Table Of ContentOperations Research in Transportation Systems
Applied Optimization
Volume 20
Series Editors:
Panos M. Pardalos
University of Florida, U.S.A.
Donald Hearn
University of Florida, U.S.A.
The titles published in this series are listed at the end oft his volume.
Operations Research in
Transportation Systems
Ideas and Schemes of Optimization Methods
for Strategic Planning and Operations Management
by
Alexander S. Belenky
SI'RlNGER-SCIENCE+BUSINESS MEDIA, B.Y.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-4803-8 ISBN 978-1-4757-6075-0 (eBook)
DOI 10.1007/978-1-4757-6075-0
Printed on acid-free paper
AII Rights Reserved
© 1998 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1998
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Contents
Preface vii
Introduction 1
References 10
Chapter 1 Mathematical Programming 13
1.1 Linear Programming 13
1.2 Convex Programming 24
1.3 Nondifferentiable Optimization 40
1.4 Special Classes of Problems 48
References 82
Chapter 2 Discrete Optimization 91
2.1 Integer Programming 91
2.2 Problems with Mixed Variables 112
2.3 Problems on Graphs 115
References 120
Chapter 3 Transportation-Oriented Optimization 125
3.1 Routing 125
References 144
3.2 Network Flows 151
References 171
3.3 Scheduling 175
References 198
3.4 The Transportation Problem 204
References 220
Chapter 4 Optimization under Uncertainty 225
4.1 Stochastic Programming 225
VI CONTENTS
4.2 Game Theory 228
4.3 Multicriteria Optimization 243
4.4 Optimization on Fuzzy Sets and under Fuzzy Goals 255
References 278
Chapter 5 Optimal Control 285
5.1 Systems Governed by Ordinary Differential Equations
(Lumped Parameter Systems) 286
5.2 Distributed Parameter Systems 297
5.3 Discrete Systems 298
5.4 Special Classes of Systems 302
5.5 Systems Described by Differential Inclusions 304
References 306
Chapter 6 Optimization in Special Methods of Strategic
Planning and Operations Management in
Transportation systems 309
6.1 Business Games 309
6.2 Simulation Modeling 312
6.3 Program-Goal Planning 316
6.4 Expert Estimates 318
References 322
Chapter 7 Mathematical Concepts and Notation Used for
Formulating and Solving Optimization Problems
of Strategic Planning and Operations
Management in Transportation systems 325
7.1 On Mathematical Tools of Optimization for Strategic
Planning and Operations Management in Transportation
systems 325
7.2 Elements of Set Theory and Theory of Functions 326
7.3 Multidimensional Geometry and Convex Analysis in Rn 331
7.4 Mappings and Differentiability in Rn 339
7.5 Algebraic and Differential Equations in Rn 345
7.6 Elements of Probability Theory 348
7.7 Numerical Optimization Methods 351
7.8 List of Useful but Undescribed Concepts 355
References 357
Conclusion 359
References 361
Appendix 365
Index 425
Preface
The scientific monograph of a survey kind presented to the reader's attention
deals with fundamental ideas and basic schemes of optimization methods that
can be effectively used for solving strategic planning and operations manage
ment problems related, in particular, to transportation. This monograph is an
English translation of a considerable part of the author's book with a similar
title that was published in Russian in 1992.
The material of the monograph embraces methods of linear and nonlinear
programming; nonsmooth and nonconvex optimization; integer programming,
solving problems on graphs, and solving problems with mixed variables; rout
ing, scheduling, solving network flow problems, and solving the transportation
problem; stochastic programming, multicriteria optimization, game theory, and
optimization on fuzzy sets and under fuzzy goals; optimal control of systems
described by ordinary differential equations, partial differential equations, gen
eralized differential equations (differential inclusions), and functional equations
with a variable that can assume only discrete values; and some other methods
that are based on or adjoin to the listed ones.
The material discussed in the monograph is oriented to problems of a trans
portation nature and is addressed, first and foremost, to experts working in the
field of operations management or strategic planning for transportation. How
ever, its major part devoted to mathematical methods is invariant with respect
to a particular application field. Therefore, the monograph may be used by stu
dents and postgraduates studying operations management, strategic planning
(and related technical and economic sciences), as well as by those studying
applied mathematics at universities, technical and economic institutes, and
colleges.
A separate chapter contains basic mathematical information (starting from the
notions of set theory) necessary for understanding the monograph's material.
It makes the presentation of the material closed and formally requires from
the reader a certain "culture of thinking" rather than a special mathematical
background. At the same time, the author hopes that any reader interested
in the acquisition of knowledge can develop such a "culture of thinking" as a
vii
viii PREFACE
result of work with the monograph, although it will take intellectual effort and
time on his part.
A few words about those who helped in giving birth to this monograph. Con
stant attention and support rendered by B. Polyak stimulated, to a considerable
degree, the author's research in optimization and its applications. O. Aven at
tracted the author's attention to control problems arising in transportation.
Fruitful research in scheduling theory applications for transportation systems
was started together with E. Levner. Collaboration with these scientists helped
the author comprehend a great deal of material in optimization and transporta
tion systems analysis and inspired the idea of writing this monograph. This
idea was supported by A. Odoni and A. Kornhauser, and productive discus
sions with them convinced the author that such a monograph could be helpful
for specialists in operations management and strategic planning that apply
mathematical methods in the field of transportation. Presentation of the ma
terial was substantially improved thanks to the advice and recommendations of
N. Tretyakov. The high quality camera-ready copy of the monograph was pro
duced through the diligence of the computer network laboratory's staff (headed
by M. Doubson) at the Central Economics and Mathematics Institute of the
USSR Academy of Sciences. The author expresses his deep appreciation to all
the above-mentioned people.
A few technical remarks on the material's presentation. Wherever possible,
the author adhered to notations of the original publications under review. It
resulted in employing different (although, traditionally used) notations for the
same mathematical subjects in some parts of the monograph. Formulae are
numbered independently by chapter. Sources are also numbered independently
by chapter and by paragraph within Chapter 3. Those paragraphs may, in fact,
be considered as small separate chapters.
Moscow, Russia
January, 1992
Boston, Massachusetts
January, 1998
Introduction
The active, rapid penetration of computers into all the spheres of strategic
planning and operations management has led to the intensive development of a
number of scientific and engineering directions in the applied theory of control.
Mathematical modeling and a part of operations research that is based on the
theory and methods of optimization should be considered as the most important
ones. Mathematical models for the description of various objects, along with
the accumulated experience in using mathematical methods that enable one
to solve problems formulated on the basis of the models, have grounded the
tools for analysis and decision making in economic and technical systems. The
tools include, in particular, mathematical models and methods for strategic
planning and operations management implemented in various decision-making
systems, which are widely used in almost all branches of the advanced economy,
transportation being one of those. However, the degree of employing the tools
in transportation still remains much lower than that in the other branches.
Three basic reasons account for the limited use of the above-mentioned tools:
the absence of mathematical models unification, the marked tendency to eschew
mathematical models for the analysis of problems in transportation systems,
and the inadequacy of models describing the objects of strategic planning and
operations management to these objects. These reasons, along with others
associated with the existing problems of employing mathematical modeling in
transportation, were analyzed in [1]. However, the difficulties in applying the
known mathematical optimization methods for solving practical problems, in
particular, in transportation, which have also substantially limited the use of
the tools, should also be discussed.
First of all, the development of personal computers, along with that of com
puter graphics means, has created premises for eclipsing mathematical meth
ods of optimization as the tool for decision making in planning and control by
widely advertised interactive and expert systems. Such systems actively incor
porate the user in the decision-making process, but they do not provide, in
fact, either finding optimal or rational decisions or analyzing, deeply enough,
the practical issues arising in strategic planning and operations management.
1
2 INTRODUCTION
Because of that, many users have fallen under the illusion that decision making
in practical situations does not require any serious mathematical analysis and
modeling and should be exclusively based on practical experience of particu
lar people using particular systems in the field. The author has many times
faced such a viewpoint in numerous conversations with representatives of trans
portation organizations both in the former USSR and abroad. The only way to
convince these people of the efficacy of mathematical modeling and operation
research methods was to present them with test examples demonstrating how
a company could lose profit and even become bankrupt (under conditions of
competition) if it did not use mathematical tools for analysis of its potential.
The examples often helped convince those who finance research and develop
ment of decision-making systems that the enormous potential of mathematical
methods for finding and analyzing planning and managing decisions should be
combined with the undoubted merits and advantages of graphic mapping and
presentation of such decisions by interactive systems.
It is well known that many mathematical problems arising in transportation
are of special mathematical structure, and it makes it, in principle, difficult
to use optimization methods for their solving; N P-hard problems, large-scale
mathematical programming problems, and nonlinear control problems should
be referred to as examples. This fact has played not a minor role in lowering the
interest in using optimization methods for solving practical planning problems
in transportation systems. Because of the complex mathematical nature of the
problems, the developers of applied systems and programs often draw an a
priori conclusion that searching for effective optimization methods for solving
such problems is inexpedient. However, a number of effective methods were
designed for particular but important classes of such problems [2]. Moreover,
classes of new problems (for instance, nonlinear programming ones on poly
hedral sets) for which optimization methods with substantial computational
potential can be designed were recently displayed and investigated [3].
The existing gap between the level of mathematical education of transporta
tion managers and graduates from engineering colleges in the transportation
field and that necessary for understanding the potential and substance of op
timization methods should also be referred to as one of the major reasons for
insufficiently employing mathematical methods in strategic planning and oper
ations management for transportation. Such a gap exists even for the standard
optimization methods, i.e., those incorporated in the standard software oriented
to solving optimization problems. The intention to "make do" with heuristic
approaches or hand-made (often incorrectly) techniques for solving the prob
lems - where the known optimization methods could be successfully used -
and the psychological distrust of optimization methods associated with the
misunderstanding of their substance and potential are also two interconnected
phenomena caused by the above-mentioned educational gap. Furthermore,
the absence of interest of mathematicians in strategic planning and operations
management problems for transportation also increases the gap, although this
situation has been mitigated, to a certain degree, by publications [1], [2], [4].
