Integer Factorization: Why Two-Item Joint Replenishment Is Hard

Abstract

Joint replenishment problems constitute an important class of models in inventory management. They exhibit aspects of possible coordination among multiple products to save costs. Their computational complexity had been open even if there are just two products that need to be synced. In “Integer factorization: Why two-item joint replenishment is hard,” Schulz and Telha present a simple framework based on integer factorization to establish the computational hardness of two variants of the joint replenishment problem with two items. Whereas difficult to solve in practice and not believed to be solvable in polynomial time, integer factorization is not as difficult as NP-complete problems. The authors show that a similar technique can be used to show even the NP-completeness of one variant of the joint replenishment problem (again with just two items).

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.…”

“…Subsequently, by exploiting the extraordinary work of Zhang (2014) on bounded gaps between successive primes, Cohen-Hillel and Yedidsion (2018) proved that the fixed-base version is in fact strongly NP-hard. The latter result has been considerably streamlined by Schulz and Telha (2022), showing that NP-hardness arises even in the presence of only two commodities. 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 has been considerably streamlined by Schulz and Telha (2022), showing that NP-hardness arises even in the presence of only two commodities. 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).…”