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A Mathematical Model and a Hybrid Algorithm for Robust Periodic Single-Track Train-Scheduling Problem


A robust periodic train-scheduling problem under perturbation is discussed in this paper. The intention is to develop a robustness index and to propose a mathematical model which is robust against perturbations. Some practical assumptions as well as the acceleration and deceleration times along with periodic scheduling in addition to a practical new robustness index are considered. The aim is to obtain timetables with minimum traveling time that are robust against minor perturbations, while the unnecessary stops are minimized. In general, the spread of delays in the railway system is called delay propagation. We show that in addition to this phenomenon, there exists a more complicated case in periodic type of scheduling that is the fact of delay propagation from one period to the next. In fact, if the delays of a period are not absorbed by the next one, the size of delays may converge to infinity. We name this as delay intensification. Furthermore, we develop a hybrid heuristic algorithm which is able to find near-optimal schedules in a limited amount of time and can absorb perturbations. To validate the algorithm, a new lower bound is introduced.

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  1. 1.

    Hansen IA (2010) Railway network timetabling and dynamic traffic management. Int J Civil Eng 8(1):19–32

    Google Scholar 

  2. 2.

    Serafini P, Ukovich W (1989) A mathematical model for periodic scheduling problems. SIAM J Discret Math 2(4):550–581

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Zhou X, Zhong M (2007) Single-track train timetabling with guaranteed optimality: branch-and-bound algorithms with enhanced lower bounds. Transp Res Part B 41:320–341

    Article  Google Scholar 

  4. 4.

    D’Ariano A, Pranzo M, Hansen IA (2007) Conflict resolution and train speed coordination for solving real-time timetable perturbations. IEEE Trans Intell Transp Syst 8(2):208–222

    Article  Google Scholar 

  5. 5.

    Castillo E, Gallego I, Ureña JM, Coronado JM (2011) Timetabling optimization of a mixed double- and single-tracked railway network. Appl Math Model 35(2):859–878

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Xu X, Li K, Ye LYJ (2014) Balanced train timetabling on a single-line railway with optimized velocity. Appl Math Model 38(3):894–909

    MathSciNet  Article  Google Scholar 

  7. 7.

    Li X, Wang D, Li K, Gao Z (2013) A green train scheduling model and fuzzy multi-objective optimization algorithm. Appl Math Model 37(4):2063–2073

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Tornquist J (2007) Railway traffic perturbation management—an experimental analysis of perturbation complexity, management objectives and limitations in planning horizon. Transp Res Part A 41:249–266

    Google Scholar 

  9. 9.

    Caprara A, Monaci M, Toth P, Guida PL (2006) A Lagrangian heuristic algorithm for a real-world train timetabling problem. Discret Appl Math 154:738–753

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Cacchiani V, Caprara A, Toth P (2008) A column generation approach to train timetabling on a corridor. 4OR 6:25–142

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Luis Espinosa-Aranda J, García-Ródenas R (2013) A demand-based weighted train delay approach for rescheduling railway networks in real time. J Rail Transp Plan Manag 3(1–2):1–13

    Google Scholar 

  12. 12.

    de Fabris S, Longo G, Medeossi G, Pesenti R (2014) Automatic generation of railway timetables based on a mesoscopic infrastructure model. J Rail Transp Plan Manag 4(1–2):2–13

    Article  Google Scholar 

  13. 13.

    Ghoseiri K, Morshedsolouk F (2004) An ant colony system heuristic for train scheduling problem. J Transp Res 2(4):257–270

    MATH  Google Scholar 

  14. 14.

    Jamili A, Kianfar F (2009) Train scheduling with the application of simulated annealing algorithm. J Transp Res 6(1):13–27

    Google Scholar 

  15. 15.

    Tormos P, Lova A, Barber F, Ingolotti L, Abril M, Salido MA (2008) A genetic algorithm for railway scheduling problems. Stud Comput Intelli 128:255–276

    MATH  Google Scholar 

  16. 16.

    Salido MA, Barber F, Ingolotti L (2008) Robustness in railway transportation scheduling. In: Proceedings of the 7th World Congress on Intelligent Control and Automation (WCICA ‘08), IEEE Press, Chongqing, China, pp 2833–2837

  17. 17.

    Shafia MA, Jamili A (2009) Measuring the train timetables robustness. In: Proceedings of 2nd international conference on recent advances in railway eng. (ICRARE-2009)

  18. 18.

    Andersson EV, Peterson A, Törnquist Krasemann J (2013) Quantifying railway timetable robustness in critical points. J Rail Transp Plan Manag 3(3):95–110

    Article  Google Scholar 

  19. 19.

    Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52:35–53

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Shafia MA, Pourseyed Aghaee M, Sadjadi SJ, Jamili A (2012) Robust Train Timetabling problem: mathematical model and Branch and bound algorithm. IEEE Trans Intell Transp Syst 13(1):307–317

    Article  Google Scholar 

  21. 21.

    Shafia MA, Sadjadi SJ, Jamili A, Tavakkoli-Moghaddam R (2013) The periodicity and robustness in a single-track train scheduling problem. Appl Soft Comput 12:440–452

    Article  Google Scholar 

  22. 22.

    Kroon LG, Dekker R, Vromans MJCM (2007) Cyclic railway timetabling: a stochastic optimisation approach. In: Geraets F, Kroon LG, Schöbel A, Wagner D, Zaroliagis C (eds.) Algorithmic methods in railway optimization. Lecture notes in computer science, vol. 4359, pp 41–66

  23. 23.

    Babar Khan M, Zhou X (2010) Stochastic optimization model and solution algorithm for robust double-track train timetabling problem. IEEE Trans Intell Transp Syst 11(1):81–89

    Article  Google Scholar 

  24. 24.

    Jamili A, Ghannadpour SF (2013) Computing the supplementary times and the number of required trains in operation plan studies under stochastic perturbations. 16th International IEEE annual conference on intelligent transportation systems, The Hague, The Netherlands

  25. 25.

    Jamili A, Pourseyed Aghaee M (2015) Robust stop-skipping patterns in urban railway operations under traffic alteration situation. Transp Res Part C 61:63–74

    Article  Google Scholar 

  26. 26.

    Chang PC, Chen SH, Fan CY (2009) A hybrid electromagnetism-like algorithm for single machine scheduling problem. Expert Syst Appl 36:1259–1267

    Article  Google Scholar 

  27. 27.

    Naderi B, Tavakkoli-Moghaddam R, Khalili M (2010) Electromagnetism-like mechanism and simulated annealing algorithms for flowshop scheduling problems minimizing the total weighted tardiness and makespan. Knowl Based Syst 23(2):77–85

    Article  Google Scholar 

  28. 28.

    Tavakkoli-Moghaddam R, Khalili M, Naderi B (2009) A hybridization of simulated annealing and electromagnetic-like mechanism for job shop problems with machine availability and sequence-dependent setup times to minimize total weighted tardiness. Soft Comput 13:995–1006

    Article  Google Scholar 

  29. 29.

    Singer M, Pinedo M (1998) A computational study of branch and bound techniques for minimizing the total weighted tardiness in job shops. IIE Trans 30(2):109–118

    Google Scholar 

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Jamili, A. A Mathematical Model and a Hybrid Algorithm for Robust Periodic Single-Track Train-Scheduling Problem. Int J Civ Eng 15, 63–75 (2017).

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  • Scheduling
  • Robustness
  • Delay intensification
  • Hybrid heuristic algorithm