Approximating the nonlinear newsvendor and single-item stochastic lot-sizing problems when data is given by an oracle

The single-item stochastic lot-sizing problem is to find an inventory replenishment policy in the presence of discrete stochastic demands under periodic review and finite time horizon. A closely related problem is the single-period newsvendor model. It is well known that the newsvendor problem admits a closed formula for the optimal order quantity whenever the revenue and salvage values are linear increasing functions and the procurement (ordering) cost is fixed plus linear. The optimal policy for the single-item lot-sizing model is also well known under similar assumptions. In this paper we show that the classical (single-period) newsvendor model with fixed plus linear ordering cost cannot be approximated to any degree of accuracy when either the demand distribution or the cost functions are given by an oracle. We provide a fully polynomial time approximation scheme for the nonlinear single-item stochastic lot-sizing problem, when demand distribution is given by an oracle, procurement costs are provided as nondecreasing oracles, holding/backlogging/disposal costs are linear, and lead time is positive. Similar results exist for the nonlinear newsvendor problem. These approximation schemes are designed by extending the technique of K-approximation sets and functions.