Critical Ratio¶

Definition¶

For newsvendor selling price $p$ , unit cost $c$ and salvage value $s$ , with overage cost $h=c-s>0$ and underage cost $b=p-c>0$ $0$ , the critical ratio is

\beta=\frac{b}{b+h}=\frac{p-c}{p-s},\qquad 0<\beta<1.

It is the cumulative-demand probability used to select the optimal stock quantile.

Interpretation¶

The critical ratio turns the two newsvendor costs into one target probability: the optimal stock covers demand with probability $\beta=b/(b+h)$ $h)$ . It compares what one unit too few costs, the underage cost $b$ , with what one unit too many costs, the overage cost $h$ .

A uniform CDF rising from 80 to 120, with the critical ratio 0.6 on the vertical axis read across to the optimal order quantity 104.

The CDF $F(y)$ gives the probability that demand is at most $y$ . For meals, $\beta=3.3/(3.3+2.2)=0.6$ $0.6$ ; reading across from height $0.6$ to the CDF and then down gives $y^*=104$ $104$ , the quantity that covers demand with probability $0.6$ .

Change $p$ , $c$ and $s$ : $\beta=b/(b+h)$ moves, and the optimal stock is the demand quantile where the CDF reaches $\beta$ .

The ratio comes from the last unit ordered. It is unsold with probability $F(y)$ , costing $h$ , and sold with probability $1-F(y)$ $F (y)$ , saving $b$ , so ordering it pays while $b(1-F(y))>hF(y)$ $h F (y)$ , that is, while $F(y)<b/(b+h)$ $h)$ . A large underage cost pushes $\beta$ towards $1$ and the stock up; a large overage cost pushes $\beta$ towards $0$ . The second form follows from $b+h=p-s$ and shows the prices directly: a higher selling price or a higher salvage value raises $\beta$ , and a higher unit cost lowers it. How far the price can move before the order changes is the subject of newsvendor price sensitivity.

A common slip is to read $\beta$ as the fraction of demand that is served on average. It is the probability that all demand is met, not a fraction of demand.

Key Results¶

For integer demand, increasing from $y$ to $y+1$ helps while $F(y)<\beta$ $β$ .
For the continuous strictly increasing CDFs used in the examples, the optimum solves $F(y^*)=\beta$ $β$ .
The ratio is not the expected fraction of demand served.

Examples¶

Newspapers with $b=1.5$ and $h=2$ have $\beta=3/7$ $3/7$ .
Meals with $b=3.3$ and $h=2.2$ have $\beta=0.6$ $0.6$ .

Prerequisites¶

Overage Cost
Underage Cost
Cumulative distribution functions

Sources¶

OR Modelling · Week 6

Aliases: Critical fractile Symbols: beta

Last updated: 20 October 2026 03:02