In this paper, we investigate a routing problem arising in the last-mile delivery of small packages. The problem, called Multi-Depot Vehicle Routing Problem with Private fleet and Common carriers (MDVRPPC), is an extension of the Multi-Depot Vehicle Routing Problem (MDVRP) where customers can either be served by the private fleet based at self-owned depots or by common carriers, i.e., subcontractors. We develop an effective Variable Neighborhood Search algorithm based on the use of cyclic-exchange neighborhoods that incorporates an adaptive mechanism to bias the random shaking step. The approach is successfully used to solve MDVRPPC as well as closely related problems, such as the MDVRP and the single-depot VRP with Private fleet and Common carriers (VRPPC), obtaining high quality solutions within short computing time. Our extensive testing on these problems shows the positive impact of the adaptive mechanism with respect to a standard VNS algorithm.

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