Table Of ContentNETWORK MODELS IN
OPTIMIZATION AND
THEIR APPLICATIONS
IN PRACTICE
NETWORK MODELS IN
OPTIMIZATION AND
THEIR APPLICATIONS
IN PRACTICE
FRED GLOVER
DARWIN KLINGMAN
NANCY V. PHILLIPS
A WILEY-INTERSCIENCE PUBLICATION
JOHN WILEY & SONS, INC.
NEW YORK · CHICHESTER · BRISBANE »TORONTO · SINGAPORE
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Library of Congress Cataloging Im Publication Data:
Glover, Fred
Network models in optimization and their applications in practice
/by Fred Glover, Darwin Klingman, Nancy V. Phillips
p. cm.
"A Wiley-Interscience publication."
Includes bibliographical references and index.
ISBN 0-471-57138-5 (cloth : alk. paper)
1. Mathematical optimization. 2. Operations research. 3. System
design. I. Klingman, Darwin, II. Phillips, Nancy V.
III. Title.
QA402.5.P47 1992
O03-dc20 91-41110
CIP
To Darwin (February 5, 1944-October 27, 1989),
whose aspirations for excellence inspired us all.
Nancy Phillips
Fred Glover
PREFACE
INTRODUCTION
In a world where terms like "ribosomal RNA," "quasars," and (around April 15)
"accelerated cost recovery system" are commonplace, the term "network flow
optimization" is perhaps not unduly forbidding. In spite of the growing
influence of the network realm on our lives, however, it is not often publicized by
the popular press, and comments about it are not to be heard rolling off the
tongue either of talk show hosts or of the public at large. What then is network
flow optimization, and what is its connection to the concerns of the modern
world?
To answer this question, it is helpful to go back to the early days of World
War II, when the eminent US economist and logistics analyst Tjalling Koop-
mans was faced with the problem of moving personnel, supplies, and equipment
from various US bases to their foreign counterparts. The goal was to do this in a
way that would optimize one or more objectives: minimize total transportation
cost, minimize total transit time, and/or maximize defensive effectiveness.
Although Koopmans was the first to analyze maritime distribution problems
in the network model format, he was not in fact the first to discover this
approach. A few years earlier, but unknown to the western world and, ironically,
largely disregarded by his own countrymen for two decades, a distinguished
Russian mathematician/economist, L. V. Kantorovich, was studying important
problems associated with the Soviet economy. One problem that concerned him
was the allocation of production levels to factories and the distribution of the
resulting products to markets. He visualized his problem in a form highly
analogous to that of Koopmans' approach, and thus developed one of the
earliest network models, within a framework relevant to practical concerns.
The significance of these models lies not in the particular applications that
VII
Vili PREFACE
first inspired them, but rather in the fact that they provide the seeds of a general
methodology for structuring and analyzing complex decision problems. Koop-
mans and Kantorovich could not have foreseen the consequences of the special
frameworks they pioneered. We now live in a world where network flow
optimization directly and indirectly influences high-level decision making
around the globe—not only in economics, but in all aspects of industry and
government planning. Whether determining the most profitable levels of
production, the best cycles of harvesting and planting, the most effective uses of
energy, the optimum allocation of scarce resources, or the best decisions in
financial planning, network flow optimization is consistently relied on to
improve our analysis capabilities and has been documented with saving millions
of dollars. Even so, relative to its full potential, we are witnessing only the
beginning.
Nobel Prize committees are often behind the times in recognizing the
importance of innovations that have changed our lives. In the case of networks,
however, they may justifiably be credited with being ahead of the times. They
bestowed the honor of the Nobel Prize on Tjalling Koopmans in 1974 and L. V.
Kantorovich in 1975. This gave recognition of the importance of network
models as a decision framework—a foresight vindicated by the explosive
development of the field and its applications since the early 1970s.
Three properties are responsible for the widespread use of network flow
optimization: (1) visual content, (2) model flexibility and comprehensiveness,
and (3) solvability.
1. Visual Content. Network flow optimization allows a problem to be
visualized by means of diagrams, thereby making it possible to capture
important interrelationships in an easily understood "pictorial" framework.
This property has allowed nontechnical and technical individuals alike to gain
valuable insights into their problems, confirming the old saying that "a picture is
worth a thousand words."
Due to recent discoveries, the visual aspect of networks may have interesting
consequences from a learning and intelligence perspective. Psychologists have
found that systems which represent concepts in a variety of visual and auditory
ways improve learning and memory and thereby enhance our cognitive
processes. Further, studies of information-processing functions of the brain
disclose that different hemispheres of the brain are specialized to perform
different mental functions, where typically the right hemisphere takes a
dominant role in functions related to spatial imagery; and the left hemisphere is
devoted more fully to processing serial, analytical, or linguistic information. By
dual coding—putting information in both verbal and visual languages—
researchers conjecture that the words, sounds, and images naturally lead the
separate parts of the brain into a highly coordinated effort. With this type of
learning device, one does not just accumulate knowledge; one actually acquires
greater facility in problem solving. If this is true, then we conjecture that
network modeling and analysis is, in addition to its other virtues, an ideal tool
PREFACE IX
for increasing the ability to grasp critical problem features and to reason more
effectively about the domains to which it is applied. Conceivably, this in-
tegration of visual and symbolic elements may stimulate "dual coding" in other
parts of our thought processes as well.
2. Model Flexibility and Comprehensiveness. A major characteristic of the
network framework is the diversity of problems that can be modeled and solved
by its application. Today it is difficult to name a field which has not benefited
from network modeling or in which network flow optimization is not an
important research component. The algorithmic aspect of the field has attracted
mathematicians for many years. In the realm of the physical sciences and
engineering, networks have found application to such things as electrical circuit
board design, telecommunications, water management, design of transportation
systems, metalworking, chemical processes, aircraft design, fluid dynamic
analysis, and computer job processing.
Networks also are used in the arts to analyze group communication
phenomena in sociology, to determine ancestral orderings in archaeology, to
probe the effect of government fiscal and regulatory policies, and to allocate
library budgets to determine what types of publications to acquire.
Businesses are using network flow optimization in practically every sector. A
wide array of problems in production, marketing, distribution, financial plan-
ning, project selection, facility location, and accounting—to mention only a
few—falls naturally into the network domain.
3. Solvability. An early network problem (although not formulated as a
network model at the time) dates back to the French Academy of Sciences in
1781, which proposed an award for developing a solution procedure for the civil
engineering problem of "cutting and filling"—the problem of creating a level
plot of land by cutting down hills and filling in valleys. The award went
unclaimed and the problem remained unsolved for more than ISO years.
Kantorovich, Koopmans, and others similarly attempted to devise solution
procedures for the problems they faced, but none were successful. The first
solution method with a theoretical guarantee of optimality appeared in 1947
when US Science Medalist G. B. Dantzig developed the Simplex Method. Even
so, the ability to solve many network problems of real-world complexity
remained undemonstrated, because the theoretical solution capability did not at
once translate into computer procedures that could solve such networks within
a reasonable length of time.
The 1950s and early 1960s saw the further development of solution al-
gorithms and the identification of new problems which could be formulated and
solved as networks. By the late 1960s, the state of the art was still beset by
notable practical limitations, however. The largest network problems to be
solved contained about 600 nodes and 2000 arcs. Government and business
applications, however, demanded the solution of much larger problems.
Research in the 1970s and 1980s resulted in computer codes 100-150 times
faster than the best codes of the 1960s. The computer software provided by this
technology is currently being used by more than 100 government agencies and
X PREFACE
200 companies, and has been credited with saving over $500 million in practical
applications. Perhaps not surprisingly, these developments have brought about
a shift of research focus on a worldwide scale during the past decade, causing
increased numbers of researchers to study networks and closely related
problems. Commenting on these events, Nobel laureate Herbert Simon notes
that these contributions have "brought computational mathematics back into
the mainstream of mathematics, as a source of fundamental new problems and
theory."
In reflecting on the resulting increase in our ability to deal with a variety of
practical problems, it is important to keep in mind that advances in modeling
can be as important as advances in solution methods. By the mid-1970s, it
became apparent that expanded problem representations were needed to handle
features beyond the scope of existing formulations. Accordingly, an approach
called netform modeling emerged that allows go/no-go problems (often called
0-1 discrete optimization problems) to be formulated in a network-related
format. A bonus of netform modeling is that it fosters insights into problem
structure that often lead to efficient specialized solution methods. This close link
between models and solution methods has led to the term "netform approach"
to describe the combined result of using a netform model and a specialized
solution procedure based upon it.
For the past several years, the netform approach has been used by several
government agencies and private corporations to solve problems which were
heretofore unsolvable. For example, one company credits this approach with
solving a problem in less than 30 min of computer time that previously had not
yielded successfully to efforts spanning 25 person years. In another application
of the netform approach, an agency was able to obtain a solution to a problem
which was $10 million better than the best solution identified by other means.
The ability to economically solve large-scale problems on many fronts opens
the door to new perspectives and options, inviting a redefinition (and re-
evaluation) of many older but still prevalent approaches to problems that range
from economics to ecology. In many respects the research challenges are just
beginning! New opportunities for practical application are bringing with them
new demands on the underlying theory—by developing further elaborations
and extensions of the models and solution approaches that have proved
successful. Although we may not expect very soon to hear late-night talk show
hosts exchanging remarks with their guests about the latest in "network flow
optimization," this is not likely to slow the continued dramatic progress of the
field.
PREREQUISITES
This book is written primarily for college students who have completed an
introductory course covering linear programming. No mathematical sophis-
tication beyond college algebra is required. The text can be used in a one-
PREFACE xi
semester course for juniors, seniors, or graduates in business or engineering
curricula.
COVERAGE AND ORGANIZATION
The book covers modeling techniques for pure, generalized, and integer
networks, equivalent formulations, and successful applications of network
models. At the end of each chapter is (in order) a synopsis of a real-world
application, one or more case studies, exercises, and references. The references,
cited throughout the text by [author, year], guide the reader to more in-
formation about the particular subject being discussed. (A list of selected
readings not cited specifically in the text appears at the end of the book in
Appendix C.) The cases and exercises are often scaled-down versions of real
problems we have encountered. More difficult exercises are marked by an
asterisk (*). Important terms appear throughout in boldface italic type. In
teaching this course, we have found in-class discussion of the cases and exercises
to be very helpful. A solutions manual is available from the publisher.
ACKNOWLEDGMENTS
We gratefully acknowledge the people who assisted in preparing this book.
Special thanks go to Dr. Peter Mevert for reviewing the manuscript and making
many valuable suggestions, and to Connie Pechal for her tireless efforts in typing
the manuscript and preparing figures.
FRED GLOVER
NANCY PHILLIPS
Boulder, Colorado
Greer, South Carolina
June, 1992
CONTENTS
1 Netform Origins and Uses: Why Modeling and
Netforms Are Important 1
1.1 Background / 1
1.2 Netform Modeling in the Context of Management
Science / 2
1.3 A Preview of Netform Applications / 8
Application: The Oil Industry / 11
Case: An Oil Company Problem / 12
Exercises /16
References / 17
2 Fundamental Models for Pure Networks 20
2.1 Fundamental Principles / 20
2.2 Formulating a Network Model from a Word Problem / 28
2.3 Intuitive Problem Solving / 30
2.4 Structural Variations / 31
2.5 More General Networks / 36
2.6 Algebraic Statement of Pure Network Model / 39
2.7 Alternative Conventions for Network Diagrams / 42
Application: US Department of the Treasury / 46
Case: Angora / 48
Exercises /51
References / 62
XIII
