and retry.

289.

STOCHASTIC PROPERTIES MAY ONLY BE ASSIGNED ONCE. You may assign stochastic properties to variable and rows one time only. Please remove multiple declarations and retry.

290.

INVALID DECLARATION OF A STOCHASTIC DISTRIBUTION. A stochastic distribution declaration was incorrectly specified. Please refer to the Declaring Distributions section of Chapter 12 for clarification.

291.

NAME IS TOO LONG. The indicated name in the model exceeds the maximum name length of 64 characters. You must shorten the name to continue.

292.

NAME ALREADY IN USE. The indicated name is already in use. To continue, you will need to change the.name so that it is unique.

293.

INVALID DISTRIBUTION PARAMETER. The indicated parameter is invalid for the chosen probability distribution. Please refer to the Declaring Distributions section of Chapter 12 for clarification.

ERROR MESSAGES 294.

853

INVALID USE OF @SPTABLEOUTC. The @SPTABLEOUTC function for declaring discrete distribution outcomes was used incorrectly. The correct usage is: @SPTABLEOUTC( 'DIST_NAME', OUTC_PROB, OUTC_VAL1[ ,..., OUTC_VALN]) For more information, refer to the Declaring Distributions Using Outcome Tables in Chapter 14.

295.

OUTCOME TABLE NAME NOT PREVIOUSLY DEFINED. You have attempted to use the designated outcome table name before defining it. For more information on defining outcome tables, please refer to the Declaring Outcome Tables section of Chapter 14. Note that the declaration of an outcome table must occur before the name is used for other purposes.

296.

INCORRECT NUMBER OF OUTCOMES FOR OUTCOME TABLE. When declaring an outcome table using @SPDISTTABLE you declare its total number of outcomes. The number of outcomes loaded via @SPTABLEOUTC does not match this total number for the specified table. Please check the declaration of the table and the calls to @SPTABLEOUTC. For more information on defining outcome tables, please refer to the Declaring Outcome Tables section of Chapter 14.

297.

INVALID USE OF @SPTABLERNDV. The @SPTABLERNDV is used to associate random variables with an instance of a discrete outcome table. The format of @SPTABLERNDV is: @SPTABLERNDV( 'DIST_INSTANCE_NAME', RANDOM_VAR_1[, ... , RANDOM_VAR_N]); Please check to see that the arguments conform to this syntax. For more information on defining outcome tables and using @SPTABLERNDV, please refer to the Declaring Outcome Tables section of Chapter 14.

298.

INCORRECT NUMBER OF RANDOM VARIABLES ASSIGNED TO DISTRIBUTION. The @SPTABLERNDV is used to associate random variables with an instance of a discrete outcome table. The format of @SPTABLERNDV is: @SPTABLERNDV( 'DIST_INSTANCE_NAME', RANDOM_VAR_1[, ... , RANDOM_VAR_N]); When the parent outcome table was defined using @SPDISTTABLE, the number of associated random variables was also declared. The number of variables associated with the specified outcome table instance does not match this declared number. For more information on defining outcome tables and using @SPTABLERNDV, please refer to the Declaring Outcome Tables section of Chapter 14.

854 APPENDIX B 299.

VARIABLES ASSIGNED TO DISTRIBUTIONS MUST BE DESIGNATED AS RANDOM VARIABLES. All variables associated with a probability distribution must first be declared as random variables using the @SPSTGRNDV function. For more information on identifying random variables and associating them with distributions, please refer to the Language Features for SP Models section in Chapter 14.

300.

RANDOM VARIABLES CAN ONLY BE ASSIGNED TO ONE DISTRIBUTION. Only one random variable may be associated with a probability distribution. For more information on identifying random variables and associating them with distributions, please refer to the Language Features for SP Models section in Chapter 14.

301.

INVALID USE @SPTABLEINST. The @SPTABLEINST function is used to declare a particular instance of an outcome table. The syntax for @SPTABLEINST is as follows: @SPTABLEINST( 'PARENT_TABLE_NAME', 'TABLE_INSTANCE_NAME') The parent table must have already been declared using @SPDISTTABLE. For more information on declaring outcome tables and using @SPTABLEINST, please refer to the Declaring Outcome Tables section of Chapter 14.

302.

INVALID USE @SPSAMPSIZE. The @SPSAMPSIZE function is used to declare a sample size for a particular stage of a stochastic programming model. The syntax for @SPSAMPSIZE is as follows: @SPSAMPSIZE( STAGE, SAMPLE_SIZE) For more information on using @SPSAMPSIZE, please refer to the Declaring Parametric Distributions section of Chapter 14.

303.

PROBABILITIES DO NOT SUM TO 1 FOR OUTCOME TABLE. Outcome tables are loaded with outcomes and their respective probabilities using the @SPDISTOUTC function. You will receive this error message if the sum of the probabilities for all outcomes for the specified outcome table do not sum to 1. You should check the probability values for the specified distribution. For more information on defining outcome tables and using @SPTABLEOUTC, please refer to the Declaring Outcome Tables section of Chapter 14.

304.

NO OUTCOMES DECLARED FOR OUTCOME TABLE. Outcome tables are loaded with outcomes and their respective probabilities using the @SPDISTOUTC function. You will receive this error message if no outcomes were declared for the specified outcome table. You will need to load at least one or more outcomes into the specified table instance. For more information on defining outcome tables and using @SPTABLEOUTC, please refer to the Declaring Outcome Tables section of Chapter 14.

ERROR MESSAGES 305.

855

NO RANDOM VARIABLES ASSIGNED TO STAGE. Each stage other than stage 0 of a stochastic programming model must have at least one random variable assigned to it. Random variables are assigned to stages using the @SPSTGRNDV function, which has the following syntax: @SPSTGRNDV( STAGE, VARIABLE_NAME); You will need to assign a random variable to the specified stage or reduce the number of stages in the model. For more information on identifying random variables and associating them with distributions, please refer to the Language Features for SP Models section in Chapter 14.

306.

SP CORRELATION TYPES CAN'T BE MIXED IN THE SAME MODEL. There are three methods available for inducing correlations between random variables Pearson, Kendall and Spearman. You may use only one method throughout an entire model. Trying to use more than one particular method will trigger this error message.

307.

STOCHASTIC PROGRAMS MUST HAVE AT LEAST TWO STAGES. Stochastic programs must have two, or more, stages. The first stage, stage 0, is the initial decision stage. The second stage, stage 1, is a recourse decision stage which occurs after the stage 1 random variable values have been revealed. Subsequent rounds of random variable revelations and recourse decisions are permitted but not required. For more information on adding random variables and stages to an SP model, please refer to the Language Features for SP Models section in Chapter 14.

308.

LINEARIZATION OPTION NOT PERMITTED ON STOCHASTIC PROGRAMS. LINGO has the capability to reformulate certain nonlinear expressions so that they become linear, while maintaining mathematical equivalency to the original model. This linearization option is only available on deterministic models. If you enable the linearization option on SP models, you will receive this error message. An option is to enable the global solver instead (assuming your license allows for this optional feature). If the SP model can be completely linearized, the global solver will realize this and perform the linearization, thereby allowing for a globally optimal solution.

309.

THE DETEQ MODEL GENERATED BY THE SP SOLVER EXCEEDS THE CAPACITY OF THIS VERSION. LINGO's SP solver generates a deterministic model to solve the original SP model. This model is called the deterministic equivalent (deteq). Depending on the number of stages, random variables and samples sizes, the deteq may get to be very large. In which case, there is a possibility that it will exceed the capacity of your LINGO license. This can occur even when the core model lies well within the capacity of your license. You will need to either reduce the size of the model (by using fewer stages, less random variables or smaller sample sizes), or a license with more capacity.

310.

INVALID SP CORRELATION MATRIX. The correlation matrix entered for the random variables using @SPCORREL is not valid. Please check the correlations to see that they lie within a range of [-1,+1]. Also check to see

856 APPENDIX B that explicit correlation values lie within any implied range, e.g., CORR( X, Y) = .9 and CORR( Y, Z) = .9 implies that CORR( X, Z) is at least .62. 311.

ATTEMPT TO CORRELATE NON-RANDOM VARIABLES. In a stochastic programming (SP) model, only random variable may be correlated via the @SPCORREL function. In addition, the correlated variables must be flagged as random variables using the @SPSTGRNDV function prior to inputting their correlation coefficient using @SPCORREL.

312.

INVALID CORRELATION COEFFICIENT. You've input an invalid correlation coefficient that lies outside [-1,+1]. Please correct the coefficient's value and try again.

313.

SCENARIO INDICES MUST BE INTEGRAL. In referring to a scenario, you've attempted to use an out-of-range index. In other words, the index is either less than 1 or greater than the total number of scenarios. Please correct the index and retry.

314.

SCENARIO INDEX OUT OF RANGE. The scenario index must be an integer greater-than-or-equal-to 1 and may not exceed the total number of scenarios in the model.

315.

THE FOLLOWING SET MEMBER: SET-MEMBER APPEARS MORE THAN ONCE IN SET: SET-NAME. The indicated set member, SET-MEMBER, appears more than one in its set SET-NAME. Please correct the duplicated set to continue.

316.

UNABLE TO GENERATE DETERMINISTIC EQUIVALENT. LINGO's SP solver generates a deterministic model to solve the original SP model. This model is called the deterministic equivalent (deteq). Depending on the number of stages, random variables and samples sizes, the deteq may get to be very large. There is a remote chance that LINGO will not be able to generate the deteq. This would be a very unusual situation. If there is no obvious reason for this problem, you may wish to LINDO Systems Technical desk for assistance.

317.

THE MODEL IS NOT A STOCHASTIC PROGRAM. You've attempted to request information relevant only to stochastic programs while running a deterministic model.

318.

THIS OPERATION REQUIRES THE STOCHASTIC PROGRAMMING OPTION. You're attempting to solve a stochastic program (SP) on a LINGO installation that does not the SP option. You will need to upgrade your license to include the option.

319.

ONE OR MORE SP SCENARIO SOLUTIONS DO NOT EXIST. LINGO was unable to generate solutions for all the models scenarios. This would be a very unusual situation. If there is no obvious reason for this problem, you may wish to LINDO Systems Technical desk for assistance.

ERROR MESSAGES

857

320.

@SPLOADSCEN() FAILED FOR SCENARIO: N. LINGO was unable to load the solution for scenario N via the @SPLOADSCEN function. This would be an unusual situation. If there is no obvious reason for this problem, you may wish to LINDO Systems Technical desk for assistance.

321.

UNABLE TO GENERATE INDIVIDUAL SCENARIO MODEL. LINGO was unable to generate a requested individual scenario for an SP mode. This would be an unusual situation. If there is no obvious reason for this problem, you may wish to LINDO Systems Technical desk for assistance.

322.

@CHART NOT ED IN LINGO API. The @CHART function for generating charts is only ed in the interactive version of LINGO. If you need charting in an application that call the LINGO API, then you will need to find a third party charting tool.

323.

THE FOLLOWING RANDOM VARIABLE IS NOT ASSIGNED TO A STAGE. Each random variable declared with @SPSTGRNDV must also be assigned to a stochastic distribution. LINGO will display the name of the first unassigned random variable. You will need to use either @SPDISTTABLE or @SPTABLERNDV to assign the variable to a distribution.

324.

THE FOLLOWING RANDOM VARIABLE ATTRIBUTE HAS ELEMENTS BELONGING TO DIFFERENT STAGES. The attribute in question was assigned to an outcome table using @SPDISTTABLE. All elements of the attribute must be random variables and must all be assigned to the same stage using the @SPSTGRNDV function. Please check to see that all the elements of the attribute have been assigned to the correct and identical stage.

325.

THE FOLLOWING PROBABILITY OR OUTCOME ATTRIBUTE CONTAINS NON-CONSTANT ELEMENTS. The attribute in question was used to construct an outcome table distribution using @SPDISTTABLE. All elements of outcome and probability attributes must be constants and may not be optimizable. Please assign appropriate values to each element of the attribute.

326.

THE SP SAMPLE SIZES ARE NOT LARGE ENOUGH TO GENERATE A MEANINGFUL SAMPLE. The sample sizes in your SP model are not large enough to allow for generation of a meaningful sample of the random variables. Note that models that induce correlation between random variable will require larger sample sizes. You will need to use the @SPSAMPSIZE function to increase the sample sizes.

327.

UNED MPI INSTRUCTION CODE: N You are attempting to read an MPI format file that contains an instruction code not currently ed by LINGO. LINGO s most of the frequently used MPI instruction codes, however, it does not 100% of the codes offered by the MPI format. In order to continue, you will need to either remove the uned function code or rewrite the expression using ed instructions codes.

858 APPENDIX B 328.

MULTIPLE WILDCARD CHARACTERS (*) NOT ED. You are attempting to use the multi-character wildcard (*) more than once in a field. You will need to either eliminate all but one asterisk or use multiple single-character wildcards (%).

329.

ERROR EXECUTING AN @CHART FUNCTION. An @CHART function failed to draw the requested chart. This is typically due to lack of memory. Try closing some windows and /or other applications and then run the model again.

330.

INVALID CHART TYPE. You've requested an uned chart type. Currently ed types are: Bar, Bubble, Contour, Curve, Histogram, Line, Pie, Radar, Scatter, or Surface.

331.

INVALID ARGUMENT FOR AN @CHART FUNCTION. You've specified an invalid argument for an @CHART function. Please refer to the @CHART documentation for information on valid argument syntax.

332.

@CHART NOT ED IN LINGO API. The @CHART function for generating charts is only ed in the interactive version of LINGO. If you need charting in an application that call the LINGO API, then you will need to find a third party charting tool.

333.

NOT USED Error code not currently in use.

334.

NO RECOURSE VARIABLES ASSIGNED TO STAGE: N You have a stochastic program without recourse variables in stage N. You will need to use the @SPSTGVAR function to assign one or more recourse variables to stage N.

335.

INVALID @SPCHANCE USAGE. Chance-constrained programs (Cs) are stochastic models that allow one or more sets of constraints to be violated up to a specified percentage of total scenarios. You identify these constraint sets with the @SPCHANCE function. Refer to the documentation on @SPCHANCE for information on proper syntax.

336.

CHANCE-CONTRAINED PROGRAMS CAN CURRENTLY HAVE ONLY ONE STAGE. Chance-constrained programs (Cs) are stochastic models that allow one or more sets of constraints to be violated up to a specified percentage of total scenarios. At present, the C solver can handle only single stage models.

337.

THE FOLLOWING CHANCE-CONSTRAINED SET IS EMPTY: SET-NAME Chance-constrained programs (Cs) are stochastic models that allow one or more sets of constraints to be violated up to a specified percentage of total scenarios. You've established a C set with the @SPCHANCE function but have assigned any constraints to the set. Refer to the documentation on @SPCHANCE for information on proper syntax.

ERROR MESSAGES

859

338.

THE CHANCE-CONSTRAINED SET NAME IS ALREADY IN USE. Chance-constrained programs (Cs) are stochastic models that allow one or more sets of constraints to be violated up to a specified percentage of total scenarios. You may establish C sets with the @SPCHANCE function but set names must be unique and may not be reused. Refer to the documentation on @SPCHANCE for information on establishing C sets.

339.

THE CHANCE-CONSTRAINED SET NAME WAS NOT PREVIOUSLY DEFINED. Chance-constrained programs (Cs) are stochastic models that allow one or more sets of constraints to be violated up to a specified percentage of total scenarios. Refer to the documentation on @SPCHANCE for information on defining a C set.

340.

THE CHANCE-CONSTRAINED ROW: ROW-NAME DOES NOT CONTAIN ANY RANDOM VARIABLES. Chance-constrained programs (Cs) are stochastic models that allow one or more sets of constraints to be violated up to a specified percentage of total scenarios. Each constraint in a C set must contain at least one random variable. You will need to use the @SPRNDVAR function to identify at least one random variable in the row.

341.

THE OBJECTIVE ROW MAY NOT BE ASSIGNED TO A CHANCECONSTRAINED SET. Chance-constrained programs (Cs) are stochastic models that allow one or more sets of constraints to be violated up to a specified percentage of total scenarios. Note that a model's objective function may not be assinged to a C set  only constraints may be assigned to a C set.

342.

THE 'ASSUME MODEL IS LINEAR' OPTION MAY NOT BE APPLIED TO STOCHASTIC PROGRAMS. The 'Assume Model is Linear' option is an option available for linear models that reduces peak memory usage. This option is currently not available for stochastic and chanceconstrained programs.

343.

STOCHASTIC MODELS MUST HAVE AT LEAST TWO SCENARIOS. A stochastic program must have at least two scenarios. Without at least two scenarios, the model would have no stochastic properties. You will need to increase the sample size using the @SPSAMPSIZE function.

344.

A NON-CHANCE-CONSTRAINED ROW CONTAINS RANDOM VARIABLES: ROW-NAME Chance-constrained programs (Cs) are stochastic models that allow one or more sets of constraints to be violated up to a specified percentage of total scenarios. Each random variable in a C must appear in a C constraint set. TYou will get this error message when a random variable is found in a row not contained in any C set. Refer to the documentation on @SPCHANCE for information on how to add a constraint to a C set.

345.

THE FOLLOWING ROW HAS MORE THAN ONE C ASSIGNMENT. Chance-constrained programs (Cs) are stochastic models that allow one or more sets of constraints to be violated up to a specified percentage of total scenarios. You may establish

860 APPENDIX B C sets with the @SPCHANCE function, but each constraint may appear in no more that one C set. Refer to the documentation on @SPCHANCE for more information on establishing C sets. 346.

AN INVALID SET REFERENCE WAS FOUND INSIDE THE MODEL'S CONSTRAINTS. There is a reference to a set or set member in a constraint. You will need to remove this reference in order to continue.

347.

AN INVALID/UNED FUNCTION CODE WAS ENCOUNTERED. The model was found to contain an uned function. In general, this error message indicates a problem in processing the model. You should technical if you encounter this error.

348.

UNABLE TO COMPLETE ALTER COMMAND -- PROCESSING TERMINATED. An ALTER command to edit the model was too complicated for execution. You should simplify the command by breaking it up into multiple, more compact commands.

349.

THE FOLLOWING VARIABLE VIOLATES ITS BOUNDS: This error message indicates that a variable whose value was set in either a calc or data section was found to violate its bounds. You can either change the variable's value or modify its bounds to include the variable's value. Another solution is to disable the Enforce bounds in CALC and DATA option on the Model Generator tab of the LINGO|Options command.

350.

THE FOLLOWING VARIABLE VIOLATES ITS INTEGRALITY CONDITION: This error message indicates that a variable whose value was set in either a CALC or DATA section was found to violate an integrality condition set with either the @BIN or @GIN functions. You can either change the variable's value to an integer or disable the Enforce bounds in CALC and DATA option on the Model Generator tab of the LINGO|Options command.

351.

THE MULTISTART OPTION MAY NOT BE USED IN CONJUNCTION WITH SLP. You have selected both the Multistart Solver and the Successive Linear Programming (SLP) Solver options. These two solvers may not be used simultaneously. In order to continue, LINGO will disable the Multistart option. You will need to disable at least one of these two options in order to eliminate this error message.

352.

NESTED CALC AND/OR PROCEDURE SECTIONS NOT ALLOWED. Calc sections and/or model procedures may not be nested inside one another. You will need to separate them into individual sections.

353.

THE LIMIT ON RECURSIVE PROCEDURE CALLS IS: N There is a limit on the number of times a procedure may call itself. You will need to reduce the number of recursive calls.

ERROR MESSAGES

861

354.

DEBUGGING CURRENTLY NOT ED FOR STOCHASTIC MODELS. The model debugger may not currently be used on stochastic models. As an alternative, you may generate the stochastic equivalent of the model and debug it. One common problem that makes stochastic models infeasible is having continuous random variables that can take on negative values. These negative values may get assigned to other variables with default lower bounds of zero, making the stochastic model infeasible.

355.

INVALID SET MEMBER FOR INSERTION INTO SET: SET-NAME An attempt has been made to insert an invalid set member into set SET-NAME using the @INSERT function. Set may only be inserted into derived sets, and the member being inserted must be a member of the derived set's parent primitive set(s). The following example illustrates: MODEL:

SETS: S1 /M1, M2, M3/; S2( S1); ENDSETS CALC !The following is valid because M2 is a member of S2's parent primitive set, S1; @INSERT( S2, M2); !The following is not valid because M4 is not a member of S1; @INSERT( S2, M4); !The following is not valid because S1 is not a derived set; @INSERT( S1, M4); ENDCALC END

862 APPENDIX B 356.

INSERTING INTO PRIMITIVE SETS IS NOT CURRENTLY ED. An attempt has been made to insert a set member into a primitive set using the @INSERT function. Set may only be inserted into derived sets, and the member being inserted must be a member of the derived set's parent primitive sets. The following example illustrates: MODEL: SETS: S1 /M1, M2, M3/; S2( S1); ENDSETS CALC !The following is valid because M2 is a member of S2's parent primitive set, S1; @INSERT( S2, M2); !The following is not valid because M4 is not a member of S1; @INSERT( S2, M4); !The following is not valid because S1 is not a derived set; @INSERT( S1, M4); ENDCALC END

357.

THE ENTIRE MODEL IS FIXED. YOU MAY NEED TO DISABLE FIXED VARIABLE REDUCTION. When LINGO exports a model to either MPS of MPI format, it first substitutes out any fixed variables in the model. The purpose is to export a model that exactly matches the one LINGO is solving internally. In this instance, all the variables were found to be fixed in value, leaving no model to export. You may disable fixed variable reduction with the LINGO | Options | Model Generator | Fixed Var Reduction option. Keep in mind, however, that turning off variable reduction may result in a nonlinear model, which may not be represented in MPS format.

358.

IMPROPER USE OF @BLOCKROW( BLOCKID, ROWNAME) FUNCTION. The @BLOCKROW function is used in conjunction with the branch-and-price (BNP) solver to assign rows to blocks. The BNP solver requires that a model be partitioned into two or more blocks of mostly independent rows. The BLOCKID argument is a non-negative integer, indicating the block index, while the ROWNAME argument is the name of a row, perhaps indexed, that appears in the model. For more information on @BLOCKROW, please refer to the BNP solver documentation.

ERROR MESSAGES

863

359.

UNABLE TO FIND A BLOCK PARTITIONING. The branch-and-price (BNP) solver requires that a model be partitioned into two or more blocks of mostly independent rows. This error typically occurs if you are explicitly providing the block structure (as opposed to letting LINGO find the block structure) and an input error was found in the block assignments. For more information on @BLOCKROW, please refer to the BNP solver documentation.

360.

INVALID PROCEDURE REFERENCE. Procedures may only be called from within calc sections and may not be embedded in other statements. Refer to the documentation on procedure calls for more information.

361.

INVALID USE OF @INTEGRAL. You have not correctly set up a call to LINGO's numeric integration function, @INTEGRAL. The correct usage is: @INTEGRAL( PROCEDURE, X, LB_X, UB_X, Y); where PROCEDURE refers to the name of the procedure containing the function to integrate, X is the variable to integreate over, LB_X and UB_X are the lower and upper bounds on X, and Y is the variable that stores the value of the function being integrated. Refer to the @INTEGRAL documentation for more information.

362-1000. NOT USED. Note:

Error messages 1001 through 1016 pertain only to the Windows version of LINGO.

1001.

NO MATCHING PARENTHESIS WAS FOUND. LINGO did not find a closing parenthesis for the selected parenthesis. Add a closing parenthesis and try again.

1002.

UNABLE TO SOLVE ... NO MODEL CURRENTLY IN MEMORY. LINGO does not have the text of a model available to attempt to solve. Select a model window and try solving again.

1003.

LINGO IS BUSY AND CAN’T COMPLETE YOUR REQUEST RIGHT NOW. TRY AGAIN LATER. You have attempted an operation that is not possible because LINGO is currently solving a model. To halt the solver, press the Interrupt Solver button on the solver status window.

1004.

ERROR WRITING TO LOG FILE. PERHAPS THE DISK IS FULL? LINGO can’t write any additional output from the command window to an open log file. Check to see if there is any free space remaining on the disk.

1005.

UNABLE TO OPEN LOG FILE. LINGO could not open the log file that you requested. Does the path exist? Do you have write access to the selected path?

1006.

COULD NOT OPEN FILE: FILENAME. LINGO could not open a requested file. Did you spell the file’s name correctly? Does the file exist?

864 APPENDIX B 1007.

NOT ENOUGH MEMORY TO COMPLETE COMMAND. LINGO was unable to allocate additional system memory while attempting to execute a command. Exit any other applications and try again. If the error still occurs, you may have set the model generator’s working memory area to be too large. You can adjust the generator’s memory allocation downward on the General tab of the LINGO|Options command's dialog box.

1008.

UNABLE TO CREATE A NEW WINDOW. LINGO was unable to create a new window. This is probably caused by insufficient system resources.

1009.

UNABLE TO ALLOCATE ENOUGH MEMORY FOR SOLUTION REPORT. LINGO could not allocate enough memory to store the solution report to the model. System resources must be very low in order to receive this message. Exit any other applications and try again. If the error still occurs, you may have set the model generator’s working memory area to be too large. You can adjust the generator’s memory allocation downward on the General tab of the LINGO|Options command's dialog box. If all else fails, you can use the LINGO|Solution command to request smaller solution reports for individual variables.

1010.

NOT USED.

1011.

UNABLE TO COMPLETE GRAPHICS REQUEST. LINGO was unable to generate a requested graph. Either LINGO is incorrectly installed or there is not enough free memory to perform the operation. Try cutting back on the amount of memory allocated to LINGO’s model generator using the LINGO|Options command. If this doesn’t help, try reinstalling LINGO.

1012.

NOT USED.

1013.

NOT USED.

1014.

NOT USED.

1015.

EDIT COMMAND NOT ED UNDER WINDOWS. USE THE FILE|OPEN COMMAND INSTEAD. The EDIT command is not ed in Windows versions of LINGO. If you need to edit the file, use the File|Open command to load the file into a window for editing.

1016.

COULD NOT SAVE FILE. LINGO was unable to save the file. Check for a valid path name and sufficient disk space.

9999.

OPTIMIZER FAILED DUE TO A SERIOUS ERROR. PLEASE REFER TO YOUR DOCUMENTATION UNDER APPENDIX B, “ERROR MESSAGES”. The optimizer was unable to continue due to a serious error. The most likely cause of this error stems from a problem in evaluating the functions within your model. Not all functions are defined for all values of their arguments. For example, @LOG(X – 1) is undefined for values of X less-than-or-equal-to 1. You should check your model for any such functions, and use the @BND function to place bounds on your variables to keep the optimizer from straying into any regions where functions become undefined.

ERROR MESSAGES

865

Another complicating factor can be nonlinear relations in a model. Linear models can be solved much more reliably and quickly than nonlinear models of comparable size. If possible, try to linearize your model by approximating nonlinear functions with linear ones, or by eliminating nonlinear equations. Another possible remedy when dealing with nonlinear models is to attempt different starting points. A starting point may be input in the model’s INIT section. Refer to Chapter 5, Windows Commands, for more details. If this problem persists after bounding any variables that could potentially lead to a problem, removing any unnecessary nonlinearities, and attempting new starting points, please LINDO systems for assistance.

Appendix C: Bibliography and Suggested Reading Anderson, Sweeney, and Williams, Introduction to Management Science, 8th ed. St. Paul, MN: West Publishing, 1997. _______________ , Quantitative Methods for Business, 6th ed. St. Paul, MN: West Publishing, 1991. Birge, J., and F. Louveaux, Introduction to Stochastic Programming, Volume 57, Springer-Verlag New York, Inc., 1997. Black, F., and M. Scholes. (1973). “The Pricing of Options and Corporate Liabilities.”, Journal of Political Economy, vol. 81, pp. 637-654. Bradley, S.P., A.C. Hax, and T.L. Magnanti, Applied Mathematical Programming. Reading, MA : Addison-Wesley Publishing Company, Inc., 1977. Cochran, W.G., Sampling Techniques. 2nd ed. New York, NY : Wiley, 1963. Conway, R.W., W.L. Maxwell, and L.W. Miller, Theory of Scheduling. Reading, MA : Addison-Wesley Publishing Company, Inc.,1967. Cox, John C. and Mark Rubinstein, Options Markets. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1985. Dantzig, G.B., Linear Programming and Extensions. Princeton, N.J. : Princeton University Press, 1963. Eppen, G.D., F.J. Gould, and Schmidt, C.P., Quantitative Concepts for Management: Decision Making Without Algorithms, 3rd ed. Englewood Cliffs, N.J. : Prentice-Hall, Inc., 1989. _______________ , Introductory Management Science, 4th ed. Englewood Cliffs, N.J. : Prentice-Hall, Inc., 1993. Gass, S., Decision Making, Models & Algorithms. New York: Wiley-Interscience, 1985. _______________ , Linear Programming, 5th ed. New York: McGraw-Hill, 1985. Geoffrion, A., The Theory of Structured Modeling, Western Management Science Institute, UCLA, 1987. Hadley, G., and T.M. Whitin, Analysis of Inventory Systems. Englewood Cliffs, N.J. : Prentice-Hall, Inc., 1963. Hillier, F., and G.J. Lieberman, Introduction to Operations Research, 9th ed. New York : McGraw-Hill, Inc., 2010. Johnson, L., and D.C. Montgomery, Operations Research in Production Planning, Scheduling, and Inventory Control. New York: John Wiley & Sons, Inc., 1974.

867

868 APPENDIX C Knowles, T., Management Science. Homewood, IL: Irwin Publishing, 1989. Lin, S., and B. Kernighan (1973), “An effective Heuristic Algorithm for the Traveling Salesman Problem.”, Operations Research, vol. 10, pp. 463-471. Markowitz, H. M., Portfolio Selection, Efficient Diversification of Investments, John Wiley & Sons, 1959. Nemha, G., and L. Wolsey, Integer and Combinatorial Optimization. New York : John Wiley & Sons, 1988. Moder, Joseph J., and Salah E. Elmaghraby (editors), Handbook of Operations Research. New York: Van Nostrand Reinhold Company, 1978. Schrage, L., Optimization Modeling with LINDO, 5th. ed. Belmont, CA: Duxbury Press, 1997. _______________, Optimization Modeling with LINGO, 6th. ed. Chicago, IL: LINDO Systems Inc., 2006. Wagner, H.M., Principles of Management Science with Applications to Executive Decisions, 2nd ed. Englewood Cliffs, N.J. : Prentice-Hall, Inc., 1975. _________, Principles of Operations Research, 2nd ed. Englewood Cliffs, N.J. : Prentice-Hall, Inc., 1975. Winston, Wayne L., Introduction to Mathematical Programming: Applications and Algorithms, 3rd ed. Belmont, CA: Duxbury Press, 1995. _____________, Operations Research: Applications and Algorithms, 2nd ed. Belmont, CA: Duxbury Press, 1995.

INDEX

Index ! !, 6, 377

@ @INTEGRAL function, 384

~ ~ (End-of-record mark), 423

A About LINGO command, 43, 281 ABS function, 383, 719 ABSINT parameter, 337 Absolute Integrality, 229 Absolute integrality tolerance, 229, 337 Absolute optimality tolerance, 232 Accelerator key, 126 Acceptance sampling, 799, 803 Access datasources, 471–76 importing, 429, 479–81 interfacing, 158 ACOS function, 383 ACOSH function, 383 , ODBC, 471–76 Airline overbooking, 774 Algebraic formulation, 143, 312 Algebraic reformulation, 359 Algorithm barrier cold start, 231, 349 error messages, 839 options, 211 parameters, 345 warm start, 231, 349 branch-and-bound, 82, 228, 347 generalized reduced gradient, 712 simplex cold start, 231, 349 options, 211

parameters, 345 warm start, 231, 349 Allowable Increase/Decrease, 320 ALTER command, 301, 323 Analysis cont, 593, 733 data envelopment, 735 range, 839 regression, 752, 797 What If, 117 Analytical derivatives, 219, 719 Anderson, D., 561 Annuity, present value, 386 APISET, 623 APISET command, 285, 844 Append output checkbox, 140 Application drop-down box, 224 AppWizard, 502, 534 Arithmetic operators, 380, 822, 826 Arrange Icons command, 277 ASCII format, 26, 130, 311, 421, 423, 439 ASIN function, 383 ASINH function, 383 Assembly line balancing, 551, 723 Assume Model is Linear, 206 ATAN function, 383 ATAN2 function, 383 ATANH function, 383 Attributes definition, 45 dynamic link library, 492 examples marketing models, 591 portfolio models, 564, 577 production models, 595 exporting, 481 importing, 478 initializing, 115 names, 819 AUTOLG.DAT, 146, 433 AutoUpdate command, 280

B Backorders, 757

869

870 INDEX Balof, J., 752 Barrier algorithm cold start, 231, 349 error messages, 839 options, 211 parameters, 345 warm start, 231, 349 Barrier Crossover, 215 BAS file, 529, 530 BASCTS, 350 BASCTS parameter, 358 BASIC, 494 Basis cuts, 226, 350, 358 Bayes rule, 725 BCROSS parameter, 361 Best bound, 234, 339 Beta Binomial distribution, 665 Beta distribution, 664 BigM Threshhold., 229 BIGMVL Paramater, 363 BIN function dual values, 99 options, 227 syntax, 88 Binary format, 440 operators, 380 Binary integer variables examples, 553 functions, 81 priority, 228, 348 syntax, 88–99 Binary Search, 628 Binding constraints, 26, 272, 307 Binomial distribution, 386, 665, 774, 778 Black & Scholes pricing model, 573, 776 Black, F., 571, 776 Blending problems, 540, 726, 731 Block Triangular, 268 BLOCKROW function, 418 BND function error messages, 825 examples, 543, 565 syntax, 105 BND(), 198 BNP Solver, 239 BNPBLK parameter, 375 BNPHEU parameter, 376 Bond portfolio optimization, 575, 780 Bounds allowable limit, 825 conflicting, 825

variables global, 359 guidelines for, 720 probing, 226 syntax, 105 Bracken, J., 801 Brahnc-and-bound global solver, 244 Branch-and-bound constraint cuts options, 224, 227 parameters, 337, 350, 358 reports, 272 definition, 82, 597, 711 direction, 347, 359 heuristics, 223, 244, 358 optimality tolerance, 232 solver, 222 strategy, 228 BRANDPR parameter, 348 BRANDR parameter, 347 BREAK, 612

C C# .NET, 490 C/C++, 490, 501–10 callback function, 524 pointer function, 494 defined function, 533, 534 Calc Section, 120 Call option, 571, 776 Callable Library, 489–533 Callback functions, 497, 499, 523–33, 529 Capacitated EOQ, 743 Capacitated lot sizing, 762 Capacitated Plant Location, 556, 729 Capacity, scarce, 762 Cartesian product, 52 Cascade command, 277 Case sensitivity, 6 Cash flow matching, 575, 780 CAT command, 283, 285 Cauchy distribution, 664 Cs, 698 Chance-Constrained Programs, 698 Charting Solutions, 164 Charts, 164 Chemical equilibrium, 732 CHESS model, 731 Chi-Square distribution, 664 Chi-squared distribution, 386

INDEX CHKDUP parameter, 343 Classification statistics, 271, 307, 713 ClassWizard, 504, 525, 526 Clipboard, 148 Close All command, 276 Close command, 137 Coefficient reduction, 350 Coefficients current, 320 matrix of, 216, 304 nonzeros, 272, 307 objective, 25 objective ranges, 320 reduction, 226, 350 right-hand side, 320 unity, 272, 307 COFCTS, 350 Cold start option, 231, 349 Colon prompt, 23, 283, 289 Coloring, 155, 188, 351 Column length, 486 COM command, 285, 830 Command scripts, 431–33 command-line, 294 DLL, 496, 509, 514 OLE automation importing, 454–58 Windows, 140–43, 274 Command Window command, 189, 274, 342 Command-line interface, 22, 23, 140, 189 Commands Command-line, 283–85, 285–377 Windows, 126–28, 129–282 Comments, 6, 377 Compiling the model generating, 58 solving, 160, 316 CompuQuick Corporation, 4, 16, 23, 83 Concavity, 717 loose, 718 strict, 718 Conditional qualifier, 54, 58 Cont analysis, 593, 733 Constraints binding, 26, 272, 307 cuts options, 224, 227 parameters, 337, 350, 358 reports, 272 defining, 4, 5, 29 examples, 543, 558, 565, 595 limits, 106

modeling language, 45 names, 38 options, 224 selective evaluation, 220, 345 two-sided, 819 types, 712–14 Constraints Cuts box, 224 ing LINDO, 44, 282 convexity, 244, 359 Convexity, 716 loose, 717 strict, 716 Convexity and Concavity, 42 Copy command, 148, 421–22, 422 Correlation, 666 COS function, 383 COSH function, 383 Covariance, 563, 564 Covering model, 740 Crashing, 220, 344, 783 Create from File, 465 Critical path, 782 Cumulative distribution, 387, 388 Current coefficients, 320 Cut command, 148 CUTAPP parameter, 337 Cutoff level, 494 CUTOFF parameter, 348 Cuts options, 224, 227, 350, 358 parameters, 337 reports, 272 Cutting Stock, 639 Cycle time, 551, 553

D Dantzig pricing, 346 Data envelopment analysis, 735 Data independence, 26, 92 Data section definition, 26 error messages, 835 attributes, 819 exporting, 832 importing, 823 names, 823, 835 question marks, 827, 836 examples, 424 syntax, 115–19 Data type, 531 Database Info command, 146

871

872 INDEX Databases, 469–87 exporting functions, 481–87 reports, 326 importing, 442, 477–81 Datasources, 470–76 linking, 146, 156, 328 ODBC function, 156, 481 DBGCLD parameter, 361 DBGLVL Parameter, 362 DBGWRM parameter, 361 DBPWD command, 146, 328 DBUID command, 146, 328 Debt defeasance, 575, 780 Debug command command-line, 314 Windows, 268–71 Debugging, 304 Decision variables defining, 4 examples, 553, 557, 568 ranges, 319 DECOMP parameter, 358 Decomposition, 216 Default button, 181 Delay field, 188 DELETE command, 325 Delphi, 490 Demand backlog model, 757 Dense set, 49 Depth first, 234, 339 DERCMP parameter, 334 Derivatives, 219, 334, 716, 719 Derived sets defining, 46, 49–52, 837 examples blending models, 66–69, 542 financial models, 568 marketing models, 594 production models, 548, 553 exporting, 481 Deg a computer cabinet, 796 Desrochers, M., 809 Devex pricing, 213, 346 Direct solver, 597, 711 Direction box, 228 Disaggregation, 226, 350 DISCTS, 350 Display As Icon button, 157 Distribution, 387, 737 binomial, 386, 774, 778 Chi-squared, 386

cumulative, 387, 388 hypergeometric, 387, 799 Multi-level, 769 Poisson, 387, 580, 745 Diversification, 563 DIVERT, 622 DIVERT command error messages, 832 printing, 24 reverting, 311 syntax, 310 DLLs, 489–533 Domain functions, 825 Double data type, 531 ing, 280 Downside risk, 567, 570, 738 Dual computations box, 196 DUAL function, 405 Dual values, 26 enabling, 196, 833 integer programming, 99 nonzeros, 826 ranges, 320 DUALCO, 346 DUALCO parameter, 319 DUALPR parameter, 346 Duplicate names, 821, 834 DYNAMB model, 560 Dynamic Link Library, 489–533 functions, 494–500 Dynamic programming, 560, 741, 767, 778

E Echo Input, 189 Echo to screen checkbox, 140 ECHOIN parameter, 343 Economic order quantity, 743, 790 Edit menu, 127, 147–59 Efficient Frontier, 632 Eigen values, 716 ELSE, 610 Embedding files, 458–67 END command, 23 End-of-file, 426 End-of-record Mark (~), 423, 426, 548, 822 ENDSUBMODEL, 597 Enforce Bounds in CALC and DATA, 207 Entering a Model Command-line, 22–23 Windows, 3–22 Eppen/Martin model, 762

INDEX Equality relation, 26, 381 Equilibrium network, 771 Erlang busy probability, 387, 792 Erlang loss probability, 387, 580, 746 ERRDLG parameter, 343 Error messages, 7, 343, 499, 501, 817–65 Error Text box, 160 Errors In Dialogs, 183 Errors Only output, 186 Excel exporting, 453 linking with LINGO, 150, 441, 495 OLE automation, 454–68 OLE function, 441, 452 TEXT function, 429 Exclamation point, 6, 377 Exercise price, 571, 776 Exit, 377 Exit command, 146 EXP function, 383 Expected rate of return, 569 Expected Value Statistics, 256 Explicit listing, 50 Exponential distribution, 664 Exponential smoothing, 580, 805 Export summary report ODBC based, 486 OLE based, 453 Exporting attributes, 481 databases functions, 481–87 reports, 326 MPI files, 144, 312 MPS files, 143–45 pasting out, 422 spreadsheets functions, 446–54 reports, 326 Expressions, 719 EXTEND command, 325 EZCOUNT model, 811 EZMREPAR model, 583, 744 EZNEWS model, 745

F F distribution, 664 Factorial function, 811 Favor Reproductibility Over Performance, 198 Feasibility tolerances, 212, 218

Feasible solution, 212 Features, new, vii File data, 439 format ASCII, 26, 130, 311, 421, 423, 439 BAS, 529, 530 binary, 440 LDT, 130, 439 LG4 definition, 130 linked objects, 158 saving, 18, 136, 439 LNG, 130, 439 LRP, 439 LTF, 130, 439 MPI, 144, 312 MPS definition, 127 exporting, 143–44, 312 header, 505, 528 log files, 863 names, 6, 38, 136 script files, 431–33 types, 130 FILE, 423 File Format box, 187 FILE function, 130, 397, 423–27 examples, 433, 547 File menu, 127, 129–46 FILFMT parameter, 342 Filter, 50 Final linear feasibility tolerance, 212, 333 Final nonlinear feasibility tolerance, 218, 334 Financial models, 386, 563–79 Find/Find Next command, 153 Finite source, Poisson, 583 Fixed costs, 94 Fixed ordering charge, 773 Fixed Var Reduction, 202 Fixed variables, 712, 824 FLFTOL parameter, 333 Floating point operations, 212, 333 FLOOR function, 383 Flow Cover, 226, 350 Flow cuts, 350 FLWCTS, 350 FNFTOL parameter, 334 Font selecting, 155 FOR, 611 FOR function constraints, 40, 57

873

874 INDEX error messages, 822, 824 nesting, 59, 154 sets, 47, 396 syntax, 54 variables, 87, 103 FORCEB parameter, 373 Forecasting, 101, 797, 805 FORMAT function, 405 Formatting, 130 FORTRAN pointer functions, 494 defined functions, 533, 534 FPA function, 386 FPL function, 386 Fragmentation ratio, 286 Free variables, 100–105 FREEZE command, 328, 336, 832 Functions, 45 callback, 523–33 DLL, 494–500 factorial, 811 interface functions, 423, 833 non-differentiable, 719 pasting, 155 probability, 386–89 set-looping, 53–60, 395 defined, 533–37 utility, 593 variable domain, 825

G Gamma distribution, 664 GCD cuts, 226, 350 GCDCTS, 350 GEN, 297, 614 GEN command, 58 GENDUAL, 615 General box, 183 General equilibrium of an economy, 747 General integer variables, 81, 82–87 General Solver tab, 191–210 Generate command, 257 Generate command, 58 Generating a model, 203 Generator Memory Limit Box, 201 Generator memory used, 286 Geoffrion, A., 769 Geometric distribution, 665 GIN function dual values, 99 examples, 83, 87

options, 227 syntax, 82 GLBBRN parameter, 359 GLBBXS parameter, 359 GLBDLT parameter, 359 GLBOPT parameter, 358 GLBREF parameter, 359 GLBUBD parameter, 359 GLBVBD parameter, 359 Global delta tolerance, 359 Global optimality tolerance, 358 Global optimum, 352, 494, 714, 716 Global solver, 354, 841 Global Solver, 182, 243, 244, 246, 248, 419, 842 Global Solver tab, 243 GO command, 23, 316 Go To Line command, 154 GOMCTS, 350 Gomory, 350 Gomory cuts, 226, 350 GP1, 268 GP2, 268 Gradient, 218, 712, 719 Graves, G., 769 Greatest common divisor, 350 Growth rate, 563 GUB cuts, 226, 272, 307, 350 GUBCTS, 350 Gumbel distribution, 664

H Handle usage, 286 Hansen, P., 771 Hayes, J.M., 732 Header file, 505, 528 Heap compacts, 286 HELP command, 285 Help menu, 128, 277–82 Help Topics command, 278 HEUCUT parameter, 372 HEURIS parameter, 337 Heuristics, 223, 358, 767, 809, 812 HIDE command, 301 Hieftje, G.M., 732 Higher Dimension Charts, 171 Holding costs, 743, 762 Home mortgage calculation, 814 HUMNTM parameter, 358 Hurdle, 233 HURDLE parameter, 337 Hypergeometric distribution, 387, 665, 799

INDEX Hyperplanes, 718

I IF function, 418 IFC, 610 ILFTOL parameter, 333 Implicit set , 839 Implied indexing, 55 Importing attributes, 478 command scripts, 431–33 databases, 442, 477–81 FILE function, 130, 423–27 pasting in, 421–22 set , 206 spreadsheets, 441–46 IN function, 822, 827 Include file, 426 Index, 303, 819, 820, 831 implied, 55 internal, 38 placeholder, 50 INDEX function error messages, 819, 824 examples, 90 parameters, 344 set names, 206, 207 Inequality relation, 381 Infeasible model, 26, 268, 269, 314, 494, 827 INFTOL parameter, 334 Init section, 119–23 importing, 442 Initial linear feasibility tolerance, 212, 333 Initial nonlinear feasibility tolerance, 218, 334 Initializing data error messages, 839 variables, 720 Insert New Object command, 156–58, 465 Installing LINGO command-line, 22 Windows, 1 Integer pre-solver tab, 222–26 Integer programming branch-and-bound constraint cuts options, 224, 227 parameters, 337, 350, 358 reports, 272 definition, 82, 597, 711 direction, 347 heuristics, 223, 358

optimality tolerance, 232 solver, 222 strategy, 228 dual values, 99 probing, 226 Integer Solver tab, 227–42 Integer variables binary (0/1) examples, 553 functions, 81 priority, 228, 348 syntax, 88–99 examples, 544 general, 81, 82–87 limits, 43 solutions, 721 variable domain functions, 81–100 INTEGRAL function, 384 Integrality tolerance, 229, 334 Integration, 384 Interest rate, 571 Interface functions, 423, 833 Interface tab, 183–90 Interfacing with other applications, 489–537 Interior point algorithm, 211, 345 Interrupting the solver, 863 invalid function, 826 Invalid switch, 827 Inventory system demand backlog, 757 newsboy model, 745, 773 PERT, 479–81, 484–87, 782 IPTOLA parameter, 338 IPTOLR parameter, 338 Iterations, 212 definition, 195, 335 dynamic link library, 525 limit, 334, 335 max es, 225 SLP direction, 221 steepest edge, 213, 221, 344, 346 ITERS function, 408 ITRLIM parameter, 335 ITRSLW parameter, 334

J Job shop scheduling, 750 Just-in-time production, 545, 765

875

876 INDEX

K K-Best Solutions, 235 KBESTS Parameter, 364 Kehoe, T.J., 747 Kendall correlation, 666 Kernighan, B., 809 KILLSC Paramater, 363 Knapsack cover, 350 Knapsack cover cuts, 226, 350 Knapsack problem, 88–94, 324, 751 KNPCTS, 350

L Labor supply, 5 Laplace distribution, 664 Laporte, G., 809 LATCTS, 350 Latin-hyper-square, 660 Lattice, 350 Lattice cuts, 226, 350 LCORE1 – LCORE4 parameters, 364 LCORES Parameter, 364 LCRASH parameter, 361 LDT format, 130, 439 Lead time, 545, 548 Learning curve, 752 LENPAG parameter, 339 LFTCTS, 350 LG4 format definition, 130 linked objects, 158 saving, 18, 136, 439 LGM function, 384, 811 Libraries (DLL), 489–533 License key global solver, 243 Windows, 145 Lift cuts, 350 Lifting cuts, 226, 350 Limits BND function, 106, 825 constraints, 4, 5, 95 cuts, 350 integers, 225 iterations, 218, 334, 335 line counts, 188, 189 page size, 190 runtimes, 195 sets, 27 software version, 43

Lin, S., 809 LINDO API, 145, 312 LINDO information, 44, 282 Line capacity, 5 Line count limits, 189 Line limit field, 188 Linear formula, 712 Linear loss function, Poisson, 387 Linear Optimality Tolerance, 215 LINEAR Parameter, 362 Linear regression, 752, 797 Linear solver, 210, 230, 597, 711 Linearization, 204, 356, 842 LINGO menu, 128, 160–273 LINGO.CNF, 832 LINGO|Picture, 266 Linking databases, 146, 156 Links command, 152, 159 LINLEN parameter, 339 Little s-big S policies, 773 LNBIGM parameter, 356 LNDLTA parameter, 356 LNG format, 130, 439 LNRISE parameter, 356 Local optimum, 244, 494, 714 Log files, 495, 498, 509, 863 LOG function, 120, 384 Log Output command, 140, 190 LOG10 function, 384 Logarithm, 573, 732, 752, 811 Logarithmic distribution, 665 Logical expressions conditions, 92 examples, 549, 791 operators, 51, 381 Logistic distribution, 664 Logistics models, 556–62 Lognormal distribution, 664 Long data type, 531 LOOK command, 273, 303, 817 Loop Optimization, 207 Looping, 611 LOOPOP parameter, 371 Lot sizing, 762 Lower Triangular, 268 LOWMEM parameter, 361 LRP format, 439 LS_DINFO_MIP_BEST_OBJECTIVE_LNG, 497 LS_DINFO_MIP_BOUND_LNG, 497 LS_DINFO_OBJECTIVE_LNG, 497 LS_DINFO_SUMINF_LNG, 497 LS_IINFO_BRANCHES_LNG, 497

INDEX LS_IINFO_CONSTRAINTS_LNG, 497 LS_IINFO_CONSTRAINTS_NONLINEAR_LNG, 497 LS_IINFO_ITERATIONS_LNG, 497 LS_IINFO_NONZEROS_LNG, 497 LS_IINFO_NONZEROS_NONLINEAR_LNG, 497 LS_IINFO_VARIABLES_INTEGER_LNG, 497 LS_IINFO_VARIABLES_LNG, 497 LS_IINFO_VARIABLES_NONLINEAR_LNG, 497 LSclearPointersLng, 495 LScloseLogFileLng, 495, 515 LSdeleteEnvLng, 496 LSERR_INFO_NOT_AVAILABLE_LNG, 501 LSERR_INVALID_INPUT_LNG, 501 LSERR_INVALID_NULL_POINTER_LNG, 501 LSERR_NO_ERROR_LNG, 501 LSERR_OUT_OF_MEMORY_LNG, 501 LSERR_UNABLE_TO_OPEN_LOG_FILE_LNG, 501 LSexecuteScriptLng, 496, 515 LSgetCallbackInfoDoubleLng, 531 LSgetCallbackInfoLng, 497, 524 LSgetCallbackInfoLongLng, 531 LSopenLogFileLng, 498 LSsetCallbackErrorLng, 499 LSsetCallbackSolverLng, 499, 523 LSsetPointerLng, 500 LTF format, 130, 439

M Machine repairman model, 583, 744, 764 Marketing models, 589–96 Markov chain model, 589, 754 Markowitz portfolio selection, 563, 567, 570, 749 Markowitz, H., 563, 749 Match Parenthesis command, 154, 188 Matching model, 756 Material requirements planning, 545, 765 Mathematical functions, 383 Mathematical notation, 26–32 Matrix decomposition, 216, 358 Matrix Decomposition, 216 Matrix form, 304, 590 Matrix Picture, 267 MAX function, 5, 54, 56, 396, 554, 719 Max es, 225 Max Scenarios limit, 255 MAXCTP parameter, 350 MAXCTR parameter, 358

McCormick, G., 801 MEM command, 286 of sets, 49 hip filter, 50 Memory allocated, 286, 336, 494 Memory Limit box, 194 usage, 286 virtual, 337 working, 817, 822 Menu commands, 125, 126, 461 edit, 127, 147–59 file, 127, 129–46 help, 128, 277–82 LINGO, 128, 160–273 window, 128, 273–77 Menu tips, 126, 342 Method, Solution, 211 METRIC model, 757 Mexican steel model, 758 MIN function, 54, 56, 719 Minimal cost queuing, 793 Minimize Memory Usage, 209 MOD function, 384 MODEL command, 23, 289, 818 Model Generator Tab, 200 Model reduction, 211, 346 Model Statistics command, 128, 271 Modeling language, 26–32, 38–41, 45 Models entering, 3–22 infeasible, 26, 268, 269, 314, 494, 827 nonlinear, 1, 119, 220, 354, 720–21 regenerating, 203 titles, 41 types, 711 unbounded, 268, 314, 494, 826 More button, 190 Mortgage calculation, 814 MPI files, 144, 312 MPS File command, 127 definition, 127 exporting, 143–44, 198, 312 format, 838 importing, 198 Multi-Core, 213 Multilevel distribution, 769 Multiprod. capacitated lot sizing, 762 MULTIS, 351 Multistart solver, 244, 351, 841 MultiStart Solver, 248

877

878 INDEX Multithreading box, 192 MXMEMB parameter, 336 MY.DLL, 534

N NAME function, 408 Naming conventions constraints, 38, 820 duplicate names, 207, 834 file names, 136, 821 MPS files, 198 parameters, 347 sets, 206, 819, 821 variables, 6 NCRASH parameter, 344 Necessary set, 268, 314 Negative Binomial distribution, 665 Negative variables, 100–105 Nemha, G., 226 Nesting, 59, 154 FOR functions, 824 TAKE command, 818 Network Charts, 174 Network equilibrium, 771 New command, 130, 818 New features, vii NEWLINE function, 408 Newsboy problem, 745, 773 No output, 186 Node selection, 234, 339 NODESL parameter, 339 Non-differentiable functions, 719 Nonlinear limits, 43 models, 1, 119, 244, 720–21 optimality tolerance, 218 solver, 597, 711 Nonlinear models, 354 Nonlinear Solver tab, 217–21 derivatives box, 219 NONNEG Parameter, 363 Nonoptimal solution, 827 Nonzeros, 272, 307 dual price, 826 NOPTOL parameter, 334 Normal distribution, 664, 737 NSLPDR parameter, 345 NSLPSV parameter, 373 NSTEEP parameter, 344 NTHRDS parameter, 373 NTMODE parameter, 374

NUMBIN parameter, 372 Numeric Integration, 384 Numeric precision, 212, 216, 494

O OBJBND function, 408 OBJCTS, 350 OBJCTS parameter, 335 Object Linking and Embedding, 440 Object Properties command, 159 Object, Insert New, 156–58, 465 Objective coefficient ranges, 320, 321 Objective coefficients, 25 Objective cuts, 350 Objective function cuts, 226, 350 definition, 4 developing, 5, 28 examples financial models, 565, 569, 577 marketing models, 595 plant location, 557 production models, 542, 553 ODBC, 469–87 , 471, 487 datasources linking, 146, 156, 328 ODBC function, 156, 481 exporting, 481–87 importing, 477–81 ODBC function, 397 OLE automation importing, 454–58 OLE function, 397 exporting reports, 453 syntax, 447–53 importing, 441–46 Omitting values, 118 One Page button, 139 On-line registration, 279 Open command, 818, 864 Open Data Base Connectivity, 469–87 Operators, 379–82, 819, 822, 826 Optimal solution, 352, 494, 714 Optimality tolerance, 218, 232 Optimization, 1, 4 Optimization Modeling with LINGO, 539 Options command, 181–273 Options pricing, 571, 778 Oracle, 476

INDEX OROUTE parameter, 342 Overbooking, Airline, 774

P PAGE command, 326 Page size limits, 190 Parameters error messages, 829, 831 pre-defined, 319, 327, 333–47 defined, 116 Paren Match checkbox, 188 Parent sets, 49 Parentheses, 154, 188, 863 Pareto distribution, 664 s databases, 146, 328 error messages, 830 HIDE command, 301 Paste command, 148, 422 Paste Function command, 155 Paste Special command, 149 PAUSE, 621 PAUSE command, 326 PBN function, 386 PCX function, 386 Pearson correlation, 666 PEB function, 387, 792 PEL function, 387, 580, 746, 793 PERT, 479–81, 484–87, 782 Peters, D.G., 732 PFD function, 387 PFS function, 387, 583 PHG function, 387, 799 PIC, 616 Picture command, 128, 266 PICTURE command, 304 Plant location, 226, 350, 556, 729 Platforms, 283 PLOCTS, 350 pLSenvLINGO, 508 Pluto Dogs, 60, 86 POINTER function, 398 data section, 492, 494 DLL, 495, 500, 508 Poisson finite source, 583 linear loss function, 387 probability distribution, 387, 580, 745 random variable, 387 Poisson distribution, 665 Portfolio selection

Markowitz, 563, 567, 570, 749 scenario based, 738, 786 Portfolio Selection, Markowitz, 632 POW function, 384 Powerstation FORTRAN pointer functions, 494 defined functions, 533, 534 PPL function, 387 PPS function, 387 PRBLVL parameter, 345 Precedence relations, 551, 554 PRECIS parameter, 370 Precision, Numeric, 212 Present value, 386 Prices Black & Scholes, 573, 776 computing, 196 options, 571, 778 strategies, 212, 346 Primal/dual interior point algorithm, 211, 345 Primitive set defining, 46–48 examples, 46, 60–65, 66 exporting, 481 names, 206, 821 PRIMPR parameter, 346 Print command, 18, 137 Print Preview command, 139 Print Setup command, 138 PRIORITY function, 99 Priority option, 228, 348 PRNCLR parameter, 351 Probabilities, 568 Probability distribution binomial, 386, 774, 778 Erlang busy, 387, 792 Erlang loss, 387, 580, 746 Erlang queuing, 580 exponential, 580 hypergeometric, 387, 799 normal, 388, 737 Poisson, 387, 580, 583, 745 steady state, 589, 754 t distributions, 737 transition, 589 Probability functions, 386–89 Probing, 226, 345 Procedures, 624 PROD function, 396 Product form inverse, 712 Production, 762 planning, 762

879

880 INDEX product mix, 4, 94–99, 141, 781, 785 rates, 743 schedules, 545, 765 Production management models, 540–55 Programming Example Binary Search, 628 Cutting Stock, 639 Markowitz Efficient Frontier, 632 Project Evaluation and Review Technique, 479–81, 484–87, 782 Prompts, 23, 283, 289 Properties command, 435 Proportionality, 712 PSL function, 387 PSMSTCDF distribution, 401 PSMSTINV distribution, 401 PSMSTPDF distribution, 401 PSN function, 388, 737 PTD function, 388, 737 Purchase transition matrix, 590

Q QRAND function, 388, 839 Quadratic functions, 788 Quadratic recognition, 220 Quasi-random numbers, 388 Question mark prompt, 23, 289 Queuing models examples Erlang, 580–88, 746, 792–93 machine repairman, 744 steady state, 794 probabilities, 387 QUIT command, 377

R R/C format names options, 198 parameter, 347 RAND function, 389, 737 Random Generator Seed, 253 Random numbers, 388, 389, 737, 754, 839 Range analysis, 839 bounding, 244 dual values, 320 names defining, 452 exporting (OLE), 448

exporting (WKX), 454 importing (OLE), 446 OLE automation, 455 reports command-line, 319 error messages, 831 Range command, 25, 26 RANGE command, 319 RANGED function, 408 RANGEU function, 409 Rate of return, 565, 569 RCMPSN parameter, 347 RCTLIM parameter, 350 Read error, 823 Redirecting I/O, 439 Redo command, 147 REDUCE parameter, 346 Reduced cost, 25 ranges, 320 REFRAQ parameter, 367 command, 279 Regression Analysis, 752, 797 Relational operators, 381, 819 Relationships, 721 Relative Integrality, 229 Relative integrality tolerance, 229, 334 Relative limit box, 225 Relative optimality tolerance, 232 RELEASE, 618 RELINT parameter, 334 REOPTB parameter, 349 REOPTX parameter, 349 Replace command, 153 Reports, 342 export summary report ODBC based, 486 OLE based, 453 generating, 310 range report command-line, 319 error messages, 831 send to command window, 189 solution report command-line, 23, 316 exporting, 486 naming constraints, 40 tolerances, 212, 333, 348 width, 327 Windows, 162 statistics, 128, 271, 307 REPROD parameter, 376 Resource limits, 4, 5, 27, 95

INDEX Right-hand side probing, 226 range report, 321 Risk downside risk, 738 portfolio selection, 563–70, 749, 786 sampling, 799, 803 RLPF command, 292 RMPI command, 289 Root filed, 225 Rounding, 82, 87, 216, 348 Row name, 820 RunLingo, 436 Runtime, 232 Runtime Limits box, 195 RVRT command, 311

S Sales forecasting, 804 Sample Continuous Only, 254 sample size, 666 Sampling, 388, 799, 801, 803 SAVE command, 25, 311 Save/Save As command, 18, 136 Scalar variables examples, 27, 29, 553, 781 initializing, 116 Scale Model, 216 SCALEM parameter, 346 SCALEW parameter, 366 Scaling, 212, 333, 720, 841 Scaling Warning Threshold, 197 Scarce capacity, 762 Scenario portfolio selection, 567, 738, 786 SCENE command, 305 Scholes, M., 571, 776 Schrage, L., 100, 539, 735 SCLRDL parameter, 351 SCLRLM parameter, 351 Script files, 431–33, 514 command-line, 294 DLL, 496, 509 OLE automation importing, 454–58 Windows, 140–43, 274 Script processor, 492 Scripting, 610 Second order derivatives, 219 SECORD Parameter, 363 Seed value, 839 SELCON parameter, 345

Select All command, 153, 422 Select Font command, 155 Selective constraint evaluation, 220, 345 Semi-variance risk, 567, 570 Send To Back command, 276 SET, 623 SET command, 329, 831 Set-looping functions, 53–60, 395 Sets, 27–37, 45–80, 839 derived sets defining, 46, 49–52, 837 dense, 553, 839 examples blending models, 66–69, 542 financial models, 568 marketing models, 594 production models, 548, 553 exporting, 481 errors, 837 implicit, 839 importing, 206 names, 819 necessary, 268, 314 primitive set defining, 46–48 examples, 46, 60–65, 66 exporting, 481 names, 821 sufficient, 268, 271, 314, 316 Sets section, 46–52, 424 Setup cost, 743, 762 Shack4Shades, 101 Shadow prices, 26, 320 Shipping problem, 27, 45, 556, 769 Shortest route problem, 560 SHOWNL command, 306 SIGN function, 385 Simplex method, 712 cold start, 231, 349 options, 211 parameters, 345 warm start, 231, 349 Simpson’s Rule, 384 SIN function, 385 SINH function, 385 Slack or Surplus, 26, 212, 333, 555 Slack variables, 272, 307, 543 Slope, 101, 719 Slow progress iteration limit, 218, 334 SLP directions, 221 SLP solver, 221 SMAX function, 385

881

882 INDEX SMIN function, 385 Smooth expressions, 580, 719, 805 SMPI, 617 SMPI command, 312 SMPS, 617 SMPS command, 312 SOLU, 619 Solution report command-line, 23, 316 exporting, 486 naming constraints, 40 tolerances, 212, 333, 348 width, 327 Windows, 162 Solutions feasible, 212 infeasible, 26, 268, 269, 314, 494, 827 non-optimal, 827 optimal, 352, 494, 714 unbounded, 268, 314, 494, 826 undefined, 827 undetermined, 494 SOLVE, 599 Solve command, 7–17, 160, 276, 467 SOLVE Time-Limit box, 196 SOLVEL parameter, 345 Solver, 597, 711 branch-and-bound constraint cuts options, 224, 227 parameters, 337, 350, 358 definition, 82, 597, 711 direction, 347 heuristics, 223, 358 optimality tolerance, 232 solver, 222 strategy, 228 global, 354 interrupting, 863 linear, 210, 230, 597, 711 method, 345, 839 multistart, 351 nonlinear, 597, 711 quadratic, 220 Solver status window, 161 error messages, 863 field definitions, 16 model classification, 712, 713 options, 185, 341 Solver Status Window, 7 SONGS model, 807 Sorting, 808

SOS variables syntax, 106 SP Sample Size/Stage, 253 SP Solver, 252 SP Solver Method, 254 SP Solver Tab, 252 Spanning tree model, 767 Sparse derived set, 69–75, 75–80, 821 Sparse set, 49 SPBIGM parameter, 372 Spearman correlation, 666 Splash screen, 185, 341 SPLOADSCENE, 625 SPMAXS parameter, 370 SPMTHD parameter, 368 SPNUMSCENE, 625 SPPRBSCENE, 625 Spreadsheets data section, 26 exporting functions, 446–54 reports, 326 importing, 441–46 interfacing with, 150, 421, 423, 441–68 SPSCON parameter, 367 SPSEED parameter, 368 SPSMSZ parameter, 367 SPXVAL parameter, 368 SQL, 476 SQR function, 385 SQRT function, 385 STABAR parameter, 342 Stack space, 819 Staff scheduling, 60, 428, 432, 501–19, 740 DLL, 489–511, 492 State of the model, 826, 827 Statistics, 128, 271, 307, 713 STATS command, 307 Status bar, 3, 126, 184, 342 STATUS function, 410 Status Window command, 185, 276 STAWIN parameter, 341 Steady state, 587, 794 equations, 585 probability, 589, 754 queuing model, 585, 794 Steepest edge, 213, 221, 344, 346 STM2DWK function, 418 STM2HR, STM2MIN functions, 416 STM2YR, STM2MON functions, 416 STMNOW function, 417

INDEX Stochastic Programming Declaring Distributions, 663 Gas Buying Core Model, 670 Gas Buying Example, 670 Introduction, 653 Investing Under, 692 Investing Under Uncertainty Example, 691 Language Features, 661 Monte Carlo Sampling, 659 Multistage Decision Making, 653 Recourse Models, 655 Scenario Tree, 657 Setting up SP Models, 660 Stock Option Core Model, 682 Stock Option Example, 681 Stock options, 571 STOP, 613 Straight line relationships, 712 Strategies, 220 Strategies box, 247 Stratified sampling design, 801 STRLEN function, 411 Strong branch, 234, 348 STRONG parameter, 348 Student t distribution, 664 SUBMODEL, 597 SUBOUT parameter, 360 Successive linear programming, 221, 712 Sufficient set, 268, 271, 314, 316 SUM function, 54, 55, 90, 396 Sum of a set, 396 Sum of squared prediction, 804 Sweeney, D., 561 Symetric Stable distribution, 664 Syntax, 6, 7, 115 coloring, 155, 188, 351

T t distribution, 737 TABLE function, 411 Table of Contents, iii TAKE command, 294 command scripts, 431 Take Commands command, 130, 140–43, 431 TAN function, 385 TANH function, 385 Tanner, K., 769 TATSLV Paramater, 364 Technical , 44 Terminal width parameter, 327, 829 TERSE command, 326

Terse output, 186 TERSEO parameter, 340 Text files, 26, 311, 421, 423, 439 TEXT function, 398 command scripts, 433 Tiered distribution, 769 Tile command, 277 TIM2RL parameter, 339 TIME command, 377 TIME function, 415 Time to relative tolerance, 232, 233, 339 Timesharing system, 583, 744 TIMLIM parameter, 335 TITLE command, 41 Tolerances, 212, 218, 229, 232 command-line, 285, 327, 329, 333–38 examples, 591 Tolerances box, 233, 247 Tool tips, 126, 342 Toolbar, 3, 125, 186, 275, 343, 461 TOOLBR parameter, 343 Traffic congestion, 772 Transfer method command line, 453, 486 Transition probability, 589, 754 Transportation model, 815 datasources, 470 exporting, 451 importing, 444 linking, 149 sets, 27 Traveling salesman problem, 809 Tree field, 225 Triangular distribution, 664 Truncating, 82, 190, 313 Two-sided constraints, 819

U Unary Minus Priority, 201 UNARYM Parameter, 362 Unbounded solution, 268, 314, 494, 826 Undefined arithmetic operation, 345, 825, 827 Undefined solution, 827 Undetermined solution, 494 Undo command, 147 Uniform distribution, 664 Unit normal linear loss, 387 Unity coefficients, 272, 307 Unix, 439 Update interval, 185, 341 Updating LINGO, 280

883

884 INDEX Updating links, 159 Use RC Format Names, 198 USEPNM parameter, 344 defined functions, 533–37 function, 420 ID, 146, 328 interrupt, 863 Utility functions, 593

V Values transferred command-line, 454 Variable domain functions, 825 Variable Upper Bound Box, 247 Variables bounds, 359 guidelines for, 720 probing, 226 syntax, 105 decision variables defining, 4 examples, 553, 557, 568 ranges, 319 defining, 5, 28 fixed variables, 712, 824 free variables, 100–105 initializing, 720 integer variables binary examples, 553 functions, 81 priority, 228, 348 syntax, 88–99 examples, 544 general, 81, 82–87 solutions, 721 variable domain functions, 81–100 limits, 43 names, 6 negative, 100–105 nonlinear, 1 production models, 542, 548 random, 387 scalar examples, 27, 29, 553, 781 initializing, 116 slack, 272, 307, 543 SOS syntax, 106 transportation models, 562 Variables Assumed Non-Negative, 198

Variance, 563, 570, 571 Vehicle routing problem, 812 VERBOSE command, 326 Verbose output, 186 Virtual memory, 337 Visual Basic, 490, 510–11 callback function, 529 macros in Excel, 454, 457 Visual Basic .NET, 490 Visual C++, 490, 501–10 callback function, 524 defined function, 534 Volatility, 571

W Warm start option, 230, 349 WARN function, 420, 591, 824, 834 Weibull distribution, 664 What If analysis, 117 WHILE, 611 WIDTH command, 327, 829 Williams, T., 561 Window menu, 128, 273–77 Windows commands in brief, 126–28 in depth, 129–282 Winston, W., 100, 805 Wireless Widgets examples, 27 importing, 424, 444 sets, 46, 69 WNLINE parameter, 342 WNTRIM parameter, 342 Wolsey, E., 226 Working memory, 822 Workstations, 551, 585, 588 Worst bound, 234, 339 WRAP function, 102 WRITE, 620 Write access, 832, 863 WRITE function, 413 WRITEFOR function, 413, 414 Writing to files, 427

X XDETEQ command, 308

INDEX

Y

YMD2STM function, 415

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