TY - JOUR

T1 - Subset-row inequalities applied to the vehicle routing problem with time windows

AU - Jepsen, Mads Kehlet

AU - Petersen, Bjørn

AU - Spoorendonk, Simon

AU - Pisinger, David

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Faculty of Science

KW - transportation

KW - Vehicle routing

KW - Programming

KW - Integer

U2 - 10.1287/opre.1070.0449

DO - 10.1287/opre.1070.0449

M3 - Journal article

VL - 56

SP - 497

EP - 511

JO - Operations Research

JF - Operations Research

SN - 0030-364X

IS - 2

ER -