Sourcing with Random Yields: Operational Hedging and Optimal Policies

What is the optimal ordering policy in a single-sourcing setting with an unreliable supplier? Specifically, given a net inventory position xt, what order quantity q(xt) should be placed, knowing that the actual quantity received in acceptable condition will be q(xt)Ut, where the yield factor Ut is uncertain? The structure of the optimal policy is known to be a threshold policy: order nothing if xt >St for some threshold St. However, the form of the optimal order quantity below this threshold remains poorly understood. As a result, prior literature has focused on heuristic approaches, notably Linear Inflation Rules (LIRs), which scale the classical order-up-to policy and have been shown to perform well in practice [Huh and Nagarajan 2010]. We present new analytical results, optimality bounds, and explicit solutions for the single-period, two- period, and infinite-horizon versions of the problem. Two new heuristics are introduced: 2-LIR, a natural extension of Linear Inflation Rules, and a robust policy designed to minimize the optimality gap when only the first two moments of the demand and yield distributions are known. We demonstrate that the optimal dynamic policy is not myopic by characterizing conditions under which operational hedging—via advance ordering or order splitting—is optimal in the two-period setting. For the infinite-horizon, average- cost problem, we establish the existence of an optimal policy by deriving a necessary and sufficient stability condition for (r,Q) policies. When demand and yield are exponentially distributed, optimizing the (r,Q) policy yields a finite upper bound on the optimal average cost.