Linear programming and its applications

390 Pages·2007·3.457 MB·English

by  H.A. Eiselt, C.-L. Sandblom

LinearProgramming
anditsApplications
·
H. A. Eiselt C.-L. Sandblom
Linear Programming
and its Applications
With71Figures
and36Tables
123
Prof.Dr.H.A.Eiselt
UniversityofNewBrunswick
FacultyofBusinessAdministration
P.O.Box4400
Fredericton,NBE3B5A3
Canada
[email protected]
Prof.Dr.C.-L.Sandblom
DalhousieUniversity
DepartmentofIndustrialEngineering
P.O.Box1000
Halifax,NSB3J2X4
Canada
[email protected]
LibraryofCongressControlNumber:2007931630
ISBN978-3-540-73670-7SpringerBerlinHeidelbergNewYork
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“A problem well stated is a problem half solved.”

Charles Franklin Kettering
PREFACE
Based on earlier work by a variety of authors in the 1930s and 1940s, the simplex
method for solving linear programming problems was developed in 1947 by the
American mathematician George B. Dantzig. Helped by the computer revolution,
it has been described by some as the overwhelmingly most significant
mathematical development of the last century. Owing to the simplex method,
linear programming (or linear optimization, as some would have it) is pervasive in
modern society for the planning and control of activities that are constrained by
the availability of resources such as manpower, raw materials, budgets, and time.

The purpose of this book is to describe the field of linear programming. While we
aim to be reasonably complete in our treatment, we have given emphasis to the
modeling aspects of the field. Accordingly, a number of applications are provided,
where we guide the reader through the interactive process of mathematically
modeling a particular practical situation, analyzing the consequences of the model
formulated, and then revising the model in light of the results from the analysis.

Closely related to the issue of building models based on specific applications is
the art of reformulating problems. Some of these models may at first appear not to
be amenable to a linear representation, and we devote an entire chapter to this
topic. A properly balanced treatment of linear programming will necessarily
require a full discussion of both duality and postoptimality, and we dedicate one
chapter to each of these two topics. As far as solution methods are concerned, we
cover the simplex method as well as interior point techniques. During the last two
decades, the latter have become serious challengers to the simplex method for
solving large scale practical problems.

This book can be seen as the last part of a trilogy. The other two volumes have
already appeared in print. "Integer Programming and Network Models" was
published in 2000, and "Decision Analysis, Location Models, and Scheduling
Problems" saw the light of day in 2004. All three volumes are similar in style,
emphasizing models, applications and formulations/reformulations. We have also
given detailed numerical illustrations for all algorithms presented, and have relied,
whenever practical, on intuitive approaches. An interesting aspect is the longevity
of a book like the present volume. It appears that descriptions of models keep their
freshness longer than discussions of algorithms, and that references to
computational aspects quickly become outdated. A statement from 1824 gives a
poignant reminder of how short the life of a book may be:
VIII Preface

“…One thousand books are published per annum in Great
Britain ... only do one hundred bring good profit ... seven
hundred are forgotten in one year, one hundred in two years, ....
not more than fifty survive seven years, and scarcely ten are
thought of after twenty years.

Of the 50,000 books published in the seventeenth century, not
fifty are now in estimation; and of the 80,000 published in the
eighteenth century not more than three hundred are considered
worth reprinting, and not more than five hundred are sought
after 1823. Since the first writings fourteen hundred years
before Christ, i.e., in thirty-two centuries, only about five
hundred works of writers of all nations have sustained
themselves against the devouring influence of time.”

(Collections, Historical and Miscellaneous; and Monthly
Literary Journal: edited by J. Farmer and J.B. Moore, Vol III,
Concord 1824)

It is our pleasure to thank all of the people who have, in one way or another,
helped to make this book a reality. Some of the typing was done by #13 (Benbin
Zhang) and the figures were produced by Dong Lin. Last, but certainly not least,
our sincere thanks go to Dr. Müller of Springer Publishers, whose gentle
reminders kept us on track and more or less on time. We are very grateful for the
assistance.

H.A. Eiselt
C.-L. Sandblom
CONTENTS
Symbols XIII
A. Linear Algebra 1
A.1 Matrix Algebra 1
A.2 Systems of Simultaneous Linear Equations 5
A.3 Convexity 23

B. Computational Complexity 31
B.1 Algorithms and Time Complexity Functions 31
B.2 Examples of Time Complexity Functions 37
B.3 Classes of Problems and Their Relations 41

1. Introduction 45
1.1. A Short History of Linear Programming 45
1.2 Assumptions and the Main Components
of Linear Programming Problems 48
1.3 The Modeling Process 53
1.4 The Three Phases in Optimization 57
1.5 Solving the Model and Interpreting the Printout 60

2. Applications 67
2.1 The Diet Problem 67
2.2 Allocation Problems 71
2.3 Cutting Stock Problems 75
2.4 Employee Scheduling 80
2.5 Data Envelopment Analysis 82
2.6 Inventory Planning 85
2.7 Blending Problems 89
2.8 Transportation Problems 91
2.9 Assignment Problems 102
2.10 A Production – Inventory Model: A Case Study 107
X Contents
3. The Simplex Method 129
3.1 Graphical Concepts 129
3.1.1 The Graphical Solution Technique 129
3.1.2 Four Special Cases 138
3.2 Algebraic Concepts 143
3.2.1 The Algebraic Solution Technique 143
3.2.2 Four Special Cases Revisited 158

4. Duality 167
4.1 The Fundamental Theory of Duality 167
4.2 Primal-Dual Relations 183
4.3 Interpretations of the Dual Problem 198

5. Extensions of the Simplex Method 203
5.1 The Dual Simplex Method 203
5.2 The Upper Bounding Technique 212
5.3 Column Generation 219

6. Postoptimality Analyses 225
6.1 Graphical Sensitivity Analysis 227
6.2 Changes of the Right-Hand Side Values 232
6.3 Changes of the Objective Function Coefficients 240
6.4 Sensitivity Analyses in the Presence of Degeneracy 245
6.5 Addition of a Constraint 248
6.6 Economic Analysis of an Optimal Solution 252

7. Non-Simplex Based Solution Methods 261
7.1 Alternatives to the Simplex Method 262
7.2 Interior Point Methods 273

8. Problem Reformulations 295
8.1 Reformulations of Variables 295
8.1.1 Lower Bounding Constraints 295
8.1.2 Variables Unrestricted in Sign 296
8.2 Reformulations of Constraints 298
8.3. Reformulations of the Objective Function 301
8.3.1. Minimize the Weighted Sum of Absolute Values 301
8.3.2 Bottleneck Problems 306
8.3.3 Minimax and Maximin Problems 313
8.3.4 Fractional (Hyperbolic) Programming 320
Contents XI
9. Multiobjective Programming 325
9.1 Vector Optimization 327
9.2 Models with Exogenous Tradeoffs Between Objectives 337
9.2.1 The Weighting Method 337
9.2.2 The Constraint Method 339
9.3 Models with Exogenous Achievement Levels 341
9.3.1 Reference Point Programming 342
9.3.2 Fuzzy Programming 346
9.3.3 Goal Programming 351
9.4 Bilevel Programming 359

References 363

Subject Index 377