Downside risk-aversion analysis for a single-stage newsvendor problem

We extend the classical newsvendor problem by introducing a downside risk constraint from the perspective of inventory control. At the beginning of a replenishment period the newsvendor will place an order, then he will review the inventory level at the end of the period. If the inventory level is positive then he will bear the holding cost and if the inventory level is negative then he will bear the backorder cost. The optimal order quantity has a simple form. We analyze the form of the optimal order quantity when we restrict that the probability that the cost level is larger than or equal to a fixed cost constant is less than a fixed value of probability. At last, we analyze the case that the fixed cost constant is equal to the expected cost.