Newsvendor

Newsvendor, or the single-period inventory problem, is a classic optimization model in operations research used to determine the optimal order quantity for a perishable or single-period product when demand is uncertain and only one ordering opportunity exists. It is often illustrated by a newspaper seller who must decide how many papers to stock before demand is realized, knowing that unsold copies have limited resale value and unmet demand yields lost profit.

Key inputs are the order quantity Q, random demand D over the period, unit cost c, selling price p, and salvage value s for any leftover items. The underage cost Cu = p − c is the profit foregone from stocking one unit too few, while the overage cost Co = c − s is the loss from stocking one unit too many. A typical assumption is p ≥ c ≥ s, with Cu, Co ≥ 0.

The optimal policy sets Q* so that the probability of demand not exceeding Q* equals the ratio Cu/(Cu + Co). In notation, F(Q*) = Cu/(Cu + Co), where F is the distribution function of D. Thus Q* is a (possibly fractional) p-quantile of D; with discrete demand one chooses the smallest Q with F(Q) ≥ Cu/(Cu + Co).

Extensions include multi-period or approximations for backorders or lost sales, uncertainty in parameters via Bayesian updating, and numerical methods for complex cost structures. The model provides a baseline for inventory decisions under single-period uncertainty and is widely taught in operations management.