Dynamic pricing inventory control under fixed cost and lost sales

This paper studies a periodic review pricing and inventory replenishment problem which encounters stochastic demands in multiple periods. In many inventory control problems, the unsatisfied demand is traditionally assumed to be backlogged but in this paper is assumed to be lost. In many practical problems, a consumer who could not buy what he/she wants in one store is not willing to wait until that store restocks it but tries to buy alternatives in other stores. Also, in this paper, the random variable for the demand function is assumed to be general, which means that any probability function for the random variable can be applied to our result. Cost terms consist of the holding cost by the leftover, the shortage cost by lost sales, and the strictly positive fixed ordering cost. The objective of this paper is to dynamically and simultaneously decide the optimal selling price and replenishment in each period by maximizing the expected profit over the finite selling horizon. We show that, under the general assumption on the random variable for the demand, the objective function is K$K$-concave, an (s,S)$(s\text{,}S)$ policy is optimal for the replenishment and the optimal price is determined based on the inventory level after the replenishment in each period.