1 / 15

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

fawzi
Download Presentation

Linear Programming

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 12000X1 + 20000X2 s.t. 2X1 + 6X2 27 (Budget) X2  2(Downtown) 3X1 + X2  19 (Managers) Both X’s  0 Both X’s INTEGER!

  8. The Linear Programming Feasible Region X2 6 5 4 3 2 1 0 Max 12X1 + 20X2 LPFeasibleRegion 2X1 + 6X2≤ 27 3X1 + 1X2≤ 19 X2≥ 2 Rounded off 3X1 + 1X2≤ 19 2X1 + 6X2≤ 27 (5,3) Roundedup 12X1 + 20X2 (6,3) X2≥ 2 (5,2) Rounded down FEASIBLEObjective Value = 100 LP Optimum(5 7/16, 2 11/16)Obj. Value = 119 5 3 1 4 6 2 X1, X2≥ 0 X1

  9. The Integer Programming Feasible Region X2 6 5 4 3 2 1 0 Max 12X1 + 20X2 ILP Optimum(4,3)OBJ. VALUE = 108 2X1 + 6X2≤ 27 X2≥ 2 2X1 + 6X2≤ 27 12X1 + 20X2 X1, X2 integer X2≥ 2 3 5 1 4 6 2 3X1 + 1X2≤ 19 3X1 + 1X2≤ 19 X1, X2≥ 0 X1

  10. Why Not Round To Get the Optimal Integer Solution? • Rounding may yield the optimal integer solution • None did in this example • But it may yield an infeasible solution • Both (5,3) and (6,3) are infeasible solutions • Or a feasible solution that is not optimal • (5,2) is feasible but not optimal • Many times a feasible rounded point gives a “good” solution (giving close to the optimal value of the objective function) -- BUT NOT ALWAYS

  11. 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

  12. 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

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

  14. Review • When to use integer models • Why rounding will not always work • Solution time • No sensitivity analysis • Objective function value cannot improve • SOLVER solution approach

More Related