Linear Programming

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Linear Programming

• Integer Linear Models

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When Variables Have To Be Integers

• Example one time production decisions
• Fractional values make no sense
• But if ongoing process, fractional values could
  represent work in progress
• Example -- building houses or planes, or
  scheduling crews
• Binary variables
• Restricted to be 0 or 1
• Example Is a plant built?

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Types of Integer Programs (ILP)

• All Integer Linear Programs (AILP)
• All the decision variables are required to be
  integers
• Mixed Integer Linear Programs (MILP)
• Only some of the variables are required to be
  integers
• Binary Integer Linear Programs (BILP)
• Variables are restricted to be 0 or 1

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Example

• Boxcar Burger will build restaurants in the
  suburbs and downtown
• Suburbs
• Profit 12000/day
• 2,000,000 investment
• Requires 3 managers
• Downtown
• Profit 20000/day
• 6,000,000 investment
• Requires 1 manager
• Constraints
• 27,000,000 budget
• At least 2 downtown restaurants
• 19 managers available

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Decision Variables/Objective

• X1 Number of restaurants built in suburbs
• X2 Number of restaurants built downtown

MAX Expected Daily Profit
MAX 12X1 20X2 (in 1000s)
MAX Expected Daily Profit
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Constraints

• Cannot invest more than 27,000,000
• At least 2 downtown restaurants
• Number of managers used cannot exceed 19

Total Amount Invested
Cannot Exceed
27
27
2X1 6X2

downtown restaurants
Must be At least
2
X2
2

Managers used
Cannot Exceed
19
19
3X1 1X2

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The Complete Model

• MAX 12X1 20X2 (in 1000s)
• s.t. 2X1 6X2 ? 27 (Budget)
• X2 ? 2 (Downtown)
• 3X1 X2 ? 19 (Managers)
• Both Xs ? 0
• Both Xs INTEGER!

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Solving ILPs Using SOLVER

• The only change in SOLVER is to add the integer
  constraints
• In the Add Constraints dialogue box, highlight
  the cells required to be integer and choose int
  from the pull down menu for the sign

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Optimal Build 4 Suburban Restaurants Build 3
Downtown Restaurants Average Daily Profit 108,000
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General Facts About Integer Models

• The solution time to solve integer models is
  longer than that of linear programs
• Because many linear programs are solved en route
  to obtaining an optimal integer solution
• For maximization models, the optimal value of the
  objective function will be less (or at least not
  greater than) the value for the equivalent linear
  model
• Because constraints have been added the integer
  constraints
• There is no sensitivity analysis
• Because the feasible region is not continuous

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Review

• When to use integer models
• Solution time
• No sensitivity analysis
• Objective function value cannot improve
• SOLVER solution approach