TY - JOUR
T1 - Integer Factorization
T2 - Why Two-Item Joint Replenishment Is Hard
AU - Schulz, Andreas S.
AU - Telha, Claudio
N1 - Publisher Copyright:
© 2022 INFORMS.
PY - 2024/5/1
Y1 - 2024/5/1
N2 - Distribution networks with periodically repeating events often hold great promise to exploit economies of scale. Joint replenishment problems are fundamental in inventory management, manufacturing, and logistics and capture these effects. However, finding an efficient algorithm that optimally solves these models or showing that none may exist have long been open regardless of whether empty joint orders are possible or not. In either case, we show that finding optimal solutions to joint replenishment instances with just two items is at least as difficult as integer factorization. To the best of the authors’ knowledge, this is the first time integer factorization is used to explain the computational hardness of any optimization problem. We can even prove that the two-item joint replenishment problem with possibly empty joint-ordering points is NP-complete under randomized reductions. This implies that even quantum computers may not be able to solve it efficiently. By relating the computational complexity of joint replenishment to cryptography, prime decomposition, and other aspects of prime numbers, a similar approach may help to establish the (integer-factorization) hardness of additional periodic problems in supply chain management and beyond, whose computational complexity has not been resolved yet.
AB - Distribution networks with periodically repeating events often hold great promise to exploit economies of scale. Joint replenishment problems are fundamental in inventory management, manufacturing, and logistics and capture these effects. However, finding an efficient algorithm that optimally solves these models or showing that none may exist have long been open regardless of whether empty joint orders are possible or not. In either case, we show that finding optimal solutions to joint replenishment instances with just two items is at least as difficult as integer factorization. To the best of the authors’ knowledge, this is the first time integer factorization is used to explain the computational hardness of any optimization problem. We can even prove that the two-item joint replenishment problem with possibly empty joint-ordering points is NP-complete under randomized reductions. This implies that even quantum computers may not be able to solve it efficiently. By relating the computational complexity of joint replenishment to cryptography, prime decomposition, and other aspects of prime numbers, a similar approach may help to establish the (integer-factorization) hardness of additional periodic problems in supply chain management and beyond, whose computational complexity has not been resolved yet.
KW - computational complexity
KW - integer factorization
KW - joint replenishment
UR - http://www.scopus.com/inward/record.url?scp=85195815604&partnerID=8YFLogxK
U2 - 10.1287/opre.2022.2390
DO - 10.1287/opre.2022.2390
M3 - Article
AN - SCOPUS:85195815604
SN - 0030-364X
VL - 72
SP - 1192
EP - 1202
JO - Operations Research
JF - Operations Research
IS - 3
ER -