nieuwsvendor

The newsvendor problem, also known as the single-period inventory problem, is a fundamental model in operations research that addresses how many units to order for a product with uncertain demand and a limited selling period. It is commonly applied to perishable or fashion items, newspapers, or seasonal goods.

In the standard formulation, demand D is a random variable with a known distribution F, and the decision variable Q is the order quantity. Costs and prices include unit purchase cost c, selling price p, and salvage value s for any unsold inventory at the end of the period. Shortage costs are represented by the foregone profit (p − c) per unit of unmet demand, while overage costs are (c − s) per unsold unit. The objective is to maximize expected profit, which can be written as E[p·min(D, Q) + s·max(Q − D, 0) − c·Q].

The optimal order quantity Q* is determined by the critical fractile: choose Q* such that F(Q*) ≥ (p − c) / (p − s). Equivalently, order up to the demand quantile that achieves this fractile. If salvage value is zero (s = 0), the fractile simplifies to (p − c) / p.

Assumptions typically include a single selling period, known demand distribution, and no replenishment within the period. The model serves as a baseline for inventory decisions under uncertainty and has numerous extensions, including multi-period replenishment, backordering, price optimization, multiple products, and stochastic programming. Limitations include its simplified demand structure and lack of dynamic competition or lead times.