Combining DC‐programming and steepest‐descent to solve the single‐vehicle inventory routing problem

The single‐vehicle cyclic inventory routing problem (SV‐CIRP) is concerned with a repeated distribution of a product from a single depot to a selected subset of retailers having stable demands. If a retailer is selected for replenishment, the supplier collects a retailer‐related fixed reward. The objective is to determine the subset of retailers to cyclically replenish, the quantities to be delivered to each, and to design the vehicle delivery routes so that the expected total distribution and inventory cost is minimized while the total collected rewards from the selected retailers is maximized. The resulting distribution plan must prevent stockouts from occurring at each retailer. In this paper, the underlying optimization problem for the SV‐CIRP is formulated as a mixed‐integer program with linear constraints and a nonlinear objective function. An optimization approach combining DC‐programming and Branch‐and‐Bound within a steepest descent hybrid algorithm, denoted by DCA‐SDHA, is developed for its solution. The approach is tested on some randomly generated problems and the obtained results are compared with results from the standard steepest descent hybrid algorithm (SDHA). These encouraging results show that the proposed approach is indeed computationally more effective and is worth being further investigated for the solution of medium to large instances.