Models and Asymptotically Optimal Algorithms for Pickup and Delivery Problems on Roadmaps

One of the most common combinatorial problems in logistics and transportation applications is the Stacker Crane problem (SCP), where commodities or customers are associated with a pickup location and a delivery location, and the objective is to find a minimum-length tour visiting all locations with the constraint that each pickup location and its associated delivery location are visited in consecutive order. While vastly many SCPs encountered in practice are embedded in road or road-like networks, very few studies explicitly consider such specific structure. In this paper, first we formulate an environment model that captures the essential features of a “small-neighborhood” road network, including a basic set of road rules. Then, we formulate a stochastic version of the SCP on such road network model, where pickup/delivery pairs are random points on network’s edges and nodes. Finally, we provide an algorithm for such problem which: (i) is asymptotically optimal, i.e., it produces a solution approaching the optimal one as the number of pickups/deliveries goes to infinity, almost surely; and (ii) is efficient in the sense that it can be computed in time polynomial in n, where n is the number of pickup/delivery pairs. Simulation results show that with a number of pickup/delivery pairs as low as 50 the proposed algorithm delivers a solution whose cost is consistently within 10% of that of an optimal solution.