Threshold results for the inventory cycle offsetting problem

In a multi-item inventory system, given the order cycle lengths and unit volumes of the items, the determination of the replenishment times (i.e. “cycle offsets”) of items, so as to minimize the resources needed to store the items, is known as the inventory cycle offsetting problem. In this paper we show that so long as the cycle times and inventory unit volumes satisfy certain mild constraints, there is an assignment of replenishment times so as to keep the resource requirements near the minimum that is theoretically possible for the time interval [0, CK 1 Q), where C1 > 1 is a certain costant, K is the number of items, and Q is the maximum of the cycle lengths. We further prove that there is a certain constant C2 > C1 > 1 so that with high probability, the resource requirements for the time interval [0, CK 2 Q] are 1 near the worst that they could be, at least when the cycle lengths and inventory volumes are chosen randomly from certain intervals with uniform distributions. Determining the best constants C1 and C2, given certain constraints, seems to be a very difficult problem, and is not something that we work out in this paper; nonetheless, at the end of the paper we present some computer experiments that suggest how C1 depends on certain problem parameters (our experiments tell us nothing about C2).