A Lower Bound for the Split Delivery Vehicle Routing Problem

Abstract

In this paper we consider the Split Delivery Vehicle Routing Problem (SDVRP), a relaxation of the known Capacitated Vehicle Routing Problem (CVRP) in which the demand of any client can be serviced by more than one vehicle. We define a feasible solution of this problem, and we show that the convex hull of the associated incidence vectors is a polyhedron (PSDVRP), whose dimension depends on whether a vehicle visiting a client must service, or not, at least one unit of the client demand. From a partial and linear description of PSDVRP and a new family of valid inequalities, we develop a lower bound whose quality is exhibited in the computational results provided, which include the optimal resolution of some known instances—one of them with 50 clients. This instance is, as far as we know, the biggest one solved so far.

Recent Developments in Modeling and Solving the Split Delivery Vehicle Routing Problem

“…The column generation algorithm is able to obtain gaps (UB − LB)/U B, which are consistently smaller than those generated by Belenguer et al [4].…”

“…The proposed column generation algorithm is tested on 11 problem instances and compared with the cutting plane approach of Belenguer et al [4]. The column generation algorithm is able to obtain gaps (UB − LB)/U B, which are consistently smaller than those generated by Belenguer et al [4].…”

“…Belenguer et al [4] develop a set of 14 random problems with 50, 75, and 100 customers where the vehicle capacity is 160 and the six scenarios of Dror and Trudeau [11] are used to randomly generate a customer's demand (we denote this problem set by BMM Recently, Chen et al [7] developed 21 test problems that range in size from eight to 288 customers (we denote this problem set by CGW). Vehicle capacity is 100 units and customer demand is either 60 or 90 units.…”

A Scalable Heuristic for Pickup-and-Delivery of Splittable Loads and Its Application to Military Cargo-Plane Routing

“…They solved its LP relaxation using column generation and applied a rounding method to get a feasible IP solution. Belenguer et al (2000) studied the polyhedron of SDVRP and identified some facets of it. Using corresponding valid inequalities, they developed a cuttingplane algorithm to generate the lower bound of the optimal value.…”

A Scatter Search Algorithm for the Split Delivery Vehicle Routing Problem

“…The procedure was tested on three small instances up to 20 clients and varying client demands. The SDVRP was studied from a polyhedral point of view in [5]. Based on the partial description of the SDVRP polyhedron, the same authors implemented a branch and cut algorithm capable of solving some medium size instances up to 51 clients.…”