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

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

Interpretation¶

The critical ratio turns the two newsvendor costs into a target cumulative-demand probability $\beta=b/(b+h)$$h)$. It compares what one unit too few costs, the underage cost $b$, with what one unit too many costs, the overage cost $h$. For a continuous strictly increasing CDF, the optimal stock $y^*$ satisfies $F(y^*)=\beta$$\beta$. For discrete demand, the CDF can jump past this target.

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 $0.6$.

The ratio comes from the last unit ordered. It is unsold with probability $F(y)$, costing $h$, and sold with probability $1-F(y)$$F(y)$, saving $b$, so ordering it pays while $b(1-F(y))>hF(y)$$hF(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. The actual probability that all demand is met is $F(y^*)$, which equals the target $\beta$ in the continuous examples but need not equal it for discrete demand. For example, if demand is $0$ or $1$ with equal probabilities and $\beta=0.6$$0.6$, the optimal integer stock is $1$ and covers all demand with probability $1$.

Key Results¶

• For integer demand, increasing from $y$ to $y+1$ helps while $F(y)<\beta$$\beta$.
• For the continuous strictly increasing CDFs used in the examples, the optimum solves $F(y^*)=\beta$$\beta$.
• 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

Related Concepts¶

• Discrete Newsvendor Rule
• Continuous Newsvendor Rule

Sources¶

• OR Modelling · Week 6

Aliases: Critical fractile Symbols: beta

Last updated: 27 October 2026 11:52