Towards Solving Very Large Scale Train Timetabling Problems by Lagrangian Relaxation

The train timetabling problem considered is to find conflict free routes for a set of trains in a given railway network so that certain time window conditions are satisfied. We deal with the very large scale problem of constructing such timetables for the German railway network. A number of restrictions on different train types like freight trains or passenger trains have to be observed, e.g., sequence dependent headway times, station capacities, and stopping times. In order to handle the enormous number of variables and constraints we employ Lagrangian relaxation of the conflict constraints combined with a cutting plane approach. The model is solved by a bundle method; its primal aggregate is used for separation and as starting point for rounding heuristics. We present some promising results towards handling a test instance comprising ten percent of the entire network.