Abstract
The task of delay management is to decide whether connecting trains should wait for delayed feeder trains or depart on time in order to minimize the passengers’ delay. To estimate the effect of the wait-depart decisions on the travel times, most delay management models assume that passengers’ routes are predefined. However, in practice, passengers can adapt their routes to the wait-depart decisions and arising changes in the timetable. For this reason, in this paper we assume that passengers’ demand is given in form of pairs of origins and destinations (OD-pairs) and take wait-depart decisions and decisions on passengers’ routes simultaneously.
This approach, called delay management with re-routing, was introduced in Dollevoet et al. (Transp. Sci. 46(1):74–89, 2012) and we build our research upon the results obtained there. We show that the delay management problem with re-routing is strongly NP-hard even if there is only one OD-pair. Furthermore, we prove that even if there are only two OD-pairs, the problem cannot be approximated with constant approximation ratio unless P=NP. However, for the case of only one OD-pair we propose a polynomial-time algorithm. We show that our algorithm finds an optimal solution if there is no reasonably short route from origin to destination which requires a passenger to enter the same train twice. Otherwise, the solution found by the algorithm is a 2-approximation of an optimal solution and the estimated travel time is a lower bound on the objective value.
Similar content being viewed by others
References
Adenso-Díaz B, Oliva González M, González-Torre P (1999) On-line timetable re-scheduling in regional train services. Transp Res, Part B, Methodol 33:387–398
Bauer R (2010) Theory and engineering for shortest paths and delay management. PhD thesis, Karlsruher Institut für Technologie
Berger A, Hoffmann R, Lorenz U, Stiller S (2011) Online railway delay management: hardness, simulation and computation. Simulation 87(7):616–629
Borndörfer R, Grötschel M, Pfetsch ME (2008) Models for line planning in public transport. In: Hickman M, Mirchandani P, VoßS (eds) Computer-aided systems in public transport. Lecture notes in economics and mathematical systems, vol 600. Springer, Berlin, pp 363–378
Cordeau JF, Toth P, Vigo D (1998) A survey of optimization models for train routing and scheduling. Transp Sci 32(4):380–404
Corman F, D’Ariano A, Pacciarelli D, Pranzo M (2012) Bi-objective conflict detection and resolution in railway traffic management. Transp Res, Part C, Emerg Technol 20(1):79–94
de Vries R, De Schutter B, De Moor B (1998) On max-algebraic models for transportation networks. In: Proceedings of the international workshop on discrete event systems, Cagliari, Italy, pp 457–462
Dollevoet T, Schmidt M, Schoebel A (2011) Delay management including capacities of stations. In: Caprara A, Kontogiannis S (eds) Proceedings of ATMOS, OASIcs, vol 20. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, pp 88–99
Dollevoet T, Huisman D, Schmidt M, Schöbel A (2012) Delay management with re-routing of passengers. Transp Sci 46(1):74–89
Fischetti M, Monaci M (2009) Light robustness. In: Ahuja RK, Möhring R, Zaroliagis C (eds) Robust and online large-scale optimization. Lecture note on computer science, vol 5868. Springer, Berlin, pp 61–84
Fischetti M, Salvagnin S, Zanette A (2009) Fast approaches to improve the robustness of a railway timetable. Transp Sci 43:321–335
Garey M, Johnson D (1979) Computers and intractability—a guide to the theory of NP-completeness. Freeman, San Francisco
Gatto M, Glaus B, Jacob R, Peeters L, Widmayer P (2004) Railway delay management: exploring its algorithmic complexity. In: Proceedings of 9th Scandinavian Workshop on Algorithm Theory (SWAT). Lecture notes in computer science, vol 3111. Springer, Berlin, pp 199–211
Gatto M, Jacob R, Peeters L, Schöbel A (2005) The computational complexity of delay management. In: Kratsch D (ed) Graph-theoretic concepts in computer science: 31st international workshop (WG 2005). Lecture notes in computer science, vol 3787. Springer, Berlin
Gatto M, Jacob R, Peeters L, Widmayer P (2007) On-line delay management on a single train line. In: Algorithmic methods for railway optimization. Lecture notes in computer science, vol 4359. Springer, Berlin, pp 306–320
Ginkel A, Schöbel A (2007) To wait or not to wait? The bicriteria delay management problem in public transportation. Transp Sci 41(4):527–538
Goerigk M, Schöbel A (2010) An empirical analysis of robustness concepts for timetabling. In: Erlebach T, Lübbecke M (eds) Proceedings of ATMOS, OASIcs, vol 14. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, pp 100–113
Goerigk M, Knoth M, Müller-Hannemann M, Schmidt M, Schöbel A (2011) The price of robustness in timetable information. In: Caprara A, Kontogiannis S (eds) Proceedings of ATMOS, OASIcs, vol 20. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, pp 76–87
Goverde R (1998) The max-plus algebra approach to railway timetable design. In: Computers in railways VI: proceedings of the 6th international conference on computer aided design, manufacture and operations in the railway and other advanced mass transit systems, Lisbon, pp 339–350
Goverde RM (2007) Railway timetable stability analysis using max-plus system theory. Transp Res, Part B, Methodol 41(2):179–201
Heidergott B, de Vries R (2001) Towards a control theory for transportation networks. Discrete Event Dyn Syst 11:371–398
Heilporn G, Giovanni LD, Labbé M (2008) Optimization models for the single delay management problem in public transportation. Eur J Oper Res 189(3):762–774
Kliewer N, Suhl L (2011) A note on the online nature of the railway delay management problem. Networks 57
Krumke SO, Thielen C, Zeck C (2011) Extensions to online delay management on a single train line: new bounds for delay minimization and profit maximization. Math Methods Oper Res 74(1):53–75
Liebchen C, Lübbecke M, Möhring RH, Stiller S (2009) The concept of recoverable robustness, linear programming recovery, and railway applications. In: Ahuja RK, Möhring R, Zaroliagis C (eds) Robust and online large-scale optimization. Lecture notes in computer science, vol 5868. Springer, Heidelberg, pp 1–27
Lusby R, Larsen J, Ehrgott M, Ryan D (2011) Railway track allocation: models and methods. OR Spektrum 33:843–883
Nachtigall K (1998) Periodic network optimization and fixed interval timetables. Deutsches Zentrum für Luft—und Raumfahrt. Institut für Flugführung, Braunschweig. Habilitationsschrift
Nachtigall K, Jerosch K (2008) Simultaneous network line planning and traffic assignment. In: Fischetti M, Widmayer P (eds) Proceedings of ATMOS, OASICS, vol 9. Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Germany
Odijk MA (1996) A constraint generation algorithm for the construction of periodic railway timetables. Transp Res, Part B, Methodol 30(6):455–464
Schachtebeck M, Schöbel A (2010) To wait or not to wait and who goes first? Delay management with priority decisions. Transp Sci 44(3):307–321
Schmidt M (2012) Integrating routing decisions in network problems. PhD thesis, Universität Göttingen
Schmidt M, Schöbel A (2010) The complexity of integrating routing decisions in public transportation models. In: Erlebach T, Lübbecke M (eds) Proceedings of ATMOS, OASIcs, vol 14. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, pp 156–169
Schöbel A (2001) A model for the delay management problem based on mixed-integer programming. Electron Notes Theor Comput Sci 50(1)
Schöbel A (2006) Optimization in public transportation. Stop location, delay management and tariff planning from a customer-oriented point of view. Optimization and its applications. Springer, New York
Schöbel A (2007) Integer programming approaches for solving the delay management problem. In: Algorithmic methods for railway optimization. Lecture notes in computer science, vol 4359. Springer, Berlin, pp 145–170
Schöbel A, Kratz A (2009) A bicriteria approach for robust timetabling? Lect Notes Comput Sci 5868:119–144
Schöbel A, Scholl S (2006) Line planning with minimal travel time. In: 5th workshop on algorithmic methods and models for optimization of railways, no. 06901 in dagstuhl seminar proceedings
Serafini P, Ukovich W (1989) A mathematical model for periodic scheduling problems. SIAM J Discrete Math 2:550–581
Suhl L, Mellouli T (1999) Requirements for, and design of, an operations control system for railways. In: Computer-aided transit scheduling. Springer, Berlin
Suhl L, Biederbick C, Kliewer N (2001a) Design of customer-oriented dispatching support for railways. In: VoßS, Daduna JR (eds) Computer-aided scheduling of public transport. Lecture notes in economics and mathematical systems, vol 505. Springer, Berlin, pp 365–386
Suhl L, Mellouli T, Biederbick C, Goecke J (2001b) Managing and preventing delays in railway traffic by simulation and optimization. In: Mathematical methods on optimization in transportation systems, pp 3–16
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the Marie Curie Action International Research Staff Exchange Scheme (IRSES) FP7-PEOPLE-2009-IRSES project OptALI under project ID 246647.
Electronic Supplementary Material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Schmidt, M. Simultaneous optimization of delay management decisions and passenger routes. Public Transp 5, 125–147 (2013). https://doi.org/10.1007/s12469-013-0069-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12469-013-0069-5