An Inventory Routing Policy Under Replenishment Lead Time

The inventory routing problem (IRP) arises in vendor-managed inventory systems, which is a combination of vehicle routing and inventory management. Differing from the traditional IRPs in which the time consumption in transportation is often ignored, in this paper, we take the time consumption into consideration, which brings the lead-time to replenishment. In this situation, the lead-time for replenishment depends on the routing decisions made by the vendor. Consequently, IRPs become more interesting because decisions on the replenishment quantity and routing depend more tightly on each other. The general case of the IRP with replenishment lead-time (i.e., one vendor and N retailers) is modeled mathematically. To solve the problem, first a simple case (i.e., one vendor and two retailers) is analyzed to obtain the theoretical optimal policy. By proving the K-convexity of the objective function, we confirm that the structure of the optimal replenishment and routing policy is of switching curve type. In this policy, the state space, which is composed of the inventory positions of the two retailers, is divided into several domains; and for inventory positions in each domain, there exists an optimal order-up-to level. This structure reveals, for managerial insight, that when lead-time is considered, the current routing decision is not independent of the previous one. Second, since the optimal policy is difficult to realize in practice, a myopic policy that is easier to implement is proposed and numerical experiments are conducted to examine the near-optimal performance of the myopic policy. Finally, the myopic policy is extended to a realistic-size IRP with replenishment lead-time (i.e., the IRP for the case of one vendor and multiple retailers) and a numerical example is provided to indicate the feasibility of the policy.