Chapter 7 Linear Programming Models: Graphical and Computer Methods

# Chapter 7 Linear Programming Models: Graphical and Computer Methods

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## Chapter 7 Linear Programming Models: Graphical and Computer Methods

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1. Chapter 7 Linear Programming Models: Graphical and Computer Methods 7-1

2. 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. 7-2

3. 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. 7-3

4. Chapter Outline 7.1 Introduction 7.2 Requirements of a Linear Programming Problem 7.3 Formulating LP Problems 7.4 Graphical Solution to an LP Problem 7.5 Solving Flair Furniture’s LP Problem using QM for Windows and Excel 7-4

5. Chapter Outline - continued 7.6 Solving Minimization Problems 7.7 Four Special Cases 7.8 Sensitivity Analysis in LP 7-5

6. 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 7-6

7. 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 7-7

8. 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. 7-8

9. Basic Assumptions of Linear Programming • Certainty • Proportionality • Additivity • Divisibility • Nonnegativity 7-9

10. Flair Furniture Company Data - Table 7.1 Hours Required to Produce One Unit Available Hours This Week T Tables C Chairs Department Carpentry Painting &Varnishing 4 2 3 1 240 100 Profit Amount \$7 \$5 Constraints: 4T + 3C 240 (Carpentry) 2T + 1C 100 (Paint & Varnishing) Objective: Max: 7T + 5C 7-10

11. Flair Furniture Company Constraints 120 100 80 60 40 20 0 Painting/Varnishing Number of Chairs Carpentry 20 40 60 80 100 Number of Tables 7-11

12. Flair Furniture Company Feasible Region 120 100 80 60 40 20 0 Painting/Varnishing Number of Chairs Carpentry Feasible Region 20 40 60 80 100 Number of Tables 7-12

13. Flair Furniture Company Isoprofit Lines 120 100 80 60 40 20 0 Painting/Varnishing 7T + 5C = 210 Number of Chairs 7T + 5C = 420 Carpentry 20 40 60 80 100 Number of Tables 7-13

14. Flair Furniture Company Optimal Solution Isoprofit Lines 120 100 80 60 40 20 0 Painting/Varnishing Solution (T = 30, C = 40) Number of Chairs Carpentry 20 40 60 80 100 Number of Tables 7-14

15. Flair Furniture Company Optimal Solution Corner Points 120 100 80 60 40 20 0 2 Painting/Varnishing Solution (T = 30, C = 40) Number of Chairs Carpentry 3 1 4 20 40 60 80 100 Number of Tables 7-15

16. Holiday Meal Turkey Ranch ( A) 2 + 3 Minimize : X X 1 2 5 + 10 ³ 90 Subject to : X X 1 2 4 + 3 ³ 48 (B) X X 1 2 ³ 1 1/2 (C) ½ X 1 7-18

17. Holiday Meal Turkey Problem Corner Points 7-19

18. Holiday Meal Turkey Problem Isoprofit Lines 7-20

19. Special Cases in LP • Infeasibility • Unbounded Solutions • Redundancy • Degeneracy • More Than One Optimal Solution 7-21

20. A Problem with No Feasible Solution X2 8 6 4 2 0 Region Satisfying 3rd Constraint 2 4 6 8 X1 Region Satisfying First 2 Constraints 7-22

21. A Solution Region That is Unbounded to the Right X2 X1 > 5 15 10 5 0 X2 < 10 Feasible Region X1 +2X2 > 10 5 10 15 X1 7-23

22. A Problem with a Redundant Constraint X2 30 25 20 15 10 5 0 Redundant Constraint 2X1 + X2 < 30 X1 < 25 X1 +X2 < 20 Feasible Region X1 5 10 15 20 25 30 7-24

23. An Example of Alternate Optimal Solutions 8 7 6 5 4 3 2 1 0 Optimal Solution Consists of All Combinations of X1 and X2 Along the AB Segment A Isoprofit Line for \$8 Isoprofit Line for \$12 Overlays Line Segment B AB 1 2 3 4 5 6 7 8 7-25

24. Sensitivity Analysis • Changes in the Objective Function Coefficient • Changes in Resources (RHS) • Changes in Technological Coefficients 7-26

25. Changes in the Technological Coefficients for High Note Sound Co. (a) Original Problem (b) Change in Circled Coefficient X2 X2 60 40 20 0 2X1 + 1X2 < 60 3X1 + 1X2 < 60 Optimal Solution Still Optimal Stereo Receivers a a 2X1 + 4X2 < 80 2X1 + 4X2 < 80 b d c e X1 30 X1 20 40 20 40 CD Players CD Players 7-27

26. Changes in the Technological Coefficients for High Note Sound Co. (a) Original Problem (c) Change in Circled Coefficient X2 X2 60 40 20 0 3X1 + 1X2 < 60 3X1 + 1X2 < 60 Optimal Solution Optimal Solution Stereo Receivers 2X1 + 5X2 < 80 a 2X1 + 4X2 < 80 f b g c c X1 20 40 20 40 X1 CD Players CD Players 7-28