Scheduling extra freight trains on railway networks

We study the problem of freight transportation in railway networks, where both passenger and freight trains are run. While the passenger trains have a prescribed timetable that cannot be changed, freight train operators send the infrastructure manager requests to insert new freight trains.For each freight train, the associated train operator specifies a preferred ideal timetable, which can be modified by the infrastructure manager in order to respect safeness operational constraints. In particular, this modification may correspond to routing the train along a path which is different with respect to the one in the ideal timetable. Roughly speaking, the objective is to introduce as many new freight trains as possible by assigning them timetables that are as close as possible to the ideal ones.For this timetabling problem on a generic railway network, we present an integer linear programming formulation, that generalizes some formulations already presented for the case of a single railway line, and a Lagrangian heuristic based on this formulation. Computational results on real-world instances are reported.