A Heuristic Algorithm for the Vehicle Routing Problem with Backhauls

Let us consider the following extension of the Capacitated Vehicle Routing Problem (VRP). We are given a set of customers partitioned in two subsets: Linehaul and Backhaul customers. Each Linehaul customer requires the delivery of a given quantity of product from the depot, while a given quantity of product must be picked up from each Backhaul customer and transported to the depot. This problem is known as the Vehicle Routing Problem with Backhauls (VRPB), or as the Linehaul-Backhaul Problem. VRPB then calls for the determination of a set of vehicle routes visiting once all customers and such that: (i) each vehicle performs one route; (ii) for each route the total load associated with Linehaul and Backhaul customers does not exceed, separately, the vehicle capacity; (iii) in each route no Backhaul customer is visited before that all Linehaul customers have been serviced. The objective is to minimize.the total traveled distance.