Optimal ordering, discounting, and pricing in the single-period problem

The single-period problem (SPP), also known as the newsboy or newsvendor problem, is to find the order quantity which maximizes the expected profit in a single-period probabilistic demand framework. Previous extensions to the SPP include, in separate models, the simultaneous determination of the optimal price and quantity when demand is price-dependent, and the determination of the optimal order quantity when progressive discounts with preset prices are used to sell excess inventory. In this paper, we extend the SPP to the case in which demand is price-dependent and multiple discounts with prices under the control of the newsvendor are used to sell excess inventory. First, we develop two algorithms for determining the optimal number of discounts under fixed discounting cost for a given order quantity and realization of demand. Then, we identify the optimal order quantity before any demand is realized. We also analyze the joint determination of the order quantity and initial price. We illustrate the models and provide some insights using numerical examples.