Flows in Transportation Networks

205 Pages·1972·2.301 MB·English

by  Renfrey B. Potts and Robert M. Oliver (Eds.)

FLOWS IN
TRANSPORTATION NETWORKS
This is Volume 90 in
MATHEMATICS IN SCIENCE AND ENGINEERING
A series of monographs and textbooks
Edited by RICHARD BELLMAN, University of Southern California
The complete listing of books in this series is available from the Publisher
upon request.
Flows in
Transportation Networks
RENFREY B. POTTS ROBERT M. OLIVER
Department of Applied Mathematics Department of Industrial Engineering
University of Adelaide and Operations Research
Adelaide, Australia University of California
Berkeley, California
ACADEMIC PRESS New YorkandLondon
COPYRIGH0 T1 972, BY ACADEMPIRCE SSI, NC.
ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY
OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM
THE PUBLISHERS.
ACADEMIC PRESS, INC.
111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NWl
LIBRAROYF CONGRECSSA TALOCA~ RDN UMBER7:0 -182673
PRINTED IN THE UNITED STATES OF AMERICA
CONTENTS
Preface ix
Acknowledgments xi
Chapter I TRANSPORTATION NETWORKS
1. Introduction 1
2. Examples of Transportation Networks 2
(a) City Street Network 2
(b) Main Road Network 4
(c) Traffic Desire Network 9
(d) Spider Web Network 9
3. Transportation Planning Process 10
4. Conclusion 13
5. Notes and References 13
Chapter I1 ELEMENTS OF NETWORK THEORY
6. Introduction 17
7. Graphs: Definitions and Notations 18
(a) Directed Graph 18
(b) Chain and Cycle 19
(c) Path and Mesh 21
(d) Accessible and Connected Nodes 22
(e) Cut-Set 22
(f) Undirected and Mixed Graphs 24
(g) Tree and Arborescence 24
V
vi Contents
8. Flows and Conservation Laws 26
(a) Link Flows and Kirchhoff’s Law 26
(b) Single 0-D Network: Link Flows 29
(c) Single 0-D Network: Chain Flows 32
(d) Multiple 0-D Network 34
(e) Compressibility and Separability 36
9. Costs and Capacities 38
(a) Link, Route, and Network Costs 38
(b) Capacitated Network 41
10. Conclusion 43
11. Notes and References 44
12. Problems 45
Chapter 111 EXTREMAL PRINCIPLES AND
TRAFFIC ASSIGNMENT
13. Introduction 49
14. Cheapest Routes 51
(a) Appraisal of Algorithms 51
(b) Tree-Building Algorithms 52
(c) Turn Penalties and Prohibitions 56
(d) Cheapest Route Assignment 63
15. Minimum Network Cost 65
(a) Link Flows 66
(b) Chain Flows 71
(c) The Out-of-Kilter Algorithm 75
16. Flow Dependent Costs 86
(a) Multicommodity Formulation 87
(b) Equilibrium Flow Patterns for Noncooperative Users 88
(c) Minimum Network Cost Flow Patterns 91
(d) Associated Traffic Assignment Problems 95
(e) A Numerical Example with Four Commodities 96
(f) Congested Assignment 100
17. Notes and References 102
18. Problems 109
Chapter IV TRIP DISTRIBUTION
19. Introduction 115
20. Model Formulation 116
21. Hitchcock Model 118
22. Entropy Models 121
(a) Network Entropy 121
(b) Proportional Model 123
(c) Mean Trip Length Model 13 0
(d) Gravity Model 133
Contents vii
23. Opportunity Models 136
(a) Intervening Opportunities Model 137
(b) Preferencing Model 138
24. Combined Distribution and Assignment 141
(a) TRC Program 142
(b) LTS Program 142
(c) Multicommodity Distribution-Assignment 143
25. Conclusion 143
26. Notes and References 144
27. Problems 149
Appendix A THEOREM FOR CHEAPEST ROUTE
ALGORITHMS 153
Appendix B DUALITY THEORY 157
Appendix C INEQUALITIES FOR MARGINAL AND
AVERAGE LINK AND CHAIN COSTS 159
Appendix D ANSWERS TO PROBLEMS 163
Index 187
This page intentionally left blank
PREFACE
Transportation problems are among the most significant being faced
by society today, and much effort is being and will be expended on the
search for transportation systems which are efficient, acceptable to
man, and compatible with his environment. Foremost in this search is
the development of sophisticated mathematical models which are
being used to analyze transportation problems and to plan for trans-
portation needs of the future.
This text is designed to provide a comprehensive formulation of the
more important transportation models; it purposely steers a middle
course between theory per se on the one hand and applications without
theorems on the other. Its aim is to bridge the gap between abstract
graph theory and its application to the analysis of large transportation
networks. The approach recognizes and emphasizes the ever increasing
role of computer algorithms and, in doing so, selects those models and
algorithms which, in the short period that they have been popularized
in the open literature, appear to warrant long-term interest. The major
portion of the text is based on fundamental conservation and extremal
principles, and only seven statements have been classified as theorems.
Readers primarily interested in transportation planning will find that
the book provides mathematical material necessary for proper under-
standing of the traffic assignment and trip distribution models widely
used in the planning process. Science and engineering students will
find that network flow theory is here motivated by applications to
significant real transportation problems.
ix