The vehicle-routing problem with delivery and back-haul options

Abstract

In this article we consider a version of the vehicle-routing problem (VRP): A fleet of identical capacitated vehicles serves a system of one warehouse and N customers of two types dispersed in the plane. Customers may require deliveries from the warehouse, back hauls to the warehouse, or both. The objective is to design a set of routes of minimum total length to serve all customers, without violating the capacity restriction of the vehicles along the routes. The capacity restriction here, in contrast to the VRP without back hauls is complicated because amount of capacity used depends on the order the customers are visited along the routes. The problem is NP-hard. We develop a lower bound on the optimal total cost and a heuristic solution for the problem. The routes generated by the heuristic are such that the back-haul customers are served only after terminating service to the delivery customers. However, the heuristic is shown to converge to the optimal solution, under mildprobabilistic conditions, as fast as N-0.5. The complexity of the heuristic, as well as the computation of the lower bound, is U(N’) if all customers have unit demand size and O(N3 log N) otherwise, independently of the demand sizes. 0 1996 John Wiley & Sons, Inc. Improvement of a physical distribution system may have crucial consequences for a business’s performance. In certain sectors of the economy, transportation costs amount to a fifth (lumber and wood products) or even a quarter (petroleum, stone, clay, and glass products) of the average sales dollar; see [ 201. A careful design of the distribution system may thus yield significant cost savings to the company, usually by exploiting joint procurement possibilities to satisfy the needs of multiple locations. This potential for savings arises in particular in systems where goods are distributed through a fleet of vehicles combining visits to distinct locations into efficient routes. The problem we study here deals with a single-period, single-warehouse distribution system with two sets of customers dispersed in the plane. The first set, denoted by D, consists of delivery customers that require a delivery of goods from the warehouse, whereas the second set, denoted by B, consists of back-haul customers that need to deliver goods from their location to the warehouse. It is possible for a customer to require both a delivery and a back haul. (Note that the literature distinguishes between back-haul customers and pickup customers: A pickup customer may deliver goods to any of the delivery customers and the warehouse. In this research we restrict ourselves to delivery and back-haul customers only.) In order to avoid excessive notation we will assume a single-commodity problem, and in Section 5 we explain the modifications needed for the multicommodity case. Each customer is characterized by its geographic location and its requirement size for either delivery or back haul. The company’s fleet of vehicles is used to deliver and back haul stock. All vehicles are assumed to have identical capacities. We use the Euclidean metric as the dis