The One Commodity Pickup and Delivery Traveling Salesman Problem with Demand Intervals

This study introduces the One-Commodity Pickup and Delivery Traveling Salesman Problem with Demand Intervals (1-PDTSP-DI), a generalization of the One Commodity Pickup and Delivery Traveling Salesman Problem (1-PDTSP). In the 1-PDTSP-DI, the vertices are required to have an inventory lying between given lower and upper bounds and initially have an inventory which does not necessarily lie between these bounds. The problem consists of redistributing the inventory among the vertices, using a single capacitated vehicle, so that the bounding constraints are satisfied and there positioning cost is minimized. An application of this problem arises in rebalancing operations for shared bicycle systems. The repositioning subproblem associated with a fixed route is shown to be a minimum cost network problem. Two integer programming formulations for the 1-PDTSP-DI are presented, together with valid inequalities adapted from constraints derived in the context of other routing problems. One of these formulations is appropriate for a Benders Decomposition scheme. Computational results for instances adapted from the 1-PDTSP are provided for both approaches.