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Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows

Department of Computer Science (DIKU), University of Copenhagen, DK-2100 Copenhagen Ø, Denmark
Department of Computer Science (DIKU), University of Copenhagen, DK-2100 Copenhagen Ø, Denmark
Department of Computer Science (DIKU), University of Copenhagen, DK-2100 Copenhagen Ø, Denmark
Department of Computer Science (DIKU), University of Copenhagen, DK-2100 Copenhagen Ø, Denmark
mitzi{at}diku.dk
bjorn{at}diku.dk
spooren{at}diku.dk
pisinger{at}diku.dk

This paper presents a branch-and-cut-and-price algorithm for the vehicle-routing problem with time windows. The standard Dantzig-Wolfe decomposition of the arc flow formulation leads to a set-partitioning problem as the master problem and an elementary shortest-path problem with resource constraints as the pricing problem. We introduce the subset-row inequalities, which are Chvatal-Gomory rank-1 cuts based on a subset of the constraints in the master problem. Applying a subset-row inequality in the master problem increases the complexity of the label-setting algorithm used to solve the pricing problem because an additional resource is added for each inequality. We propose a modified dominance criterion that makes it possible to dominate more labels by exploiting the step-like structure of the objective function of the pricing problem. Computational experiments have been performed on the Solomon benchmarks where we were able to close several instances. The results show that applying subset-row inequalities in the master problem significantly improves the lower bound and, in many cases, makes it possible to prove optimality in the root node.

Subject classifications: transportation; vehicle routing; programming; integer.
History: Received March 2006; revision received October 2006; accepted November 2006.