A constraint relaxation-based algorithm for the load-dependent vehicle routing problem with time windows

We introduce the load-dependent vehicle routing problem with time windows (LDVRPTW) in this paper. Transportation costs in this new problem, unlike those in the classical vehicle routing problem with time windows (VRPTW), are calculated based on not only the travel distances but also the vehicular loads on travel arcs. To solve this challenging NP-hard problem, we design a new constraint relaxation-based algorithm. In the proposed algorithm, a new constraint relaxation is introduced, i.e., some clients are not visited by a real vehicle and instead are entrusted to an additional virtual vehicle. Based on this relaxation, we present an effective execution scheme of local search procedures. The proposed algorithm is tested on benchmark instances of several special cases of the LDVRPTW, including the VRPTW. Numerical results for different variant problems demonstrate that the algorithm consistently yields impressive results: in particular, for one special variant, namely the fuel consumption rate considered vehicle routing problem (FCR-VRP), the algorithm improves the best-known solutions found by existing state-of-the-art methods.