Abstract
Technical restrictions and challenging details let railway traffic become one of the most complex transportation systems. Routing trains in a conflict-free way through a track network is one of the basic scheduling problems for any railway company, also known as the train timetabling problem (TTP). This article focuses on a robust extension of the TTP, which typically consists in finding a conflict free set of train routes of maximum value for a given railway network. Timetables are, however, not only required to be profitable. Railway companies are also interested in reliable and robust solutions. Intuitively, we expect a more robust track allocation to be one where disruptions arising from delays are less likely to propagate and cause delays to subsequent trains. This trade-off between an efficient use of railway infrastructure and the prospects of recovery leads us to a bi-criteria optimization approach. On the one hand, we want to maximize the profit of a schedule, that is the number of routed trains. On the other hand, if two trains are scheduled with a minimum gap the delay of the first one will affect the subsequent train. We present extensions of the standard integer programming formulation for solving the TTP. These models incorporate both aspects with additional track configuration variables. We discuss how these variables reflect a certain robustness measure. These models can be solved by column generation techniques. We propose scalarization techniques to determine efficient, i.e., the decisions Pareto optimal, solutions. We prove that the LP-relaxation of the TTP including an additional ε-constraint remains solvable in polynomial time. Finally, we present some preliminary computational results on macroscopic real-world data of a part of the German long distance railway network.
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Notes
- 1.
In reality, train conflicts are more complex. For simpler notation, though, we avoid the introduction of headway matrices and train types.
- 2.
This scenario can be downloaded as part of the TTPlib 2008, see Erol et al. (2008), at ttplib.zib.de, i.e., hakafu_simple_37_120_6_req02_0285_0331_6.xml.
- 3.
Furthermore, we slightly penalize deviations from certain desired departure and arrival times at visiting stations.
- 4.
In addition CPLEX MIPSolve needs only some minutes and a few hundred branch and bound nodes to find an IP solution with an optimality gap of at most 2 %.
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Acknowledgements
This work was funded by the German Federal Ministry of Economics and Technology (BMWi), project Trassenbörse, grant 19M4031A and 19M7015B. Furthermore, we want to thank the two anonymous referees and in particular Hans-Florian Geerdes for improving this paper by their valuable comments.
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Schlechte, T., Borndörfer, R. (2010). Balancing Efficiency and Robustness – A Bi-criteria Optimization Approach to Railway Track Allocation. In: Ehrgott, M., Naujoks, B., Stewart, T., Wallenius, J. (eds) Multiple Criteria Decision Making for Sustainable Energy and Transportation Systems. Lecture Notes in Economics and Mathematical Systems, vol 634. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04045-0_9
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