Large Neighborhoods with Implicit Customer Selection for Vehicle Routing Problems with Profits

We consider several Vehicle Routing Problems (VRP) with profits, which seek to select a subset of customers, each one being associated with a profit, and to design service itineraries. When the sum of profits is maximized under distance constraints, the problem is usually called team orienteering problem. The capacitated profitable tour problem seeks to maximize profits minus travel costs under capacity constraints. Finally, in the VRP with private fleet and common carrier, some customers can be delegated to an external carrier subject to a cost. Three families of combined decisions must be taken: customers selection, assignment to vehicles, and sequencing of deliveries for each route. We propose a new neighborhood search for these problems, which explores an exponential number of solutions in pseudo-polynomial time. The search is conducted on “exhaustive” solutions visiting all customers, while an efficient “Select” algorithm, based on resource-constrained shortest paths, is repeatedly used to select customers and to evaluate routes. Speed-up techniques are introduced to solve more efficiently the shortest paths and prune unpromising moves. The remarkable performance of these neighborhood structures is demonstrated by extensive computational experiments with a local search, an iterated local search and a hybrid genetic algorithm. Intriguingly, even a local-improvement method to the first local optimum of this neighborhood achieves an average gap of 0.09% on classic team orienteering problem instances, rivaling with the current state-of-the-art metaheuristics. For all three problems, the proposed methodology leads to solutions of higher quality than previous algorithms in similar CPU time. Promising research avenues on hybridizations with standard routing neighborhoods are also open.