Scheduling extra trains to an existing timetable along a railway corridor: An ADMM-based optimisation approach

This paper addresses the problem of scheduling extra train paths (SETP) onto an existing timetable along railway corridors. Each extra train has a pre-assigned departure time window: the train can be scheduled to depart at any time within the window, or else a failure to schedule would be registered. Modifications to both existing and extra train paths are permitted, including adjustments to departure and dwell times, the introduction of unplanned stops, and overtaking at stations. The objective function minimises a weighted combination of total travel time of scheduled extra trains and schedule deviations of existing trains. Adjustable weighting factors enable operators to balance punctuality requirements against operational flexibility. We formulate the problem as an Integer Linear Programme (ILP) on a space–time network and develop an Alternating Direction Method of Multipliers (ADMM) decomposition framework. The approach decomposes the problem into train-specific subproblems, each solved using a time-dependent shortest-path algorithm. Coupling constraints are enforced through iterative updates to Lagrange multipliers and penalty parameters. Computational experiments on a real-world corridor demonstrate that the ADMM-based approach achieves feasible solutions within reasonable computation time, with an average optimality gap of 5.43% compared with Lagrangian relaxation lower bounds.