An Exact Algorithm for the Single-vehicle Cyclic Inventory Routing Problem

The single-vehicle cyclic inventory routing problem (SV − CIRP ) consists of a repetitive distribution of a product from a single depot to a selected subset of customers. For each customer that is selected for replenishments, the supplier collects a corresponding fixed reward. The objective is to determine the subset of customers to replenish, the quantity of the product to be delivered to each, and to design the vehicle route so that the resulting profit (difference between the total reward and the total logistical cost) is maximized while preventing stockouts at each of the selected customers. In this paper, the SV − CIRP is formulated as a mixed-integer program with a nonlinear objective function. After an efficient analysis of the problem, an exact algorithm for its solution is proposed. This exact algorithm requires only solutions of linear mixed-integer programs. Values of an insertion-based heuristic for this problem are compared to the optimal values obtained for a set of some test problems. In general the gap may get as large as 25%, which justifies the effort to continue exploring and developing exact and approximation algorithms for the SV −CIRP .