Network_Analysis.pdf

Unit - V: Network Analysis
Dr. Tushar Bhatt
Ph. D, Mathematics
Assistant professor
Department of Science and Humanities
Faculty of Engineering and Technology
Atmiya University
Rajkot - 360005
March 3, 2023
Dr. Tushar Bhatt (AU) SM & OT March 3, 2023 1 / 45

Table of Contents
1 Introduction
Phases of CPM and PERT
Some Important Definitions
Project management or representation by a network diagram
Types of activities
Types of events
Common Errors
Rules of network construction
Numbering the events
Time analysis
Determination of Floats and Slack times
Critical activity and Critical path
2 Critical Path Method - CPM
3 Program Evaluation and Review Technique - PERT

1. Introduction
The techniques of operations research used for planning, scheduling
and controlling large and complex projects are often referred as
network analysis. All these techniques are based on representation of
the project as a network of activities.
A network is a graphical plan consisting of a certain configuration of
arrows and nodes for showing the logical sequence of various activities
to be performed to achieve project objectives.
PERT (Program Evaluation and Review Technique) and CPM
(Critical Path Method) are two of the well known techniques belong
to the family of network analysis.

1.1 Phases of CPM and PERT
Project management by CPM and PERT can be based on the following
three phases.
1 Planning
In this phase, the whole project is broken down into smaller projects
and further into activities.
The time estimates for these activities are then determined.
Network diagram is constructed.
Study the network diagram in detail and incorporate any modification
at the initial or execution stage.
2 Scheduling
Scheduling phase involves time related activities.
A time chart is constructed showing the start and finish times of each
activity.
The chart also shows or relates one activity to other activities of the
project.
Identify non critical activities and show the amount of slack or float
times.This is very essential to take the advantages of delay in execution
of activities or using limited available resources.

1.1 Phases of CPM and PERT
3 Control
Controlling includes careful supervision of progress of the project with
the help of network diagram and time chart.
Continuously analyze and update all activities involved and if necessary
reschedule the project for the remaining portion of the project.

1.2 Some Important Definitions
1 Activity: It represents some action and is a time consuming effort
necessary to complete a particular part of overall project.

2 Event: Beginning and end points of an activity are called events or
nodes.

nodes.
3 Critical activity: An activity is said to be critical if any delay in its
start further delays the completion of the whole work.

nodes.
4 Amount of Slack or Float time: The difference between the
earliest time and the latest time is called as activity slack.

nodes.
5 Total amount of slack or total float time: Total slack is the
difference of latest finish time and the duration.

nodes.
6 Free slack or free float time: Free slack is the amount of time a
job can be delayed without affecting the early start of any other job.

nodes.
6 Free slack or free float time: Free slack is the amount of time a
job can be delayed without affecting the early start of any other job.
7 Independent slack: Independent slack = Free slack −(Lf − Es)
where Lf the amount of latest finish time of a job is and Es is the
amount of earlier start of a job.

1.3 Project management or representation by a network
diagram
Network diagram is the graphical representation of logically and
sequentially connected arrows and node, representing activities and events
respectively in a project.

diagram
Network diagram is the graphical representation of logically and
sequentially connected arrows and node, representing activities and events
respectively in a project.
Figure: Events and activity

diagram
Now observe the following figure i and j indicates event represented by the
nodes in which node i known as tail and j is known as head moreover the
symbol A means an arrow in between i and j indicates an activity.
Figure: Network diagram

1.4 Types of activities
1 Preceding activity:
Activity that must be accomplished before a given event can occur.
2 Succeeding activity:
Activity that cannot be accomplished until an event has occurred.
3 Concurrent activity:
Activity taking place at same time or in the same location.
4 Dummy activity:
Activity which neither consumes time nor resources but is used simply
to represent a connection or a link between the events is known as
dummies. It is shown in network by a dotted line.
5 In above mentioned figure, A is a preceding activity of B, B is a
preceding activity of C, D is a preceding activity of E and E is a
preceding activity of F.
6 Similarly B is a succeeding activity of A, C is a succeeding activity of
B, E is a succeeding activity of C and F is a succeeding activity of E.
7 Moreover the activities A and D arise from even 1 at same time of
same location therefore A and D are becomes concurrent activities.

1.4 Types of activities
Now for understanding the dummy activity observe the following figure:
Figure: Network diagram and dummy activity
Here according to the rule of network theory, there is only one starting
point and one ending point. But in above mentioned figure, we observe
that there are two ending points node 4 and node 5. So it is necessary to
add a dummy activity between node 3 and node 4 because after that
procedure event 4 is not an end of the network. Therefore the activity E in
above figure is called dummy activity and it is denoted by a dotted line.

1.5 Types of events
1. Merge Event
An event is said to be Merge event if two or more than two activities are
end at the same node (event).
Figure: Merge Event
In above figure an event i is Merge event.

1.5 Types of events
2. Burst Event
An event is said to be Burst event if it is a starting point of two or more
than two activities.
Figure: Burst Event

1.6 Common Errors
1. Looping (Cycling)
Drawing an endless loop in a network is known as error of looping.
Figure: Loop or Cycle
In above figure double arrow for activity B create an endless loop in
between the events 2, 3 and 4.

1.6 Common Errors
2. Dangling
To disconnect an activity before the complication of all activities called
error of dangling.
Figure: Dangling
In above figure an activity F is disconnected before completion of
remaining all activities.

1.6 Common Errors
3. Redundancy
Unnecessarily inserting the dummy activity in a network diagram is known
as error of redundancy.
Figure: Redundancy
In above figure an activity F is a dummy activity which is unnecessary
because there is a path between events 2 to 4 is already exists. Therefore
it creates an error namely redundancy.

1.7 Rules of network construction
Try to avoid the arrow the cross each other.
Use straight arrow.
No event can occur until every activity preceding it has been
completed.
An event can’t occur twice.
Dummies should be introduced only if it is extremely necessary.
Network has only one entry point called start event and one end point
(emergence).
Use arrow left to right. Avoid mixing two directions.

1.8 Numbering the events
Number must be unique.
Number should be carried out on a sequential basis from left to right
and top to bottom.
Initial event must be numbered as 1.
Number of all new starts events 2, 3 and so on.

1.9 Time analysis
1 Tij = tij = Estimate completion time of activity (i, j).
2 ESij = Earliest starting time of activity (i, j).
3 EFij = Earliest finishing time of activity (i, j).
4 LSij = Latest starting time of activity (i, j).
5 LFij = Latest finishing time of activity (i, j).
Forward pass computation (Left to Right)
Zero is the starting time for the project.
(EF)ij = (ES)ij + tij
Ej = maxi (Ej + tij )
Backward pass computation (Right to Left)
For ending event assume E = L.
(LFij ) = (LF)ij − tij .
Lj = minj (Lj − tij ).

1.10 Determination of Floats and Slack times
1. Float
It is defined as the difference between the latest and earliest activity time.
There are three types of floats.
(a) Total Float
(TF)ij = (LS)ij − (ES)ij OR (Li − Ei ) − tij
Where Ei = Earliest time for tail event
Li =Latest time for tail event
tij = Normal time for activity (i, j).
(b) Free Float
FFij = (Ej − Ei ) − tij
tij =Normal time for activity (i, j).

1.10 Determination of Floats and Slack times
(c). Independent Float
IFij = (Ej − Li ) − tij
Li = Latest time for tail event
tij = Normal time for activity (i, j).
2. Slack
It is defined as the difference between latest and earliest event time.

1.11 Critical activity and Critical path
An activity (i, j) is said to be critical, if it has total float TFij = 0.
A path throughout the network is said to be critical if it satisfies the
following three conditions.
1 ESi = LFi
2 ESj = LFj
3 ESj = ESi = LFj − LFi = tij
Where ESi = Earliest starting time of initial event.
LFi = Latest finishing time of initial event.
ESj = Earliest starting time of final event.
LFj = Latest finishing time of final event.

2. CPM
Ex-1: Construct a network for a project whose activities and precedence
relationships are as given below:
Activity A B C D E F G H I J K
Predecessor - - - A B B C D E H, I F, G
Figure: Given Network

2. CPM - Ex - 1: Solu...
Figure: Required Network

2. CPM - Ex - 2

2. CPM - Ex - 2: Solu...
Solution:

2. CPM - Ex - 3
Ex-3: A project schedule has the following properties. According to that
(1) Construct a network.
(2) Compute the earliest event time and latest event time.
(3) Determine the critical path and total project duration.
(4) Compute total and free float for each activity.
Activity 1-2 1-3 2-4 3-4
Time (days) 14 1 1 1
Solution:
(i) Network diagram

2. CPM - Ex - 3: Solu...

2. CPM - Ex - 3: Solu...
The critical path = 1 − 2 − 4
Total project duration = 4 + 1 = 5 days

2. CPM - EX - 4
Ex - 4: Determine the early start and late start in respect of all node
points and identify critical path for the following network

2. CPM - Ex - 4: Solu...
Solution : Calculation of E and L for each node is shown in the network

2. CPM - Ex - 4: Solu...

2. CPM - Ex - 4: Solu...
From the table, the critical nodes are
(1, 2), (2, 5), (5, 7), (5, 8), (7, 10)and(8, 10).
From the table, there are two possible critical paths
(i)1 → 2 → 5 → 8 → 10
(ii)1 → 2 → 5 → 7 → 10

2. CPM - Ex - 5
Ex - 5: Find the critical path and calculate the slack time for the
following network

2. CPM - Ex - 5: Solu...
Solution: The earliest time and the latest time are obtained below

2. CPM - Ex - 5: Solu...
Solution: From the above table, the critical nodes are the activities
(1, 3), (3, 5)and(5, 9)
The critical path is 1 → 3 → 5 → 9.

3. PERT
The main objective in the analysis through PERT is to find out the
completion for a particular event within specified date. The PERT
approach takes into account the uncertainties. The three time values are
associated with each activity.
1 Optimistic time: It is the shortest possible time in which the activity
can be finished. It assumes that everything goes very well. This is
denoted by t0.
2 Most likely time: It is the estimate of the normal time the activity
would take. This assumes normal delays. If a graph is plotted in the
time of completion and the frequency of completion in that time
period, then most likely time will represent the highest frequency of
occurrence. This is denoted by tm.

3. PERT
3 Pessimistic time: It represents the longest time the activity could
take if everything goes wrong. As in optimistic estimate, this value
may be such that only one in hundred or one in twenty will take time
longer than this value. This is denoted by tp.
4 Expected time: It is the average time an activity will take if it were
to be repeated on large number of times and is based on the
assumption that the activity time follows Beta distribution, this is
given by
te =
(t0 + 4tm + tp)
6
5 The variance for the activity is given by
σ2
=

tp − t0
6
2
Dr. Tushar Bhatt (AU) SM OT March 3, 2023 38 / 45

3. PERT - Ex - 1
Ex - 1: For the project
Task A B C D E F G H I J K
Least time 4 5 8 2 4 6 8 5 3 5 6
Greatest time 8 10 12 7 10 15 16 9 7 11 13
Find the earliest and latest expected time to each event and also critical
path in the network.

3. PERT - Ex - 1: Solu...
Solution:

Solution:

Solution:
The critical path is A → C → E → H → K.

3. PERT - Ex - 2
Ex - 2: A project has the following characteristics
Construct a PERT network. Find the critical path and variance for each
event.

Solution:

Solution: The network is constructed as shown below
The critical path: 1 → 2 → 4 → 6 → 7 → 9 → 10.

Network_Analysis.pdf

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