A stochastic inventory problem with piecewise quadratic costs

In this paper we consider a periodic review inventory problem with stochastic demand. The deviations from ideal order and inventory levels are penalized using a quadratic cost structure which includes a “cost-free” interval. If the order quantity or the inventory level is within the desired “cost-free” interval, no cost is incurred. It is assumed that the order cost is a linear function of the order quantity. We derive an optimal order-up-to policy which is similar to the classical (s,S) policy except that the quantity S is now a nonlinear function of the entering inventory level.