Quantitative Analysis for Management

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Chapter 9

Learning Objectives
After completing this chapter, students will be able to: 1. Convert LP constraints to equalities with slack, surplus, and artificial variables 2. Set up and solve LP problems with simplex tableaus 3. Interpret the meaning of every number in a simplex tableau 4. Recognize special cases such as infeasibility, unboundedness, and degeneracy 5. Use the simplex tables to conduct sensitivity analysis 6. Construct the dual problem from the primal problem © 2009 Prentice-Hall, Inc. 9–2

Linear Programming: The Simplex Method

© 2008 Prentice-Hall, Inc.

Chapter Outline
9.1 Introduction 9.2 How to Set Up the Initial Simplex Solution 9.3 Simplex Solution Procedures 9.4 The Second Simplex Tableau 9.5 Developing the Third Tableau 9.6 Review of Procedures for Solving LP Maximization Problems 9.7 Surplus and Artificial Variables 9.8 9.9 9.10 9.11 9.12 9.13

Chapter Outline
Solving Minimization Problems Review of Procedures for Solving LP Minimization Problems Special Cases Sensitivity Analysis with the Simplex Tableau The Dual Karmarkar’s Algorithm

© 2009 Prentice-Hall, Inc.

9–3

© 2009 Prentice-Hall, Inc.

9–4

Introduction
 With only two decision variables it is possible to

Introduction
 Why should we study the simplex method?  It is important to understand the ideas used to

use graphical methods to solve LP problems
 But most real life LP problems are too complex for  

 

simple graphical procedures We need a more powerful procedure called the simplex method The simplex method examines the corner points in a systematic fashion using basic algebraic concepts It does this in an iterative manner until an optimal solution is found Each iteration moves us closer to the optimal solution © 2009 Prentice-Hall, Inc. 9–5

produce solutions
 It provides the optimal solution to the decision

variables and the maximum profit (or minimum cost)  It also provides important economic information  To be able to use computers successfully and to interpret LP computer printouts, we need to know what the simplex method is doing and why

© 2009 Prentice-Hall, Inc.

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How To Set Up The Initial Simplex Solution
 Let’s look at the Flair Furniture Company from

Converting the Constraints to Equations
 The inequality constraints must be converted into

Chapter 7  This time we’ll use the simplex method to solve the problem  You may recall T = number of tables produced C = number of chairs produced

equations
 Less-than-or-equal-to constraints (≤) are

converted to equations by adding a slack variable to each  Slack variables represent unused resources  For the Flair Furniture problem, the slacks are S1 = slack variable representing unused hours in the painting department S2 = slack variable representing unused hours in the carpentry department

and
Maximize profit = $70T + $50C subject to 2T + 1C ≤ 100 4T + 3C ≤ 240 T, C ≥ 0 (objective function) (painting hours constraint) (carpentry hours constraint) (nonnegativity constraint) © 2009 Prentice-Hall, Inc. 9–7

 The constraints may now be written as

2T + 1C + S1 = 100 4T + 3C + S2 = 240
© 2009 Prentice-Hall, Inc. 9–8

Converting the Constraints to Equations
 If the optimal solution uses less than the

Converting the Constraints to Equations
 Each slack variable must appear in every

available amount of a resource, the unused resource is slack  For example, if Flair produces T = 40 tables and C = 10 chairs, the painting constraint will be 2T + 1C + S1 = 100 2(40) + 1(10) + S1 = 100 S1 = 10  There will be 10 hours of slack, or unused

constraint equation
 Slack variables not actually needed for an

equation have a coefficient of 0
 So

2T + 1C + 1S1 + 0S2 = 100 4T + 3C +0S1 + 1S2 = 240 T, C, S1, S2 ≥ 0  The objective function becomes

painting capacity
© 2009 Prentice-Hall, Inc. 9–9

Maximize profit = $70T + $50C + $0S1 + $0S2

© 2009 Prentice-Hall, Inc.

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