Optimality and Optimization of a Joint Pricing and Inventory-Control Policy for a Periodic-Review System

This paper considers a concurrent pricing and inventory control problem in a stochastic and periodic review setting. Demand in a period is random and depends on the posted price. Other than holding and backlogging costs, the system incurs stock-replenishment costs that consist of both variable and fixed components. At the beginning of each period, a simultaneous decision has to be made on stock replenishment and the selling price for the period. The objective is to find an optimal pricing and stock replenishment policy that maximizes the long-run average profit. We show that the optimal decision in each period is to follow a so-called (s, S, p) policy. An (s, S, p) policy works as follows. Whenever the inventory level falls to s or below, an order is placed to bring it up to S, and when the inventory level is at x, s < x ≤ S, no order is issued and the product is sold at price p(x), where p(x) depends on the inventory level x. We develop an efficient algorithm to find an optimal (s, S, p) policy.