PERIODIC TIMETABLE OPTIMIZATION IN PUBLIC TRANSPORT

vorgelegt von Dipl.-Math. oec. Christian Liebchen Von der Fakultat II - Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischei^ Grades Doktor der Naturwissenschaften - Dr. rer. nat. genehmigte Dissertation

Berichter: Prof. Dr. Rolf H. Mohring Prof. Dr. Karl Nachtigall Zusiitzlicher Gutachter: PD Dr. Frank H. Geraets Vorsitzender: Prof. Dr. Rolf Dieter Grigorieff

Tag der wissenschaftlichen Aussprache: 22. Marz 2006

Berlin August 2006 D83

Contents

1

Introduction

1

Part I General Properties of Timetabling 2

Why Timetabling?

9

3

The Planning Process in Public Transport

19

4

Strategies for Timetabling 4.1 Definitions 4.2 Specialization Causes Suboptimality 4.3 Major Properties

25 25 27 36

5

Scope

47

Part II Modeling Periodic Timetables 6

Three Ways to Model Periodic Timetables 6.1 Synchronizing Individual Trips

57 58

6.2 6.3

QUADRATIC SEMI-ASSIGNMENT PROBLEM PERIODIC EVENT SCHEDULING PROBLEM (PESP)

59 62

7

The Modeling Power of the PESP 7.1 Timetabling Requirements Covered by the PESP 7.2 Timetabling Requirements Not Covered by the PESP 7.3 Further Planning Steps Covered by the PESP 7.4 Conclusion

77 78 89 95 106

8

Complexity of the PESP 8.1 Inapproximability of MAX-T-PESP 8.2 An (Almost) Constant Factor Approximation Algorithm

109 110 Ill

XII

9

Contents

Integer Programming Formulations for the PESP 9.1 IP Formulation Based on Vertex Variables 9.2 IP Formulation Based on Arc Variables 9.3 IP Formulation Based on Cycle Variables 9.4 Transformations Between Variables 9.5 Additional Modeling Capabilities in IP Context

115 116 117 121 124 124

Part i n Cycle Bases of (Directed) Graphs 10 Classification of Cycle Bases 10.1 Notation 10.2 Classes of Cycle Bases 10.3 Characterizations

,,

133 133 134 137

11 Examples of Cycle Bases 11.1 Elementary Cases 11.2More Challenging Examples 11.3 Map of Directed Cycle Bases

141 141 143 150

12 Minimum Cycle Basis Problem 12.1 Minimizing Among Strictly Fundamental Cycle Bases 12.2 Minimizing Among 2-Bases 12.3 Minimizing Among Undirected Cycle Bases 12.4Minimizing Among Directed Cycle Bases 12.5 Other Cases

151 152 155 157 178 182

13 Summary and Applications

197

Part IV Computing Periodic Timetables 14 Preprocessing

205

15 Valid Inequalities for the IP models 15.1 General Results 15.2 Cycle Inequalities 15.3 Cycle Inequalities and Minimum Integral Cycle Bases 15.4Change Cycle Inequalities 15.5 Valid Inequalities Inherited from Linear Ordering

209 211 216 225 229 233

16 Other Deterministic Solution Approaches 16.1 Constraint Programming (CP) 16.2 Heuristics

237 237 242

Contents

Xin

17 Local Search Techniques 17.1 Advanced Evaluation of a Periodic Timetable 17.2 Neighborhoods 17.3 A Genetic Algorithm for Periodic Timetabling

247 248 252 256

18 Computational Results 18.1Data Sets 18.2Used Software and Parameter Tuning 18.3 Comparison of the Algorithms 18.4 Miscellaneous

261 261 265 276 280

19 The First Optimized Periodic Timetable in Practice v 19.1 Timetabling at Berlin Underground . 19.2 Detailed Requirements of Berlin Underground 19.3 Optimization Results

285 285 287 292

Notation Index

295

^

References

299

Index

313