## Programme Evaluation and Review Technique (PERT)

In most of the projects the activity times are not known with certainty and they may be assumed as random variable. In such cases where activity times are not known with certainty PERT can be used for planning, scheduling and controlling the project. It was developed in 1950 by US Navy to control large defense projects.

The Programme Evaluation and Review Technique (PERT) makes the following assumptions:

- Activity times are statistically independent and usually associated with beta distribution.
- There are enough activities involved in the network that sum of activity times based on their means and variance will be normally distributed.
- In PERT, for each activity three time estimates can be obtained.
- Optimistic time estimate (t
_{o}): it is the minimum time in which an activity can be completed under favourable conditions. - Most Likely time estimate (t
_{m}): it is the time required to complete an activity under normal conditions or it is the time required to complete an activity most often if it is repeated number of times. - Pessimistic time estimate (t
_{p}): it is the maximum time in which an activity can be completed under unfavourable conditions.

- Optimistic time estimate (t

After determining three time estimates of an activity its expected time can be computed on the basis of beta distribution.

Expected (Mean) time = (t_{o }+ 4 t_{m }+ t_{p}) / 6

Where t_{o }= optimistic time estimate

t_{m} = Most Likely time estimate

t_{p} = Pessimistic time estimate

t_{e }= expected time

Standard Deviation s^{2} = (tp – to/6)^{2}

Steps to apply Programme Evaluation and Review Technique (PERT)

- Identify the activities to complete the project and their predecessor requirement.
- Calculate the expected time and standard variation for each activity.
- Construct the network and find out the critical path considering the expected time as the activity time.
- Find out the expected time for project completion.

Example: A small project consists of nine activities, the details of which are given below:

Programme Evaluation and Review Technique (PERT)

- Draw the network diagram, number the nodes.
- Find out the critical path and the expected project completion time.
- What is the probability of completing the project within 36 days?
- What is the probability of completing the project within 31 days?
- What project duration will have 95% confidence of completion?

Alternative Paths:

1-2-4-8 = 24 days

1-3-6-8 = 32 days

- = 34 days

- Critical path is 1-3-5-7-8 = 34 days

Variance of critical path = 4+16+4+1 = 25 days

- Probability of completing the project in 31 days

Z = (due date – expected date of completion)/√variance of critical path

= 36 – 34/ √25 = 2/5 = 0.40

Probability = .5+.1554 (area under normal distribution table for z=.1554)

= .6554 = 65.54%

- Probability of completing the project in 31 days

Z = (due date – expected date of completion)/√variance of critical path

= 31 – 34/ √25 = -3/5 = -0.6

= .5 – .2257 (area under normal distribution table for z=.2257)

= .2743 = 27.43%

For 95% confidence level, the value of z = 95% – 50% = 45% or 0.45

Z = (due date – expected date of completion)/√variance of critical path

1.65 (z value from the table equivalent to 0.45) = due date – 34 / 5

Due date = 34+1.65*5 = 42.25 days

Example: A small project consisting of eight activities has the following characteristics:

- Draw the Programme Evaluation and Review Technique (PERT) network.
- Determine the critical path and project duration.
- If a 30 weeks deadline is imposed, what is the probability that the project will be finished within the time limit?
- If the project manager wants to be 99% sure that the project is completed on the schedule date, how many weeks before that date should he start the project work?

Alternative Paths are

1-2-4-5-6

1-2-3-5-6

1-3-5-6

The critical path of the project is 1-2-4-5-6 and project duration is

Example: A small project consisting of eight activities has the following characteristics:

Alternative Paths

Critical Path B-E-G-H

Expected duration of project is 19 days.

The variance of the critical path is = 1+4+4+0.108 = 9.108

Std deviation = 3.02

Z= due date – expected date of completion/ √variance of critical path

Z = due date – 19 / 3.02 ( for 95% confidence level z = 1.65 from normal distribution table)

1.65 = due date – 19 / 3.02

Due date = 24 days

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