Abstract
In this work we consider the single track train scheduling problem. The problem consists of scheduling a set of trains from opposite sides along a single track. The track has intermediate stations and the trains are only allowed to pass each other at those stations. Traversal times of the trains on the blocks between the stations only depend on the block lengths but not on the train. This problem is a special case of minimizing the makespan in job shop scheduling with two counter routes and no preemption. We develop a lower bound on the makespan of the train scheduling problem which provides us with an easy solution method in some special cases. Additionally, we prove that for a fixed number of blocks the problem can be solved in pseudo-polynomial time.
Similar content being viewed by others
References
Babushkin, A. I., Bashta, A. A., & Belov, I. S. (1974). Scheduling for the problem with oppositely directed routes. Kibernetika, 7, 130–135.
Babushkin, A. I., Bashta, A. A., & Belov, I. S. (1977). Scheduling for problems of counterroutes. Cybernetics and Systems Analysis, 13(4), 611–617.
Balas, E. (1969). Machine sequencing via disjunctive graphs: An implicit enumeration algorithm. Operations Research, 17(6), 941–957.
Bersani, C., Qiu, S., Sacile, R., Sallak, M., & Schn, W. (2015). Rapid, robust, distributed evaluation and control of train scheduling on a single line track. Control Engineering Practice, 35, 12–21.
Borndörfer, R., Erol, B., Graffagnino, T., Schlechte, T., & Swarat, E. (2014). Optimizing the simplon railway corridor. Annals of Operations Research, 218(1), 93–106.
Brucker, P., Heitmann, S., & Knust, S. (2005). Scheduling railway traffic at a construction site. In G. Hans-Otto & H. K. Kap (Eds.), Container terminals and automated transport systems. Berlin: Springer.
Butler, S., & Karasik, P. (2010). A note on nested sums. Journal of Integer Sequences, 13(2), 3.
Cacchiani, V., & Toth, P. (2012). Nominal and robust train timetabling problems. European Journal of Operational Research, 219(3), 727–737.
Carlier, J. (1982). The one-machine sequencing problem. European Journal of Operational Research, 11(1), 42–47.
Castillo, E., Gallego, I., Ureña, J. M., & Coronado, J. M. (2009). Timetabling optimization of a single railway track line with sensitivity analysis. TOP, 17(2), 256–287.
Chen, B., Potts, C., & Woeginger, G. (1998). A review of machine scheduling: Complexity, algorithms and approximability. In Handbook of combinatorial optimization (pp. 1492–1641). Springer.
Cordeau, J.-F., Toth, P., & Vigo, D. (1998). A survey of optimization models for train routing and scheduling. Transportation Science, 32(4), 380–404.
Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2001). Introduction to algorithms (Vol. 2). Cambridge: MIT Press.
Disser, Y., Klimm, M., & Lübbecke, E. (2015). Scheduling bidirectional traffic on a path. In M. M. Halldórsson, K. Iwama, N. Kobayashi, & B. Speckmann (Eds.), Automata, languages, and programming: 42nd international colloquium, ICALP 2015, Kyoto, Japan, July 6–10, 2015, Proceedings, Part I (pp. 406–418). Berlin: Springer.
Dushin, B. I. (1988). An algorithm for the solution of the two-route johnson problem. Cybernetics and Systems Analysis, 24(3), 336–343.
Frank, O. (1966). Two-way traffic on a single line of railway. Operations Research, 14(5), 801–811.
Garey, M. R., Johnson, D. S., & Sethi, R. (1976) .The complexity of flowshop and jobshop scheduling. Mathematics of Operations Research, 1(2), 117–129.
Gonzalez, T. (1982). Unit execution time shop problems. Mathematics of Operations Research, 7(1), 57–66.
Higgins, A., Kozan, E., & Ferreira, L. (1996). Optimal scheduling of trains on a single line track. Transportation Research Part B: Methodological, 30(2), 147–161.
Higgins, E., Kozan, A., & Ferreira, L. (1997). Heuristic techniques for single line train scheduling. Journal of Heuristics, 3(1), 43–62.
Hromkovič, J., Mömke, T., Steinhöfel, K., & Widmayer, P. (2007) . Job shop scheduling with unit length tasks: Bounds and algorithms. Algorithmic Operations Research, 2(1), 1–14.
Jackson, J. R. (1956). An extension of Johnson’s result on job IDT scheduling. Naval Research Logistics Quarterly, 3(3), 201–203.
Janić, M. (1984). Single track line capacity model. Transportation Planning and Technology, 9(2), 135–151.
Kraay, D., Harker, P. T., & Chen, B. (1991). Optimal pacing of trains in freight railroads: Model formulation and solution. Operations Research, 39(1), 82–99.
Lenstra, J. K., & Kan, A. R. (1979). Computational complexity of discrete optimization problems. Annals of Discrete Mathematics, 4, 121–140.
Lübbecke, E. (2015). On-and offline scheduling of bidirectional traffic. Berlin: Logos Verlag Berlin GmbH.
Lübbecke, E., Lübbecke, M. E., & Möhring, R. H. (2014). Ship traffic optimization for the kiel canal. Technical Report 4681. (Optimization Online).
Manne, A. S. (1960). On the job-shop scheduling problem. Operations Research, 8(2), 219–223.
Mesa, J. A., Ortega, F. A., & Pozo, M. A. (2014). Locating optimal timetables and vehicle schedules in a transit line. Annals of Operations Research, 222(1), 439–455.
Petersen, E. R. (1974). Over-the-road transit time for a single track railway. Transportation Science, 8(1), 65–74.
Rahman, S. A. A. (2013). Freight train scheduling on a single line network. Ph.D. thesis, The University of New South Wales.
Šemrov, D., Marsetič, R., Žura, M., Todorovski, L., & Srdic, A. (2016). Reinforcement learning approach for train rescheduling on a single-track railway. Transportation Research Part B: Methodological, 86, 250–267.
Sevast’janov, S. V. (1994). On some geometric methods in scheduling theory: a survey. Discrete Applied Mathematics, 55(1), 59–82.
Simons, B. (1978). A fast algorithm for single processor scheduling. In IEEE symposium on foundations of computer science (FOCS) (pp. 246–252).
Sotskov, Y. N., & Gholami, O. (2012). Shifting bottleneck algorithm for train scheduling in a single-track railway. IFAC Proceedings Volumes, 45(6), 87–92.
Yang, L., Li, K., & Gao, Z. (2009). Train timetable problem on a single-line railway with fuzzy passenger demand. IEEE Transactions on Fuzzy Systems, 17(3), 617–629.
Yang, L., Gao, Z., & Li, K. (2010). Passenger train scheduling on a single-track or partially double-track railway with stochastic information. Engineering Optimization, 42(11), 1003–1022.
Zhou, X., & Zhong, M. (2007). Single-track train timetabling with guaranteed optimality: Branch-and-bound algorithms with enhanced lower bounds. Transportation Research Part B: Methodological, 41(3), 320–341.
Acknowledgements
The authors were partially funded by the DFG under Grant Number SCHO1140/3-2 and by the European Union Seventh Framework Programme (FP7-PEOPLE-2009-IRSES) under Grant Number 246647 with the New Zealand Government (project OptALI). We also thank the Simulationswissenschaftliches Zentrum Clausthal-Göttingen (SWZ) for financial support.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Harbering, J., Ranade, A., Schmidt, M. et al. Complexity, bounds and dynamic programming algorithms for single track train scheduling. Ann Oper Res 273, 479–500 (2019). https://doi.org/10.1007/s10479-017-2644-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-017-2644-7