A newsvendor inventory model with an emergency order to supply a non-increasing fraction of shortage

This paper generalizes the newsvendor inventory model when the possibility of an emergency order to satisfy the excess of demand exists. In this situation, we assume that there are impatient customers who do not wait for the emergency order and other customers who are willing to wait to satisfy their demand. We consider that the fraction of shortage that is satisfied with delay through the emergency order is described by a function, which is non-increasing with respect to the amount of shortage. Our objective is to determine the optimal order quantity, which maximizes the relevant expected total profit for the period, when the demand follows a uniform probability distribution. As is well known, this distribution is usually used to denote the results of a demand whose values are unknown, except for the fact that such results belong to an interval. The uniqueness and existence of optimal decisions are proved, and a procedure to determine the optimal policy and the maximum expected profit is developed. Illustrative examples, which help us to understand the theoretical results, are also given.