Linking advertising and quantity decisions in the single-period inventory model

The single-period problem (SPP), also known as the newsboy or news-vendor problem, is to find the order quantity which maximizes the expected profit in a single period probabilistic demand framework. Interest in the SPP remains unabated and many extensions to it have been proposed in the last decade. An important aspect of the SPP is the effect of advertising on sales. In this paper, we extend the SPP to a case in which advertising leads to increases in sales. We assume that the mean demand is incrreasing and concave in advertising expenditure and address three cases of demand variance as a function of advertising expenditure: (1) demand has a constant variance, (2) demand has constant coefficient of variation, and (3) demand has an increasing coefficient of variation. We deal with two objectives: (a) maximizing the expected profit and (b) maximizing the probability of achieving a target profit. We provide closed or near-closed form solutions to the problem under uniform, exponential, and normal demand distributions. We illustrate the results using numerical examples and provide some insights into the effect of advertising.