The train frequency compatibility problem

How should trains that arrive at a railway station in constant intervals be scheduled so that the safety interval between two trains is maximum? This is a timetabling problem which we call the train frequency compatibility (TFC) problem that can be mathematically stated as follows. Given a collection A of N (possible repeated) positive integers a_i (the train frequencies), find δ_i≥0 (the starting time of the periodic trip i), for i=1,…,N, such that z=min|na_j+δ_j−(ma_i+δ_i)|, with i,j=1,…,N,i≠j and m,n∈N₀, is maximum. We present Mixed Integer Linear Programming (MILP) formulations for the problem, describe a procedure to obtain bounds on maximum z, and report computational results. We also consider a restricted version of the TFC which yields more realistic solutions regarding the train scheduling applications. For the restricted version we give a formulation based on matchings in bipartite graphs, compare the computational performance of this formulation with the MILP model, and prove that the problem is NP-hard.