Stochastic Single Period Inventory Decisions: Based on Full Quadratic Cost Functions

This study addresses a general class of decision situations whose solutions are directly applicable to inventory acquisitions and or disposals. Although optimal solutions are well known when subsequent costs are linear to the amount of surplus or shortage, the perhaps more realistic case of non-linear costs has not been extensively studied. The results of this study suggest the optimal solutions i.e., acquisition quantity or supply for both conditions of risk and uncertainty about demand when the associated cost function is non- linear, i.e., quadratic. For conditions of risk optimal solutions are found which will yield minimum expected costs for the two-piece cost function where surplus and shortage costs are quadratic. This is done for both discrete and continuous demand variable. When future need for the item is unknown and only the maximum value can be estimated, optimal solutions are obtained for goals of minimaxing cost, minimaxing regret, and the Laplace criteria using a uniform probability distribution. It is shown that these different approaches to determining acquisition quantities under conditions of uncertainty lead, for this general class of decision problems, to the same optimal result. Hopefully, this information will aid in the decision process while making affordability assessments of new acquisitions. Affordability assessments, Single period inventory, Newsboy problem, Quadratic cost functions.