The Nonlinear Dynamics of Order-Up-To Inventory Systems with Lost Sales

We study the dynamic consequences of lost sales when there is insufficient inventory to satisfy demand. Demand is assumed to be independently and identically distributed and drawn from a normal distribution. We consider the industrially popular order-up-to policy with unit lead time is used to make replenishment orders. In this scenario, we obtain expressions for the order and inventory distributions, allowing us to quantify the Bullwhip and Net Stock Amplification ratios. We show that both these metrics are equivalent. We also determine the mean inventory levels held, and the achieved fill rate. We do this when the lost sales are fully observable, and when the lost sales are unobservable