Linear Programming and its Applications

Linear Programming and its Applications

H. A. Eiselt C.-L. Sandblom Linear Programming and its Applications With 71 Figures and 36 Tables 123

Prof. Dr. H. A. Eiselt University of New Brunswick Faculty of Business Administration P.O. Box 4400 Fredericton, NB E3B 5A3 Canada haeiselt@unb.ca Prof. Dr. C.-L. Sandblom Dalhousie University Department of Industrial Engineering P.O. Box 1000 Halifax, NS B3J 2X4 Canada carl-louis.sandblom@dal.ca Library of Congress Control Number: 2007931630 ISBN 978-3-540-73670-7 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Production: LE-TEXJelonek, Schmidt &Vöckler GbR, Leipzig Cover-design: WMX Design GmbH, Heidelberg SPIN 12092093 42/3180YL - 5 4 3 2 1 0 Printed on acid-free paper

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

SYMBOLS This part introduces the reader to some of the support methodology used in this book. We have made every possible attempt to keep the exposition as brief and concise as possible. Readers who are interested in more in-depth coverage are referred to the pertinent literature. Notation ù = {1, 2,...}: Set of natural numbers ù 0 = {0, 1, 2,...}: Set of natural numbers including zero ú: Set of real numbers ú + : Set of nonnegative real numbers ú n : n-dimensional real space : Element of : Subset : Proper subset : Union of sets : Intersection of sets : Empty set : Implies : There exists at least one : For all S : Cardinality of the set S inf: infimum sup: supremum

XIV Symbols x [a, b]: a x b x [a, b[: a x < b x ]a, b]: a < x b x ]a, b[: a < x < b x : Ceiling of x, the smallest integer greater or equal to x x : Floor of x, the largest integer smaller or equal to x x : Absolute value of x a:= a + b: Valuation, a is replaced by a + b f ( x) : partial derivative of the function f(x) with respect to x x