Underage Cost¶

Definition¶

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

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.

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¶

• Newsvendor Model
• Positive Part

Related Concepts¶

• Overage Cost
• Critical Ratio
• Shortage Cost

Sources¶

• OR Modelling · Week 6

Aliases: understocking cost Symbols: beta

Last updated: 20 October 2026 03:02