ZLC Seminar Series: Optimal Inventory and Demand Management with Dynamic Pricing

by Sang Jo KIM and Youyi FENG 

We consider a manufacturer or a retailer who orders and sells product 1, 2 and 3 to maximize sales profit. In each period of a finite planning horizon, the seller replenishes the three products and controls the price of product 3 after the other products’ prices are fixed or exogenously determined. We assume that unmet demand is backlogged and unsold items are carried over to the next period. In addition, customer demand for each product is random and dependent on the prices of the three products. Products may be interrelated through certain economic properties, e.g., product 1 is a complement to product 2 while product 1 and 2 each are substitutes to product 3. 

We reveal that the optimal profit-to-go functions in the dynamic program are all L-natural concave, which implies supermodularity and concavity as well as diagonal dominance in the Hessian matrix if the functions are differentiable. This functional property is instrumental in characterizing the optimal decisions and performing comparative statics. Our analysis shows that elegant transformations of variables and the demand function can turn the value functions to be L-natural-concave although the original functions do not have such property. Through this procedure, we find the structure of the optimal policies and show the optimal ordering and pricing decisions are either monotone increasing or decreasing in the problem states such as inventory positions and market shares already determined by the prices of product 1 and 2. Further, it is shown that each optimal decision has bounded sensitivity to a change of any state, which, together with the monotone property, suggests intuitive and efficient decision making rules to cope with the changing operational environment. These findings also lay the foundation for further competitive analysis by showing a player’s best response to others’ inventory positions and prices.