Journal of Scheduling

, Volume 11, Issue 4, pp 279–297

# Exact train pathing

Article

## Abstract

Suppose we are given a schedule of train movements over a rail network into which a new train is to be included. The origin and the destination are specified for the new train; it is required that a schedule (including the path) be determined for it so as to minimize the time taken without affecting the schedules for the old trains. In the standard formulations of this single train pathing problem, the time taken by the train to traverse any block (track segment guarded by a signal) in the network is deemed to be a fixed number, independent of how the train arrived onto that block. In other words, the standard formulations of train pathing do not accurately model the acceleration/deceleration restrictions on trains.

In this paper we give an algorithm to solve the single train pathing problem, while taking into account the maximum allowed acceleration and deceleration as well as explicitly modeling signals. For trains having ‘large’ maximum acceleration and deceleration, our algorithm runs in polynomial time. On the other hand, if the train to be pathed is capable of only very small acceleration so that it must take a long time to reach full speed, our algorithm takes exponential time. However, we prove that the pathing problem is NP-complete for small acceleration values, thus justifying the time required by our algorithm.

Our algorithm can be used as a subroutine in a heuristic for multiple train pathing. If all trains have large (but possibly different) accelerations this algorithm will run in polynomial time.

## Keywords

Train pathing Dynamics Signaling Train simulation Train scheduling

## References

1. Blendinger, C., Mehrmann, V., Steinbrecher, A., & Unger, R. (2002). Numerical simulation of train traffic in large networks via time-optimal control. Technical Report, Institute of Mathematics, Technical university of Berlin, February 2002. Google Scholar
2. Cai, X., & Goh, C. J. (1994). A fast heuristic for the train scheduling problem. Computers in Operations Research, 21(5), 499–510.
3. Carey, M., & Lockwood, D. (1995). A model, algorithms and strategy for train pathing. Journal of the Operational Research Society, 46, 988–1005.
4. Higgins, A., Kozan, E., & Ferreira, L. (1996). Optimal scheduling of trains on a single line track. Transportation Research B, 30(2), 147–161.
5. Kraay, D., Harker, P. T., & Chen, B. (1991) Optimal pacing of trains in freight railroads: model formulation and solution. Operations Research 39(1). Google Scholar
6. Lu, Q. Dessouky, M., & Leachman, R. (2004). Modelling train movements through complex rail networks. ACM Transactions on Modelling and Computer Simulation, 14(1), 48–75.
7. Madhusudan, K., & Gopi Singh, B. (2002). Personal communication. Google Scholar
8. Malde, S. (2001). Train scheduling. BTech Project Report, Department of Computer Science and Engineering, I.I.T. Bombay. Google Scholar
9. Mees, A. I. (1991). Railway scheduling by network optimization. Mathematical and Computer Modelling, 15(1), 33–42.
10. Moreira, A. B. S., & Oliveira, R. C. (2000). A decision support system for operational planning in railway networks. In 8th conference on computer-aided scheduling of public transport, June 2000. Google Scholar
11. Garey, M. R., Johnson, D. S., & Sethi, R. (1976). The complexity of flowshop and jobshop scheduling. Mathematics Operation Research, 1, 117–129.
12. Olivera, E., & Smith, B. M. (2000). A job-shop scheduling model for the single track railway scheduling model. Research Report, School of Computer Studies, University of Leeds, August 2000. Google Scholar
13. Papadimitriou, C. H., & Kanellakis, P. C. (1980). Flowshop scheduling with limited temporary storage. Journal of the ACM, 27, 533–549.
14. Raghuram, G., & Rao, V. (1991). A decision support system for improving railway line capacity. Public Enterprise, 11, 64–72. Google Scholar
15. Rangaraj, N., Ranade, A., Moudgalya, K., Naik, R., Konda, C., & Johri, M. (2003). A simulator for estimating railway line capacity. In Proceedings of the sixth international conference of the association of Asia–Pacific operational research societies within IFORS (pp. 347–355). Google Scholar