Robust Dynamic Programming for Inventory Control: Joint Supply and Demand Uncertainty

We study a multiperiod robust inventory problem with joint demand and supply uncertainty, in which the quantity received is an uncertain fraction of the quantity ordered, and an adversary selects demand and yield after the order is placed. Existing robust inventory methods gain tractability either from static reformulations, which discard the sequential order--adversary interaction and so protect against shocks that cannot co-occur, or from affine decision rules, which lose exactness once yield enters multiplicatively. We instead solve the exact sequential min--max dynamic program. The difficulty is that, unlike additive demand, the order quantity itself determines the decision maker's exposure to the worst-case yield, a decision-dependent uncertainty in which a larger order widens the range of adverse outcomes, so which realization is adverse depends jointly on the order and the inventory state. Preserving this decision-first timing lets us characterize the optimal policy exactly: the optimal order combines multiple linear inflation rules (order-up-to targets scaled by a factor that compensates for yield loss), selected according to the inventory state, and collapses to a single base-stock rule only when yield is deterministic. This characterization yields closed-form short-horizon policies, a fixed-point terminal valuation under which a myopic inflation rule remains optimal over a certified state region at every horizon, and worst-case representation bounds showing that the number of affine pieces can be exponential under yield uncertainty but grows only linearly when yield is deterministic. Exploiting the same geometry, we develop a decomposition algorithm in the spirit of robust dual dynamic programming whose stage subproblems are solved by exact algebraic oracles rather than linear programs. In experiments, the exact robust DP policy achieves lower cost than static and affine-rule robust benchmarks, with the largest gains when demand or supply is extreme.