Uniform Demand Distribution¶

Definition¶

The course's uniform demand model assumes $D$ is uniformly distributed on a known interval $[a,u]$ , with $0\le a<u$ $u$ . Its CDF inside the interval is $F(y)=(y-a)/(u-a)$ $a)$ , with values 0 below $a$ and 1 above $u$ . Thus the continuous newsvendor quantity is

y^*=a+(u-a)\beta.

Interpretation¶

Uniform demand says that every demand level in $[a,u]$ is equally likely: only the range is known. The probability that demand is at most $y$ then grows in proportion to $y-a$ $a$ , so the critical quantile is found by a linear interpolation between the two endpoints.

A flat uniform demand density between 80 and 120 with the area up to 104 shaded, equal to the critical ratio 0.6.

The density is flat at height $1/40$ between $a=80$ and $u=120$ $120$ , so the shaded rectangle from $80$ to $y$ has area $(y-80)/40$ $80) /40$ , which is $F(y)$ . Setting this equal to $\beta=0.6$ gives width $24$ and $y^*=104$ $104$ : the order lies $60\%$ of the way from $a$ to $u$ .

In $y^*=a+(u-a)\beta$ $a) β$ , the lower endpoint $a$ is demand that is certain to occur, $u-a$ is the uncertain part, and the critical ratio $\beta$ is the fraction of the uncertain part worth covering. With $\beta=\tfrac{1}{2}$ the order is the midpoint; with $\beta$ near $1$ it is close to $u$ , and with $\beta$ near $0$ close to $a$ . Since $0<\beta<1$ $1$ , the order always lies inside $[a,u]$ , so the continuous newsvendor rule needs no boundary case here.

A common slip is to compute $u\beta$ instead of $a+(u-a)\beta$ $a) β$ . For the meals this gives $120\cdot0.6=72$ $72$ , below the smallest possible demand; the two formulas agree only when $a=0$ $0$ .

Key Results¶

The distribution is used as an application of probability prerequisites; the course does not develop a general distribution theory.
For $\beta=1/2$ $1/2$ , the quantity is the midpoint.
The quantile lies within the demand interval.

Examples¶

For meal demand on $[80,120]$ and $\beta=0.6$ $0.6$ , order 104.
For that same interval, $F(100)=0.5$ and $F(104)=0.6$ $0.6$ .

Prerequisites¶

Continuous Newsvendor Rule
Uniform distribution

Normal Demand Distribution

Sources¶

OR Modelling · Week 6

Aliases: uniform demand model Symbols: beta

Last updated: 20 October 2026 03:02