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.

供应商通常会规定最低订购量 (MOQ)，以确保每个特定项目的生产运行和运输数量在经济上是可行的。由于各种项目分担高昂的联合成本（例如海外集装箱运输的成本）并需要协调其补货政策，可能会产生额外的规模经济。买方需要为每个单独的项目找到联合订购间隔和补货政策，以在满足数量限制的同时最小化系统范围的订购和库存成本。我们专注于没有积压的持续需求的情况。给定一个固定的联合重新订购间隔，我们描述了每个项目的最佳库存订购策略，我们将其称为多补货 (MR) 库存订购策略，并推导出单位时间最优平均库存成本的封闭式表达式。与之前的文献相比，MR 库存订购策略不遵循零库存订购 (ZIO) 规则，并允许随时间推移不同大小的订单以最佳地适应 MOQ 限制。数值方法用于确定最佳联合重新排序间隔，后来扩展到考虑空补货的存在。一项广泛的计算研究表明 1) 与 MR 库存策略和空补货相关的总库存和设置成本降低；2）各种参数对成本和政策绩效的影响；3) 与在不同粒度级别（周、天、小时、分钟、