An exact algorithm for the vehicle routing problem with backhauls

The Vehicle Routing Problem with Backhauls is an extension of the capacitated Vehicle Routing Problem where the customers' set is partitioned into two subsets. The first is the set of Linehaul, or Delivery, customers, while the second is the set of Backhaul, or Pickup, customers. The problem is known to be NP-hard in the strong sense and finds many practical applications in distribution planning. In this paper we consider, in a unified framework, both the symmetric and the asymmetric versions of the vehicle routing problem with backhauls, for which we use an integer linear programming model and a Lagrangian lower bound which is strengthened in a cutting plane fashion. The Lagrangian lower bound is then combined according to the additive approach, with a lower bound obtained by dropping the capacity constraints, thus obtaining an effective overall bounding procedure. A branch-and-bound algorithm, reduction procedures and dominance criteria are also described. Computational tests on symmetric and asymmetric instances from the literature, involving up to 100 customers, are given, showing the effectiveness of the proposed approach.