Theory and Methodology A heuristic algorithm for the symmetric and asymmetric vehicle routing problems with backhauls Paolo Toth * , Daniele Vigo DEIS, Universit  a di Bologna, Viale Risorgimento, 2-40136 Bologna, Italy Received 1 July 1997; accepted 1 January 1998 Abstract 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. Keywords: Vehicle routing; Lagrangian relaxation; Heuristic algorithms; Local search 1. Introduction 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. An example of this partition of the customer set, which is extremely frequent in practical situa- tions, is represented by the grocery industry. In this case, supermarkets and shops are the Linehaul customers, and grocery suppliers are the Backhaul customers. In recent years it has been widely European Journal of Operational Research 113 (1999) 528±543 * Corresponding author. Tel.: +39 51 6443028; fax: +39 51 6443073; e-mail: ptoth@deis.unibo.it. 0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 0 8 6 - 1