Looking Upstream: Optimal Policies for a Class of Capacitated Multi‐Stage Inventory Systems

We consider a multi‐stage inventory system with stochastic demand and processing capacity constraints at each stage, for both finite‐horizon and infinite‐horizon, discounted‐cost settings. For a class of such systems characterized by having the smallest capacity at the most downstream stage and system utilization above a certain threshold, we identify the structure of the optimal policy, which represents a novel variation of the order‐up‐to policy. We find the explicit functional form of the optimal order‐up‐to levels, and show that they depend (only) on upstream echelon inventories. We establish that, above the threshold utilization, this optimal policy achieves the decomposition of the multidimensional objective cost function for the system into a sum of single‐dimensional convex functions. This decomposition eliminates the curse of dimensionality and allows us to numerically solve the problem. We provide a fast algorithm to determine a (tight) upper bound on this threshold utilization for capacity‐constrained inventory problems with an arbitrary number of stages. We make use of this algorithm to quantify upper bounds on the threshold utilization for three‐, four‐, and five‐stage capacitated systems over a range of model parameters, and discuss insights that emerge.