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

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-2 2© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 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.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-3 3© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 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.

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-4 4© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 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

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-5 5© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 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

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-6 6© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 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.

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

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna 7-8 8© 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 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 Profit Amount $7 $5 Constraints: 4T + 3C  240 (Carpentry) 2T + 1C  100 (Paint & Varnishing) Objective: Max: 7T + 5C

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

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ Flair Furniture Company Feasible Region Number of Chairs Number of Tables Painting/Varnishing Carpentry Feasible Region

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

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

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ Flair Furniture Company Optimal Solution Number of Chairs Number of Tables Painting/Varnishing Carpentry Solution (T = 30, C = 40) Corner Points

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

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

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

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

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

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ A Problem with No Feasible Solution X2X2 X1X Region Satisfying 3rd Constraint Region Satisfying First 2 Constraints

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ A Solution Region That is Unbounded to the Right X2X2 X1X Feasible Region X 1 > 5 X 2 < 10 X 1 + 2X 2 > 10

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ A Problem with a Redundant Constraint X2X2 X1X Feasible Region 2X 1 + X 2 < 30 X 1 < 25 X 1 + X 2 < 20 Redundant Constraint

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ An Example of Alternate Optimal Solutions 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

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

To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ Changes in the Technological Coefficients for High Note Sound Co. Stereo Receivers X1X CD Players 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 < X1X1 30 CD Players

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