An Exact Algorithm for the Pickup and Delivery Problem with Time Windows

  • Authors:

  • Roberto Baldacci;Enrico Bartolini;Aristide Mingozzi

  • Affiliations:

  • Department of Electronics, Computer Science, and Systems (DEIS), University of Bologna, 47521 Cesena, Italy;Department of Computer Science, University of Bologna, 40127 Bologna, Italy;Department of Mathematics, University of Bologna, 47521 Cesena, Italy

  • Venue:
  • Operations Research
  • Year:
  • 2011

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Abstract

The pickup and delivery problem with time windows (PDPTW) is a generalization of the vehicle routing problem with time windows. In the PDPTW, a set of identical vehicles located at a central depot must be optimally routed to service a set of transportation requests subject to capacity, time window, pairing, and precedence constraints. In this paper, we present a new exact algorithm for the PDPTW based on a set-partitioning--like integer formulation, and we describe a bounding procedure that finds a near-optimal dual solution of the LP-relaxation of the formulation by combining two dual ascent heuristics and a cut-and-column generation procedure. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between a known upper bound and the lower bound achieved. If the resulting problem has moderate size, it is solved by an integer programming solver; otherwise, a branch-and-cut-and-price algorithm is used to close the integrality gap. Extensive computational results over the main instances from the literature show the effectiveness of the proposed exact method.