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.

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This paper proposes lower and upper bounds for the split-delivery capacitated arc-routing problem (SDCARP), a variant of the capacitated arc-routing problem in which an edge can be serviced by several vehicles. Recent papers on related problems in node routing have shown that this policy can bring significant savings. It is also more realistic in applications such as urban refuse collection, where a vehicle can become full in the middle of a street segment. This work presents the first lower bound for the SDCARP, computed with a cutting plane algorithm and an evolutionary local search reinforced by a multistart procedure and a variable neighborhood descent. Tests on 126 instances show that the new metaheuristic outperforms on average a published memetic algorithm; achieves small deviations to the lower bound; and finds 44 optima, including 10 new ones.