New Approximation Guarantees for the Inventory Staggering Problem

Since its inception in the mid-1960s, the inventory staggering problem has been explored and exploited in a wide range of application domains, such as production planning, stock control systems, warehousing, and aerospace/defense logistics. However, even with a rich history of academic focus, cornerstone computational questions and related structural characterizations around inventory staggering remain unresolved, with our methodological toolbox remaining quite limited. The central contribution of this paper consists of devising a host of algorithmic techniques and analytical ideas—some being entirely novel and some leveraging well-studied concepts in combinatorics and number theory—for improving upon existing approximation guarantees for the inventory staggering problem. In particular, our work demonstrates that numerous structural properties enable designing polynomial-time approximation schemes, including polynomially bounded cycle lengths, constantly many distinct time intervals, so-called nested instances, and pairwise coprime settings. These findings offer substantial improvements over currently available constant-factor approximations and resolve several open questions in their respective contexts. In parallel, we develop new theory around a number of yet-uncharted questions, related to the sampling complexity of peak inventory estimation as well as to the plausibility of groupwise synchronization. Interestingly, we establish the global nature of inventory staggering, proving that there are n-item instances where, for every subset of roughly $\sqrt{n}$ items, no policy improves on the worst possible one by a factor greater than $1+ϵ$, whereas for the entire instance, there exists a policy that outperforms the worst-possible one by a factor of nearly 2, which is optimal.

Funding: N. Alon was supported in part by the National Science Foundation Division of Mathematical Sciences [Grant DMS-2553988]. D. Segev was supported by the Israel Science Foundation [Grant 1407/20].