Scheduling of multiple vehicle types : the allocation of locomotives to trains

Each year railway network organisations concern themselves with the complete specification of a commercial timetable. Once a timetable is established for a region and a limited number of locomotives of different types are allocated to the region, the stock diagramming exercise commences. The problem is to schedule locomotives to work the timetabled trains such that every train is worked on a daily basis and no more than the available locomotives are used. The stock diagramming problem can therefore be characterized as a scheduling problem. The objective is to minimize the light-run time incurred by locomotives having to perform unproductive runs, i.e. not working a train, between stations in the region. The proposed solution procedure uses a column generation technique. The problem is decomposed into two parts, the master problem and the subproblems. The master problem, formulated as a set-partitioning problem, selects a subset of locomotive schedules from a set of known feasible schedules. The subproblems generate the feasible schedules. The linear programming relaxation of the master problem is solved and a constraint branching technique used to locate an integer solution. Due to the size of the problems encountered in practice, the solution methodology is essentially a heuristic. At the branch and bound stage, an optimal solution is not pursued. Instead, a ‘good’ integer solution, strongly based on the optimal solution to the linear programming relaxation of the set-partitioning problem, is found. The computational results presented refer to problems drawn from real data.