Delft University of Technology Faculty of Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics Resolving infeasibilities in the PESP model of the Dutch railway timetabling problem A thesis submitted to the Delft Institute of Applied Mathematics in partial fulfillment of the requirements for the degree MASTER OF SCIENCE in APPLIED MATHEMATICS by GERT-JAAP POLINDER Delft, the Netherlands August 2015 Copyright c 2015 by Gert-Jaap Polinder. All rights reserved. MSc THESIS APPLIED MATHEMATICS \Resolving infeasibilities in the PESP model of the Dutch railway timetabling problem" Gerrit Jacob Polinder Delft University of Technology Daily supervisor Responsible professor Prof. dr. L.G. Kroon Prof. dr. ir. K.I. Aardal Other thesis committee members Dr. M. Molinaro August 2015 Delft, the Netherlands Preface In this master thesis the results of the final research I did as part of my studies in Applied Mathematics is described. It is written to obtain a masters degree in Applied Mathematics and marks the end of the period as a student at Delft University of Technology. This research clearly shows what applied mathematics is about. What for many people seems to be a dull subject, is very useful in practice. This is only one example, many more can be given. From the beginning of my studies, I was interested in logistics and trains. When the time was there to look for a thesis project, I came in contact with Leo Kroon from Rotterdam School of Management, Erasmus University Rotterdam, thanks to my supervisor Karen Aardal. Professor Kroon has been doing research on railway related problems for a long time and has been involved in the team that received the Franz Edelman Award for their work on train scheduling problems. Leo has introduced me at the Innovation Department of Netherlands Railways to carry out my masters research there. In this research, I have developed a method that resolves infeasibilities in the railway timetabling problem. During this research, I got the help of quite a few people. First of all, I would like to thank my colleagues from NS for their help on my research. A few have to be mentioned by name. First of all, my thanks go to G´abor. Although there is no free lunch, you gave many useful advices how to get the `most cheapest' that is possible. Next, thanks go to Jo¨el,Pieter-Jan and Leon for helping in a lot of DONS-related issues. Further, I would like to thank Stefan Schuurman (ORTEC) for his advices and answers to a lot of questions on the CADANS software. Most important, I would like to thank my supervisors for their help on this thesis, Leo, Karen and Marco. Many thanks for your comments, your help and your patience during this project. It was a lot of fun working on this research under your supervision. Leo, many thanks for introducing me to the fascination world of optimisation in public transport. A lot of interesting problems arise in this field and still a lot of research can be done in this field, I am looking forward to that. Karen and Marco, many thanks for your comments and advices during this research project. You clearly showed me that things that became more or less obvious for me are not that obvious at all and are challenging to be explained in a clear way. Last but not least, many thanks to my wife Chantine. Thanks for listening to all my `very interesting' train facts and my programming issues. You put some constraints to my pastime optimisation model that were very relevant, also to the success of this thesis. If those constraints were violated any time as well, you turned my attention to keep me on track, thanks! One final note to all of you: I hope you enjoy reading this thesis as much as I did working on it, have fun! Gert-Jaap Polinder Rotterdam, August 2015 i ii Contents Preface i Table of contents v List of Figures v List of Tables viii 1 Introduction 1 1.1 A description of the Dutch railway network . .3 1.2 The need for timetabling . .4 1.3 The planning process in practice . .5 1.4 Designing a timetable . .6 1.4.1 Time space diagram . .6 1.4.2 Platform occupation chart . .6 1.4.3 Timetable planning . .9 1.5 Problem description . .9 1.5.1 Conflicts . 10 1.5.2 Problem statement . 13 1.6 Literature review . 13 1.7 Outline of the thesis . 14 2 Optimisation techniques 15 2.1 Computational Complexity . 15 2.2 Constraint programming . 16 2.3 Mixed Integer Linear Programming . 16 2.3.1 Linear Programming . 16 2.3.2 Integer Linear Programming . 17 2.3.3 Mixed Integer Linear Programming . 19 2.3.4 Branch and cut . 20 3 The model and the current solver 23 3.1 Introduction . 23 3.2 The current model . 24 3.2.1 Trip time and dwell time constraints . 27 3.2.2 Frequency and synchronisation constraints . 27 3.2.3 Fixed arrival or departure times . 28 3.2.4 Safety constraints: headway . 29 3.2.5 Safety constraints: incoming-outgoing and collission . 30 3.2.6 Safety constraint: overtaking . 32 3.2.7 Safety constraints: hindrance . 32 3.2.8 Connection constraints . 35 3.3 Properties of the model . 37 3.3.1 Constraint graph . 37 3.3.2 Reducing the problem . 37 3.3.3 If the p-values are given . 38 3.3.4 Safety clique . 39 3.3.5 Cycles . 39 iii iv CONTENTS 3.3.6 Total unimodularity . 43 3.3.7 Sequencing . 44 3.4 The solver CADANS . 44 3.4.1 Conflex . 45 3.4.2 Optimize, Bepaal speling, Options and Verify . 49 4 Solution 51 4.1 Introduction . 51 4.2 Relaxation of constraints . 52 4.2.1 Relaxation . 53 4.2.2 A further look into the flexible constraints . 53 4.2.3 Dependent constraints . 54 4.3 A mathematical model . 56 4.4 Solving the model . 57 4.4.1 On the conflict graph . 57 4.4.2 Use total unimodularity . 58 4.4.3 Adding cuts . 58 4.4.4 Branching strategy . 59 4.4.5 Using special order sets . 60 4.4.6 Quadratic objective . 60 4.4.7 Symmetry breaking . 60 4.4.8 CPLEX settings . 61 4.5 Algorithmic approach . 61 4.5.1 Determine best conflict graph . 63 4.5.2 Use a cut generation approach . 64 5 Results 65 5.1 Description of test instances . 65 5.1.1 Hdr30 : Single track network Hdr-Sgn . 65 5.1.2 NL13 : Complete Dutch network 2013-2018, adapted by planners . 65 5.1.3 NL13new: Complete Dutch network 2013-2018, no changes . 66 5.2 Computational results . 66 5.2.1 Results on Hdr30 . 66 5.2.2 Results on NL13 . 67 5.2.3 Results on NL13new . 68 5.3 Summary of the findings . 69 6 Conclusion 71 7 Discussion and recommendations 73 Appendices 75 A Abbreviations train path points 77 B Reducing a graph 81 C Detailed results 85 C.1 Results on finding best conflict graph . 85 C.1.1 Result on NL13 . 85 C.1.2 Result on NL13new . 90 C.2 Results on cut generation algorithm . 92 CONTENTS v D Practical example Amf - Zl - Dv 99 D.1 Problem description . 99 D.1.1 Overview of the network . 99 D.1.2 Requirements on the timetable . 100 D.2 Derivation of the constraints . 101 D.2.1 Trip-dwell constraints . 101 D.2.2 Headway constraints . 101 D.2.3 Constraints around Deventer . 102 D.2.4 Single track . 102 D.2.5 Transfers . 103 D.2.6 Rolling stock constraints . 103 D.2.7 Absolute constraints . 103 D.3 Calculating.