The Joint Replenishment Problem with General Joint Cost Structures

We consider inventory systems with several distinct items. Demands occur at constant, item specific rates. The items are interdependent because of jointly incurred fixed procurement costs: The joint cost structure reflects general economies of scale, merely assuming a monotonicity and concavity (submodularity) property. Under a power-of-two policy each item is replenished with constant reorder intervals which are power-of-two multiples of some fixed or variable base planning period. Our main results include a proof that, depending upon whether the base planning period is fixed or variable, the best among all power-of-two policies has an average cost which comes within either 6% or 2% of an easily computable lower bound for the minimum cost value. We also derive two efficient algorithms to compute an optimal power-of-two policy. The proposed algorithms generate as a by-product, a specific cost allocation of the joint cost structure to the individual items. With this specific allocation, the problem with separable costs is in fact equivalent to the original problem with nonseparable joint costs in the sense that the two problems share the same sets of optimal power-of-two policies with identical associated long-run average costs.