Revisiting the newsboy problem–optimization with a little help from the airline industry

In a typical inventory planning problem with a life cycle of only one planning period we incur the cost of production per unit produced, profit per unit sold, loss per unit not sold, and lost revenue per unit ordered but not matched due to the lack of availability. The goal is to find the inventory level that maximizes the expected net profit. Textbooks often use the newsboy problem to illustrate the inventory management paradigm. The derivation of the formulas for the optimal level is usually done on an ad hoc basis, by dull and rote mathematical manipulations, for each modification of the simple basic model. The only purpose of this note is to give a simple transparent proof of the fact that quite surprisingly the lost revenue can be combined with the profit by reducing the general problem to a well known simplified case with no lost revenue. The reduction uses an airline analogy and thus, with some tweaking, it places the proof into a classical revenue management paradigm. We also provide an alternative derivation of the optimal solution for the discrete case which integrates the problem into a much broader class of optimization problems. Keywords– newsboy problem, inventory and revenue management, optimization