This paper introduces a variant of the vehicle routing problem with time windows where a limited number of vehicles is given (m-VRPTW). Under this scenario, a feasible solution is one that may contain either unserved customers and/or relaxed time windows. We provide a computable upper bound to the problem. To solve the problem, we propose a tabu search approach characterized by a holding list and a mechanism to force dense packing within a route. We also allow time windows to be relaxed by introducing the notion of penalty for lateness. In our approach, customer jobs are inserted based on a hierarchical objective function that captures multiple objectives. Computational results on benchmark problems show that our approach yields solutions that are competitive to best-published results on VRPTW. On m-VRPTW instances, experiments show that our approach produces solutions that are very close to computed upper bounds. Moreover, as the number of vehicles decreases, the routes become more densely packed monotonically. This shows that our approach is good from both the optimality as well as stability point of view.
