# Linear Optimization in Applications

Publication Year: 1999

Published by: Hong Kong University Press, HKU

#### Cover

#### Title Page

#### Copyright

#### CONTENTS

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pp. v-vi

#### PREFACE

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pp. vii

This book is not written to discuss the mathematics of linear programming. It is designed to illustrate, with practical examples, the applications of linear optimization techniques The simplex method and the revised simplex method, therefore, are included as appendices only in the book ...

#### 1. INTRODUCTION

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pp. 1-8

Linear programming is a powerful mathematical tool for the optimization of an objective under a number of constraints in any given situation. Its application can be in maximizing profits or minimizing costs while making the best use of the limited resources available. Because it is a mathematical tool, it is best explained using a practical example ...

#### 2. PRIMAL AND DUAL MODELS

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pp. 9-18

The shadow price (or called opportunity cost) of a resource is defined as the economic value (increase in profit) of an extra unit of resource at the optimal point. For example, the raw material available in Example 1.1 of Chapter 1 is 800 kg; the shadow price of it means the increase in profit (or the increase in Z, the objective function) if the raw material is increased by one unit, to 801 kg ...

#### 3. Formulating Linear Optimization Problems

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pp. 19-50

In Chapter 1, we have seen how a Iinear programming model for maximizing profit under limited resources is formulated. In Chapter 2, we have also seen how its dual model is formulated. This chapter contains more examples on the formulation of linear programming problems. ...

#### 4. Transporlalion Problern and Algorithm

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pp. 51-74

In Example 3.1 of Chapter 3, we have seen what a typical transportation problem is. Examples 3.3, 3.4 and 3.5 are also transportation problems although they are less obvious than example 3.1. Transportation problems can be solved, of course, by the simplex method. However, there is an algorithm which can also solve transportation problems without using the techniques of ...

#### 5. Integer Programming Formulation

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pp. 75-104

In many linear optimization problems, we require that the decision variables be integers. Strictly speaking , Example 1.1 in Chapter 1 is an integer programming problem , because the optimal number of pipes must be whole numbers. Example 1.1 , fortunately, has an optimal solution with decision variables in integers, and so we did not worry about the process of converting ...

#### 6. Integer Programming Solution

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pp. 105-116

The most powerful method of finding solutions for integer linear programming
problems is the **branch and bound** method. In this chapter, we will consider
two examples using the branch and bound method. One example is a problem
with decision variables greater than zero but which must be integers. Another
example is a problem with zero-one variables. ...

#### 7. Goal Programming Formulation

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pp. 117-130

In previous chapters, all examples involved the maximization or minimization of
one single objective under constraints. In **goal programming**, it is possible for
us to optimize more than one objective in a problem. In fact, an ordinary linear
programming model (with only one objective) can also be formulated as a goal
programming model. The first example in this chapter will iIlustrate this. From ...

#### 8. Goal Programming Solution

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pp. 131-144

In this chapter we will see how a goal programming model is solved. The revised simplex method (Appendix B) will be used as the too1. The example given below wi1l i1lustrate the solutioning process. The model formulated in Example 7.2 of the previous chapter is used. ...

#### Appendix A. Examples on Simplex Method

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pp. 145-152

#### Appendix B. Examples on Revised Simplex Method

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pp. 153-160

#### Appendix C. Use of Slack Variables, Artifical Variables and Big-M

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pp. 161

#### Appendix D. Examples of Special Cases

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pp. 162-164

E-ISBN-13: 9789882202085

Print-ISBN-13: 9789622094833

Page Count: 172

Publication Year: 1999