Inventory Planning with Supply Yield Uncertainty: On the Optimality of Linear Policies

We study a single-period inventory planning problem under demand and supply yield uncertainty. We present new optimality conditions and explicit solutions for the associated single-period optimal inventory policy for two yield models: stochastically proportional and binomial yield. Some of the new exact results solve the case where demand and yield are normally distributed; and the case where demand is uniformly distributed and yield is stochastically proportional with uniformly distributed stochastic factor. For this random yield problem, the literature has proposed Linear Inflation Rules (LIRs) that inflate the classic order-up-to policy. However, the stochastically-optimal order policy generally is a non-linear function of the starting inventory level. We prove that LIRs are only stochastically optimal in the degenerate setting where either demand or supply yield factor is uncertain, but not both. Nevertheless, we show that LIRs work well and further investigate their performance under two robust formulations, for which we provide bounds. We show that LIRs are robustly optimal when the only information available includes the supports of the demand and the proportional yield distributions. Yet when the only information available includes the first two moments of these distributions, we show that the optimal distribution-free policy is again non-linear. Our models provide novel, explicit order rules that are easy to compute and hence useful in practice.