Path inequalities for the vehicle routing problem with time windows

In this paper we introduce a new formulation of the VRPTW involving only binary variables associated with the arcs in the underlying digraph. The new formulation is based on the formulation of the ATSPTW by Ascheuer et al. [1] and has the advantage of avoiding additional variables and linking constraints. In the new formulation of the VRPTW time windows are modeled using path inequalities. The path inequalities eliminate time or capacity infeasible paths. We present a new class of strengthened path inequalities based on the polyhedral results obtained by Mak [2] in the context of the TSP with replenishment arcs. We study the VRPTW polytope and determine the polytope dimension. We show that the lifted path inequalities are facet defining under reasonable assumptions. We also introduce precedence constraints in the context of the VRPTW. Computational experiments are performed with a branch-and-cut algorithm on the longhorizon Solomon test problems. The outcome is based on 25-node problems that the algorithm shows promising results compared to leading algorithms in the literature. The conclusion is therefore that the path formulation of Desrochers et al. [3] is no longer the unchallenged winning strategy for solving the VRPTW. Finally, we report a solution to a previously unsolved 50-node Solomon test problem R208.