GAMS: General Algebraic Modeling System Part 4 Integer Programming

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GAMS General Algebraic Modeling SystemPart
4 Integer Programming

• Tanya Borisova and Xiaobing Zhao
• For Graduate Seminar ARE 696 by Dr. Fletcher
• Tatiana.Borisova_at_mail.wvu.edu
• xbzhao_at_mail.wvu.edu

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Outline

• Integer Programming (IP)
• Model Scaling
• GAMS Seminar Evaluation

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References

• IP
• McCarl, Bruce A., and Thomas H. Spreen. Applied
  Mathematical Programming Using Algebraic Systems.
  
• http//agecon.tamu.edu/faculty/mccarl/regbook.htm
• Model Scaling
• McCarl, Bruce A. GAMS User Guide.
• (GAMS Menu Help-McCarl Guide)

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IP When use it?

• discontinuous decision variables
• machines, workers
• fixed costs
• investment portfolio optimization fixed costs
  if buy a stock
• logical conditions
• Complementary products if A is produced, then B
  must be produced
• Mutually exclusive products if A is produced,
  than B can not be produced
• discrete levels of resources
• acres of crop limited by area of 1, 2, or 3 fields

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Examples of IP Problems

• hiker selects the most valuable items to carry,
  subject to a weight or capacity limit
• Partial items are not allowed, thus choices are
  depicted by zero-one variables.
• minimum cost route for a salesman visiting N
  cities then returning home
• decision variable Xij equals one if the salesman
  goes from city i to city j, and zero otherwise.
• machinery selection when one maximizes profits,
  trading off the annual costs of machinery
  purchase with the extra profits obtained by
  having that machinery
• decision variables Yk are integer numbers of
  units of the k type machinery purchased

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? General Types of Mathematical Programming
Problems

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Classes of IP Problems

• Linear or nonlinear
• In the class, we focus on linear problems
• Pure IP all variables are integers
• Mixed IP integers and continuous variables
• Zero-one IP integer variables are restricted to
  equal either 0 or 1
• Pure zero-one IP problems
• Mixed zero-one IP problems

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IP in GAMS

• Variable declaration
• Binary Variables or Integer Variables
• Solve statement (example for linear model)
• Solve ModelName using mip minimizing
  VariableName
• Solvers ZOOM (in GAMS student version), OSL,
  LAMPS, XA, CPLEX

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IP Solution

• IP problems are difficult to solve
• there is no rule for the number of feasible
  solutions or their location
• most IP algorithms enumerate all possible integer
  solutions requiring substantial search effort
• duality is not well-defined
• discontinuous feasible solution region
• many of the shadow prices reported by GAMS are
  not relevant to the original problem, but are
  rather relevant to a problem transformed during
  the solution process

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Linear IP Example
Linear objective function
Linear constraints
Continuous variables
X 0 and integer Y 0 or 1
Integer variables
Zero-one variables
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Warehouse location

• Extension of transportation problem
• An example of modeling fixed costs and logical
  condition
• Mutual exclusivity - only one warehouse is
  constructed
• Construction costs are incurred only if warehouse
  is constructed

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Warehouse location (cont.)

• Locate warehouses within a transportation system
  to minimize total transportation costs
• Tradeoff between fixed warehouse construction
  costs and transportation costs
• Closely related examples location of processing
  facility, store or distribution center

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GAMS Program

• Open GAMS
• Create a new project in a convenient location
  (Desktop or MyDocuments)
• Open warehouse.gms file from the Desktop

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Warehouse location (cont.)

• Indexes
• i producers
• j markets
• k warehouse locations
• Data
• Producers production capacities si
• Demand on the markets dj
• Warehouses storage capacities CAPk
• Fixed costs of construction warehouses fk
• Total number of warehouses b

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Warehouse location (cont.)

• Variables
• qik shipment from producers to warehouses
• ykj shipment from warehouses to markets
• xij shipment directly from producers to markets
• vk indicate if warehouse k is constructed
  (binary)

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Warehouse Location Objective Function

• Minimize total costs in the system, which include
• fixed costs of facility construction
• transportation costs from producers to warehouses
• transportation costs from warehouses to markets
• transportation costs directly from producers to
  markets

Producers-warehouses
Producers-markets
Fixed costs
Warehouses-markets
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Warehouse location constraints

• Subject to supply constraints
• Total supply from ith producer to all warehouses
  qik and directly to all markets xij is no greater
  than the total production capacities si
• Demand constraints
• For jth market, total supply from all warehouses
  ykj and directly from all producers xij, is no
  less than the demand on the market dj

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Warehouse location constraints (cont.)

• Balance constraint 1
• Total supply of goods from all producers to kth
  warehouse is no less than total delivery from
  this warehouse to all markets
• Balance constraint 2
• Total shipment of goods from kth warehouse is no
  greater than the warehouses storage capacities,
  given that warehouse is constructed

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Warehouse location constraints (cont.)

• Configuration constraint
• total number of warehouses is no greater than b

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Warehouse location Results

• Which warehouse location is selected?
• What are the total transportation costs?
• Which producers ship to warehouse?

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Fixing Improperly Working Models Scaling

• Model solutions within GAMS - manipulation of
  large matrices and many computations
• Poorly scaled models can cause excessive time to
  be taken in solving or can cause the solver to
  fail

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Well-Scaled model

• Each of the following model components are
  ranging in absolute value from 0.01 to 100
• Optimal variable values
• Constants in the model equations
• Absolute values of the nonzero constraint
  marginals
• Derivatives of nonlinear terms (Jacobian
  elements) in equations

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Scaling

• Scaling using aggregate units
• (tons instead of pounds or thousand instead of
  )
• We can scale
• objective function
• constraints
• variables
• combination of the above

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Scaling (cont.)

• By yourself
• divide both sides of constraints by a scalar
• divide an objective function by a scalar
• change measurement units for variables and adjust
  variables coefficients
• note changes in reduced costs and marginals
• Using GAMS assistance
• variablename.scale(setdependency)k1
• equationname.scale(setdependency)k2

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GAMS Program Example

• Open the file scaling.gms from your desktop
• Note three parts of the program
• Initial problem
• Scaling with GAMS help
• Scaling by yourself

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Example Initial LP Model
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• Step 1 constraint scaling
• divide the 1st constraint by 10,000
• divide the 3rd constraint by 1,000
• divide the 4th constraint by 50
• Result

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• Step 2 scale variables X1 and X4
• X1 X1 / 10,000 -- aggregate units
• X4 X4 50 -- smaller units
• Coefficients in objective function and
  constraints
• X1 coefficients 10,000 X1 coefficients
• X4 coefficients X4 coefficients / 50

Result
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• Step 3 scale the objective function
• divide the objective function by 10,000

Result
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Program Output

• Run the program
• Find the outputs for the three models
• Compare number of iterations and computer
  resource use for the models
• Compare marginal values and reduced costs

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GAMS Seminar Summary

• Basics of GAMS language
• Linear Programming (Primal and Dual)
• Nonlinear Programming
• Integer Programming

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Course Evaluation

• Operations Research (OR)
• Scientific approach to analyzing problems and
  making decisions
• Involves understanding and structuring of complex
  situations
• Focus on optimization of system performance
• Use analytical and numerical techniques to
  develop and manipulate mathematical and computer
  models of organizational systems composed of
  people, machines, and procedures.