Models and efficient algorithms for pickup and delivery problems on roadmaps

One of the most common combinatorial problems in logistics and transportation-after the Traveling Salesman Problem-is the Stacker Crane Problem (SCP), where commodities or customers are associated each with a pickup location and a delivery location, and the objective is to find a minimum-length tour `picking up' and `delivering' all items, while ensuring the number of items on-board never exceeds a given capacity. While vastly many SCPs encountered in practice are embedded in road or road-like networks, very few studies explicitly consider such environments. In this paper, first, we formulate an environment model capturing the essential features of a “small-neighborhood” road network, along with models for omni-directional vehicles and directed vehicles. Then, we formulate a stochastic version of the unit-capacity SCP, on our road network model, where pickup/delivery sites are random points along segments of the network. Our main contribution is a polynomial-time algorithm for the problem that is asymptotically constant-factor; i.e., it produces a solution no worse than κ+o(1) times the length of the optimal one, where o(1) goes to zero as the number of items grows large, almost surely. The constant κ is at most 3, and for omni-directional vehicles it is provably 1, i.e., optimal. Simulations 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.