David SIMCHI-LEVI

Coordinating Inventory Control and Pricing Strategies
with Random Demand and Fixed Ordering Cost

We analyze a single product, periodic review model in which pricing and production/inventory decisions are made simultaneously. Demands in different periods are random variables that are independent of each other and their distributions depend on the product price. Pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged. Ordering cost includes both a fixed cost and a variable cost proportional to the amount ordered. We consider both the finite and infinite horizon models. In the finite horizon model the objective is to find an inventory policy and a pricing strategy maximizing expected discounted profit over the finite horizon. In the infinite horizon the objective is to maximize expected discounted, or expected average profit. For the finite horizon case, we show, by employing the classical k-convexity concept, that an (s,S,p) policy is optimal when the demand functions are additive. In such a policy, the period inventory is managed
based on the celebrated (s,S) policy and price is determined based on the inventory position at the beginning of each period. For the model with more general demand functions, we show that an (s,S,p) policy is not necessarily optimal. We introduce a new concept, the symmetric k-convex functions, and apply it to provide a characterization of the optimal policy. Surprisingly, in the infinite horizon case, the concept of symmetric k-convex functions allows us to show that a stationary (s,S,p) policy is optimal for both discounted and average profit models even for general demand functions.