Working with GAMS

Dr. Hussam Alshraideh

Mathematical Modeling

Real-world

problem

Mathematical

model

assumptions,

abstraction, data

makes sense?

Solution to

model

Solution to

real-world problem

optimization

algorithm

interpretation

makes sense?

change the model,

assumptions?

General Algebraic Modeling System (GAMS)

• High-level modeling system for mathematical programming and optimization

• Algebraic formulation : closeness to mathematical notation

• Calling appropriate Algorithms• Calling appropriate Algorithms

• Efficient handling of mathematical optimization

problems

• Simple model building and solution process

• Increase productivity and maintainable models

GAMS Modeling Process

GAMS

Compilation

of Model

Input file:

MODEL

Output file:

RESULTS

Optimization

SOLVER

ExampleProblem Statement

Several forms of gasoline are produced during the petroleumrefining process, and a last step combines them to obtain marketproducts with specified quality measures. Suppose the followingfour gasoline products are available:

Type Quality Index-1 Quality Index-2 Cost($/Barrel)Type Quality Index-1 Quality Index-2 Cost($/Barrel)

1 99 210 48

2 70 335 43

3 78 280 58

4 91 265 46

Determine the minimum cost blend which has quality index-1

between 85 and 90 and quality index-2 between 270 and 280.

Mathematical Formulation

Mathematical Formulation

)1(909178709985

)(46584348

4321

4321

−≤+++≤

+++

ionSpecificatIndexQualityxxxx

toSubject

blendofCostxxxxMin

)2941.45,4706.0,0000.0,3529.0,1765.0:(

10101010

1

)2(280265280335210270

*

4

*

3

*

2

*

1

4321

4321

4321

=====

≤≤≤≤≤≤≤≤

=+++

−≤+++≤

CostxxxxSolution

xxxx

xxxx

ionSpecificatIndexQualityxxxx

sets

i set of gasoline /G1*G4/

j set of quality indices /Q1,Q2/;

parameters

cost(i) cost of gasoline set i

/G1 48

G2 43

G3 58

G4 46/

quality_spec_min(j) quality specifications -minimum

/Q1 85

Q2 270/

quality_spec_max(j) quality specifications-maximum

/Q1 90

equations

objective objective or cost function

min_spec_cons(j) minimum Quality indexconstraint

max_spec_cons(j) maximum Quality index constraint

tot_frac_cons total fraction=1 constraint ;

***********Model Definition********************

objective ..

tot_cost =e= sum(i,cost(i)*x(i));

min_spec_cons(j) ..

sum(i,quality_val(i,j)*x(i)) =g= quality_spec_min(j);

max_spec_cons(j) ..

sum(i,quality_val(i,j)*x(i)) =l= quality_spec_max(j);

tot_frac_cons ..

sum(i,x(i)) =e= 1;Q2 280/;

Table quality_val(i,j) quality indices

Q1 Q2

G1 99 210

G2 70 335

G3 78 280

G4 91 265;

variables

tot_costtotal cost in $

x(i) fraction of each Gasoline set;

sum(i,x(i)) =e= 1;

x.lo(i) = 0; x.up(i) = 1;

***********************************************

model blend /

objective,min_spec_cons,max_spec_cons,tot_frac_cons/;

***********Solve Statement*********************

solve blend using lp minimizing tot_cost;

***********************************************

displayx.l;

Page 1 Page 2

Structure of a GAMS Model

Inputs Outputs

• Sets

Declaration

Assignment of members

• Data

(Parameters, Tables, Scalar)

Declaration

Assignment of values

• Echo Print

• Reference Maps

• Equation Listings

• Status Reports

• Results

Assignment of values

• Variables

Declaration

Assignment of type

• Assignment of bounds and/or initial values

(optional)

• Equations

Declaration

Definition

• Model andSolve statements

• Display statement(optional)

sets

Sets are the basic building blocks of a GAMS

model, corresponding exactly to the indices in

the algebraic representations of models.

Text is optional. It’s like a sets

i set of gasoline /G1,G2,G3,G4/

j set of quality indices /Q1,Q2/;

•Any name of your choice

• No space in any name

• G1*G4 is same as G1,G2,G3,G4

Text is optional. It’s like a

comment

Mathematically:

i = {G1, G2,G3,G4}

j = {Q1,Q2}

parameters

parameters are for inserting given data

parameters

cost(i) cost of gasoline set i

/G1 48

G2 43 Name of the G2 43

G3 58

G4 46/

quality_spec_min(j) quality specifications -minimum

/Q1 85

Q2 270/

quality_spec_max(j) quality specifications-maximum

/Q1 90

Q2 280/;

Name of the

parameterGiven data

Optional text, useful

for understanding

Tables

Table quality_val(i,j) quality indices

Q1 Q2

G1 99 210

G2 70 335

G3 78 280G3 78 280

G4 91 265;

• Very useful for larger datasets

• GAMS will perform domain checking to make sure

that the row and column names of the table are

members of the appropriate sets.

General Comments on Data Entry

• Direct Assignment

e. g cost(‘Q1') = 48;

• Zero is the default value for all the parameters.

• A scalar is regarded as a parameter that has no domain. e.g..

Scalar f cost in dollars per mile/90/ ;Scalar f cost in dollars per mile/90/ ;

• The same parameter can be assigned a value more than once.

Each assignment statement takes effect immediately and

overrides any previous values.

• The same parameter may not be declared more than once

• Element-value can be separated by commas or entered as

separate lines

• Entire list must be enclosed in slashes

variables

variables

tot_cost total cost in $

x(i) fraction of each Gasoline set;

Every GAMS optimization problem must contain at least one variable which is

minimized or maximized (tot_cost in this example)

Variable Type Allowed range of variables

Free -∞ to +∞

Positive 0 to +∞

Negative -∞ to 0

Binary 0 or 1

Integer 0, 1, 2,….100

e. g Positive variable x;

Bounds

GAMS is designed with a small database system in which records

are maintained for the variables and equations.

.lo = lower bound

.l = level or primal value.l = level or primal value

.up = upper bound

.m = marginal or dual value

.fx = fixed value for the variable.

x.lo(i) = 0;

x.up(i) = 1;

Equations

equations

objective objective or cost function

min_spec_cons(j) minimum Quality index constraint

max_spec_cons(j) maximum Quality index constraint

tot_frac_cons total fraction=1 constraint;

Equations must be declared and defined in separate statements.

The power of algebraic modeling languages like GAMS is most

apparent in the creation of the equations and inequalities that

comprise the model under construction. This is because

whenever a group of equations or inequalities has the same

algebraic structure, all the members of the group are created

simultaneously, not individually.

Equation declaration :Summation

Format : Sum(index of summation, summand)

As a simple example, let us consider that the model has summation of x over i

1≥∑i

ix Sum(i,x(i)) =g=1

∑∑i

ij

j

ijxc Sum(i, Sum(j, c(i,j)*x(i,j)))

Sum((i,j), c(i,j)*x(i,j))

or

Other types of indexed operations

1≥∏i

ix prod(i,x(i)) =g=1

Product over controlling index

Maximum value over controlling indexMaximum value over controlling index

max_demand = smax((i,j), demand_array(i,j))

Minimum value over controlling index

min_distance = smin((p,q), distance(p,q))

Equation Declaration

• The format of the declaration is the same as

for other GAMS entities.

objective.. tot_cost =e= sum(i,cost(i)*x(i));objective.. tot_cost =e= sum(i,cost(i)*x(i));

1. The name of the equation being defined

2. The symbol '..’

3. Left-hand-side expression

4. Relational operator: =l=, =e=, or =g=

5. Right-hand-side expression

Equations

min_spec_cons(j) .. sum(i,quality_val(i,j)*x(i)) =g= quality_spec_min(j);

max_spec_cons(j) .. sum(i,quality_val(i,j)*x(i)) =l= quality_spec_max(j);

tot_frac_cons .. sum(i,x(i)) =e= 1;

1

)2(280265280335210270

)1(909178709985

4321

4321

4321

=+++

−≤+++≤

−≤+++≤

xxxx

ionSpecificatIndexQualityxxxx

ionSpecificatIndexQualityxxxx

Model

model blend /all/ ;

model blend /

objective,min_spec_cons,max_spec_cons,tot_frac_cons/;

or

• Model is a collection of Equations.

• Like other GAMS entities, it must be given a name in a

declaration.

• The format of the declaration is the keyword Model followed by

the name of the model, followed by a list of equation names

enclosed in slashes.

• If all the defined equations are to be used /all/ can be used

Solve

The format of the solve statement is as follows:

1. The keyword solve

2. The name of the model to be solved

3. The keyword using

4. An available solution procedure. Examples are

solve blend using lp minimizing tot_cost;

4. An available solution procedure. Examples are

• lp for linear programming

• nlp for nonlinear programming

• mip for mixed integer programming

• rmip for relaxed mixed integer programming

• minlp for mixed integer nonlinear programming

5. The keyword minimizing or maximizing

6. The name of the variable to be optimized

Display

• The solve statement will cause several things to happen when executed.

– The specific instance of interest of the model will be generated

– the appropriate data structures for inputting this problem to the solver will be createdsolver will be created

– the solver will be invoked

– the output from the solver will be printed to a file. (*.lst)

– To get the optimal values of the primal and/or dual variables, we can look at the solver output, or, if we wish, we can request a display of these result from GAMS as follows;

display x.l; for final values of x

Comments in the code

• Comments can be added in two ways;• Use the * sign at the beginning of the line

* This is a comment line

• Use the commands• Use the commands– $ontext

– $offtext

$ontext

This is a comment line

$offtext

GAMS output

• Echo print

• Error messages

• Reference maps

• Model statistics• Model statistics

• Status reports

• Solution reports

Echo Print and Error messages

• Echo Print:

– Copy of your input file with line numbers

– Irrespective of errors

• Error messages:• Error messages:

– Coded error message inside the echo print on the line

immediately following the scene of offense

– Messages start with **** and contain a $ followed

by a numerical error code

– Explanation of numerical code after the echo prints

Reference maps

• Include $onsymxrefas the first line in the script

• Cross reference maps

– This lists the named items (Sets, Parameters, Variables, Equations, Models, Files, Acronyms) in alphabetical order

– identifies them as to type, shows the line numbers where the symbols appear, and classifies each appearancesymbols appear, and classifies each appearance

SYMBOL TYPE REFERENCES

blend MODEL declared 55 defined 55 impl-asn 59 ref 59

cost PARAM declared 8 defined 9 ref 41

• List of model entities

– All the model entities are grouped by type

– Also lists documentary text associated with entities

Status and Solution reportsS O L V E S U M M A R Y

MODEL blend OBJECTIVE tot_cost

TYPE LP DIRECTION MINIMIZE

SOLVER CPLEX FROM LINE 59

**** SOLVER STATUS 1 Normal Completion1 Normal Completion**** MODEL STATUS 1 Optimal

**** OBJECTIVE VALUE 45.2941

RESOURCE USAGE, LIMIT 0.000 1000.000

ITERATION COUNT, LIMIT 4 2000000000

---- EQU min_spec_cons minimum Quality index constraint

LOWER LEVEL UPPER MARGINAL

G1 . 0.1765 1.0000 .

G2 . 0.3529 1.0000 .

G3 . . 1.0000 13.0000

G4 . 0.4706 1.0000 .

Useful links/Reference

• Online GAMS documentation

http://www.gams.com/docs/document.htm

• GAMS World : http://www.gamsworld.org/• GAMS World : http://www.gamsworld.org/

Exercise

• Assume a fifth product called Kerosene “K1” is

added to the list of products. The cost of this

product is $42 and it has quality index 1 of 80

and quality index 2 of 270. Modify the GAMS and quality index 2 of 270. Modify the GAMS

code to find the optimal mix that minimizes

the total cost.