Discrete Newsvendor Rule¶

Proposition¶

Let demand be a nonnegative integer-valued random variable with CDF $F$ , finite mean, and newsvendor underage cost $b>0$ and overage cost $h>0$ $0$ . Over nonnegative integer order quantities, a smallest optimal quantity is

\begin{gathered} y^*=\min\{y\in\mathbb{Z}_{\ge0}\colon F(y)\ge\beta\},\\ \beta=\frac{b}{b+h}. \end{gathered}

For the expected cost $G(y)$ of ordering $y$ , the marginal difference is

G(y+1)-G(y)=(b+h)F(y)-b.

Interpretation¶

With demand in whole units, the order is raised one unit at a time. Going from $y$ to $y+1$ changes the expected cost by $(b+h)F(y)-b$ $b$ , which is negative exactly when $F(y)<\beta$ $β$ . So keep adding units while $F(y)<\beta$ $β$ , and stop at the first $y$ with $F(y)\ge\beta$ $β$ , where $\beta$ is the critical ratio.

A staircase CDF of discrete demand with a horizontal line at the critical ratio 0.75; the CDF first reaches it at 10, the optimal order quantity.

The CDF of integer demand is a staircase that jumps at the possible demand values. The level $\beta=0.75$ lies between $F(9)=0.70$ and $F(10)=0.85$ $0.85$ . Going from $9$ to $10$ still lowers the expected cost, because $0.70<0.75$ $0.75$ , while going from $10$ to $11$ raises it, because $0.85>0.75$ $0.75$ ; hence $y^*=10$ $10$ .

With integer demand, increase $y$ while $F(y)<\beta$ $β$ : the first $y$ with $F(y)\ge\beta$ is optimal.

The difference formula compares the two outcomes for unit $y+1$ $1$ . It is left over when $D\le y$ $y$ , with probability $F(y)$ and cost $h$ , and it is sold when $D\ge y+1$ $1$ , with probability $1-F(y)$ and saving $b$ . Because $F$ never decreases, the differences never decrease either, so the first stop is a global optimum and not only a local one. If $F(y)=\beta$ exactly, the step from $y$ to $y+1$ costs nothing and both quantities are optimal; the rule returns the smaller one. Usually no integer $y$ has $F(y)=\beta$ $β$ , which is why the rule uses $\ge$ instead of solving an equation as the continuous rule does.

A common slip is to stop at the last $y$ with $F(y)<\beta$ $β$ , here $9$ . That quantity still gains from one more unit.

Key Results¶

If $F(y)=\beta$ $β$ , quantities $y$ and $y+1$ have equal expected cost.
All integer optimal quantities satisfy $F(y-1)\le\beta\le F(y)$ $F (y)$ , taking $F(-1)=0$ $0$ .
Compute the expected cost with the actual discrete probabilities, not only the critical ratio.

Examples¶

The drawn CDF steps other than $F(9)$ and $F(10)$ are schematic; they do not specify additional probabilities for the dress example.

For the newspaper distribution, $F(16)=0.25<3/7\le F(17)=0.50$ $0.50$ , so the smallest optimum is 17; expected cost is 1.975.
The dress example has $\beta=0.75$ $0.75$ , optimal quantity 10, and expected cost 138.

Prerequisites¶

Sources¶

OR Modelling · Week 6

Symbols: beta, element of, integers

Last updated: 20 October 2026 03:02