The integrated lot-sizing and vehicle routing problem

We consider the problem of scheduling both a production distributed on several sites and the transportation of items between those sites. By doing so, the objective is to synchronize the two components and to build a better overall solution. The production system on each site is modeled as a Capacitated Lot-Sizing Problem where stock both on resources and produced items is available. The inter-site items transportation is a simplified version of the Vehicle Routing Problem where time is discretized. We first propose a mixed integer linear programming formulation. Then we present two heuristics. The first one is based on production order propagation over the sites. Then, at each iteration, it computes a compatible transportation schedule and it tries to improve the solution by modifying the production on the sites. The second heuristic is an adaptation of the Relax and Fix strategy. Computational results are presented to evaluate the efficiency of the two heuristics.