PERT (Program Evaluation and Review Technique) estimates activity durations using three time estimates: optimistic, most likely, and pessimistic. It produces an expected time and variance for each activity, enabling probabilistic project scheduling. This page explains the formulas with diagrams and numerical examples.

1. PERT Time Estimates

• Optimistic time (a) — best-case scenario.
• Most likely time (m) — mode of the distribution.
• Pessimistic time (b) — worst-case scenario.

PERT assumes a Beta distribution for activity durations.

2. PERT Expected Time Formula

\[ t_e = \frac{a + 4m + b}{6} \]

This weighted average emphasizes the most likely time.

3. PERT Variance Formula

\[ \sigma^2 = \left(\frac{b – a}{6}\right)^2 \]

Variance depends only on the spread between optimistic and pessimistic times.

4. Numerical Example

Activity estimates:

• a = 2 days
• m = 5 days
• b = 14 days

Expected time

\[ t_e = \frac{2 + 4(5) + 14}{6} = \frac{36}{6} = 6 \text{ days} \]

Variance

\[ \sigma^2 = \left(\frac{14 – 2}{6}\right)^2 = 4 \]

Standard deviation

\[ \sigma = 2 \]

5. PERT Network Diagram Example

   (Start)
      |
   [A: 6 days]
      |
   [B: 4 days]----\
      |            \
   [C: 7 days]      >--> (Finish)
      |            /
   [D: 5 days]----/

Each activity uses its PERT expected time for critical path analysis.

6. Project Duration Mean & Variance

For a critical path with activities A, C, D:

• A: \(t_e=6\), \(\sigma^2=4\)
• C: \(t_e=7\), \(\sigma^2=1\)
• D: \(t_e=5\), \(\sigma^2=0.25\)

Total expected project duration

\[ T = 6 + 7 + 5 = 18 \text{ days} \]

Total variance

\[ \sigma_T^2 = 4 + 1 + 0.25 = 5.25 \]

Standard deviation

\[ \sigma_T = \sqrt{5.25} = 2.29 \]

7. Using PERT for Probability of Completion

To find the probability of finishing by a deadline \(D\):

\[ Z = \frac{D – T}{\sigma_T} \]

Then use the standard normal table.

Example

Deadline = 20 days \[ Z = \frac{20 – 18}{2.29} = 0.87 \] \[ P = \Phi(0.87) = 0.807 \approx 80.7\% \]