PERT. ppts for operation researchs assgn.pptx

Presenters:
Mr. Faisal Ahmed
Mr.Abdul Ghani
PERT
(Program Evaluation and Review
Techinque)
Operations Research Presentation Dated: 22/06/2024

Network Representation
Network representation refers to the depiction of a
system, process, or problem as a collection of nodes and
arcs that connect them.This visual representation is used
to:
1. Model complex relationships and interactions
2. Analyze and optimize performance
3. Identify patterns and connections
4. Make informed decisions

Rules for Constructing the Network
 Rule 1. Each activity is represented by one, and only one arc.
Example: consider the nodes
A(start)
B(activity 1)
C(activity 2)
D(end)
Arcs:
A → B (activity 1 starts after start)
B → C ( activity 2 starts after activity 1)
C → D (end follows activity 2)
In this example:
 Each activity ( B,C) is represented by only one arc.
 There are no duplicate arcs for the same activity.
 Each arc represents a unique relationship between nodes.

Continue…
 Rule 2. Each activity must be identified by two distinct end nodes.
Example: consider the nodes
A(start), B(activity 1 start), C(activity 1 end), D(activity 2 start), E(activity 2
end), F(activity 3 start), G(activity 3 end), H(end)
Arcs:
A → B(activity 1 starts after start), B →C(activity 1 ends), C →D(activity
2 starts), D →E(activity 2 end), E →F(activity 3 starts), F →G(activity
3 ends), G →H(end follows activity 3)
In this Example:
 Each activity (activity 1, activity 2) is identified by two distinct end
nodes.
 Activity 1: Nodes B(start) and C (end)
 Activity 2: Nodes D(start) and E(end)
 Activity 3: Nodes F(start) and G(end)
 Each arc represents a unique relationship between nodes

Continue….
 Rule 3. To maintain the correct precedence relationships, the
following questions must be answered as each activity is added to
the network.
1. What activities immediately precede the current activity?
2. What activities immediately follow the current activity ?
3. What activities are concurrent with the current activity?
Consider the following segment of a project:
a) Activity C starts immediately after activities A and B have been
completed.
b) Activity E can start after activity B is completed.
A →○→C
↑D
B →○→E

PERT( Program Evaluation and Review
Technique)
PERT ( Program Evaluation and Review
Technique) is a project management tool
used to plan, organize, and coordinate tasks
in a project. It is a method for analyzing
and representing the tasks involved in a
project, and for determining the minimum
duration required to complete the project.

Continue…
 PERT is used to:
1. Identify critical tasks and their dependencies.
2. Determine the minimum project duration.
3. Develop a project schedule.
4. Identify potential bottlenecks and delays.
5. Optimize resource allocation.
6. Monitor progress and adjust the project plan as needed.
 PERT is a network-based approach that uses the following
elements:
1. Activities
2. Dependencies
3. Durations
4. Events

How PERT works?
1. Task Identification: A task is identified by a unique ID,
task name, task description, dependencies, duration, resources,
start and end dates or time.
2. Time Estimation: It is based on three different time
estimates made for each activity.
 Optimistic time: which occurs when execution goes
extremely well.
 Most Likely time: which occurs when execution is done
under normal conditions.
 Pessimistic time: which occurs when execution goes
extremely poor.
3. Expected time Calculation → 6
4 t
t
t
t
p
m
o
e




Continue….
4. Network diagram: Visual representation of the
project schedule, showing the relationship between
tasks and activities.
5. Critical Path Analysis: Identifies the longest
sequence of tasks in a project that must be completed
on time for the whole project to be completed on
time.
6. Project Schedule: Timeline that outlines the start
and finish dates for each task and milestone,
coordinating resources and deadlines.

Example:
For the given activities determine:
1. Draw the project network.
2. Find the expected duration and variance of each activity .
3. Calculate the earliest and latest occurrence for each
event.
4. Calculate expected project length.
5. Calculate the variance and standard deviations of project
length.
6. Find the probability of the project completing in 26 days.

Given the table:
Activities Optimistic
time ( )
Most Likely
time( )
Pessimistic
Time ( )
1-2 6 9 12
1-3 3 4 11
2-4 2 5 14
3-4 4 6 8
3-5 1 1.5 5
2-6 5 6 7
4-6 7 8 15
5-6 1 2 3
to tm tp

Solution:
1. First we draw the Network Diagram for activities
in the table
1
2
3
4
5
6
2. Now we find expected duration( ) and variance(σ )
of the data given in table.
where
and





 

6
2
O
P
 2
te
6
4 t
t
t
t
p
m
o
e








 

6
2
2 t
t o
p


After calculating expected duration ( ) and
variance( ), we get

2
Activities Optimisti
c time
( )
Most
Likely
Time( )
Pessimisti
c time
( )
Expected
time( )
Variance
( )
1-2 6 9 12 9 1
1-3 3 4 11 5 1.78
2-4 2 5 14 6 4
3-4 4 6 8 6 0.44
3-5 1 1.5 5 2 0.44
2-6 5 6 7 6 0.11
4-6 7 8 15 9 1.78
5-6 1 2 3 2 0.11

2
te
te
to
tm tp

3. Now for earliest and latest occurrence of even.
We use where in upper column we write latest and in lower we write
earliest, by using forward pass computation and we take initial as zero
1
2
3
4
5
6
9
5
6
2
6
6
9
2
0
0
9
5
9
9
22
7
24
24
15
15
Earliest
Latest
Minimum
Minimum value
Maximum
Maximum
value
Minimum

4. For project length, first we find critical path from
network diagram
1
2
3
4
5
6
9
5
6
2
6
6
9
2
0
0
9
5
9
9
22
7
24
24
15
15
The double lines in the network diagram represent the
critical path
So
1-2-4-6 is the critical path

The shaded rows represent the data for critical path
Activities Optimisti
c time
( )
Most
Likely
Time( )
Pessimisti
c time
( )
Expected
time( )
Variance
( )
1-2 6 9 12 9 1
1-3 3 4 11 5 1.78
2-4 2 5 14 6 4
3-4 4 6 8 6 0.44
3-5 1 1.5 5 2 0.44
2-6 5 6 7 6 0.11
4-6 7 8 15 9 1.78
5-6 1 2 3 2 0.11

2
Expected project duration = = 9+6+9 = 24 days
te
to tm tp
te

5. Now we calculate the variance and standard
deviation of project length.
we have found that the critical length is 1-2-4-6
so project length variance ( )
= 1+4+1.78 = 6.78
and project length standard deviation will be
σ = 2.603

2

2

6. To find the probability of the project completing
in 26 days.
we have
where is the schedule time to complete the
project and = 26 days
= normal expected project length
and = 24 days
σ = 2.603

t
t e
s
D


ts
ts
te
te

Therefore,
D = 0.768 ≈ 0.8
Now for probability
p(Z ≤ D) = p(Z ≤ 0.8)
From standard normal distribution table we get,
p(Z ≤ 0.8) = 0.7881
Or
p(Z ≤ 0.8) = 78.81%
603
.
2
24
26 

D

PERT. ppts for operation researchs assgn.pptx

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