(Modified) PERT distribution

The United States navy developed in the 1950’s a program evaluation research task (PERT). It was designed to analyze the duration of a project and the tasks within the project. Each task in the project is given the following properties:

  • Name of the task
  • Predecessor, the list of tasks that have to be completed before the task can start.
  • The amount of time it will take for a task to be finished.

The duration of a task is often not fixed. It can vary between a minimum and maximum. The PERT distribution was created to give a good estimate of what the probability is of the duration of a task. It uses the same three parameters as the Triangular distribution, namely, the minimum (\text{min}), the most likely (\text{mode}) and the maximum (\text{max}). The probability density function (PDF) is given by

    \begin{equation*} f(x) = \frac{1}{B(\alpha_1,\alpha_2)}\frac{(x - \text{min})^{\alpha_1 - 1} (\text{max} - x)^{\alpha_2 - 1}}{(\text{max} - \text{min})^{\alpha_1 + \alpha_2 - 1}} \end{equation*}

where

    \begin{equation*} \alpha_1 = 6 \left( \frac{\mu - \text{min}}{\text{max} - \text{min}} \right), \quad \alpha_2 = 6 \left( \frac{\text{max} - \mu}{\text{max} - \text{min}} \right), \quad \end{equation*}

with

    \begin{equation*} \mu = \frac{\text{min} + 4\text{mode} + \text{max}}{6} \end{equation*}

being the mean.

An additional shape parameter \gamma might be added, in which case we are dealing with the modified PERT (MPERT) distribution. The parameter influences the peakness of the distribution. The only difference with PERT is the definition of \alpha_1 and \alpha_2. These are namely defined as:

    \begin{equation*} \alpha_1 = 1 + \gamma \left( \frac{\mu - \text{min}}{\text{max} - \text{min}} \right), \quad \alpha_2 = 1 + \gamma \left( \frac{\text{max} - \mu}{\text{max} - \text{min}} \right), \quad \end{equation*}

I created some Matlab code which includes functions for the PDF, the (inverse) cumulative distribution function (CDF). The inverse CDF is important for when you want to generate random numbers. The code can be download from here.

Finally, below is an interactive example of the PDF of PERT (blue) and MPERT (red) distribution.