1. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-1 1© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Chapter 7 Linear Programming Models: Graphical and Computer Methods

2. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-2 2© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Learning Objectives Students will be able to: Understand the basic assumptions and properties of linear programming (LP). Formulate small to moderate- sized LP problems. Graphically solve any LP problem with two variables by both the corner point and isoline methods.

3. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-3 3© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Learning Objectives - continued Understand special issues in LP - infeasibility, unboundedness, redundancy, and alternative optima. Understand the role of sensitivity analysis. Use Excel spreadsheets to solve LP problems.

4. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-4 4© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Examples of Successful LP Applications 1. Development of a production schedule that will satisfy future demands for a firm’s production and at the same time minimize total production and inventory costs 2. Selection of the product mix in a factory to make best use of machine- hours and labor-hours available while maximizing the firm’s products

5. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-5 5© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Examples of Successful LP Applications 3. Determination of grades of petroleum products to yield the maximum profit 4. Selection of different blends of raw materials to feed mills to produce finished feed combinations at minimum cost 5. Determination of a distribution system that will minimize total shipping cost from several warehouses to various market locations

6. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-6 6© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Requirements of a Linear Programming Problem All problems seek to maximize or minimize some quantity (the objective function). The presence of restrictions or constraints, limits the degree to which we can pursue our objective. There must be alternative courses of action to choose from. The objective and constraints in linear programming problems must be expressed in terms of linear equations or inequalities.

7. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-7 7© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Basic Assumptions of Linear Programming Certainty Proportionality Additivity Divisibility Nonnegativity

8. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-8 8© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture Company Data - Table 7.1 Hours Required to Produce One Unit Department T Tables C Chairs Available Hours This Week Carpentry Painting &Varnishing 4242 3131 240 100 Profit Amount $7 $5 Constraints: 4T + 3C  240 (Carpentry) 2T + 1C  100 (Paint & Varnishing) Objective: Max: 7T + 5C

9. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-9 9© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture Company Constraints Number of Tables 120 100 80 60 40 20 0 Number of Chairs 20 40 60 80 100 Painting/Varnishing Carpentry

10. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-10 10© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture Company Feasible Region 120 100 80 60 40 20 0 Number of Chairs 20 40 60 80 100 Number of Tables Painting/Varnishing Carpentry Feasible Region

11. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-11 11© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture Company Isoprofit Lines Number of Tables Number of Chairs 120 100 80 60 40 20 0 20 40 60 80 100 Painting/Varnishing Carpentry 7T + 5C = 210 7T + 5C = 420

12. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-12 12© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture Company Optimal Solution Number of Chairs 120 100 80 60 40 20 0 20 40 60 80 100 Number of Tables Painting/Varnishing Carpentry Solution (T = 30, C = 40) Isoprofit Lines

13. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-13 13© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture Company Optimal Solution Number of Chairs 120 100 80 60 40 20 0 20 40 60 80 100 Number of Tables Painting/Varnishing Carpentry Solution (T = 30, C = 40) Corner Points 1 2 3 4

14. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-14 14© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Flair Furniture - Excel

15. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-15 15© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Holiday Meal Turkey Ranch (C) (B) toSubject :Minimize               ½ X XX A)( XX: XX

16. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-16 16© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Holiday Meal Turkey Problem Corner Points

17. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-17 17© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Holiday Meal Turkey Problem Isoprofit Lines

18. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-18 18© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Special Cases in LP Infeasibility Unbounded Solutions Redundancy Degeneracy More Than One Optimal Solution

19. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-19 19© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 A Problem with No Feasible Solution X2X2 X1X1 8642086420 24682468 Region Satisfying 3rd Constraint Region Satisfying First 2 Constraints

20. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-20 20© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 A Solution Region That is Unbounded to the Right X2X2 X1X1 15 10 5 0 5 10 15 Feasible Region X 1 > 5 X 2 < 10 X 1 + 2X 2 > 10

21. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-21 21© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 A Problem with a Redundant Constraint X2X2 X1X1 30 25 20 15 10 5 0 510 15 20 25 30 Feasible Region 2X 1 + X 2 < 30 X 1 < 25 X 1 + X 2 < 20 Redundant Constraint

22. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-22 22© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 An Example of Alternate Optimal Solutions 876543210876543210 1 2 3 4 5 6 7 8 Optimal Solution Consists of All Combinations of X 1 and X 2 Along the AB Segment Isoprofit Line for $12 Overlays Line Segment Isoprofit Line for $8 A B AB

23. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-23 23© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Sensitivity Analysis Changes in the Objective Function Coefficient Changes in Resources (RHS) Changes in Technological Coefficients

24. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-24 24© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Changes in the Technological Coefficients for High Note Sound Co. Stereo Receivers X1X1 60 40 20 0 CD Players 20 40 X2X2 (a) Original Problem 3X 1 + 1X 2 < 60 Optimal Solution a 2X 1 + 4X 2 < 80 b c X2X2 (b) Change in Circled Coefficient Still Optimal a 2X 1 + 4X 2 < 80 d e 2X 1 + 1X 2 < 60 20 40 X1X1 30 CD Players

25. To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-25 25© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458 Changes in the Technological Coefficients for High Note Sound Co. X1X1 Stereo Receivers 60 40 20 0 CD Players 20 40 X2X2 (a) Original Problem 3X 1 + 1X 2 < 60 Optimal Solution a 2X 1 + 4X 2 < 80 b c 20 40 X2X2 X1X1 (c) Change in Circled Coefficient 3X 1 + 1X 2 < 60 Optimal Solution f 2X 1 + 5X 2 < 80 g c CD Players