The traveling salesman problem with delivery and backhauls

The problem that we consider here deals with a single vehicle of a given capacity that needs to serve N customers: some of the customers require a delivery of stock from the warehouse, whereas others need to deliver stock from their location to the warehouse. The objective is to find a shortest feasible tour visiting all customers, emanating from and ending at the warehouse. We introduce an efficient (O(N²)) heuristic for this problem. The heuristic improves the worst-case bound known in the literature from 2.5 to 2. However, its average performance is shown in our numerical study to be slightly worse than that of a previously published (O(N³)) solution method.