Lagrangian based heuristics for the multicommodity network flow problem with fixed costs on paths

We study the multicommodity network flow problem with fixed costs on paths, with specific application to the empty freight car distribution process of a rail operator. The classification costs for sending a group of cars do not depend on the number of cars in the group, as long as the group is kept together as one unit. Arcs correspond to trains, so we have capacity restrictions on arcs but fixed costs on the paths corresponding to routes for groups of cars. As solution method, we propose a Lagrangian based heuristic using dual subgradient search and primal heuristics based on path information of the Lagrangian subproblem solutions. The method illustrates several ways of exploiting the specific structures of the problem. Computational tests indicate that the method is able to generate fairly good primal feasible solutions and lower bounds on the optimal objective function value.