A combined DC programming and Branch-and-Bound to solve the single-vehicle cyclic inventory routing problem with a fixed cycle time

The single-vehicle cyclic inventory routing problem (SV-CIRP) consists of distributing a given product, on a cyclical basis, from a single depot to a selected subset of retailers. A fixed retailer-related reward is collected if a retailer is selected for the cyclic replenishment. The objective is thus to select a subset of retailers to be replenished and to develop an optimal cyclic distribution plan that minimizes expected distribution and inventory costs and in the same time maximizes the sum of rewards collected from the selected retailers. This distribution plan must prevent stockout from occurring at any of the selected retailers during each cycle. The underlying optimization problem for this SV-CIRP can be formulated as a mixed-integer program with linear constraints and a nonlinear objective function. In this paper, a DC-programming approach combined with branch-and-bound is used to solve the SV-CIRP with a fixed cycle time. The approach is then tested on some cases given in the literature, and the obtained results are compared with the results from the usual branch-and-bound. These preliminary results show that DC-programming combined with branch-and-bound is more effective that the usual branch-and-bound based LP-relaxations. The approach is therefore worth being investigated further for the solution of the cyclic inventory routing problem (CIRP).