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Linear Programming

Linear Programming. Integer Linear Models. 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

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Linear Programming

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  1. Linear Programming Integer Linear Models

  2. 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?

  3. 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

  4. 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

  5. Decision Variables/Objective • X1 = Number of restaurants built in suburbs • X2 = Number of restaurants built downtown MAX Expected Daily Profit MAX 12X1 + 20X2(in $1000’s) MAX Expected Daily Profit

  6. Constraints In $1,000,000’s • 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 2 X2 ≥ # Managers used Cannot Exceed 19 19 3X1 + 1X2 ≤

  7. The Complete Model MAX 12X1 + 20X2 (in $1000’s) s.t. 2X1 + 6X2 27 (Budget) X2  2(Downtown) 3X1 + X2  19 (Managers) Both X’s  0 Both X’s INTEGER!

  8. Solving ILP’s 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

  9. Optimal Build 4 Suburban Restaurants Build 3 Downtown Restaurants Average Daily Profit $108,000

  10. 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

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

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