Hybrid IP/CP Methods for Solving Sports Scheduling Problems Rasmus Vinther Rasmussen PhD dissertation July 2006 Hybrid IP/CP Methods for Solving Sports Scheduling Problems Rasmus Vinther Rasmussen, Department of Operations Research, University of Aarhus, Denmark. PhD dissertation, July 2006. Dissertation committee: - Jørgen Aase Nielsen, University of Aarhus. - Dominique de Werra, Ecole Polytechnique F´ed´eralede Lausanne. - Jørgen Bang-Jensen, University of Southern Denmark. Advisor: - Kim Allan Andersen, Aarhus School of Business. Rasmus Vinther Rasmussen Department of Operations Research University of Aarhus Ny Munkegade Building 1530 DK-8000 Aarhus C Denmark [email protected] Subject classification: - MSC2000 : Primary: 90B35; secondary: 90C10, 90C27, 90C90. - OR/MS: Scheduling, Combinatorics, Integer programming. Preface This dissertation presents the outcome of the research I have done during my PhD program at the Department of Operations Research at the University of Aarhus. The objective of my work has been to develop solution methods integrating in- teger programming (IP) and constraint programming (CP) and to find practical applications for which such methods can be successfully applied. At the beginning of my PhD programme a Danish net-operator Sonofon pre- sented a job scheduling problem. Since hybrid IP/CP methods have previously performed very well on similar applications, I started working on the problem together with Christian Roed Pedersen and Kim Allan Andersen. Unfortunately, we discovered that the size of the problem prevented the use of an exact solution method and instead a tabu search was developed. A paper [67] describing the solution method has subsequently been accepted for publication in Computers & Operations Research. During the work for Sonofon I discovered the field of sports scheduling, which is a research area containing numerous small, highly constrained and very hard optimization problems. These problems turn out to be well suited for hybrid IP/CP methods since they often contain a very hard feasibility aspect and at the same time have an objective function. For this reason solution methods for sports scheduling problems became the main focus in the rest of my work. In Denmark, soccer is the number one sport and developing a solution method for scheduling the best Danish soccer league quickly became a natural goal. How- ever, before this goal could be achieved a number of preliminary steps had to be taken. During a three months visit to Carnegie Mellon University at the beginning of 2005 I worked together with Michael A. Trick. At first we developed a hybrid IP/CP method for mirrored double round robin tournaments taking advantage of a known decomposition approach. Compared to previous methods our approach decreases computation time by following an iteration scheme instead of solving one step at a time. The solution method gave a lot of insight into how the strengths of IP and CP can be utilized, however, the real breakthrough came when we combined the solution method with a pattern generating approach. This second i ii Preface approach is now known as the pattern generating Benders approach and a paper [72] describing this approach has been accepted for publication in the European Journal of Operational Research. In the fall of 2005 the approach was further developed in order to handle the numerous constraints present for the best Danish soccer league. This research has led to a solution method capable of finding high quality solutions for the tournament and the Danish Football Association has used it for scheduling the 2006/2007 season. A paper [70] presenting the application and the solution method has been submitted to an international journal of operations research. In addition to the work relating to the Danish soccer league, I have also consid- ered sports scheduling problems in which the travel distance must be minimized. This problem applies for many leagues in the USA, since the teams travel from one away game to the next without returning home. In a paper [73] presented at the CP-AI-OR 2006 conference and published in Lecture Notes of Computer Science, Michael A. Trick and I define a new problem called the timetable con- strained distance minimization problem (TCDMP) and we evaluate a number of solution methods for this problem. Furthermore, an extended version of the paper has been submitted to Annals or Operations Research in which we present a new solution method denoted the circular traveling salesman approach for a problem called the traveling tournament problem. Finally, I have used the knowledge on sports scheduling obtained during my PhD programme to write a survey paper on round robin tournament schedul- ing. Again this is a joint work with Michael A. Trick and the paper [74] will be submitted to an international journal of operations research. Acknowledgements During my PhD programme several people have contributed with valuable assis- tance and suggestions. First of all I would like to thank my advisor Kim Allan Andersen for countless meetings at which his insightful comments and suggestions enabled continuing progress. At the beginning of 2005 I had the pleasure of visiting Michael A. Trick at Carnegie Mellon University for three months. This visit led to some of the major contributions presented in this dissertation and I am deeply indebted for his sup- port and guidance - not only during my stay in Pittsburgh - but during the rest of my PhD programme. I would also like to thank my colleagues Trine Krogh Kristoffersen and Chris- tian Roed Pedersen. They have provided an ideal working atmosphere and their ability to come up with qualified suggestions and ideas when things have seemed stuck has been a huge help. In the writing process Randi Mosegaard has provided invaluable linguistic support for which I am very thankful. Acknowledgements iii During my work I have found it important to relate the theoretical contri- butions to practical applications. Therefore I would like to thank Benny Olsen, Peter Ebbesen, Allan G. Hansen (the Danish Football Association), Morten Bech Kristensen, Lars Jørgensen (Sonofon) and Lars Grynderup (DM-Data) for mak- ing this possible. Without their data supply and patience when explaining the problems, it would have been impossible to test the solution methods on practical applications. I thank the Department of Mathematical Sciences for sponsoring a number of conferences. These conferences have made it possible to establish contact with researches from outside University of Aarhus. Finally, a very special thanks goes to Line Fiil Tarp for her love, support and friendship. Without her and her incredible faith in me this work would never have been possible. Arhus,˚ June, 2006 Rasmus Vinther Rasmussen Contents Preface i Acknowledgements . ii 1 Introduction 1 2 Solution Methods 5 2.1 Integer Programming . 5 2.1.1 Branch and Bound . 6 2.1.2 Branch and Cut . 6 2.1.3 Branch and Price . 7 2.1.4 Benders Decomposition . 8 2.2 Constraint Programming . 9 2.2.1 Constraints . 10 2.2.2 Backtracking . 11 2.2.3 Consistency . 13 2.2.4 Constraint Propagation . 13 2.3 Combining IP and CP . 14 2.3.1 Logic-Based Benders Decomposition . 15 2.4 Metaheuristic Solution Methods . 17 2.4.1 Tabu Search . 18 3 Round Robin Scheduling 21 3.1 Terminology . 22 3.2 Constraints . 25 3.3 Minimizing Breaks . 26 3.3.1 Constructive Methods . 27 3.3.2 The Constrained Minimum Break Problem . 30 3.4 Minimizing Travel Distance . 41 3.4.1 Practical Applications . 41 3.4.2 The Traveling Tournament Problem . 44 v vi Contents 4 A Benders Approach for Sports Scheduling 49 4.1 The Algorithm . 49 4.1.1 Master Problem . 51 4.1.2 Separation Procedure . 52 4.1.3 Benders Cuts . 53 4.1.4 CP Subproblem . 55 4.1.5 Cut Pool . 57 4.2 Computational Results . 57 5 The Pattern Generating Benders Approach 61 5.1 Problem Formulation . 62 5.2 Methodology . 63 5.3 Generating Patterns . 64 5.4 Pattern Sets . 65 5.5 Feasibility Check and Benders Cuts . 66 5.5.1 Team Allocation . 66 5.5.2 Diversity of Patterns . 67 5.5.3 Game Separation . 68 5.5.4 Game Assignment . 70 5.6 The Algorithm . 70 5.7 Computational results . 74 5.7.1 The Constant Distance Traveling Tournament Problem . 76 6 Scheduling SAS Ligaen 79 6.1 Constraints for SAS Ligaen . 79 6.2 Methodology . 82 6.3 The Algorithm . 83 6.3.1 Generating Patterns . 83 6.3.2 Pattern Set . 85 6.3.3 Feasibility Checks . 87 6.3.4 Timetable . 91 6.4 Computational Results . 93 7 Minimizing Travel Distance 99 7.1 The Timetable Constrained Distance Minimization Problem . 100 7.1.1 Integer Programming Formulation . 100 7.1.2 Constraint Programming Formulation . 101 7.2 Hybrid IP/CP Approach . 102 7.2.1 Phase 1 . 102 7.2.2 Phase 2 . 103 7.3 Branch and Price . 104 7.3.1 Initial Feasible Pattern Set . 105 Contents vii 7.3.2 Node Selection Strategy . 105 7.3.3 Master Problem . 106 7.3.4 Pricing Problem . 107 7.3.5 Branching Strategy . 108 7.4 Computational Results for the TCDMP . 109 7.5 The Circular Traveling Salesman Approach . 112 8 Job Scheduling 115 8.1 Problem Formulation . 116 8.2 Tabu Search . 120 8.2.1 Preprocessing . 121 8.2.2 Initial Solution . 121 8.2.3 Completing a Solution . 124 8.2.4 Neighbourhood . 125 8.2.5 Tabu List . 127 8.2.6 Intensification Strategy . 127 8.2.7 Diversification Strategies . 127 8.3 Computational Results . 128 8.3.1 The Practical Application . 130 8.3.2 General Large-scale Scheduling Instances . 131 Bibliography 135 Index 145 Chapter 1 Introduction The field of sports scheduling comprises a challenging research area with a great variety of problems and applications.