Coordinating order cycles under joint replenishments of multi-item inventories

A recent work obtained closed-form expressions for the problem of optimal partitioning multi-item inventories into a given number of groups, where each group has a common order cycle for all the items in the group, and a Pareto function characterizes the distributions by value of the inventory. The optimal order points so obtained for the groups may have no synchronization among themselves at all. This note extends the closed-form analysis to the requirement that the order cycles be coordinated so that each cycle is an integral multiple of the shortest order cycle. The extent of suboptimality which thus follows is investigated. Several alternatives to achieve integrality have been considered and compared through an illustrative example within the framework of the Pareto function.