Integrated train timetabling and locomotive assignment

Train timetabling and locomotive assignment are often performed separately in a sequential manner. One obvious disadvantage of such hierarchical planning process is that it often results in poor coordination between the train schedule and the locomotive schedule. This paper focuses on modeling and solving an integrated train timetabling and locomotive assignment problem. To solve this integrated problem, we first construct a three-dimensional state-space-time network in which a state is used to indicate which train a locomotive is serving. We then formulate the problem as a minimum cost multi-commodity network flow problem with incompatible arcs and integer flow restrictions. We present a Lagrangian relaxation heuristic for solving this network flow problem. We conduct a computational study to test the effectiveness of our Lagrangian relaxation heuristic, compare the performance of our heuristic with that of two benchmark solution methods, and report the benefits obtained by integrating train timetabling and locomotive assignment decisions.