A Benders decomposition approach for the locomotive and car assignment problem

One of the many problems faced by rail transportation companies is to optimize the utilization of the available stock of locomotives and cars. In this paper, we describe a decomposition method for the simultaneous assignment of locomotives and cars in the context of passenger transportation. Given a list of train legs and a fleet composed of several types of equipment, the problem is to determine a set of minimum cost equipment cycles such that every leg is covered using appropriate equipment. Linking constraints, which appear when both locomotives and cars are treated simultaneously, lead to a large integer programming formulation. We propose an exact algorithm, based on the Benders decomposition approach, that exploits the separability of the problem. Computational experiments carried out on a number of real-life instances indicate that the method finds optimal solutions within short computing times. It also outperforms other approaches based on Lagrangian relaxation or Dantzig–Wolfe decomposition, as well as a simplex-based branch-and-bound method.