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Distribution networks with periodically repeating events often hold greatpromise to exploit economies of scale. Joint replenishment problems are afundamental model in inventory management, manufacturing, and logistics thatcapture these effects. However, finding an efficient algorithm that optimallysolves these models, or showing that none may exist, has 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 withjust two products is at least as difficult as integer factorization. To thebest of the authors' knowledge, this is the first time that integerfactorization is used to explain the computational hardness of any kind ofoptimization problem. Under the assumption that Riemann's Hypothesis iscorrect, we can actually prove that the two-item joint replenishment problemwith possibly empty joint ordering points is NP-complete under randomizedreductions, which implies that not even quantum computers may be able to solveit efficiently. By relating the computational complexity of joint replenishmentto cryptography, prime decomposition, and other aspects of prime numbers, asimilar approach may help to establish (integer factorization) hardness ofadditional open periodic problems in supply chain management and beyond, whosesolution has eluded standard methods.

Integer factorization and Riemann's hypothesis: Why two-item joint replenishment is hard