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Quantitative Analysis for Management

Quantitative Analysis for Management. Chapter 7 Linear Programming Models: Graphical and Computer Methods. 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

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Quantitative Analysis for Management

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  1. Quantitative Analysis for Management Chapter 7 Linear Programming Models: Graphical and Computer Methods 7-1

  2. 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 7-2

  3. Chapter Outline - continued 7.6 Solving Minimization Problems 7.7 Summary of the Graphical Solution Methods 7.8 Four Special Cases 7.9 Sensitivity Analysis in LP 7-3

  4. 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. 7-4

  5. 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, 7-5

  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 • 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 7-6

  7. Requirements of a Linear Programming Problem • 1. All problems seek to maximize or minimize some quantity (the objective function). • 2. The presence of restrictions or constraints, limits the degree to which we can pursue our objective. • 3. There must be alternative courses of action to choose from. • 4. The objective and constraints in linear programming problems must be expressed in terms of linear equations or inequalities. 7-7

  8. Basic Assumptions of Linear Programming • Certainty • Proportionality • Additivity • Divisibility • Nonnegativity 7-8

  9. Flair Furniture Company Data - Table 7.1 4 + 3 £ 240 X X (carpentry ) 1 2 2 + 1 £ 100 X X (painting & varnishing ) 1 2 7 + 5 X X 1 2 Hours Required to Produce One Unit Available Hours This Week X1 Tables X2 Chairs Department Carpentry Painting/Varnishing 4 2 3 1 240 100 Profit/unit Constraints: $7 $5 Maximize: Objective: 7-9

  10. 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 7-10

  11. 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 7-11

  12. Flair Furniture Company Isoprofit Lines 120 100 80 60 40 20 0 Painting/Varnishing 7X1 + 5X2 = 210 Number of Chairs 7X1 + 5X2 = 420 Carpentry 20 40 60 80 100 Number of Tables 7-12

  13. Flair Furniture Company Optimal Solution 120 100 80 60 40 20 0 Painting/Varnishing Solution (X1 = 30, X2 = 40) Number of Chairs Carpentry 20 40 60 80 100 Number of Tables Isoprofit Lines 7-13

  14. Flair Furniture Company Optimal Solution 120 100 80 60 40 20 0 Painting/Varnishing Solution (X1 = 30, X2 = 40) Number of Chairs Carpentry 20 40 60 80 100 Number of Tables Corner Points 2 3 1 4 7-14

  15. Flair Furniture - QM for Windows 7-15

  16. Flair Furniture - Excel 7-16

  17. Holiday Meal Turkey Ranch 2 + 3 Minimize : X X 1 2 5 + 10 ³ 90 Subject to : X X ( A) 1 2 4 + 3 ³ 48 X X (B) 1 2 1 1 ³ 1 X (C) 1 2 2 7-17

  18. Holiday Meal Turkey Problem Corner Points 7-18

  19. Holiday Meal Turkey Problem Isoprofit Lines 7-19

  20. Special Cases in LP • Infeasibility • Unbounded Solutions • Redundancy • Degeneracy • More Than One Optimal Solution 7-20

  21. 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 7-21

  22. A Solution Region That is Unbounded to the Right X1 > 5 X2 < 10 X2 15 10 5 0 Feasible Region X1 +2X2 > 10 5 10 15 X1 7-22

  23. 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 7-23

  24. 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 7-24

  25. Sensitivity Analysis • Changes in the Objective Function Coefficient • Changes in Resources (RHS) • Changes in Technological Coefficients 7-25

  26. 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 a a 2X1 + 4X2 < 80 2X1 + 4X2 < 80 b d c e X1 20 40 30 X1 20 40 CD Players CD Players Stereo Receivers 7-26

  27. 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 7-27

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