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

• Published in: Operations research Vol. 56; no. 2; pp. 497 - 511
• Main Authors: Jepsen, Mads, Petersen, Bjorn, Spoorendonk, Simon, Pisinger, David
• Format: Journal Article
• Language: English
• Published: Linthicum, MD INFORMS 01.03.2008; Institute for Operations Research and the Management Sciences
• Subjects: Algorithms; Analysis; Applied sciences; Branch & bound algorithms; Exact sciences and technology; integer; Integer programming; Integers; Logistics; Operational research and scientific management; Operational research. Management science; Polynomials; Product labeling; programming; Routing; Studies; Time windows; Traffic congestion; Traffic flow; Transportation; Vehicle capacity; vehicle routing; Vehicles; Vertices; United States; Lower bound; Fleet; Vehicle routing problem; Shortest path; Dominance; Routing; Set partitioning; Cut generation; Integer programming; Time window; Branch and cut method; Resource constraint; Pricing; transportation: vehicle routing; Objective function; programming: integer
• ISSN: 0030-364X; 1526-5463
• DOI: 10.1287/opre.1070.0449

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.