Continuous Newsvendor Rule¶

Proposition¶

For continuous demand with finite mean and CDF $F$ , the newsvendor expected cost $G(y)$ of ordering $y$ , with underage cost $b$ and overage cost $h$ , satisfies

G'(y)=(b+h)F(y)-b.

If $F$ is continuous and strictly increasing through $\beta=b/(b+h)$ and that quantile lies in the allowed quantity domain, the unique optimum is

y^*=F^{-1}(\beta).

Interpretation¶

The rule says that the best order is a demand level, not the average demand: order the quantity $y^*$ that demand stays at or below with probability exactly $\beta$ , the critical ratio. Which level is chosen depends on the costs only through $\beta$ .

The derivative of the expected cost plotted against the order quantity, negative below 104 and positive above it, with the optimal order quantity 104 where it crosses zero.

For meal demand uniform on $[80,120]$ with $b=3.3$ and $h=2.2$ $2.2$ , the line $G'(y)=5.5F(y)-3.3$ rises from $-3.3$ at $y=80$ to $2.2$ at $y=120$ $120$ . Where it is negative, ordering a little more lowers the expected cost; where it is positive, ordering more raises it. It crosses zero at $y^*=104$ $104$ , where $F(104)=0.6=\beta$ $β$ .

Where $G'(y)=(b+h)F(y)-b$ is zero the CDF equals $\beta$ ; read $y^*=F^{-1}(\beta)$ off the right panel.

The formula for $G'(y)$ compares the two risks of one extra unit. It is left over with probability $F(y)$ , at cost $h$ , and is sold with probability $1-F(y)$ $F (y)$ , saving $b$ ; the net effect is $hF(y)-b(1-F(y))=(b+h)F(y)-b$ $b$ . Because $F$ never decreases, $G'$ never decreases either, so $G$ first falls and then rises and a zero of $G'$ is a global minimum. Strict increase of $F$ at $\beta$ makes the zero unique: if $F$ stays equal to $\beta$ on a whole interval, every point of that interval is optimal, so continuity alone is not enough. The quantile must also lie in the allowed range of $y$ ; a normal quantile can be negative, and then the best order is $0$ .

The discrete newsvendor rule makes the same comparison for whole units, with a difference $G(y+1)-G(y)$ instead of a derivative.

Key Results¶

For uniform demand on $[a,u]$ , the optimum is $a+(u-a)\beta$ $a) β$ .
For the course normal demand approximation $\mathcal{N}(\mu,\sigma^2)$ , it is $\max\{0,\mu+\sigma\Phi^{-1}(\beta)\}$ $σ Φ^{- 1} (β)}$ , where $\Phi$ is the standard normal CDF.
A continuous demand approximation may give a noninteger answer even when actual items are indivisible.

Examples¶

Meal demand uniform on $[80,120]$ with $\beta=0.6$ gives 104 meals.
Normal meal demand with mean 100 and standard deviation 20 gives about 105.07.

Prerequisites¶

Critical Ratio
Continuous distributions and differentiation

Sources¶

OR Modelling · Week 6

Symbols: beta, mu, phi, sigma

Last updated: 20 October 2026 03:02