Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Finite Horizon Case

TLDR: This work shows that when the demand model is additive, the profit-to-go functions arek-concave and hence an ( s, S, p) policy is optimal and introduces a new concept, the symmetrick-con cave functions, and applies it to provide a characterization of the optimal policy.

Abstract: We analyze a finite horizon, 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. The objective is to find an inventory policy and a pricing strategy maximizing expected profit over the finite horizon. We show that when the demand model is additive, the profit-to-go functions arek-concave and hence an ( s, S, p) policy is optimal. In such a policy, the period inventory is managed based on the classical ( s, S) policy and price is determined based on the inventory position at the beginning of each period. For more general demand functions, i.e., multiplicative plus additive functions, we demonstrate that the profit-to-go function is not necessarilyk-concave and an ( s, S, p) policy is not necessarily optimal. We introduce a new concept, the symmetrick-concave functions, and apply it to provide a characterization of the optimal policy.