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Branch and Cut and Price for the Pickup and Delivery Problem with Time Windows

Department of Transport, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
HEC Montréal, 3000, chemin de la Côte-Sainte-Catherine, Montréal H3T 2A7, Canada
sr{at}transport.dtu.dk
jean-francois.cordeau{at}hec.ca

In the pickup and delivery problem with time windows vehicle routes must be designed to satisfy a set of transportation requests, each involving a pickup and delivery location, under capacity, time window, and precedence constraints. This paper introduces a new branch-and-cut-and-price algorithm in which lower bounds are computed by solving through column generation the linear programming relaxation of a set partitioning formulation. Two pricing subproblems are considered in the column generation algorithm: an elementary and nonelementary shortest path problem. Valid inequalities are added dynamically to strengthen the relaxations. Some of the previously proposed inequalities for the pickup and delivery problem with time windows are also shown to be implied by the set partitioning formulation. Computational experiments indicate that the proposed algorithm outperforms a recent branch-and-cut algorithm.

Key Words: pickup and delivery; column generation; branch and price; branch and cut; valid inequalities
History: Received: November 2007; revised: July 2008; accepted: October 2008.