A comparison of approaches to modeling train scheduling problems as job-shops with blocking constraints

A train scheduling problem is considered, where a set of trains travels through a given railway network consisting of single tracks, sidings and stations. For every train a fixed route and travel times, an earliest departure time at the origin and a desired arrival time at the destination are given. To increase customer satisfaction and planning certainty in adjacent networks, the goal is to avoid trains arriving later than the desired time. The train scheduling problem is interpreted as a job-shop scheduling problem, where jobs represent trains and machines constitute tracks or track sections. For every job a technological order, in which its operations are to be processed, is given by the train route. The release times and due times of the jobs as well as the processing times for all operations are known by the input data of the problem. A feasible schedule minimizing the total tardiness of all jobs is to be determined. Blocking constraints are additionally included to account for a train blocking a track until the succeeding track is free to travel on. Dependent on the transformation approach applied a job-shop scheduling problem with or without additional routing flexibility in stations and sidings is to be solved. For this NP-complete problem several mixed integer programming formulations based on different transformation approaches using distinct types of decision variables are derived and compared in terms of total tardiness values, formulation size and computation time.