Thesis for the degree of Doctor of Philosophy The Tail Assignment Problem Mattias Gronkvist¨ Department of Computer Science and Engineering Chalmers University of Technology and Goteborg¨ University SE-412 96 G¨oteborg, Sweden G¨oteborg, August 2005 The Tail Assignment Problem Mattias Gronkvist¨ ISBN 91-7291-645-1 Copyright °c Mattias Gronkvist,¨ 2005 Doktorsavhandlingar vid Chalmers Tekniska H¨ogskola Ny serie nr 2327 ISSN 0346-718X Technical Report 8D Department of Computer Science and Engineering Algorithms Research Group Department of Computer Science and Engineering Chalmers University of Technology and G¨oteborg University SE-412 96 G¨oteborg Sweden Telephone +46 (0)31-772 1000 Cover: An Iran Air Boeing 747SP-86 (tail number EP-IAA, cn 20998/275) blowing up a cloud of snow on take off. Landvetter Airport, G¨oteborg, Sweden, December 12, 1998. Photo: °c Jan Mogren – AirPixPro.com, [email protected]. Chalmers Reproservice G¨oteborg, Sweden, 2005 Till mamma The Tail Assignment Problem Mattias Gr¨onkvist Department of Computer Science and Engineering Chalmers University of Technology and G¨oteborg University Abstract The Aircraft Assignment problem is the problem of assigning flights to aircraft in such a way that some operational constraints are satisfied, and possibly that some objective function is optimized. We propose an approach to aircraft assignment which captures all op- erational constraints, including minimum connection times, airport curfews, maintenance, and preassigned activities. It also allows multiple fleet optimiza- tion, and can model various types of objective functions, which sets it apart from most other aircraft assignment approaches. The name we have chosen for our approach is the Tail Assignment problem. The tail assignment problem is general in the sense that it can model the entire aircraft planning process, from fleet assignment to the day of operation. We develop solution methods based on mathematical programming and constraint programming, and also show how the methods can be combined. The resulting hybrid solution method is general in the sense that it can be used both to quickly find initial solutions and to find close to optimal solutions, if longer running times are allowed. We present a mathematical programming approach based on column gen- eration, conduct thorough computational experiments to show the impact of various implementation choices on running time and convergence, and present heuristics to find integer solutions. We show how constraint programming can be used stand-alone to quickly produce solutions to the tail assignment problem, and to substantially improve the computational performance of the column generation algorithm. Prepro- cessing algorithms based on constraint programming are presented that can reduce the size of the problem substantially more than standard balance-based preprocessing, resulting in major speedups and increased solution quality. Our complete solution approach combines column generation, constraint program- ming and local search. Finally, as proof of concept, we present modeling examples and computa- tional results for a selection of real-world tail assignment instances, demon- strating how our model and solution methods can be used to increase op- erational robustness, enforce equal aircraft utilization, and decrease aircraft leasing costs. Our tail assignment system is currently in use at two medium- sized airlines. Keywords: fleet planning, airline optimization, column generation, constra- int programming, hybrid optimization, tail assignment, aircraft routing. Acknowledgments I would like to start by thanking my supervisor during my years as a Ph.D. student, Dag Wedelin, who has been a great help and motivation. His deep insights in algorithms as well as the real-world problems I have studied have really been useful. The other members of my Ph.D. advisory committee, Devdatt Dubhashi and Michael Patriksson, have been a great help in directing and supervising my work. I would also like to thank the people at the Department of Computing Science at Chalmers for providing a stimulating and fruitful working envi- ronment. I would especially like to thank Birgit Grohe and Erik Kilborn for the interesting discussions, mostly about cost propagation and constraint programming, over the years. Since the work presented in this thesis has been done in collaboration with, and is partially funded by, Carmen Systems AB, I would like to extend my gratitude to the people at Carmen Systems who gave me the opportunity to embark on the long journey of Ph.D. studies. Special thanks to Erik Andersson for continued support and encouragement. The one person, except myself, who has had the most impact on the work presented in this thesis, is Sami Gabteni. Sami has always had great confidence in my work, and has never failed to support and help me. I would also like to thank Darren Dejun Hang and Magnus Larsson, members of the original Carmen fleet optimization team, for the stimulating cooperation. I also want to thank Curt Hjorring, Rastislav ‘Rastjo’ Galia and Stefan Karisch for excellent cooperation, help and continued encouragement. The Carmen Matador team, consisting of Hamid Kharraziha, Marek Ozana, Johan Andersson, Geoff Bache, Claes Verdoes, Sami Spjuth and Jens Kjerrstr¨om, have provided a truly remarkable working environment. Working together with the matador team is never boring. Special thanks to Hans Eriksson, Oscar Rodriguez, Christopher Scott and Karine Cousineau-Ouimet, who have put my work to the ultimate test – reality. Finally, I thank my family. You have always supported me and believed in me. I owe you much more than words can ever express. Jessica and Nova, you are the light of my life. Everything, including writing a doctoral thesis, is much easier with you. Nova – kan du s¨aga‘doktorsavhandling’? Mattias Gr¨onkvist G¨oteborg, August 2005 Contents List of Figures v List of Tables vii List of Algorithms ix List of Models xi I Introduction to Airline Planning and Tail Assignment 1 1 Introduction 3 1.1 Modeling the Aircraft Assignment Problem ........... 4 1.2 Solving the Tail Assignment Problem .............. 6 1.3 An Engineering Perspective ................... 9 1.4 Main Contributions ........................ 11 1.5 Outline of the Thesis ....................... 12 2 Airline Planning 15 2.1 Airline Network Types ...................... 16 2.2 Timetable Construction and Yield Management . 17 2.3 Fleet Assignment ......................... 18 2.4 Crew Pairing and Rostering ................... 22 2.5 Aircraft Assignment ........................ 22 2.6 Recovery Planning ........................ 25 2.7 Integrated Planning ........................ 26 3 Tail Assignment 29 3.1 Connection Constraints ...................... 30 3.2 Maintenance Constraints ..................... 33 3.3 Flight Restriction Constraints . 35 3.4 Other Constraints ......................... 36 3.5 Optimization Criteria ....................... 37 3.6 Multiple Fleets .......................... 38 i Contents 3.7 Related Problems and Literature Review . 39 3.8 Rolling Stock Assignment .................... 43 II A Mathematical Programming Approach 45 4 Mathematical Models and Solution Methods 47 4.1 Model Tas ............................. 47 4.2 The Multi-Commodity Network Flow Problem . 49 4.3 Solution Methods for the Multi-Commodity Network Flow Problem .............................. 50 4.4 Model Path-Tas ......................... 58 4.5 Mixed Fleet Models ........................ 60 4.6 Lower Bounds ........................... 60 5 Column Generation 63 5.1 Dantzig-Wolfe Decomposition . 63 5.2 The Restricted Master Problem . 66 5.3 The Pricing Problem ....................... 68 5.4 Solving the Resource Constrained Shortest Path Problem . 71 5.5 The Integer Problem ....................... 76 5.6 Summary .............................. 76 6 Solving the Linear Programming Relaxation 77 6.1 Test Instances and Testing Methodology . 79 6.2 Initialization of the RMP ..................... 82 6.3 Number of Columns per Iteration . 84 6.4 Column Deletion ......................... 86 6.5 Dual Value Re-Evaluation .................... 87 6.6 Ordering of Pricing Subproblems . 89 6.7 RMP Algorithm Choice ...................... 91 6.8 Dual Stabilization ......................... 94 6.9 Constraint Aggregation ...................... 97 6.10 Summary ..............................102 7 Solving the Integer Program 107 7.1 A Variable Fixing Heuristic . 111 7.2 A Connection Fixing Heuristic . 114 7.3 A Hybrid Fixing Heuristic . 119 7.4 Backtracking ............................121 7.5 Summary ..............................123 ii III A Constraint Programming Approach 125 8 Introduction to Constraint Programming 127 8.1 Variables and Domains . 127 8.2 Constraints ............................128 8.3 The Constraint Satisfaction Problem . 128 8.4 Consistency and Propagation . 129 8.5 Searching for Solutions . 133 8.6 Constraint Optimization . 137 8.7 A Constraint Solver Implementation . 139 9 Constraint Programming Models 145 9.1 Literature Review . 146 9.2 Model Csp-Tasrelax . 148 9.3 Model Csp-Tas . 152 9.4 Reachability Propagation: Reducing Thrashing . 152 9.5 The pricing Constraint . 157 9.6 Ordering Heuristics for Csp-Tas . 158 9.7 An Alternative Way to Handle Cumulative Constraints . 159 9.8 Computational Results . 166 9.9 Summary ..............................168 10 Preprocessing with Constraint Programming 171 10.1 Simple Aircraft Balance Preprocessing . 172 10.2 Consistency-Based Preprocessing . 173 10.3 Cost-Based Preprocessing