Arc Routing: Theory, Solutions and Applications

496 Pages·2000·41.11 MB·English

by  H. A. Eiselt

ARC ROUTING
ARC ROUTING
Theory, Solutions and Applications
Edited by
MOSHE DROR
University of Arizona
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Arc routing: theory, solutions, and applications / edited by Moshe Dror.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4613-7026-0 ISBN 978-1-4615-4495-1 (eBook)
DOI 10.1007/978-1-4615-4495-1
1. Operations research. 2. Graph theory. 1. Dror, Moshe.
T57.6 A 73 2000
658.4 '034--dc21
00-056135
Copyright © 2000 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers, New York in 2000
Softcover reprint of the hardcover 1st edition 2000
All 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.
in memory of
Chaja & Asher
Frydman
Contents
Preface
XVll
Contributing Authors xiii
1
A Historical Perspective on Arc Routing 1
H.A. Eiselt, Gilbert Laporte
1.1 Introduction 1
1.2 The Chinese Postman Problem 2
1.2.1 The Undirected CPP 3
1.2.2 The Directed CPP 6
1.2.3 The Mixed CPP 7
1.2.4 The Windy CPP 8
1.2.5 The Hierarchical CPP 9
1.3 The Rural Postman Problem 9
1.3.1 The Undirected RPP 10
1.3.2 The Directed RPP 10
1.3.3 The Mixed RPP 11
1.4 The Capacitated Arc Routing Problem 11
1.5 Research Outlooks 12
Part I THEORY
2
Traversing Graphs: The Eulerian and Hamiltonian Theme 19
Herbert Fleischner
2.1 Introductory Remarks 19
2.2 Basics ofGraph Theory 20
2.2.1 Graphs and Their Parts 20
2.2.2 Walks, Trails, Paths, Cycles; Connectedness 23
2.2.3 Bipartite Graphs, Trees, Blocks, Mappings 27
2.3 Connectivity, Menger's Theorem, the Splitting Lemma, and
Factors 32
2.4 Eulerian Graphs and Covering Walks, Cycle Decompositions
and Cycle Covers 40
2.4.1 Algorithms for Constructing Eulerian Trails 47
2.4.2 Mazes 48
2.5 Hamiltonian Cycles and Vertex-Covering Walks 50
2.6 Elements ofMatching Theory 58
2.7 The Chinese Postman Problem, The TravelingSalesman Prob-
lem, and Related Problems 69
2.8 Elements of Network Theory 77
viii ARCROUTING
3
Matching: Arc Routing and the Solution Connection 89
Ulrich Derigs
3.1 Introduction 89
3.2 Matching: Applications 91
3.2.1 Team Selection 91
3.2.2 Task Scheduling 92
3.2.3 Processor Scheduling 93
3.2.4 Route Connection 94
3.2.5 Arc Routing 95
3.2.6 Node Routing 98
3.2.7 General Routing 101
3.2.8 Set Partitioning 104
3.3 Matching: Combinatorial Aspects 107
3.4 Matching: Polyhedral Aspects 112
3.5 Matching Algorithms: Linking Combinatorial and Polyhedral
Results 116
3.6 Matching Algorithms: Implementation Issues 119
3.6.1 StartProcedures: ConstructingtheInitialExtremeMatch-
~ 1W
3.6.2 Organization ofDual Updates 122
3.6.3 Price and Reoptimize 123
4
Arc Routing: Complexity and Approximability 133
Moshe Dror
4.1 Introduction: Easy and Hard Problems 133
4.2 CPP as a Problem in P 141
4.3 NP-Hard Generalizations ofthe CPP 143
4.3.1 The Mixed CPP 143
4.3.2 The MCPP NP-Completeness 144
4.3.3 The Rural Postman Problem 147
4.3.4 The Windy Postman Problem 148
4.3.5 Non-intersecting Eulerian Circuits and A-trails in Eule-
rian Graphs 149
4.3.6 Dominating Trails 151
4.3.7 Precedence in Arc Routing 152
4.3.8 Capacitated Arc Routing 154
4.4 Approximation Algorithms 156
4.5 Approximation Results for Arc Routing 159
4.5.1 The Mixed CPP 159
4.5.2 The Windy CPP 161
4.5.3 The RPP and Other Variants 161
4.5.4 The CARP 162
4.6 Conclusions 164
5
Chinese Postman and Euler Tour Problems in Bi-directed Graphs 171
Ellis L. Johnson
5.1 Bi-directed Graphs, Euler Tours, and Postman Tours 171
5.2 Aircraft Routing in a Space-time Network 176
5.3 The Chinese Postman Problem in an Undirected Graph 181
5.4 Binary Group Problems and Blocking Problems 189
5.5 Ideal Binary Matrices 192
5.6 Four Problems on Planar Graphs 193
Part II Solutions
Contents ix
6
Polyhedral Theory for Arc Routing Problems 199
Richard W. Eglese, Adam N. Letchford
6.1 Introduction 199
6.2 The Basics ofPolyhedral Theory 200
6.3 The Routing Problems Defined 203
6.4 Variants ofthe Chinese Postman Problem 205
6.4.1 The CPP 205
6.4.2 The DCPP 205
6.4.3 The MCPP 206
6.4.4 The WPP 208
6.5 Variants of the Rural Postman Problem 209
6.5.1 The RPP 209
6.5.2 The GRP 212
6.5.3 The DRPP 216
6.5.4 The MRPP 217
6.6 The Capacitated Arc Routing Problem 219
6.6.1 Preliminaries 219
6.6.2 Sparse Formulations ofthe CARP 220
6.6.3 The Dense and Supersparse Formulations ofthe CARP 224
6.7 Conclusions 226
7
LinearProgrammingBasedMethodsfor SolvingArcRoutingProblems 231
E. Benavent, A. Corberan, J.M. Sanchis
7.1 Introduction 232
7.2 Chinese Postman Problems 236
7.2.1 The Undirected CPP 236
7.2.1.1 Odd-Cut Separation 237
7.2.2 The Directed CPP 238
7.2.3 The Mixed CPP 239
7.2.3.1 A Cutting Plane Algorithm for the MCPP 240
7.2.3.2 Odd-Cut Separation 241
7.2.3.3 Balanced Set Separation 242
7.2.4 The Windy Postman Problem 243
7.3 Rural Postman Problems 244
7.3.1 The Undirected RPP 244
7.3.1.1 Connectivity Separation 246
7.3.1.2 R-odd cut Separation 246
7.3.1.3 K-C Inequalities Separation 247
7.3.1.4 CuttingPlaneand Branch & Cut Algorithms for the RPP249
7.3.2 The General Routing Problem 250
7.3.2.1 Honeycomb Separation 251
7.3.2.2 Path-Bridge Separation 253
7.3.2.3 CuttingPlaneand Branch& Cut Algorithmsfor the GRP255
7.3.3 The Directed RPP 256
7.3.4 The Mb:ed RPP 257
7.3.4.1 Connectivity Separation 258
7.3.4.2 R-odd Cut and Balanced Set Separation 259
7.3.4.3 K-C and Path-Bridge Separation 259
7.4 The Capacitated Arc Routing Problem 259
7.4.1 Sparse Formulations 260
7.4.1.1 Connectivity Separation 261
7.4.1.2 Parity Separation 262
7.4.1.3 Obligatory Cutset Separation 263
7.4.1.4 Separation ofConstraints from the Knapsack Problem 263
7.4.2 Supersparse Formulation for the CARP 264
x ARCROUTING
7.4.2.1 Capacity Constraints Separation 267
7.4.2.2 Disjoint Path Inequalities Separation 267
7.4.3 Exact Methods Based on the Sparse Formulation 268
7.4.4 A CuttingPlaneAlgorithmfor the CARP Basedonthe
Supersparse Formulation 269
7.5 Other Problems 269
7.6 Conclusions 270
8
Transformations and Exact Node Routing Solutions by Column 277
Generation
Mashe Drar, Andre Langevin
8.1 Introduction 278
8.2 Transformations to Node Routing: Why? 279
8.2.1 The Capacitated Rural Postman Problem 279
8.2.2 Mathematical Formulation ofthe CARP 280
8.2.2.1 Time Window Constraints for Arc Routing 281
8.2.3 When to Transform to Node Routing 282
8.3 Arc Routing Transformations: How 283
8.3.1 Transformations of Uncapacitated Arc Routing Prob-
~~ 2M
8.3.2 Transformations of Capacitated Arc Routing Problems 285
8.3.3 Transformation ofCARP with Time Windows to
VRPTW 287
8.3.4 Split Delivery Arc Routing with Time Windows 289
8.4 Column Generation for Routing Problems with Non-split De-
livery 290
8.4.1 RevisedSimplex: Chvatal'sIntroductiontoColumnGen-
eration 290
8.4.2 Set Covering, Vehicle Routing, and Column Generation 292
8.4.3 The Shortest Path Subproblem 294
8.4.4 An Algorithmfor theShortest Pathwith Resource Con-
straints 295
8.4.5 Solving SPRCP with only Elementary Paths 297
8.4.5.1 Overlooking an Optimal Elementary Path 299
8.4.5.2 An SPRCP - Improved Elementary Path Algorithm 300
8.4.5.3 Description ofthe Algorithm 302
8.5 Column Generation for Routing Problems with Split Delivery 305
8.5.1 Properties ofSplit Deliveries with Triangle Inequality 306
8.5.2 Formulation 308
8.5.3 Set Covering Approach for Split Deliveries 309
8.5.4 The Subproblem for Generating Feasible Columns for
Split DelIvery 311
8.5.4.1 Formulating the Subproblem 312
8.5.4.2 Alternating between the Master Problem and the Sub-
problem 313
8.5.5 The Computational Phase: A MixedInteger Solver and
a DynamIC Programming Algoritlun 314
8.5.5.1 DiscretizingtheSplitDeliveriesintheSPPRCSubprob-
lem 315
8.5.5.2 Discussion of Computational Experiments for the SD-
VRPTW 316
8.5.6 Another Set Covering Formulation for SDVRPTW 317
8.5.6.1 The (lvIP3) Model and Column Generation Approach 318
8.5.6.2 The Branching Scheme for the SDVRPTW 321
8.6 Conclusion 322
Contents xi
9
Heuristic Algorithms 327
Alain Hertz, Michel Mittaz
9.1 Introduction 327
9.2 Heuristics for Uncapacitated Arc Routing Problems 329
9.2.1 The Chinese Postman Problem 329
9.2.1.1 The undirected chinese postman problem 329
9.2.1.2 The directed chinese postman problem 331
9.2.1.3 The mixed chinese postman problem 332
9.2.1.4 The windy postman problem 336
9.2.2 The Rural Postman Problem 338
9.2.2.1 The undirected rural postman problem 338
9.2.2.2 The directed rural postman problem 340
9.2.2.3 The mixed rural postman problem 344
9.2.2.4 The stacker crane problem 349
9.2.2.5 Additional algorithmic tools 355
9.3 Heuristics for Capacitated Arc Routing Problems 361
9.3.1 Simple Constructive Methods for the CARP 361
9.3.2 Two-Phase Constructive Methods for the CARP 373
9.3.2.1 Route first-cluster second algorithms 373
9.3.2.2 Cluster first-route second algorithms 377
9.3.3 Meta-Heuristics for the CARP 379
9.4 Conclusion 383
Part III Applications
10
Roadway Snow and Ice Control 389
James F. Campbell, Andre Langevin
10.1 Introduction 389
10.2 BriefHistory ofRSIC 390
10.3 Characteristics ofArc Routing for RSIC 392
10.4 Solution Approaches 395
10.5 Early Work 396
10.6 Recent Work 400
10.6.1 CASPER 400
10.6.2 GeoRoute 406
1O.6.2.10ttawa, Canada 408
1O.6.2.23uffolk County, UK 410
10.6.3 Other Recent Work 411
10.7 Future 413
11
Scheduling of Local Delivery Carrier Routes for the United States 419
Postal Service
Lawrence Bodin, Laurence Levy
11.1 Introduction 420
11.2 The Route Adjustment Process 422
11.2.1 Inspection 422
11.2.2 Route Adjustment 423
11.2.3 Auxiliary (or Remnant) Routes 424
11.3 Types ofDelivery Routes 424
11.3.1 Park and Loop Routes 425
11.3.2 Curblinej Dismount Routes 426
11.3.3 Combined Park and Loop and
Curblinej Dismount Routes 426
xii ARCROUTING
11.3.4 Relay Box Routes 427
11.4 Lines ofTravel for the Four Types of
Postal Delivery Routes 427
11.4.1 Walking Line ofTravel 428
11.4.2 Driving Line ofTravel 428
11.5 Examples of the Lines ofTravel for the Four
Types ofPostal Delivery Routes 429
11.5.1 Park and Loop Routes 430
11.5.2 Curbline/ Dismount Routes 431
11.5.3 Combined Park and Loop and
Curbline/ Dismount Routes 432
11.5.4 Relay Box Routes 433
11.5.5 Discussion 434
11.6 Algorithm for Route Adjustment 435
11.6.1 Estimate the Number ofRoutes to Form 436
11.6.2 Partition the AOI 437
11.6.3 Form the Line ofTravel for each Partition 438
11.6.4 Is the Solution Balanced? 439
11.7 Manual Intervention 440
11.8 Conclusions 440
12
Livestock Feed Distribution and Arc Traversal Problems 443
Moshe Dror, lanny M. Y. Leung, Paul A. Mullaseril
12.1 Introduction 444
12.1.1 The Cattle Industry 444
12.1.2 The Cattle Yard Operations 444
12.2 Livestock Feed Distribution as Arc Traversals 447
12.2.1 Arc-Routing Models for Pen Inspectionand Feed Deliv-
erv 448
12.2.1.1MathematicalFormulationoftheCapacitatedRural Post-
man Problem (CRPP) 448
12.2.2 Split Deliveries 450
12.2.3 Time Windows 451
12.3 A Heuristic Approach for Trip Generation 452
12.3.1 Generating a Non-split Feasible Solution 452
12.3.1.1Extended Path Scanning Algorithm for Feeding-Time
Feasibility 453
12.3.1.2Modified Augment-Merge Algorithm 454
12.3.2 Arc Swapping 455
12.3.3 Generating Split Delivery Routes 456
12.3.4 Route Addition 456
12.4 Route-First Cluster-Second Generalized TSP Heuristic 457
12.4.1 The Generalized Traveling Salesman Problem 457
12.4.2 Transforming the CRPP to an Equivalent GTSP 457
12.4.3 Solving the Equivalent GTSP 458
12.5 Computational Results 458
12.6 Epilogue 461