Table Of ContentTRAFFIC CONTROL
AND TRANSPORT PLANNING:
A Fuzzy Sets and Neural Networks Approach
INTERNATIONAL SERIES IN
INTELLIGENT TECHNOLOGIES
Prof. Dr. Dr. h.c. Hans-Jiirgen Zimmermann, Editor
European Laboratory for Intelligent
Techniques Engineering
Aachen, Germany
Other books in the series:
Fuzzy Logic and Intelligent Systems
edited by Hua Li and Madan Gupta
Fuzzy Set Theory and Advanced Mathematical
Applications
edited by Da Ruan
Fuzzy Databases: Principles and Applications
by Frederick E. Petry with Patrick Bose
Distributed Fuzzy Control ofM ultivariable Systems
by Alexander Gegov
Fuzzy Modt;lling: Paradigms and Practices
by Witold Pedrycz
Fuzzy Logic Foundations and Industrial Applications
by DaRuan
Fuzzy Sets in Engineering Design and Configuration
by Hans-Juergen Sebastian and Erik K. Antonsson
Consensus Under Fuzziness
by Mario Fedrizzi, Janusz Kacprzyk, and Hannu Nunni
Uncertainty Analysis in Enginerring Sciences:
Fuzzy Logic, Statistices, and Neural Network Approach
by Bilal M. Ayyub and Madan M. Gupta
Fuzzy Modelingfor Control
Robert Babuska
TRAFFIC CONTROL
AND TRANSPORT PLANNING:
A Fuzzy Sets and Neural Networks Approach
Dusan Teodorovic, Ph.D.
Katarina Vukadinovic
Faculty of Transport and Traffic Engineering
University of Belgrade
Belgrade, Yugoslavia
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SPRINGER SCIENCE+BUSINESS MEDIA, LLC
ISBN 978-94-010-5892-6 ISBN 978-94-011-4403-2 (eBook)
DOI 10.1007/978-94-011-4403-2
Library of Congress Cataloging-in-Publication Data
Copyright © 1998 by Springer Science+Business Media New York
OriginaHy published by Kluwer Academic Publishers in 1998
Softcover reprint of the hardcover 1s t edition 1998
AH rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer Science+Business Media, LLC
Printed on acid-free paper.
CONTENTS
PREFACE XI
ACKNOWLEDGMENTS XV
ABOUT THE AUTHORS xvii
1. BASIC DEFINITIONS OF THE FUZZY SETS THEORY 1
1.1. The Concept of Fuzzy Sets 1
1.2. The Equality of Fuzzy Sets 4
1.3. Subsets of Fuzzy Sets 4
1.4. The Intersection of Fuzzy Sets 5
1.5. The Union of Fuzzy Sets 7
1.6. Fuzzy Set Height 9
1. 7. Support of Fuzzy Set 9
1.8. The Scalar Cardinality of Fuzzy Set 10
1.9. Complement of Fuzzy Sets 11
1.10. Convex Fuzzy Sets 11
1.11. Linguistic Hedges 12
1.12. Fuzzy Sets as Points in Hypercubes 17
1.13. Fuzzy Relations 19
1.13.1. Cartesian product 19
1.13.2. Definition of a fuzzy relation 20
1.13.3. The Cartesian product of fuzzy sets 24
1.14. Max-min Composition 26
1.15. Extension Principle 27
1.16. Alpha Cut 32
1.17. The Concept 01 a Fuzzy Number 33
1.18. Adding Fuzzy Numbers 35
1.19. Subtracting Fuzzy Numbers 40
1.20. Multiplying and Dividing Fuzzy Numbers 45
1.21. MUltiplying a Fuzzy Number by a Constant 49
1.22. Triangular and Trapezoidal Fuzzy Numbers 51
VI Contents
1.23. Kaufinann and Gupta's Method for Comparing
Fuzzy Numbers 53
1.24. The Basic Elements of Fuzzy Systems 56
1.24.1. Graphical interpretation of fuzzy logic inference 64
2. PRESENTATION OF FUZZY MODELS DEVELOPED
IN TRANSPORTATION APPLICATIONS 69
2.1. Vehicle Routing Model Based on the Rules of Fuzzy
Arithmetic 69
2.1.1. Basic assumptions of the vehicle routing problem 69
2.1.2. Calculating and ranking "savings" using fuzzy
arithmetic 71
2.1.3. Vehicle routing algorithm when travel times are
fuzzy numbers 73
2.2. Scheduling Algorithm for a Static Dial-a-ride Problem
Based on Fuzzy Arithmetic Rules 74
2.3. Fuzzy Multiple-attribute Decision Making: The
Choice of the Best Alternative for Organizing
Air Shuttle Service 86
2.3.1. The choice of the best alternative for organizing
air shuttle service 89
2.4. Controlling an Isolated Signalized Intersection Using
Fuzzy Logic 95
2.5. A Fuzzy Traffic Control System on an Urban
Expressway 99
2.6. Solving the Route-choice Problems in Urban Networks
Using Fuzzy Logic 101
2.6.1. A fuzzy logic route-choice model 104
2.6.1.1. Network "loading" algorithm based on
hybrid numbers 109
2.6.2. Route-choice behavior by multistage fuzzy reasoning 111
2.7. Modeling Route Choice with Advanced Traveler
Information by Fuzzy Logic 112
2.7.1. Interactive computer simulation for data collecting 115
2.7.2. Analysis of the factors that influence route choice
usingATIS 116
2.7.3. Fuzzy logic and the route-choice problem 119
2.8. Approximate Reasoning Route-choice Model
on an Airline Network 125
2.9. A Fuzzy Logic Approach to the Vehicle Routing
Problem when Demand at Nodes Is Uncertain 133
Contents vii
2.9.1. Problem assumptions 133
2.9.2. Proposed solution to the problem 134
2.9.2.1. Determining route length 139
2.10. Air Traffic Flow Management Model Based on
Fuzzy Logic 144
2.10.1. Approximate reasoning model to manage air
traffic flows when there is airport congestion 145
2.11. Controlling a Fleet in River Traffic Using Fuzzy Logic 150
2.11.1. Approximate reasoning model to control the process
of loading, transporting, and unloading gravel 151
2.12. A Fuzzy Logic Approach to the Aircrew Rostering
Problem 157
2.12.1. Mathematical formulation of the problem of
assigning crew members to planned rotations 158
2.12.1.1. Modified "day-by-day" method for solving
the aircrew rostering problem 163
2.12.1.2. Approximate reasoning algorithm for
assigning a specific pilot to one ofthe
rotations starting on the day in question 164
2.13. A Fuzzy Logic Approach to the Vehicle Assignment
Problem 170
2.13.1. Statement of the problem 171
2.13.2. Proposed solution to the problem 173
2.13.3. Approximate reasoning model for calculating the
dispatcher's preference when only one type of
vehicle is used to meet every transportation request 175
2.13.4. Calculating the dispatcher's preference when
several types of vehicle are involved in meeting
requests 178
2.13.5. Heuristic algorithm to assign vehicles to
transportation requests 179
2.13.6. Numerical example 181
3. A FUZZY MATHEMATICAL PROGRAMMING
APPROACH TO TRANSPORTATION 185
3.1. The Basic Premises of Fuzzy Mathematical Programming 185
3.2. Fuzzy Linear Programming 188
3.3. Solving a Fuzzy Linear Programming Problem when
Constraint Coefficients Are Fuzzy 190
3.4. Solving a Fuzzy Linear Programming Problem when
Objective Function Coefficients Are Fuzzy 196
Vlll Contents
3.5. Solving a Fuzzy Linear Programming Problem
when the Constraints Are Fuzzy 198
3.6. Airline Network Design by Fuzzy Linear Programming 200
3.6.1. Statement of the problem 201
3.6.2. Proposed solution to the problem 202
3.6.3. Choosing air traffic route candidates 203
3.6.4. Determining flight frequencies on route candidates 206
3.7. Airline Network Seat Inventory Control: a Fuzzy
Mathematical Programming Approach 219
3.7.1. Statement of the problem 220
3.7.2. Heuristic algorithm to solve the ANSIC problem 226
3.7.3. Numerical example 230
4. APPLICATIONS OF ARTIFICIAL NEURAL
NETWORKS IN TRANSPORTATION 235
4.1. Basic Concepts of Artificial Neural Networks 236
4.1.1. Neurons 237
4.1.1.1. Biological neuron 237
4.1.1.2. An artificial neuron 238
4.1.2. Characteristics of neural networks 240
4.1.3. Classification of neural networks 241
4.1.4. A multilayered feedforward neural network 242
4.1.5. Training of a neural network 243
4.1.5.1. Generalization (of the acquired knowledge) 245
4.1.5.2. Consideration of a network's reliability and
stability 246
4.1.5.3. Validation ofa neural model 247
4.1.5.4. Network transparency 247
4.2. Transportation Demand Forecasting with Neural
Network Models 248
4.3. Backpropagation Neural Network for Estimation of an
Origin-destination Matrix 250
4.4. Backpropagation Neural Network for Estimating a
Real-time Origin-destination Matrix from Traffic Counts 257
4.5. Neural Network Approach to Transportation Network
Improvement Problem 263
4.6. Routing and Scheduling Algorithm for the Dynamic
Dial-a-ride Problem Based on Neural Networks 267
4.7. A Neural Network Approach to Mitigation of Vehicle
Schedule Disturbances 274
Contents ix
4.7.1. Statement of the problem 275
4.7.2. Proposed solution to the problem 278
4.7.2.1. Network configuration 278
4.7.2.2. Pattern of connectivity 279
4.7.2.3. Learning (Least mean sum of absolute
errors learning algorithm) 280
4.7.3. Numerical example 282
4.8. A Neural Network Approach to Vessel Dispatching
Problem 284
4.8.1. Statement of the problem 285
4.8.1.1. Analysis ofthe characteristics of the
dispatcher's assignment policy 287
4.8.2. Proposed solution to the problem 288
4.8.2.1. Network configuration 289
4.8.2.2. Pattern of connectivity 290
4.8.2.3. Learning (Least mean sum of absolute
errors learning algorithm) 291
4.8.2.4. Network application in the case of
disturbance in a daily schedule 294
4.8.3. Numerical example 295
4.8.3.1. Analysis of the trained network solution 297
5. GENERATING AND TUNING THE FUZZY LOGIC
SYSTEMS DEVELOPED IN TRANSPORTATION
APPLICATIONS 303
5.1. Membership Functions Design 303
5.2. Membership Value Assignment Process Based
on Intuition 305
5.3. Membership Value Assignment Process Based
on Facts and Knowledge 307
5.4. Membership Value Assignments as a Combinatorial
Optimization Problem 308
5.4 .1. Tuning membership functions using simulated
annealing 309
5.4 .1.1. Tuning membership functions using
simulated annealing: Traffic signal
coordination by fuzzy logic 3 10
5.4.2. Tuning membership functions using genetic
algorithms 316
x Contents
5.4.2.1. Tuning membership functions using
genetic algorithms: Traffic signal
coordination by fuzzy logic 319
5.4.3. Tuning membership functions using tabu search 322
5.4.3.1. Tuning membership functions using tabu
search: Traffic signal coordination by
fuzzy logic 330
5.5. Method To Generate Fuzzy Rules from Numerical and
Linguistic Information 332
5.5.1. Route choice problem in air transportation 333
5.5.2. Solving modal split problem by fuzzy rules
generated by learning from examples 340
5.6. Simultaneous Design of Membership Functions and Fuzzy
Rules from Numerical Information by Genetic Algorithms 345
5.7. An Application ofNeurofuzzy Modeling: The Vehicle
Assignment Problem 349
5.7.1. The basic concepts of neurofuzzy modeling 349
5.7.2. Tuning the fuzzy logic systems: neurofuzzy
modeling 352
5.7.3. Statement of the problem 353
5.7.3.1. Initial fuzzy logic systems for assignment
of vehicles to transportation requests 354
5.7.4. Tuning the fuzzy logic systems for assignment
of vehicles to transportation requests: Neurofuzzy
modeling 355
5.7.4.1. Network configuration 356
5.7.4.2. Pattern of connectivity 356
5.7.4.3. Supervised learning (A simulated annealing
algorithm) 358
5.7.5. Numerical example 360
REFERENCES 367
INDEX 385
