Setting lines frequency and capacity in dense railway rapid transit networks with simultaneous passenger assignment

We propose a Mixed Integer Non-Linear Programming (MINLP) model in order to determine optimal line frequencies and capacities in dense railway rapid transit (RRT) networks in which typically several lines can run over the same open tracks. Given a certain demand matrix, the model determines the most appropriate frequency and train capacity for each line taking into account infrastructure capacity constraints, allocating lines to tracks while assigning passengers to lines. The service provider and the user points of view are simultaneously taken into account. The first one is considered by selecting the most convenient set of frequencies and capacities and routing passengers from their origins to their destinations while minimizing the average trip time. The second one by minimizing operation, maintenance and fleet acquisition costs. Due to the huge number of variables and constraints appearing in real size instances, a preprocessing phase determining the best k-paths linking origin and destination stations is followed. Then, the best paths are used to define sparse index sets in order to drastically reduce the size of the problem. As illustration, the model is applied to a simplified version of the Madrid Metropolitan Railway network.