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

In this paper, we provide an alternative approach for proving the preservation of concavity togetherwith submodularity, and apply it to finite-horizon non-stationary joint inventory-pricing modelswith general demands. The approach characterizes the optimal price as a function of the inven-tory level. Further, it employs the Cauchy-Schwarz and arithmetic-geometric mean inequalitiesto establish a relation between the one-period profit and the profit-to-go function in a dynamicprogramming setting. With this relation, we demonstrate that the one-dimensional concavity ofthe 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 forthe one-period profit and the profit-to-go function in conventional approaches. As a result, wederive the optimality condition for a base-stock, list-price (BSLP) policy for joint inventory-pricingoptimization models with general form demand and profit functions. With examples, we extendthe optimality of a BSLP policy to cases with non-concave revenue functions in mean demand. Wealso 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 expectedsales is greater than the percentage change in the slope of the expected one-period profit. Theconcavity preservation conditions for the additive, generalized additive, and location-scale demandmodels in the literature are unified under this framework. We also obtain the conditions underwhich a BSLP policy is optimal for the logarithmic and exponential form demand models.