OR Spectrum

, Volume 37, Issue 4, pp 903–928 | Cite as

Integrated timetabling and vehicle scheduling with balanced departure times

  • Verena Schmid
  • Jan Fabian Ehmke
Regular Article


Sequential planning of public transportation services can lead to inefficient vehicle schedules. Integrating timetabling and vehicle scheduling, the vehicle scheduling problem with time windows (VSP-TW) aims at minimizing costs of public transport operations by allowing small shifts of service trips’ departure times. Within the scope of tactical planning, a larger flexibility of departure times following predefined departure time windows may be desirable. However, with increasing degrees of freedom, conventional solution approaches for the VSP-TW become computationally prohibitive. Furthermore, a sole focus on cost minimization might produce timetables of insufficient quality, while public transport agencies expect high-quality timetables with service trips scheduled at times reasonable from a passenger’s point of view. Extending the VSP-TW, we propose the vehicle scheduling problem with time windows and balanced departure times (VSP-TW-BT). In addition to the cost-efficiency objective of the VSP-TW, our objective function considers the quality of a timetable from a passenger’s point of view. Timetables are generated by balancing consecutive departures on a line according to predefined departure time intervals. We use a weighted sum approach to combine both objectives, namely costs of operation and quality of timetables. Our mathematical model and solution approach are based on efficient techniques known from the area of vehicle routing with time windows. A hybrid metaheuristic framework is proposed, which decomposes the problem into a scheduling and a balancing component. Real-world-inspired instances allow for the evaluation of quality and performance of the solution approach. The proposed solution approach is able to outperform a commercial solver in terms of run time and solution quality.


Vehicle scheduling Timetabling Balancing of departure times  Hybrid metaheuristic Problem decomposition Collaborative solution approach 


  1. Blum C, Blesa Aguilera MJ, Roli A, Sampels M (eds) (2008) Hybrid metaheuristics. An emerging approach to optimization, studies in computational intelligence, vol 114. Springer, Berlin. doi: 10.1007/978-3-540-78295-7
  2. Bräysy O, Gendreau M (2005) Vehicle routing problem with time windows, part II: metaheuristics. Transp Sci 39:119–139CrossRefGoogle Scholar
  3. Bunte S (2009) Lösungen für Anwendungsfälle der Fahrzeugeinsatzplanung im öffentlichen Personennahverkehr. PhD Thesis. Universität PaderbornGoogle Scholar
  4. Bunte S, Kliewer N (2009) An overview on vehicle scheduling models. Public Transp 1:299–317CrossRefGoogle Scholar
  5. Cacchiani V, Toth P (2012) Nominal and robust train timetabling problems. Eur J Oper Res 219(3):727–737Google Scholar
  6. Ceder A, Wilson N (1986) Bus network design. Transp Res Part B 20:331–344CrossRefGoogle Scholar
  7. Desaulniers G, Hickman MD (2007) Public transit. In: Barnhart C, Laporte G (eds) Handbooks in operations research and management science: transportation, vol 14. Elsevier, pp 69–127Google Scholar
  8. Desrochers M, Soumis F (1989) A column generation approach to the urban transit crew scheduling problem. Transp Sci 23:1–13CrossRefGoogle Scholar
  9. Doerner K, Schmid V (2010) Survey: matheuristics for rich vehicle routing problems. In: Blesa M, Blum C, Raidl G, Roli A, Sampels M (eds) Hybrid metaheuristics, vol 6373. Lecture notes in computer science. Springer, Berlin Heidelberg, pp 206–221Google Scholar
  10. Forbes MA, Holt JN, Watts AM (1994) An exact algorithm for multiple depot bus scheduling. Eur J Oper Res 72(1):115–124CrossRefGoogle Scholar
  11. Kliewer N, Amberg B, Amberg B (2012) Multiple depot vehicle and crew scheduling with time windows. Public Transp 3:213–244CrossRefGoogle Scholar
  12. Kliewer N, Mellouli T, Suhl L (2006) A time–space network based exact optimization model for multiple-depot bus scheduling. Eur J Oper Res 3(175):1616–1627CrossRefGoogle Scholar
  13. Leithäuser N (2012) Algorithms and complexity of timetable synchronization and vehicle scheduling problems in an integrated approach. Verlag Dr. Hut, MunichGoogle Scholar
  14. Liebchen C (2006) Periodic timetable optimization in public transport. PhD Thesis. Technische Universität BerlinGoogle Scholar
  15. Liebchen C, Möhring R (2007) The modeling power of the periodic event scheduling problem: railway timetables—and beyond. Algorithmic methods for railway optimization, vol 4359. Lecture notes on computer science. Springer, Berlin, pp 3–40Google Scholar
  16. Liebchen C, Schachtebeck M, Schöbel A, Stiller S, Prigge A (2010) Computing delay resistant railway timetables. Comput Oper Res 37(5):857–868Google Scholar
  17. Loebel A (1999) Solving large-scale multiple-depot vehicle scheduling problems. In: Lecture notes in economics and mathematical systems (LNEMS). Computer-aided transit scheduling, vol. 471. Springer, pp 69–127Google Scholar
  18. Michaelis M, Schöbel A (2009) Integrating line planning, timetabling, and vehicle scheduling. Public Transp 1:211–232CrossRefGoogle Scholar
  19. Parragh SN, Schmid V (2013) Hybrid column generation and large neighborhood search for the dial-a-ride problem. Comput Oper Res 40(1):490–497CrossRefGoogle Scholar
  20. Pisinger D, Ropke S (2010) Large neighborhood search. In: Gendreau M, Potvin JY (eds) Handbook of metaheuristics. Springer, pp 399–419Google Scholar
  21. Puchinger J, Raidl GR (2005) Combining metaheuristics and exact algorithms in combinatorial optimization: a survey and classification. In: Mira J, Álvarez JR (eds) Artificial intelligence and knowledge engineering applications: a bioinspired approach, vol 3562. Lecture notes in computer science. Springer, Berlin Heidelberg, pp 41–53Google Scholar
  22. Raidl G, Puchinger J, Blum C (2010) Metaheuristic hybrids. In: Gendreau M, Potvin JY (eds) Handbook of metaheuristics, 2nd edn, vol 146. Springer, pp 469–496. doi: 10.1007/978-1-4419-1665-5_16
  23. Reinboth M, Breme B (2014) Gestaltung des SPNV-Angebots auf der Südharzstrecke und Süd-Westharzstrecke von 2015 bis 2029.
  24. Ribeiro C, Soumis F (1994) A column generation approach to the multiple-depot vehicle scheduling problem. Oper Res 42(1):41–52CrossRefGoogle Scholar
  25. Ropke S, Pisinger D (2006) An adaptive large neighborhood search heuristic for the pickup and delivery problem with time windows. Transp Sci 40:455–472CrossRefGoogle Scholar
  26. Savelsbergh M (1992) The vehicle routing problem with time windows: minimizing route duration. INFORMS J Comput 4:146–154CrossRefGoogle Scholar
  27. Schmid V (2014) Hybrid large neighborhood search for the bus rapid transit route design problem. Eur J Oper Res 238(2):427–437CrossRefGoogle Scholar
  28. Schmid V, Doerner K (2013) Examination and operating room scheduling including optimization of intra-hospital routing. Transp Sci 48(1):59–77Google Scholar
  29. Schrimpf G, Schneider J, Stamm-Wilbrandt H, Dueck G (2000) Record breaking optimization results using the ruin and recreate principle. J Comput Phys 159(2):139–171CrossRefGoogle Scholar
  30. Verkehrsverbund-Südniedersachsen (2014) Timetables for lines 120, 154, 180, 185, 220.
  31. Vidal T, Crainic TG, Gendreau M, Prins C (2014) A unified solution framework for multi-attribute vehicle routing problems. Eur J Oper Res 234(3):658–673CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Christian Doppler Laboratory for Efficient Intermodal Transport Operations, Department of Business AdministrationUniversity of ViennaViennaAustria
  2. 2.Advanced Business Analytics Group, Department Information SystemsFreie Universität BerlinBerlinGermany

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