INTRODUCTION TO GAMS 1 Daene C. McKinney CE385D Water Resources Planning and Management The University of Texas at Austin

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1 INTRODUCTION TO GAMS 1 Daene C. McKinney CE385D Water Resources Planning and Management The University of Texas at Austin Table of Contents 1. Introduction GAMS Installation GAMS Operation Examples Algebraic Equation Background and GAMS Code What's Going On in the Model? Least Squares Parameter Estimation Background and GAMS Code What's Going On in the Model? A Brief Overview of the GAMS Language Model Structure Sets Scalars, Parameters and Tables Variables Equations Assembling a Model Solving a Model Model Output IF-ELSE Operator References This Introduction to GAMS Modeling benefited from research by Andrey G. Savitsky. The use and distribution of GAMS is made possible by the GAMS Development Corporation ( and it greatly appreciated. 1

2 1. INTRODUCTION The General Algebraic Modeling System (GAMS) facilitates the creation of mathematical models, particularly optimization models, and allows the modeler to build models using the same logical structures as the mathematical equations that need to be solved. In this section, the river basin model described in Annex 3 will be described using the GAMS language to allow users to understand the model and to develop additional models. This section should be supplemented by the official GAMS Guide and Tutorials (Brooke et al., 2006), the monograph on optimization models for water and energy management (McKinney and Savitsky, 2006), and the GAMS website ( 2. GAMS INSTALLATION A. Run setup.exe: Run the file setup.exe directly from the CD. Use the Windows Explorer to browse the CD and double click on the file setup.exe. The setup program will prompt you for the name of the directory in which to install GAMS. Accept the default choice which the install program offers. Then the GAMS software will be installed. B. License file: You will be prompted for a license file during the installation. You do not have one, so choose No when asked if you wish to copy a license file. GAMS will function in demonstration mode and will only solve small problems. 3. GAMS OPERATION A. Start GAMS: Start GAMS by selecting Start All Programs GAMS GAMSIDE from the Windows Desktop 2

3 Figure 1. Starting GAMS. B. Create new GAMS project: To create a new project in GAMS, choose File Project New project from the GAMSIDE. Figure 2. Creating a new project in GAMS. Navigate from My Documents to the C\: drive. Create a new directory by pressing the 3

4 folder icon. Name the new directory C:\Example. Double click on the Example folder. Type Eq1 in the File name box and press Open. Figure 3. Naming a new project in GAMS. The GAMS window should now show the new Shapley.gpr project window. Figure 4. Naming a new project in GAMS. 4

5 C. Create a new GAMS code file: To create a new GAMS code file, select File New from the GAMSIDE. This will create the file Untitled_1.gms in the C:\Example directory. We will then copy the EQ1 GAMS code into this file from a text file. Figure 5. Creating a new GAMS code file. D. Open GAMS code file: Select File Open. Select Text files as the Files of type type. Navigate to the directory where you stored the and open the file EQ1.txt. Click on this files. Now it is automatically copied to GAMS. Figure 6. Opening the EQ1 text file. E. Tabs and Files: Now the window shows that you have two files open. This is indicated by the two Tabs along the top of the window in GAMS. The first tab displays the file Eq1.txt. This file, which we just opened, is shown in the active window. The second tab displays the untitled_1.gms file that we recently created. This is the GAMS Code file that 5

6 we are creating. To move back and forth between the two files, simply click on the appropriate tab at the top of the window. Figure 7. Multiple files open in GAMS. F. Copy text file: Now we will copy all of the text from the Eq1.txt file to the Untitled_1.gms file. Click on the Eq`.txt tab. Click the cursor in the Eq1.txt file window. Select Edit Select All. The entire file should become highlighted. Select Edit Copy. Figure 8. Selecting GAMS code from text file. 6

7 G. Paste into GAMS Code file: Now we will paste the text we just copied into the Untitled_1.gms file. Click the Untitled_1.gms tab. Click the cursor in the Untitled_1.gms file window. Select Edit Paste. Now the text is pasted into the file. Figure 9. Pasting GAMS code into the Untitled_1.gms file. H. Save GAMS Code file: Now we will save the new GAMS code file Untitled_1.gms in the new project directory C:\Example as Eq1.gms. Select File Save As. Navigate to the directory you created earlier: C:\Example. Name the file Untitled_1.gms as Eq1.gms. Save the file Eq1.gms. Figure 10. Saving the new GAMS file Eq1.gms. Now the files Eq1.txt and Eq1.gms are displayed in the window. 7

8 Figure 11. Window showing Eq1.txt and Eq1.gms. I. Run the model: Now we are ready to run the Shapley GAMS model. Press the red arrow button at the top of the GAMSIDE window. Alternatively, you can select File Run from the main menu Figure 12. Running the GAMS model. J. GAMS Model Results: The results should appear as below. There are two windows displayed by GAMS now. One is titled No active process and this window displays all of 8

9 the control commands and information that were produced by GAMS when it ran the model. The most important information is the line that reads Optimal Solution., which indicates to us that the model was solved successfully. Figure 13. Control screen from running the GAMS model. K. Viewing the GAMS model results: Near the bottom of this window, there is a line that reads: Putfile result C:\Example\Eq1.txt. The file Eq1.txt now contains the GAMS model results. Double clicking on this line will open the results file automatically in another window. Figure 14. Viewing GAMS model results. 9

10 L. Rerunning the model: You can make modifications to the Eq1 GAMS model and rerun the model to see the results of your changes. Select the Eq1.gms tab. Make your changes to the GAMS code in that window. Press the red arrow button at the top of the GAMSIDE window or select File Run from the main menu. Once the model is finished running, you will see the same window as in Figure 12 and you can again double click on the line with the name of the output file to display the new results. When you do this you will be asked if you want to Reload the Results. Select Yes. Figure 14. Rerunning the model. or Figure 15. Reloading the results. M. Exiting and Restarting GAMS: Select: File Exit to quit the GAMS program. When you restart GAMS, you will be returned to the place where you exited the program. 10

11 f1(x), f2(x), f1(x)-f2(x) 4. EXAMPLES 4.1. Algebraic Equation Background and GAMS Code The problem is to find the roots of the following nonlinear equation: (10x 2 10x 10) ( 10x 100) 0 One can see the roots in the following graph: f1(x) f2(x) f1(x)-f2(x) 10x 2 +10x x x 2 +10x +10-(-10x+100) x The GAMS code is: VARIABLES x, e, obj; EQUATIONS Eq1, Objective; Eq1.. e =E= 10*x*x+10*x+10-(-10*x+100); 11

12 Objective.. obj =E= e*e; MODEL Eq /ALL/ SOLVE Eq USING NLP MINIMIZING obj; FILE res /Eq1.txt/ PUT res; put "Solution x = ", put x.l, put /; The steps to enter this model into the GAMS/IDE are shown in the file Introduction to GAMS.pdf What's Going On in the Model? A. VARIABLES Variables can be zero-dimensional or multidimensional. The dimensions are determined by the availability and number of indices. The declaration of variables is based on the following principle. For example, consider the variables from the nonlinear equation model presented above: VARIABLES x, e, obj; These are all aero-dimensional (scalar) variables. All these variables are allowed to take on any vales, positive or negative. Variables are included in equations along with parameters and constants. Variables are always defined in the solution process. B. EQUATIONS The block of equations in GAMS models consists of two parts: 1. Declaration The syntax of declaring equation names is as follows: EQUATION name comment ; EQUATION is a keyword which must appear before the name of each equation if there is semicolon at the end of the line. It is used only once at the beginning of a list of equation names. The name of an equation may include no more than 10 symbols and must always begin with a letter. Consider the equation declaration in the model above EQUATIONS Eq1, Objective; This declares two scalar equations, the first equation is named Eq1 and the second objective. 12

13 2. Definition The definition of equations is where the mathematical form of the model is defined. The syntax for defining equations in GAMS is as follows: name.. expression; name is the name of the equation as previously defined in the list of declared equation names, then two periods.. follow separating the name of the equation and the mathematical expression (algebraic statement) of the equation. The equation must contain a relation operator, such as: =E= (equality) =G= (greater than or equal to) =L= (less than or equal to) Again, consider the equations from the model above Eq1.. e =E= 10*x*x+10*x+10-(-10*x+100); Objective.. obj =E= e*e; C. MODEL After all the VARIABLES and EQUATIONS of a model have been declared, a system (called a MODEL) can be formed from the equations and given a name. When all the declared equations are included in the model, the syntax of declaring the model is as follows: MODEL name comment /ALL/; For example: MODEL Eq /ALL/ The model is named "Eq", and the keyword ALL means that the model includes all of the equations. D. SOLVE Once a model has been defined, the next step is to solve it. This is ordered by including a SOLVE statement in the model. Consider the example of the river system management model in Section 3.3 SOLVE Eq USING NLP MINIMIZING obj; The SOLVE statement contains 6 parts: 1. The keyword SOLVE; 2. The previously defined name of the model (Eq); 3. The keyword USING; 13

14 4. The type of solver to use in the solution (NLP in this case since the model is nonlinear); 5. The keyword MINIMIZING; and 6. The name of a scalar variable (obj in this case) to be optimized. E. OUTPUT GAMS allows writing information about the model and its solution to text files. In addition, a text output file model_name.lst will always be produced. This file is created automatically, but its format is not entirely useful for analysis purposes. The following is the full syntax for defining output text files FILE pointer comments / filename / where FILE is a keyword used to define files; pointer is a pointer to direct output to the external text file filename; filename is an external text file which receives the output. Consider the example above FILE res /Eq1.txt/ where res is the pointer to the output file; and Eq1.txt is the output file. The PUT operator is used to write information into the output file. Consider the example: PUT put "Solution x = ", put x.l, put /; res; This example shows how the pointer res is associated with the output file Eq1.txt. In the first line, the output is directed into this file. In the next line, some text and the value (level) of the variable x is inserted into the file. 14

15 4.2. Least Squares Parameter Estimation Background and GAMS Code Independent variables (x 1, x 2, x 3 ) are related to the dependent variable (y) through the following function 2 2 B C y( t) y( t) Ax1 ( t) e x3( t) x2( t) where A, B, and C are unknown coefficients. Data have been observed at times t on the variables x 1 (t), x 2 (t), x 3 (t), and y(t). The measured values of y are designated as yˆ ( t) The residual between the model value and the observed value is e( t) y( t) yˆ( t) Numerical values of the coefficients are to be be determined by minimizing the squared residuals Minimize e( t) y( t) yˆ( t) t 1 t 1 In this case we need to introduce the notion of sets and parameters into the GAMS model. Sets are a way of defining indices in GAMS and we need one for time in this problem. Consider the following model developed in the GAMS language: SETS t / 1, 2, 3, 4, 5, 6, 7, 8 /; PARAMETER x1(t) /1 2, 2 3, 3 3, 4 3, 5 5, 6 5, 7 6, 8 7/; PARAMETER x2(t) /1 30, 2 60, 3 70, 4 60, 5 80, 6 90, 7 100, 8 100/; PARAMETER x3(t) /1 1, 2 6, 3 7, 4 3, 5 5, 6 9, 7 8, 8 17/; PARAMETER y_hat(t) /1 10, 2 20, 3 30, 4 20, 5 40, 6 50, 7 60, 8 70/; VARIABLES a, b, c, y(t), e(t), obj; EQUATION mod(t), residual(t), objective; mod(t).. a*x1(t)*x1(t)-b/x3(t)-c/x2(t)+exp(-y(t)*y(t)) =E= y(t); residual(t).. e(t) =E= y(t)-y_hat(t); objective.. obj=e=sum(t,power(e(t),2)); MODEL Leastsq / ALL /; SOLVE Leastsq USING NLP MINIMIZING obj; FILE res /Eq2.txt/ PUT res; 15

16 PUT "t x1(t) x2(t) x3(t) y(t) y_hat(t)"/; LOOP(t, PUT t.tl, x1(t), x2(t), x3(t), y.l(t), y_hat(t)/;); PUT /" a b c"/; PUT a.l, b.l, c.l; The steps to enter this model into GAMS and solve it are as follows: 1. Use the same subdirectory Example on the c:/ drive of your computer as you created in the previous example. 2. In GAMS, create a new project named Eq2. 3. Within this new project create a new GAMS file named Eq2.gms. 4. Open the file EQ1.txt and copy and paste the GAMS code into the new file Eq2.gms and save it in the "Example" directory. 5. Run the "Eq2.gms" model to ensure that it is working and examine the results file "Eq2.txt" What's Going On in the Model? A. SETS SETS, the equivalent of indices in a typical programming language, are used to make connections between variables and equations in models. In the model above, a SET is used to denote the number of the measurements: SETS t / 1, 2, 3, 4, 5, 6, 7, 8 /; B. PARAMETERS Digital data are contained in arrays (zero, scalar, or multidimensional matrices called parameters in GAMS). The SETs, described above, can play the role of indices for these arrays. To declare an array to contain data values, we use the functional words: SCALAR PARAMETER TABLE (zero-dimensional), (one-dimensional), and (multidimensional). Parameters are declared in a free format with the following general structure Parameter a(i,j) input-output matrix where Parameter data-type-keyword (required) 16

17 a identifier (required) (i,j) domain list (optional) input-output matrix text (optional) Scalars are defined similarly but without the domain list. Tables are simply multi-dimensional versions of parameters. For example, in the model above parameters are used to store the measurements: PARAMETER x1(t) /1 2, 2 3, 3 3, 4 3, 5 5, 6 5, 7 6, 8 7/; C. LOOPS The LOOP operator is often used with the PUT operator for printing out and displaying information on the screen or in files. The operator has the following syntax: For example LOOP (control_ area operator [; operator]); LOOP(t, PUT t.tl, x1(t), x2(t), x3(t), y.l(t), y_hat(t)/;); 5. A BRIEF OVERVIEW OF THE GAMS LANGUAGE 5.1. Model Structure GAMS models have a structure shown in Table 1. Using the GAMS Integrated Design Environment (GAMS-IDE, a graphic user interface for creating and running GAMS models), a GAMS model is prepared in text form in a file with the extension.gms. During model execution, information is displayed on the screen in the GAMS interface related to the compilation and solution characteristics and progress. The results of the model and computations are automatically sent to a special "LIST" file with the same name as the model file, but with an extension.lst. In addition, as discussed below, the user can program output to be directed to separate output files. 17

18 Table 1. Structure of a GAMS Model. 1. SETS Structures consisting of indices or names 2. DATA SCALARS (zero-dimentional), PARAMETERS (one-dimentional), and TABLES (multi-dimensional) Determination of values of input parameters 3. VARIABLES Variables or arrays of variables Declaration with assigning a type of variable Declaration of limits for possible changes, initial level 4. EQUATIONS Equations or complexes and arrays of equations (includes both declaration and definition) 5. MODEL Model declaration (which equations to include) 6. SOLVE Method of solution (which algorithm to use) 7. OUTPUT Output of information to files 5.2. Sets SETS are a major item in the GAMS language. SETS can be used to construct connections between variables and equations in models. SETS are the equivalent of indices in a typical programming language. Numerical values or text values can be used as indices, e.g., 1, 2, 3 can be used or first, second, etc. A set name must begin with either a letter or a digit, but the next symbol can be a letter, digit or the marks + and -. For example: SET t Months /Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec/; where the set is named t, the word Months is descriptive text to identify the purpose of the set, and the values contained between the / symbols are the indices or elements of the set. The elements in a SET are numbered sequentially or instance here, Feb is second element and Sep is the ninth element of the set. Elements of a set are separated by commas. LAG and LEAD operators are used to relate an element of a set with the next or preceding 18

19 element of the set. In circular LAG and LEAD operators (++, --), after the final element of the set, the first is repeated again. Sometimes this can be useful in reservoir modeling, especially if one wants the beginning values of variables in a time period (say, one year) to be the same as the ending values. Linear LAG and LEAD operators (+, -) are very convenient for modeling time periods that are not cycled. For example, if the initial value of reservoir storage over a time period is not to be identical to the final value. It is unlikely that the initial storage of a large, multi-year storage reservoir will be identical to its ending storage. As an example, consider the basic reservoir volume balance equation St St 1 Qt Rt beg _ S Qt Rt t 2,... T t 1 where S t and S t-1 are the storage in a reservoir at the end and beginning of a time period, respectively, Q t is the inflow to the reservoir during period t, and R t is the reservoir release during period t. Here we can write the equation in general, for all elements of the set t=1,2,, using a single equation and making special provision for the first month: S(t) =E= beg_s$(ord(t) EQ 1) + S(t-1)$(ord(t) GT 1) + Q(t) - R(t); If we do not want the starting and ending storage values to be identical, we need to specify a starting value for the previous month s storage when we write the equation for January, since t- 1 is undefined when the equation is written for t = 1 (i.e., Jan). In this case, when t = 1, the initial storage (beg_s) of the reservoir is used in the balance equation, but when the index t > 1 the variable storage is used in the calculation. The ordinal (ord) operator is used in this equation to return an ordinal of t for a specific instance of the equation. In the equation above, we have used the conditional ($) operator which provides the ability modify equations according to logical conditions. The general format of the operator is item$(condition) 19

20 This is understood to mean "fulfill the item if the condition is true". So in our reservoir storage equation example, the $ operator is operating on the conditions ord(t) EQ 1 and ord(t) GT 1. Thus, in the first case, if t takes on the value of the first element in the set, the value beg_s is added to the equation, otherwise it is skipped. In the second case, it means that if t takes on any value of the set greater than the first value, then S(t-1) is added to the equation, otherwise it is skipped Scalars, Parameters and Tables Data in GAMS models are contained in arrays (zero-, one-, or multi-dimensional matrices called scalars, parameters, and tables). SETs can play the role of indices for these arrays. An example of declaring a scalar and assigning a value to it is: SCALAR beg_s Beginning Storage of Reservoir (million m3) /14000/ where the scalar is beg_s and its value is All of the text between beg_s and is simply documentation of the purpose of the scalar. An example of declaring a parameter and assigning values to all of its elements is the reservoir inflow in a river basin model. This can be written in one of two ways: or PARAMETER Q(t) inflow to Reservoir A (million m3) /Jan 236, Feb 284, Mar 315, Apr 445, May 1388, Jun 2306, Jul 2749, Aug 1538, Sep 676, Oct 497, Nov 394, Dec 391/; PARAMETER Q(t) inflow to Reservoir A (million m3) / Jan 236 Feb 284 Mar 315 Apr 445 May 1388 Jun 2306 Jul 2749 Aug 1538 Sep

21 Oct 497 Nov 394 Dec 391 /; Often the second method is preferred since this facilitates cutting and pasting from spreadsheets Variables There are five types of variables that can be used in GAMS models: VARIABLE - Can take on any real values; POSITIVE VARIABLE - Can take on only positive real values; NEGATIVE VARIABLES - Can take on only negative real values; INTEGER VARIABLES - Can take on only integer values; and BINARY VARIABLES - Can take on only the integer values 0 and 1 Variables can be zero- or multi-dimensional. The dimensions are determined by the number of indices used in the definition of the variable. For example, consider the following variable declaration statement for the zero-dimensional (scalar) objective function value a model: VARIABLE obj Objective function value ($); Everything after the variable obj is documentation to explain the purpose of the variable. An example of a positive one-dimensional (vector) variable declaration is the reservoir storage value from a river basin model: POSITIVE VARIABLE S(t) Storage in Reservoir (million m3) The variable S is defined over the range of the set t. That is, is has elements for all the members of the index set. In this case S(t) is declared to be a positive variable, since we do not want to allow negative storage volumes and this saves us having to make specific lower bond declarations for the variables. Boundaries can be defined for variables with the help of specific suffixes. For example, in the 21

22 river basin model, several bounds on storage are set: S.UP(t) = K; S.LO(t) = S_min; S.LO('Dec') = end_s; The first example defines the maximum storage allowed in the reservoir (its capacity ), hence the use of the suffix.up ; the second defines the minimum storage volume allowed in the reservoir (known as dead storage), hence the use of the suffix.lo ; and the third defines the volume at the end of the year Equations Equations are defined in two parts in GAMS models: (1) Declaration of the names of equations; and (2) Definition of the equation structure. An example of declaring equations names is: EQUATIONS Res_balance(t) objective Reservoir mass balance Objective Function; Note that the first equation is declared to have the name Res_balance and GAMS will create an instance of the equation for each member of the set t, that is, it is a vector equation. The second equation objective is a scalar equation and only one instance of this equation is created by GAMS. The name of an equation must begin with a letter and contain less than 10 symbols. The definition of the structure of an equation follows the syntax: Equation_name.. expression1 =operator= expression2; First, we have the name of the equation (Equation_name) as declared in the list of equation names, then two periods.. separate the name of the equation and the mathematical definition of the equation structure. The expression of the equation is an algebraic statement using allowed GAMS algebraic operations. The equation must contain a GAMS relation operator, and these 22

23 are: =E= =G= =L= right hand side is equal to the left hand side; right hand side is greater than or equal to the left hand side; and right part is less than or equal to the left hand side An example of an equation definition from a river basin model is: Res_balance(t).. S(t) =E= beg_s$(ord(t) EQ 1) + S(t-1)$(ord(t) GT 1) + Q(t) - R(t); Equations can be continued over as many lines as necessary and they end with a semicolon. This equation is written in the same manner as we would write the mathematical equation. GAMS will include an instance of the equation for each element of the set t. An example of an equation using the less than or equal to relation operator is: LimitB(t).. D(t) =L= alpha*r(t); This equation sets the upper limit of water diversion in a model to be less than or equal to the flow in the river R(t) times the fraction alpha of the flow that is allowed to be diverted (the remaining portion being available to downstream users) Assembling a Model After the SETS, PARAMETERS, VARIABLES, and EQUATIONS of a model have been declared a system (called a MODEL) can be formed from a collection of equations and given a name. When all of the declared equations are included in the model, the syntax of declaring the model is as follows (Note this is the usual situation): MODEL name comment /ALL/; 23

24 For example: MODEL Reservoir / ALL /; The model is named "Reservoir", and the keyword ALL means that the model includes all of the equations. Sometimes, the user may want to create numerous variants of equations and unite them in groups, forming different models with distinct names and they may have various optimization objectives Solving a Model Once a model has been defined, the next step is to solve it. This is ordered by including a SOLVE statement in the model. Consider the following example: SOLVE Reservoir USING NLP MAXIMIZING obj; The SOLVE statement contains several parts: The keyword SOLVE; The previously defined name of the model (Reservoir in our example); The keyword USING; The type of solver to use in the solution (NLP or nonlinear programming, in this case). The keyword MAXIMIZING (in out case, but other models will use MINIMIZING); and The name of a scalar variable (obj in this case) to be optimized 5.8. Model Output The GAMS software can write information about the model and its solution to text files. The syntax for defining output text files is: FILE pointer_name comments / external_filename / 24

25 where FILE is a keyword used to define files; pointer_name is a pointer used by GAMS model to direct output to the external text file external_ filename; and external_filename is an external text file which receives the output from GAMS. In a river basin model, we may have FILE result /Basin_Results.txt/; PUT result where result is a pointer to the external file named Basin_Results.txt that will receive the model output. The pointer to the output file is named result in the model and it directs information into the file Basin_Result.txt. The PUT operator is used for both assigning the name of the current_file to a declared text file and writing information into this file. This example shows how the pointer res is associated with the external text file Basin_Result. Then, the output is directed into the file associated with the pointer result. Consider the following Put statements that occur later on in the river basin model: PUT /"Weights "; PUT /" wa = ", wa:7:3 PUT /" wb = ", wb:7:3; PUT /" wc = ", wc:7:3/; In the first line, some text (Weights) is inserted into the file. Then, the values of the objective function weights are written to the file preceded by text stating what each value corresponds to and the format for printing (field seven spaces wide with three places to the right of the decimal point). Sometimes, it is necessary to output the value of parameters or tables. This usually requires the use of the LOOP function. Consider the following example: PUT /"Reservoir (Mm3)"; PUT /" Storage Input Release "; PUT /"Initial ", beg_s:9:0/; LOOP(t,PUT t.tl, S.L(t):9:0, Q(t):9:0, R.L(t):9:0/;); 25

26 The first line prints a title for the table, and then, on the second line, a header at the top of the table. The third line prints the initial value of storage used in the calculation. The fourth line loops over all the elements of the set t and prints the value of t, S(t), Q(t), and R1(t). Note that the variables S(t) and R(t) need the suffix.l, but the parameter Q(t) does not. This is because Q(t) is a defined data value whose level is not determined by GAMS. However, the others are variables and their levels are determined in the solution process IF-ELSE Operator The IF-ELSE operator is useful for transferring from one operator to another. In some cases, it can be written down as a set of $ conditions. The IF operator can be used for making GAMS code more understandable. The optional "ELSE" part allows formulating the traditional construction "IF-THEN-ELSE". The following syntax is for the "IF-THEN-ELSE" operator: if (condition, operators; {condition ELSEIF, operators} [operators ELSE;] ); Note, the braces and brackets are not required, but can be used. "Condition" means logical conditions described in the paragraph on conditional operations of GAMS. Declaration or definition of equations can not be performed within the IF operator. Consider the following expressed through the IF-ELSEIF-ELSE operators: IF (f <= 0, p(i) = -1; 26

27 q(j) = -1; ELSEIF ((f > 0) AND (f < 1)), p(i) = p(i)**2; q(j) = q(j)**2; ELSE p(i) = p(i)**3; q(j) = q(j)**3; ); REFERENCES Brooke, A., D. Kendrick, A. Meeraus, and R. Raman (2006). GAMS Language Guide. Gams Development Corporation. Washington D.C. McKinney, D.C., and A. Savitsky, Basic Optimization Models for Water and energy management, Technical Report, Center for Research in Water Resources, University of Texas at Austin,