Eingereicht von Arevik Hakobian Angefertigt am Institut fur Produktions- und Logistik- management Betreuung Univ.-Prof. Dr. Karl D¨orner Mitbetreuung Mag. Michael Schilde, PhD Februar 2017 The Tabu Search Heuristic for the Vehicle Routing Problem with Soft Time Windows Diplomarbeit zur Erlangung des akademischen Grades Magistra der Sozial- und Wirtschaftswissenschaften im Diplomstudium Wirtschaftswissenschaften JOHANNES KEPLER UNIVERSITAT¨ LINZ Altenbergerstraße 69 4040 Linz, Osterreich¨ www.jku.at DVR 0093696 Acknowledgements First, I would like to thank Univ.-Prof. Dr. Karl D¨ornerand Mag. Michael Schilde, PhD for introducing me in the field of Vehicle Routing Problems and supervising me during the work on this thesis. I also would like to thank Mag. Stefanie Kritzinger, PhD for providing me with her code and helping me whenever questions arose. Further, I wish to thank the Institute of Production and Logistics Management for their support, especially Mag. Gabriele Traugott and Sabine Frank. I also wish to thank my family for their patience and motivating words. Most of all, I would like to thank my parents for always encouraging me in my studies and life decisions. I wish to thank my friends for accompanying me during my studies. When recalling the study years, all I can remember are delicious moments I had with you, specially during lunch and coffee breaks. University would have been quite monotonous without you guys, especially without you Gayaneh! Thanks for being a great friend and a very good listener. I would like to take off my hat to a good friend and cerebrum, Dominik, who deserves to be thanked a lot for helping me without hesitation whenever technical difficulties arose and encouraging me in my work. Finally, I would like to thank the unibrat for crossing my ways with Sebastian, who is not only a very good friend but also my family. Thanks for picking me up. i Eidesstattliche Erkl¨arung Ich erkl¨arean Eides statt, dass ich die vorliegende Diplomarbeit selbstst¨andigund ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die w¨ortlich oder sinngem¨aßentnommenen Stellen als solche kenntlich gemacht habe. Die vorliegende Diplomarbeit ist mit dem elektronisch ¨ubermittelten Textdokument identisch. Linz, 09.02.2017 Arevik Hakobian iii Abstract In this work metaheuristics are introduced for the solution of the vehicle routing problem with soft time windows. The tabu search algorithm is implemented in this context along with the iterated local search and tested on different parameter settings and methods. During the tests on tabu search, intensification, diversification and route minimization strategies are gradually developed and implemented. The tabu search algorithm is further tested on different types of soft time windows. Computational test results of benchmark problems are calculated and compared to results from literature. Results show that in comparison with other tabu search algorithms, the tabu search generated in this work could outperform several paper results. v Kurzfassung Diese Arbeit befasst sich mit Metaheuristiken f¨urdie L¨osungdes Tourenpla- nungsproblems mit weichen Zeitfenstern, welches das Ziel hat kosteng¨unstigste Routen von einem Depot zu geographisch verteilte Kunden zu entwerfen, wobei jeder Kunde innerhalb eines vorgegebenen Zeitintervalls durch ein Fahrzeug bedient werden muss bzw. kann. Als L¨osungsverfahren, wird der Tabu Suchalgorithmus zusammen mit der Iterierten Lokalen Suche implementiert und mit verschiedenen Parametern und Methoden getestet. W¨ahrendder Tests zur Tabu Suche werden Intensivierungs-, Diversifizierungs- und Routenminimierungsstrategien entwickelt und umgesetzt. Das Tabu Suchver- fahren wird mit verschiedene Arten von weichen Zeitfenstern getestet. Rechnerische Testergebnisse von Benchmark-Problemen werden berechnet und mit Ergebnissen aus der Literatur verglichen, die zeigen, dass der entwickelte Tabu Suchalgorithmus Publikationen zur selben Thematik ¨ubertreffen konnte. vii Contents Abstract v Kurzfassung vii List of Tables xi List of Figures xiii List of Algorithms xv 1. Introduction 1 1.1. Related Work . .2 1.2. Structure . .4 2. The Vehicle Routing Problem 5 2.1. Introduction . .5 2.2. Problem Definition . .6 2.3. Solution Methods . .6 2.3.1. Classical Heuristics . .7 2.3.2. Metaheuristics . .7 3. The Vehicle Routing Problem with Time Windows 11 3.1. Introduction . 11 3.2. Problem Definition . 12 3.3. Solution Method . 13 3.3.1. Initial Solution . 13 3.3.2. Neighborhood Operators . 14 3.3.3. Tabu Search . 16 4. Computational Experiments 21 4.1. Test Instances . 21 4.2. HeuristicLab Experiment . 22 4.3. Simulation Software . 23 4.3.1. The Iterated Local Search Algorithm . 24 4.3.2. The Tabu Search Algorithm . 25 4.3.3. Infeasible Solutions . 26 4.3.4. Minimizing the Number of Routes . 26 ix Contents 4.4. Computational Test Results . 27 4.4.1. Tests on Iterated Local Search . 27 4.4.2. Tests on Tabu Search . 30 4.5. Performance Analysis . 46 5. Conclusion 51 Appendices 53 A. Appendix A 55 A.1. VRPTW . 55 A.1.1. Problem Definition . 55 A.1.2. An Example . 56 A.2. VRPSTW . 57 A.2.1. Experiment on Minimizing the Number of Vehicles . 57 B. Appendix B 59 B.1. Computational Experiments . 59 List of Abbreviations 95 Bibliography 97 x List of Tables 1. Related Work on TS and VRPTW . .3 2. Cross Exchange . 17 3. Results HL-experiment for the VRPSTW . 23 4. Overview Parameters . 27 5. Test on Acceptance Strategy with HTW and TO-NNB Heuristic . 28 6. Test on Acceptance Strategy with HTW and RIS . 28 7. Test on Acceptance Strategy with STW and TO-NBB Heuristic . 29 8. Test on Acceptance Strategy with STW and RIS . 29 9. Overview Tests on ILS . 30 10. Overview of Results of TT . 31 11. Overview Results of Intensification Strategies . 32 12. Comparison of TS Implementations of Chiang & Russell [8] . 33 13. Comparison of TS Implementations . 34 14. Comparison Test Results of Minimizing Number of Routes . 35 15. Test on Initial Solution . 36 16. Comparison of the Results of Type a of VRPSTW . 38 17. Comparison of CPU Time on Type a of VRPSTW . 39 18. Comparison of the Results of Type b of VRPSTW . 40 19. Comparison of CPU Time on Type b of VRPSTW . 40 20. Comparison of the Results of Type c of VRPSTW (Pmax = 0) . 42 21. Comparison of the Results of Type c of VRPSTW (Pmax = 5) . 43 22. Comparison of the Results of Type c of VRPSTW (Pmax = 10) . 44 23. Comparison of CPU Time on Type c of VRPSTW . 45 24. Comparison of the Results of VRPTW . 45 25. Comparison to the Optimal Solution . 46 26. Comparison ILS and TS Algorithm with Best Known Solutions . 48 27. Comparison of Tabu Search Algorithms . 49 A1. Average Test Results of toleranceV alue ................ 57 B1. Test on Acceptance Strategy with HTW and TO-NNB Heuristic . 59 B2. Test on Acceptance Strategy with HTW and RIS . 59 B3. Test on Acceptance Strategy with STW and TO-NNB Heuristic . 59 B4. Test on Acceptance Strategy with STW and RIS . 59 B5. Overview Tests on ILS . 60 xi List of Tables B6. Detailed Results HL-experiment for the VRPSTW . 61 B7. Detailed Test Result ILS with TO-NNB Heuristic and FF for the VRPTW . 62 B8. Detailed Test Result ILS with TO-NNB Heuristic and BF for the VRPTW . 63 B9. Detailed Test Result ILS with RIS and FF for the VRPTW . 64 B10. Detailed Test Result ILS with RIS and BF for the VRPTW . 65 B11. Detailed Test Result ILS with TO-NNB Heuristic and FF for the VRPSTW . 66 B12. Detailed Test Result ILS with TO-NNB Heuristic and BF for the VRPSTW . 67 B13. Detailed Test Result ILS with RIS and FF for the VRPSTW . 68 B14. Detailed Test Result ILS with RIS and BF for the VRPSTW . 69 B15. Overview Tests on TT . 70 B16. Detailed Test Result TS with Tabu Tenure =1 . 71 B17. Detailed Test Result TS with Tabu Tenure =2 . 72 B18. Detailed Test Result TS with Tabu Tenure =3 . 73 B19. Detailed Test Result TS with Tabu Tenure =4 . 74 B20. Detailed Test Result TS with Tabu Tenure =5 . 75 B21. Detailed Test Result TS with Tabu Tenure =6 . 76 B22. Detailed Test Result TS with Tabu Tenure =7 . 77 B23. Detailed Test Result TS with Tabu Tenure =8 . 78 B24. Detailed Test Result TS with Tabu Tenure =9 . 79 B25. Detailed Test Result TS with Tabu Tenure =10 . 80 B26. Detailed Test Result TS with Tabu Tenure =11 . 81 B27. Detailed Test Result TS with Tabu Tenure =12 . 82 B28. Detailed Test Result TS with Random Tabu Tenure =[1,10] . 83 B29. Detailed Test Result TS with Random Tabu Tenure =[11,20] . 84 B30. Detailed Test Result TS on Intensification with Or-Opt . 85 B31. Detailed Test Result TS on Intensification with S/F Heuristic . 86 B32. Detailed Test Result TS on Intensification with C/R Heuristic . 87 B33. Detailed Test Result TS on Intensification and Diversification . 88 B34. Detailed Test Result TS on Minimizing Number of Routes (toleranceV alue = 1:2) .................................... 89 B35. Detailed Test Result TS on Minimizing Number of Routes . 90 B36. Detailed Test Result TS on RIS . 91 B37. Detailed Test Result TS on VRPTW . 92 B38. Detailed Test Result TS-VRPSTW - Type b ............. 93 B39. Detailed Test Result ILS with TO-NNB Heuristic and BF for the VRPSTW (toleranceV alue = 1:2) ................... 94 xii List of Figures 1. The Vehicle Routing Problem . .5 2. Soft Time Windows [23] . 11 3. 2-Opt and Or-Opt [6] . 15 4. 2-Opt* and Relocate Operator [6] . 16 5. Cross Operator [6] . 16 6. Tabu Search Functioning [80] . 17 7. Tabu List . 18 8. Test Instances C101, R101, RC101 [78] . 21 A1. VRPTW - a simple example . 56 xiii List of Algorithms 1. Basic VNS [35] .