Underage Cost¶

Definition¶

The underage cost in the newsvendor model is the foregone margin on one unit of demand that cannot be served:

b=p-c>0,

where $p$ is selling price and $c$ is unit cost. Expected underage cost is $b\mathbb{E}[(D-y)^+]$ $y)^{+}]$ .

Interpretation¶

The underage cost is what one unit of unmet demand costs: the sale is lost, and with it the margin $p-c$ that the unit would have earned. It is the price of ordering one unit too few.

A vertical money scale with the selling price 5, purchase cost 3.50 and salvage value 1.50; the gaps between them are the underage cost 1.50 and the overage cost 2.

On the money scale for newspapers, the upper gap between the purchase cost $c=3.50$ and the selling price $p=5$ is $b=1.50$ $1.50$ ; the lower gap, down to the salvage value $s=1.50$ $1.50$ , is the overage cost $h=2$ $2$ . Here $b<h$ $h$ : running short is the cheaper mistake, and the critical ratio $\beta=1.5/3.5=3/7$ is below $\tfrac{1}{2}$ .

The loss is the margin, not the full price, because the purchase cost $c$ of a unit never bought is saved; using $p$ as the underage cost is a common slip. The positive part $(D-y)^+$ counts the units of unmet demand, and $b$ is charged once for each of them, unlike the EOQ shortage cost, which is charged per unit of waiting time.

Key Results¶

Only demand above the available quantity contributes to underage loss.
Increasing selling price increases underage cost when unit cost is fixed.
The course assumes no additional per-unit stockout penalty beyond the lost margin.

Examples¶

The illustrated newspaper data give overage cost $h=2$ and underage cost $b=1.5$ $1.5$ , hence critical ratio $\beta=1.5/3.5=3/7$ $3/7$ . These are single-period per-unit losses, rather than EOQ cost rates.

Newspapers sold for 5 and purchased for 3.50 have $b=1.50$ $1.50$ .
Meals sold for 5.50 and costing 2.20 have $b=3.30$ $3.30$ .

Prerequisites¶

Sources¶

OR Modelling · Week 6

Aliases: understocking cost Symbols: beta

Last updated: 20 October 2026 03:02