What is PERT (Program evaluation and review technique)

PERT (Program evaluation and review technique) is a project management technique used to schedule, monitor, and control large and complex projects. PERT is used to examine the tasks in a project schedule and to determine Critical Path and variations. 

PERT is used in projects where the completion time of each activity is not determined exactly and variations may happen in completion of each activity; hence 3 possible times are given for each activity and a probability method is followed in PERT. Then PERT analyzes the time required to complete each task and its associated dependencies to determine the minimum time to complete a project.

 PERT employs three-time estimates for each activity. They are:

Optimistic time (a ) = The time an activity will take if everything goes as planned.

Pessimistic time (b ) = The time an activity will take assuming very unfavorable conditions.

Most likely time (m ) = The most realistic estimate of the time required to complete an activity.

Example

A project consisting of eight activities has the following characteristics:

Time Estimates (In Weeks)
Activity	Preceding Activity	Most Optimistic Time (To)	Most Likely Time (Tm)	Most Pessimistic Time (Tp)
A	None	2	4	12
B	None	10	12	26
C	A	8	9	10
D	A	10	15	20
E	A	7	7.5	11
F	B,C	9	9	9
G	D	3	3.5	7
H	E,F,G	5	5	5

(i) Prepare the activity schedule for the project.

(ii) Draw the PERT network for the project.

(iii) Determine the critical path.

(iv) If a 30- week deadline is imposed, what is the probability that the project will be finished within the time limit?

Solution:

Step 1: Find The expected time and variance of each activity.  

The formula for the Expected Time:

Where:

Te = Expected Time

To= Optimistic Time Estimate

T_m = Most Likely Time Estimate

T_p = Pessimistic Time Estimate

The formula for finding variance of each activity:

 

Where:

B = Pessimistic Time Estimate

A = Optimistic Time Estimate

Expected time and Variance of each activities
Activity	Preceding Activity	Most Optimistic Time (To)	Most Likely Time (Tm)	Most Pessimistic Time (Tp)	Expected Time (te)	Variance
A	None	2	4	12	5	2.78
B	None	10	12	26	14	7.12
C	A	8	9	10	9	0.12
D	A	10	15	20	15	2.78
E	A	7	7.5	11	8	0.45
F	B,C	9	9	9	9	0
G	D	3	3.5	7	4	0.45
H	E,F,G	5	5	5	5	0

Step 2: Draw the Network Diagram, and find the Earliest time and Latest Time

The network Diagram includes: 

1. Events: An event is depicted in the form of a Nod which includes activity number on top (1,2,3,4,5,6), Early Time in the lower left, and latest time on the lower right side. 

2. Activity: An activity that is the actual performance of the task and requires time and resources for its completion, is depicted using arrows connecitng the events. Arrow bears the activity name and expected time of the activity.

Time Estimates (In Weeks)
Activity	Preceding Activity	Most Optimistic Time (To)	Most Likely Time (Tm)	Most Pessimistic Time (Tp)	Expected Time (te)	Variance	Earliest Time		Latest Time
							Start	Finish	Start	Finish
A	None	2	4	12	5	2.78	0	5	0	5
B	None	10	12	26	14	7.12	0	14	1	15
C	A	8	9	10	9	0.12	5	14	6	15
D	A	10	15	20	15	2.78	5	20	5	20
E	A	7	7.5	11	8	0.44	5	13	16	24
F	B, C	9	9	9	9	0	14	23	15	24
G	D	3	3.5	7	4	0.45	20	24	20	24
H	E,F,G	5	5	5	5	0	24	29	24	29

Step 3: Find the critical path

The critical path includes the activities whose slack time is zero, ie, Earliest Finish - Latest Finish = 0

The critical path of the project is 1-2-4-5 -6, with critical activities being A, D, G, and H.

The expected project length is the sum of duration of each critical activity. Expected project length = 5 + 15 + 4 + 5 = 29 weeks.

Step 4: Find the variance

Variance project length is obtained by summing the variance of each critical activity. 

ie, the sum of the variance of A +D+G+H,

2.78+2.78+0.44+0 = 6

(vi)  Find the probability of completing the project within the 30-week deadline:

The probability that the project will be completed within 30 weeks is given by 

σ = √variance

z = (30 - 29) / √6 

= 0.41

Use the Standard Normal Distribution tables for Z score: 

The probability of completing the project within the 30-week deadline is 65%

References:

1) Heizer, J. (2016). Operations management, 11/e. Pearson Education India.

2) https://www.engineeringenotes.com/networking/network-analysis/top-4-problems-on-pert-network-analysis-networking/15725