Solution strategies for the vehicle routing problem with backhauls

The paper concerns the classical vehicle routing problem (VRP) with backhauls (VRPB), which can be seen as a special case of the asymmetric VRP with mixed backhauls (AVRPMB). We tackle the VRPB by: (i) directly applying a state-of-the-art AVRPMB matheuristic for the problem, in which a VRPB instance is transformed into an AVRPMB instance, i.e., the infeasible VRPB arcs are penalized; (ii) adapting the same matheuristic for the VRPB itself, in particular, preventing infeasible moves to be unnecessarily evaluated during the local search and also by only allowing feasible solutions to be explored in all steps of the algorithm; (iii) modifying the set partitioning formulation used in the matheuristic to specifically tackle the VRPB. The three approaches were capable of obtaining all best known solutions for the traditional benchmark instances and the result of two instances were improved. We also compare the performance and scalability of the three strategies for instances with up to 1000 customers.