The one-commodity pickup-and-delivery traveling salesman problem: Inequalities and algorithms

This article concerns the “One-commodity Pickup-and-Delivery Traveling Salesman Problem” (1-PDTSP), in which a single vehicle of fixed capacity must either pick up or deliver known amounts of a single commodity to a given list of customers. It is assumed that the product collected from the pickup customers can be supplied to the delivery customers, and that the initial load of the vehicle leaving the depot can be any quantity. The problem is to find a minimum-cost sequence of the customers in such a way that the vehicle’s capacity is never exceeded. This article points out a close connection between the 1-PDTSP and the classical “Capacitated Vehicle Routing Problem” (CVRP), and it presents new inequalities for the 1-PDTSP adapted from recent inequalities for the CVRP. These inequalities have been implemented in a branch-and-cut framework to solve to optimality the 1-PDTSP that outperforms a previous algorithm [12]. Larger instances (with up to 100 customers) are now solved to optimality. The classical “Traveling Salesman Problem with Pickups and Deliveries” (TSPPD) is a particular case of the 1-PDTSP, and this observation gives an additional motivation for this article. The here-proposed algorithm for the 1-PDTSP was able to solve to optimality TSPPD instances with up to 260 customers.