Finite-horizon joint inventory-pricing optimization with non-concave demand and lost sales

In this paper, we study a periodic-review joint inventory and pricing optimization problem in a stochastic price-sensitive demand framework with lost sales. We contribute to the existing literature from three aspects: generalizing the form of demand functions, developing the technique of proving concavity preservation, and extending the scope of optimality conditions. Specifically, we set the demand in our model as a general function. In order to obtain the optimality condition under general demands, we first introduce the concept of Price Elasticity of the Slope (PES) and transform the original PES condition between sales and profit in literature into a PES condition between sales and demand. Then, we obtain the lower bound of the PES difference between sales and profit by utilizing our transformed condition. With the non-negativity of the lower bound, we obtain two groups of easy-to-verify conditions which consist of the mean demand and the probability distribution of the demand. Under specific demand forms, the obtained conditions are reduced to constraints on the distribution of price free uncertainty and the deterministic functions of price, respectively. Compared with the existing literature, our study therefore extends the scope of optimality conditions and includes non-concave demand functions when adapting a base-stock list-price (BSLP) policy in a joint inventory-pricing problem. Our conditions ensure that more types of products (e.g., the products of which the price elasticity of demand is constant) meet the optimality of the BSLP strategy, and more retailers can make pricing and replenishment decisions based on this strategy.