The Continuous Joint Replenishment Problem is Strongly NP-Hard

Abstract

The Continuous Periodic Joint Replenishment Problem (CPJRP) has been one of the core and most studied problems in supply chain management for the last half a century. Nonetheless, despite the vast effort put into studying the problem, its complexity has eluded researchers for years. Although the CPJRP has one of the tighter constant approximation ratio of 1.02, a polynomial optimal solution to it was never found.Recently, the discrete version of this problem was finally proved to be N P-hard. In this paper, we extend this result and finaly prove that the CPJRP problem is also strongly N P-hard.

The Continuous-Time Joint Replenishment Problem: $ε$-Optimal Policies via Pairwise Alignment

“…Computational characterization of the joint replenishment problem. Complementing previously-mentioned intractability results due to Schulz and Telha (2022) and Tuisov and Yedidsion (2020), our approximation scheme for the joint replenishment problem resolves the long-standing open question regarding the computational complexity of this setting.…”

“…Finally, for the joint replenishment problem with arbitrarily-structured periodic policies, which is precisely the topic of our work, Schulz and Telha (2022) have recently extended their original findings to derive its polynomial-relatability to integer factorization. The latter result was further lifted to a full-blown strong NP-hardness proof by Tuisov and Yedidsion (2020). Given this state of affairs, yet another repeatedly-occurring open question is whether the above-mentioned evidence for intractability is best-possible or not, or put differently:…”

“…As previously mentioned, the work of Tuisov and Yedidsion (2020) classifies joint replenishment as being strongly NP-hard, thereby ruling out the potential existence of a fully polynomial-time approximation scheme (FPTAS). Consequently, due to the inevitable exponential dependency on the accuracy level ǫ, Theorem 1.3 provides the best possible form of such dependency, up to perhaps replacing 2 Õ(1/ǫ 3 ) by lower-order exponential terms.…”