A heuristic algorithm for the symmetric and asymmetric vehicle routing problems with backhauls

We consider an extension of the capacitated Vehicle Routing Problem (VRP), known as the Vehicle Routing Problem with Backhauls (VRPB), in which the set of customers is partitioned into two subsets: Linehaul and Backhaul customers. Each Linehaul customer requires the delivery of a given quantity of product from the depot, whereas a given quantity of product must be picked up from each Backhaul customer and transported to the depot. VRPB is known to be NP-hard in the strong sense, and many heuristic algorithms were proposed for the approximate solution of the problem with symmetric or Euclidean cost matrices. We present a cluster-®rst-route-second heuristic which uses a new clustering method and may also be used to solve problems with asymmetric cost matrix. The approach exploits the information of the normally infeasible VRPB solutions associated with a lower bound. The bound used is a Lagrangian relaxation previously proposed by the authors. The ®nal set of feasible routes is built through a modi®ed Traveling Salesman Problem (TSP) heuristic, and inter-route and intra-route arc exchanges. Extensive computational tests on symmetric and asymmetric instances from the literature show the e€ectiveness of the proposed approach. Ó 1999 Elsevier Science B.V. All rights reserved.