Balancing Efficiency and Robustness – A Bi-criteria Optimization Approach to Railway Track Allocation

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.