Joint Inventory-Pricing Optimization with General Demands: An Alternative Approach for Concavity Preservation

In this study, we provide an alternative approach for proving the preservation of concavity together with submodularity, and apply it to finite-horizon non-stationary joint inventory-pricing models with general demands. The approach characterizes the optimal price as a function of the inventory level. Further, it employs the Cauchy–Schwarz and arithmetic-geometric mean inequalities to establish a relation between the one-period profit and the profit-to-go function in a dynamic programming setting. With this relation, we demonstrate that the one-dimensional concavity of the price-optimized profit function is preserved as a whole, instead of separately determining the (two-dimensional) joint concavities in price (or mean demand/risk level) and inventory level for the one-period profit and the profit-to-go function in conventional approaches. As a result, we derive the optimality condition for a base-stock, list-price (BSLP) policy for joint inventory-pricing optimization models with general form demand and profit functions. With examples, we extend the optimality of a BSLP policy to cases with non-concave revenue functions in mean demand. We also propose the notion of price elasticity of the slope (PES) and articulate the condition as that in response to a price change of the commodity, the percentage change in the slope of the expected sales is greater than the percentage change in the slope of the expected one-period profit. The concavity preservation conditions for the additive, generalized additive, and location-scale demand models in the literature are unified under this framework. We also obtain the conditions under which a BSLP policy is optimal for the logarithmic and exponential form demand models. © 2019 Production and Operations Management Society