New Approximation Guarantees for The Inventory Staggering Problem

TL;DR

This paper introduces new algorithmic techniques and analytical insights that significantly improve approximation guarantees for the inventory staggering problem, addressing longstanding computational challenges and structural questions.

Contribution

It develops novel algorithms and theoretical results that surpass existing approximation bounds and establish structural properties enabling polynomial-time approximation schemes.

Findings

Improved approximation guarantees for various structural instances.

Existence of polynomial-time approximation schemes under certain conditions.

Proof of the global nature of inventory staggering with near-optimal policies.

Abstract

Since its inception in the mid-60s, 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, we are still very much in the dark when it comes to cornerstone computational questions around inventory staggering and to related structural characterizations, with our methodological toolbox being severely under-stocked. The central contribution of this paper consists in 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 surpassing essentially all known approximation guarantees for the inventory staggering problem. In particular, our work demonstrates that numerous structural properties open the door for 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 outstanding 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 $1+\epsilon$, whereas for the entire instance, there exists a policy that outperforms the worst-possible one by a factor of nearly $2$, which is optimal.