Vom Fachbereich fur¨ Mathematik und Informatik der Technischen Universitat¨ Braunschweig genehmigte Dissertation zur Erlangung eines Doktors der Naturwissenschaften (Dr.rer.nat.) Michael Bussieck Optimal Lines in Public Rail Transport 21. Dezember 1998 1. Referent: Prof. Dr. Uwe T. Zimmermann 2. Referent: Prof. Dr. Robert E. Bixby, Prof. Dr. Michael L. Dowling eingereicht am: 9. September 1998 Meinen Eltern gewidmet Acknowledgements In 1994 the Federal Ministry for Education, Science, Research and Technology (BMBF) in Ger- many started the program Application Oriented Joint Projects in Mathematics. People from in- dustry and research work together in about 70 projects on real life problems. I had the great plea- sure of being involved in one of these projects, namely, the project Optimal Line- and Routeplan- ning in Traffic Systems (Railroad Traffic). The outstanding atmosphere of this research program by means of scientific and commercial experience of the participants and a financial equipment provided an environment that made this kind of applied and scientific work possible. I am grateful to the industrial partners for providing me with real-world data and support. I am especially thankful to SMA und Partner AG, Leo Kroon (Nederlandse Spoorwegen) and Matthias Krista (Adtranz). Matthias is an inexhaustible source for railroad related information which helped me to overcome many obstacles in chapter 2 and 3. I would like to thank Peter Kreuzer who set the ball “line optimization” rolling. His diploma thesis [42] formed the basis of the research proposal submitted to the BMBF as well as the fundamentals of the direct traveler approach (chapter 5). For the comparison of different MIP solvers (cf. section 5.5) I required access to several commercial software products. I would like to thank Robert Bixby (CPLEX) for solving the huge linear programs arising from the (LOP) formulation in his garage. Furthermore I am grateful to the license team of Dash Associates (XPRESS), Franz Nelissen (GAMS), Uwe Suhl (MOPS), Martin Savelsbergh (MINTO), and Knut Haase who provided me with a user account on his machine running OSL. I had the pleasure of working with my colleague Thomas Lindner who was intensely involved in the cost optimal line planning (chapter 6). He developed a passion for solving the sp97ic instance. The working group of Professor Uwe Zimmermann provided an outstanding environment for my work. I am grateful to him for supporting my work in many aspects. I give my gratitude to my colleagues who never got tired of asking, “Isn’t it done yet?”. Moreover, I would like to thank our secretary Heidemarie Pf¨ortner who firstly persuaded me to take on the BMBF-position. I have restricted to mention people from academia and commerce, although there are many others, whose support was at least as important. I am especially very thankful to my parents Margrete and Bruno Bussieck and to my wife Susanne for her patience, encouragement, and love. Braunschweig, September 1998 Michael Bussieck vi Contents 1Preface 1 2 Public rail transport planning 5 2.1Passengerdemand.................................. 7 2.2Lineplanning.................................... 8 2.3Trainscheduleplanning............................... 8 2.4 Circulation of rolling stock and personnel . ................... 8 3 Line planning 11 3.1 Definition of supply networks . .......................... 12 3.2Systemsplit..................................... 14 3.3 Line optimization . ................................. 15 3.4Simulationandvaluation.............................. 17 4 Models for line planning 19 4.1Graphs........................................ 19 4.2Computationalcomplexity............................. 20 4.3Polyhedraltheory.................................. 22 4.4Alinearedgeformulation.............................. 23 4.5Complexityresults................................. 25 4.5.1 Polynomially solvable cases . ................... 28 4.6Alinearpathformulation.............................. 29 4.7 Linear programming based branch-and-bound ................... 30 4.7.1 Node selection . .......................... 34 4.7.2 Partition . ................................. 34 4.8 Improving the branch-and-bound algorithm . ................... 35 4.9 Improving the linear programming relaxation ................... 36 4.9.1 Preprocessing and probing . ................... 36 4.9.2 Constraint generation . .......................... 38 5 Line planning with respect to direct travelers 43 5.1 Introduction . ................................. 43 5.2Problemdescription................................. 44 vii viii CONTENTS 5.3 A branch-and-bound algorithm . .......................... 45 5.4Areviseddirecttravelerapproach......................... 47 5.5Arelaxationof(LOP)................................ 51 5.5.1 Preprocessing . .......................... 54 5.5.2 Constraint generation . .......................... 56 5.6Polyhedralaspects................................. 60 5.7Backtothe(LOP)model.............................. 66 5.8Extensionsofthemodels.............................. 70 5.8.1 The software LOP ............................. 70 5.8.2 A weighted version of (LOP) and (lop) . ............... 70 5.8.3 Flexibility versus hardness . ................... 71 5.8.4 Related problems . .......................... 72 5.8.5 Line planning with delayed column generation . ........ 74 6 Cost optimal line plans 81 6.1 Introduction . ................................. 81 6.2Problemdescription................................. 82 6.3Anonlinearformulation.............................. 83 6.4LinearizationI.................................... 85 6.4.1 Reducing the size of the problem . ................... 86 6.4.2 Improving lower bounds .......................... 88 6.4.3 The branch-and-bound algorithm . ................... 88 6.4.4 Features and limitations of (COSTBLP) . ............... 89 6.5LinearizationII................................... 92 6.5.1 Preprocessing and lower bounding derived from (COSTBLP) . 93 6.5.2 New preprocessing and lower bounding techniques . ........ 94 6.6Computationalinvestigation............................102 6.7Extensionofthemodels...............................113 7 Conclusions and suggestions for further research 117 List of Figures List of Tables Bibliography Index Deutsche Zusammenfassung 133 Chapter 1 Preface This thesis deals with the line planning problem for public transportation networks based on periodic schedules. The models and algorithms represented in this thesis take care of peculiarities of public rail transport. However, the ideas mentioned in this monograph can be easily adapted to the line planning problem for other transportation systems with periodic schedules, e.g. busses. A comprehensive discussion of the line planning problem including its modeling and solution applying mathematical programming methods, constitutes the core of this thesis. Beyond the practical aspects we concentrate on structural properties of the problems. For instance, we prove that the line planning problem belongs to the class of the hardest optimization problems. For a particular line planning problem we analyze the polyhedral structure of the corresponding integer linear program. These investigations represent the theoretical background of the methods we apply to the different models. Moreover, they prove why certain techniques improve the solution of the models by means of shorter computation time of the corresponding algorithms. The theoretical characteristics without the peculiarities of the practical problem being under consideration permit a tractable adaptation of models and methods to related problems. This level of problem abstraction which is in fact higher than in most engineering sciences, provides the construction of optimal or provably good solutions. This type of solution quality can be frequently transfered to the real life problem. Particularly, in strategic planning with a planning period of 10 – 20 years this approach becomes most important. The most commonly used way of comparing the new and the current solution is inaccessible at this point of planning. These advantages of a mathematical approach to practical problems face certain difficulties outside mathematics. Practitioners are sceptical about non-intuitive methods and use a complete different language than mathematicians. The Federal Ministry for Education, Science, Research and Technology in Germany started the program Application Oriented Joint Projects in Mathe- matics in 1994 to overcome these obstacles. People from industry and research work together in about 70 projects on real life problems. The project Optimal Line- and Routeplanning in Traffic Systems (Railroad Traffic) which forms the fundamentals of this thesis, was carried out together with engineers from Adtranz Signal GmbH. Former academic research on real life problems con- sists of constructing models and algorithms and proving their efficiency by testing small random problem instances. In contrast to that, the results elaborated in these projects, must stand the test 1 2 CHAPTER 1. PREFACE of (large scale) real life instances. In case of the line planning problem the input data consists of information about the infrastructure (including the network topology) and the customers (usually givenbyanorigin-destination matrix). A proper set of real life data instances is difficult to get hold of. The data, particularly the origin-destination matrix, represents a trade secret. The uti- lization of data is laid down by certain contracts granting uses which