1. LPX-1

2. Linear Programming Models in Services Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

3. LPX-3  Describe the features of constrained optimization models.  Formulate LP models for computer solution.  Solve two-variable models using graphics.  Explain the nature of sensitivity analysis.  Solve LP models with Excel Add-in Solver and interpret the results.  Formulate a goal programming model. Learning Objectives

4. LPX-4 Let x = number of receivers to stock y = number of speakers to stock Maximize 50x + 20y gross profit Subject to 2x + 4y 400 floor space 100x + 50y 8000 budget x 60 sales limit x, y 0 Stereo Warehouse

5. LPX-5 Let E = units of egg custard base in the shake C = units of ice cream in the shake S = units of butterscotch syrup in the shake Minimize Subject to cholesterol fat protein calories Diet Problem Lakeview Hospital

6. LPX-6 Let x i = number of officers reporting at period i for i =1, 2, 3, 4, 5, 6 Minimize x 1 + x 6 6 period 1 x 1 + x 2 4 period 2 x 2 + x 3 14 period 3 x 3 + x 4 8 period 4 x 4 + x 5 12 period 5 x 5 + x 6 16 period 6 Shift-Scheduling Problem Gotham City Police Patrol

7. LPX-7 Let T t = number of trainees hired at the beginning of period t for t = 1,2,3,4,5,6 A t = number of tellers available at the beginning of period t for t = 1,2,3,4,5,6 Minimize subject to A 1 = 12 For t = 2,3,4,5,6 A t, T t 0 and integer for t = 1,2,3,4,5,6 Workforce-Planning Problem Last National Drive-in Bank

8. LPX-8 Let x ij = number of cars sent from city i to city j for i = 1,2,3 and j = 1,2,3,4 Minimize 439x 11 + 396 x 12 +... +479x 33 + 0x 34 subject to x 11 + x 12 + x 13 + x 14 = 26 x 21 + x 22 + x 23 + x 24 = 43 x 31 + x 32 + x 33 + x 34 = 31 x 11 + x 21 + x 31 = 32 x 12 + x 22 + x 32 = 28 x 13 + x 23 + x 33 = 26 x 14 + x 24 + x 34 = 14 x ij 0 for all i, j Transportation Problem Lease-a-Lemon Car Rental

9. LPX-9 Z=2000 Z=3000 Z=3600 Z=3800 A B C D E Optimal solution ( x = 60, y = 40) Graphical Solution Stereo Warehouse

10. LPX-10 Let s 1 = square feet of floor space not used s 2 = dollars of budget not allocated s 3 = number of receivers that could have been sold Maximize Z = 50x + 20y subject to 2x + 4y + s 1 = 400 (constraint 1) 100x + 50y + s 2 = 8000 (constraint 2) x + s 3 = 60 ( constraint 3) x, y, s 1, s 2, s 3 0 Model in Standard Form

11. LPX-11 Extreme Nonbasic Basic Variable Objective-function point variables variables value value Z A x, y s 1 400 0 s 2 8000 s 3 60 B s 3, y s 1 280 3000 s 2 2000 x 60 C s 3, s 2 s 1 120 3800 y 40 x 60 D s 1, s 2 s 3 20 3600 y 80 x 40 E s 1, x s 3 60 2000 y 100 s 2 3000 Stereo Warehouse Extreme-Point Solutions

12. LPX-12 z = 50x + 20y (constraint 3 ) (constraint 1) (constraint 2) A B C D Sensitivity Analysis Objective-Function Coefficients

13. LPX-13 (constraint 3 ) (constraint 2) A BI C D H Sensitivity Analysis Right-Hand-Side Ranging

14. LPX-14 Let x = number of receivers to stock y = number of speakers to stock = amount by which profit falls short of $99,999 = amount by which profit exceeds $99,999 = amount by which floor space used falls short of 400 square feet = amount by which floor space used exceeds 400 square feet = amount by which budget falls short of $8000 = amount by which budget exceeds $8000 = amount by which sales of receivers fall short of 60 = amount by which sales of receivers exceed 60 = priority level with rank k Minimize subject to profit goal floor-space goal budget goal sales-limit goal Goal Programming Stereo Warehouse Example

15. LPX-15 Topics for Discussion  How can the validity of LP models be evaluated?  Interpret the meaning of the opportunity cost for a nonbasic decision variable that did not appear in the LP solution.  Explain graphically what has happened when a degenerate solution occurs in an LP problem.  Is LP a special case of goal programming? Explain.  What are some limitations to the use of LP?