Multimodal multicommodity routing problem with scheduled services

Abstract

We study a multicommodity network flow problem faced by a third party logistics company that has the possibility of using ground and maritime transportation. We are given a set of commodities which should be picked up from their origins at given release times and should be delivered to their destinations no later than their duedates. The commodities may be carried directly from their origins to their destinations on trucks, or they may be carried on trucks to a seaport, may visit several seaports using maritime services, and then to be carried to their destinations on trucks. There is no capacity and time limitation on the use of ground transportation. However, the maritime services are scheduled in advance and the company has limitations on the amounts of volume that it can use on each service. The aim is to determine routes for commodities in order to minimize the sum of transportation cost and stocking costs at seaports, respecting the capacity and time related constraints. We call this problem the “Multimodal Multicommodity Routing Problem with Scheduled Services (MMR-S)”. We first prove that the problem is NP-hard. Next, we propose a first mixed integer programming formulation and strengthen it using variable fixing and valid inequalities. We relax the capacity constraints in a Lagrangian manner and show that the relaxed problems decompose into a series of shortest path problems defined on networks augmented by time for each commodity. The corresponding Lagrangian dual yields a lower bound, which may be stronger than that of the linear programming relaxation of our first formulation. Then, we provide an extended formulation whose linear programming relaxation gives the same bound as the Lagrangian dual. Finally, we use the Lagrangian relaxation to devise heuristic methods and report the results of our computational study.