Efficient algorithms for the joint replenishment problem with minimum order quantities

Suppliers often impose a minimum order quantity (MOQ) to ensure that production runs and shipping quantities of each specific item are economically viable. Additional economies of scale may arise as various items share high joint costs (e.g. the cost of an overseas container shipment) and require coordination of their replenishment policies. The buyer needs to find the joint ordering interval and replenishment policies for each of the individual items to minimize the system-wide ordering and inventory costs while satisfying the quantity restrictions. We focus on the case of constant demand without backlogging. Given a fixed joint reorder interval, we characterize the optimal inventory ordering strategy for each item, which we refer to as Multi-Replenishment (MR) inventory ordering policy, and derive a closed-form expression for the optimal average inventory costs per unit of time. In contrast to the previous literature, the MR inventory ordering policy does not follow the zero-inventory-ordering (ZIO) rule and allows for orders of different sizes over time to optimally accommodate the MOQ restriction. A numerical approach is used to determine the optimal joint reorder interval, and is later extended to account for the presence of empty replenishments. An extensive computational study shows 1) the total inventory and setup cost reduction associated with the MR inventory policies and empty replenishments; 2) the impact of various parameters on cost and policy performance; and 3) the loss associated with discretizing time at different levels of granularity (weeks, days, hours, minutes, seconds) to accommodate restrictions in the timing of item deliveries.