Derivation of Weibull Distribution Parameters When the Minimum, Most Likely Value, and a Percentile are Known

A key part of using CB Pro is creating assumptions from qualitative information such as expert opinion. Expert opinion is sometimes expressed by assigning low, most likely, and high values to a random variable with unknown distribution parameters. The Weibull distribution is useful because it can be shaped to match low, most likely, and high values, yet still allow some probability of exceeding the high value. The Weibull distribution has a long tail, which the analyst can use to specify a distribution with, say, a 10% chance of exceeding the high value.

Recall that the Weibull distribution is given by:

(1)

where L is the location, a is the scale, and b is the shape. The cumulative distribution is

(2)

To fit the expert opinion, Weibull distribution parameters are specified in the following manner:

1. The output deviates must be greater than the expert opinion for low value,
2. The mode (most probable value), x_m, must be equal to the expert opinion most likely value,
3. The high value, x_b, must corresponded to some particular percentile point (e. g., 95%, or 90%)

For the Weibull distribution, output deviates are always greater than the location parameter, L, which allows the analyst to simply set L equal to the expert opinion for low value.

To find an expression for mode, x_m, of the Weibull distribution, differentiate equation (1) and set it equal to zero to obtain:

(3)

The high, x_b, is easily determined using the cumulative distribution (2),

(4)

where F(x_b) is the desired percentile point.

The scale parameter (a) can be eliminated to obtain an equation for the shape parameter (b);

(5)

because x_b, x_m, L, and F(x_b) are specified.

This equation is easily solved using the Goal Seek or Solver utilities available in Excel. Once b is known, a can be found from equation (3) or (4).

This solution can be checked using CB Pro, of course. Specify a Weibull distribution with the calculated parameters (location, scale, and shape) in an assumption cell and generate a forecast that equals the assumption. Forecast histogram and statistics verify that the Weibull distribution matches the desired shape.