GAMS General Algebraic Modeling Software A brief introduction.

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GAMS General Algebraic Modeling SoftwareA
brief introduction.

• GAMS homepage
• http//www.gams.com
• Additional Reference
• http//gilbreth.ecn.purdue.edu/rardin/gams/notes.
  html

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Introduction

• A high level modeling system
• Components
• Language compiler
• Integrated high performance solvers
• Models supported
• Linear Programs
• Non-Linear Programs
• Mixed Integer Optimization
• Platforms
• PC
• Workstations
• Mainframes

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Introduction

• Advantages/Uses
• Easy representation of models
• Allows users to concentrate on modeling
• Programming similar to commonly used programming
  languages
• Solvers and models can be switched easily
• Easy to identify errors
• Available Solvers
• CPLEX
• XPRESS
• OSL
• CONOPT
• MINOS

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Basics

• How to compile a GAMS code ?
• Models encoded in filename.gms
• Compilation from command line on Unix/MSDOS
  gams filename
• Windows Environment
• GAMS creates input file to the solver
• Output generated in filename.lst

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Transportation Problem An Example

• A set of supply points with a finite amount of
  supply
• A set of demand points with a finite amount of
  demand
• Cost of satisfying the demand using the available
  supply
• Objective Satisfy all the demand at a minimum
  cost
• Issues Balance of supply and demand

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Mathematical Formulation
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Mathematical Formulation
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Sample Data
Cost of transportation 10 per mile
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GAMS Code
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GAMS CODE Continued..
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Results
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Structure of GAMS Model
Inputs
Outputs

• Set
• Declaration
• Assignment - indices
• Data (Parameters, Tables, Scalars)
• Declaration
• Assignment - values
• Variables
• Declaration
• Type Assignment
• Assignments of Bounds (optional)
• Equations
• Declaration
• Definition
• Model and Solve Statements
• Display Statements (optional)

• Echo Print
• Errors (if present)
• Reference Maps
• Equation Listings
• Status Reports
• Results

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Remarks

• An entity cannot be referenced before declaration
• Lines beginning with are treated as a comments
  / Statements between declarations and assignments
• Upper and Lower case letters are not
  distinguished
• Names of entities should start with a letter and
  can be followed up to 9 letters or digits
• Duplication not allowed

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Sets

• Basic building blocks Correspond to indices in
  Mathematical Formulation
• Different ways of declaration
• sets
• i supply points /1, 2/
• j demand points /1, 2, 3/
• set j demand points /13/
• sets j
• include j.inc
• In file inclusion all the indices should be
  enclosed between /./
• Alias(i,k)
• Multidimensional sets
• Dynamic Sets

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Data

• Different formats for data entry
• Lists
• Tables
• Direct Assignments
• Assignments from files
• Data entry by lists
• parameter d(j) demand at point j
• /1 100
• 2 200
• 3 300/
• List must be enclosed within slashes, entries
  separated by commas or separate lines
• Domain Checking
• Default value is zero
• Scalar is considered as a parameter with no
  domain
• Parameters with multi-dimensional domains can
  also be entered by lists

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Data

• Data entry by tables
• table dist(i,j) distance from i to j
• 1 2 3
• 1 10.2 5.3 50
• 2 30 21.2 4.5
• Declares parameter dist and specifies its domain
  as set of ordered pairs of Cartesian product of
  i, j
• Blanks are interpreted as zeros
• Domain checking
• Entering tables with more than one dimension

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Data

• Data entry by direct assignment
• parameter c(i,j) cost of transportation from i to
  j
• c(i,j) M dist_1(i,j)10
• Semicolon between two statements
• Parameters used in computation have to be
  previously declared
• Domain checking
• Specific elements in the domain can be assigned
  by
• c(1,3) 100
• Different possible operations possible using
  functions available in GAMS
• csq sqr(c)
• e m csq
• w l/lamda
• eoq(i) sqrt( 2 demand(i) ordcost(i) /
  holdcost(i))
• t(i) min(p(i), q(i)/r(i), log(s(i)))
• euclidean(i,j) qrt(sqr(xi(i) - xi(j)
  sqr(x2(i) - x2(j)))
• present(j) future(j)exp(-interesttime(j))

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Data

• Data entry by assignment from files
• Files can have a structure of list or tables
• parameter s(i)
• SUPPLY DATA
• include supply.inc
• Format //
• parameter c(i,j)
• COST DATA
• include cost.inc
• Format
• Table c(i,j)
• j
• i -

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Variables

• Decision variables must be declared
• free variable z cost of transportation
• positive variable x(i,j) amount transported
  from i to j
• Or
• Variables
• x(i,j) amount transported from i to j
• z cost of transportation
• positive variable x
• Default type is free
• Different permissible types for variables

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Variables

• z is a scalar quantity serves to model
  objective value either maximization or
  minimization
• Assignment of bounds to variables
• Different possible fields of a variable
• .lo lower bound
• .l level or primal value
• .up upper bound
• .m marginal or dual value
• Set automatically according to variable type
• Can be overwritten by
• x.up(i,j) 100
• x.lo(i,j) 10

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Summation and Product in GAMS

• Not possible to have standard mathematical
  notation for summation and product
• Summation
• sum(index of summation, summand)
• sum(j, x(i,j)) is equivalent to
• sum((i,j), c(i,j)x(i,j)) is equivalent to
• Product
• prod(j, x(i, j)) is equivalent to
• Product and Summation can be used to make
  assignments and in implementing equations
• Conditional summation and product

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Equations

• Must be declared and defined in separate
  statements
• Declaration
• equations obj, const1, const2
• Definition
• The name of the equation being defined
• The domain
• Domain restriction condition (optional)
• The symbol '..'
• Left-hand-side expression
• Relational operator l, e, or g
• Right-hand-side expression
• Multiple equations are created based on the
  domain
• dollar and such-that operators
• Difference between and e
• Variables can appear in the right hand side

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Equations

• obj..
• zesum((i,j),c(i,j)x(i,j))
• const1(i)..
• sum(j,x(i,j)) e s(i)
• const2(j)..
• sum(i,x(i,j)) e d(j)

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Model and Solve Statements

• model is a keyword in GAMS collection of
  equations
• Syntax
• model modelname /all/
• model formulation /all/
• model formulation /obj, const1, const2/
• Specific constraints can be implemented
• Option Statements
• option lp cplex

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Model and Solve Statements

• Solve Statement
• solve formulation using lp minimizing z
• Format of Solve statement
• The key word solve
• The name of the model to be solved
• The key word using
• An available solution procedure. The complete
  list is
• 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
• rminlp for relaxed mixed integer nonlinear
  programming
• The keyword "minimizing" or "maximizing"
• The name of the variable to be optimized

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Display Statements

• display x.l,z.l
• Displays the primal values (solution) of x(i,j)
  and objective z in filename.lst file
• Printing solution in customized format
• file soln/assign.out/
• put soln
• soln.sw20
• PRINTS OUT THE SOLUTION
• loop((i,j,t)(x.l(i,j,t) gt 0), put _at_2 i.te(i),
  _at_18, j.te(j), _at_38 x.l(i,j,t) 30 /)

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GAMS Output

• Echo Print
• Errors (if present)
• Example 160
• Summary of the error messages is given at the end
  of the filename.lst file
• Reference Maps
• Equation Listings
• Status Reports
• Results

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Questions ?