This document provides an overview of network analysis techniques like CPM and PERT. It begins with basic concepts of networks including nodes, branches, events and activities. It then discusses rules for constructing networks and uses examples to illustrate key points. The document explains the critical path method for determining the longest path and critical activities in a network. It also covers concepts like dummy activities. Finally, it provides a high-level introduction to the Programme Evaluation and Review Technique (PERT) which incorporates uncertainties into project scheduling.

1. Chapter six
Network Analysis with
CPM and PERT
By;-Dagnaygebaw Goshme (MSc.)

2. 6.1.Basic concepts on net work
A 'network' is defined as a graphic representation with a
flow of some type in its branches. It represents
nodes and branches.
A node is the intersection of the two branch lines. It is
denoted by a circle.
Each node represents an event, which is a specific
definable accomplishment recognizable at a particular
instant of time.
Each branch represents an activity.
The arrowheads indicate the sequence in which events
must be achieved.
Thus an event is the completion of all the activities leading
into that node and this event must precede the
initiation of the activity leading out of the node.

3. node Branch node

4. Example 1: A company is interested in preparing a
budget. The details of the activities and the
departments involved are given in the Table.

5. The project of budgeting can be displayed in a
network or project graph by an arrow diagram. Jobs
are shown as arrows leading from one node to the
other as in Figure
From the arrow diagram, we infer that:
activity A is the first job.
Jobs B and C start only after A is over.
A is called the predecessor of B and C and
B and C the successors of A.

6. Rules for Construction of Network
(a) Each activity is represented by one and only one arrow.
 This means that no single activity can be
represented twice in a network.
(b) No two activities can be identified by the same end events.
This means that there should not be loops in the network.
(c)Time follows from left to right.
All the arrows point in one direction.Arrows pointing in
opposite direction must be avoided.
(d) Arrows should not cross each other.
(e) Every node must have at least one activity preceding it and at
least one activity following it, except for the nodes at the very
beginning and at the very end of the network

7. Dummy Activities
 There is a need for dummy activities when the
project contains groups of two or more jobs which
have common predecessors.
 The time taken for the dummy activities is zero.
 Suppose we have the following project of jobs with
their immediate predecessors.

8. Activity B is the common immediate predecessor of both D
and E.
A is the immediate predecessor of D alone and C is the
predecessor of E.
Let B lead into two dummy jobs D1 and D2 and let D1 be
an immediate predecessor of D and D2 of E as shown in
Figure

9. Dummy arrow represents an activity with zero time duration.
It is represented by a dotted line and is introduced in a network
to clarify activity pattern under the following situations.
(i) It is created to make activities with common starting and
finishing events distinguishable.
(ii)To identify and maintain the proper precedence relationship
between activities those are not connected by events.
Consider an example where A and B are parallel (concurrent
activities). C is dependent on A and D is dependent on both A
and B.
Such a situation can be represented easily by use of dummy
activity as shown in Figure

10. Example 2: A project consists of the following activities
whose precedence relationship is given below. Draw an arrow
diagram to represent the project.

11. Since B & C are the immediate predecessor of H

13. CPM(Critical Path Method)
It is also a network technique but it is concerned with
obtaining the trade-off, between cost and
completion date for large projects.
In any project consisting of several activities each
activity can be completed in a normal duration with
normal cost.
If we employ more persons or skilled people or given
overtime to the workers, the activity could be
completed in a reduced duration known as crash
duration.
But this involves an increased cost in the form of
additional resources.
With CPM the amount needed to complete the
various activities is assumed to be known with
certainty.

14.  So, the direct costs for the activities increase and hence the cost of the
project also increases.
 By reducing the activity duration of some or all possible
completed ahead of the schedule.
 This will naturally reduce the overhead cost for the entire
project.
 On one hand the direct expenses increase, if we shorten the activity
duration, but, the indirect expenses for the project are reduced.
 We have to strike a balance or an optimum time schedule, or
a least cost schedule is to be obtained.This is the purpose of
the Critical Path Method.
 CPM is mostly used in construction projects, or in situations
already handled, so that the details like
the normal completion time,
crash duration and
cost of crashing are already known.

15.  The critical path method (CPM) aims at:-
 The determination of the time to complete a project
 The important activities on which a manager shall focus
attention.
 From the start event to the end event, the time required to
complete all the activities of the project in the specified sequence
is known as the project completion time.
 A continuous sequence, consisting of nodes and activities
alternatively, beginning with the start event and stopping at the end
event of a network is called a path in the network.
 The path with the largest time is called the critical path and
the activities along this path are called the critical activities
or bottleneck activities.
 The activities are called critical because they cannot be
delayed.
 In CPM, it is assumed that precise time estimate is available
for each activity.

16. To FindThe Critical Path
 After listing all the activities with their precedence relationship we
project these activities in a project graph represented by arrows or
Activity On Node diagram (AON).
 Now we have to find the minimum time required for completion of
the entire project.
 For this we must find the longest path with the sequence of
connected activities through the network. This is called the critical
path of the network and its length determines the time for
completion of the project.
 The activities in the critical path are so critical that, if they are
delayed, the project completion date cannot be met and the project
finish time will have to be extended.
 We shall now see how to identify the critical path, the critical
activities and the duration of the project.
 The meaning of path and length of a path should first be made clear.

17. D
We have five activities A, B, C, D and E with the time of
completion of the activities 5, 6, 7, 3 and 6 days respectively.
In this network, there are three ways to get from the
starting point at node 1 and travel through the network to
end at node 4.
These ways are called paths.Thus, a path is defined as "a
set of nodes connected by lines which begin at the initial
node of a network and end at the terminal node".
There are three paths namely, 1-3-4,1-2-3-4 and 1-2-4
where the numbers represent the nodes.

18.  The length of a path in a network is the total time it takes to
travel along the path.
 This time is calculated by adding the individual times between the
connected nodes on the path.
 A path is called a critical path if it is the longest the path in a
project network.
 We have the times of the three distinct paths has shown below.
 Path Time (days)
1-3-4 5 + 3 = 8 days
1-2-3-4 6 + 7 + 3 = 16
1-2-4 6 + 6 = 12
 The path connecting the nodes 1,2,3 and 4 constitutes the longest
path and hence 1-2-3-4 is the critical path.
 The minimum time to complete the project is the time taken for
the longest path namely 16 days.
 The activities B (1-2), C (2-3) and D(3-4) constitute the critical
activities.
 These jobs are critical in determining the project's duration.

19. Determine the critical path, the critical activities and the
project completion time.

20. Consider the paths, beginning with the start node
and stopping with the end node.There are two such
paths for the given project.
A- B- D- F= time of 3 + 5 + 10 + 4 = 22 weeks.
A -C –E- F= time of 3 + 7 + 5 + 4 = 19 weeks.
Compare the times for the two paths. Maximum of
{22,19} = 22.
We see that path I has the maximum time of 22
weeks.Therefore, path I is the critical path.The
critical activities are A, B, D and F.
The project completion time is 22 weeks.
We notice that C and E are non- critical activities.

21. Find out the completion time and the critical activities for the following
project
we identify 4 paths, beginning with the start node of 1 and terminating at the
end node of 10.They are as follows:
Path I ,A D G K=Time for the path = 8 + 20 + 8 + 6 = 42 units of time
Path II, B E H K=Time for the path = 10 + 16 + 11 + 6 = 43 units of time
Path III, B E I L=Time for the path = 10 + 16 + 14 + 5 = 45 units of time.
Path IV, C F J L=Time for the path = 7 + 25 + 10 + 5 = 47 units of time.
Maximum of {42, 43, 45, 47} = 47 which is a critical path.And critical activities
are C, F, J and L.The non-critical activities are A, B, D, E, G, H, I and K.The
project completion time is 47 units .

22. PERT "Programme Evaluation and
ReviewTechnique
It was created as a means to plan and accelerate the
development of the Polaris Ballistic Missile.
In USA the defense department developed a nuclear
missile to be launched from beneath the ocean's
surface by a mobile submarine, which would be an
effective deterrent against aggression by an enemy.
This paved way to plan how to design, develop and
plan the different stages in the production of a missile
and how quickly this task could be completed.
A planning and scheduling technique named PERT gave
the answer to these questions.

23. In any new venture, uncertainties are bound to creep
in.
PERT incorporated these uncertainties into a
model, which provides a reasonable answer to these
uncertainties.
There are certain statistical aspects scheduling large
projects consisting of numerous activities whose
completion times are uncertain and are
independent of one another.
PERT is an event-oriented technique.
By 'event' we mean reaching a certain stage of
completion of the project.
PERT is useful in research and developmental
projects,

24. Programme Evaluation and Review Technique
(PERT) is a tool that would help a project manager
in project planning and control.
It would enable him in continuously monitoring a
project and taking corrective measures wherever
necessary.
This technique involves statistical methods.
In PERT, we assume that it is not possible to have
precise time estimate for each activity and instead,
probabilistic estimates of time alone are possible.
A multiple time estimate approach is followed here.
In probabilistic time estimate, the following 3 types
of estimate are possible:

25. 1. Pessimistic time estimate ( P t )
2. Optimistic time estimate ( O t )
3. Most likely time estimate ( M t )
The optimistic estimate of time is based on the
assumption that an activity will not involve any
difficulty during execution and it can be completed
within a short period.
On the other hand, a pessimistic estimate is made
on the assumption that there would be unexpected
problems during the execution of an activity and hence
it would consume more time.
The most likely time estimate is made in between
the optimistic and the pessimistic estimates of time.

26. Thus the three estimates of time have the relationship
to < tm < tp
 Practically speaking, neither the pessimistic nor the
optimistic estimate may hold in reality and it is the most
likely time estimate that is expected to prevail in almost
all cases.
 Therefore, it is preferable to give more weight to the most
likely time estimate.
 We give a weight of 4 to most likely time estimate and a
weight of 1 each to the pessimistic and optimistic time
estimates.
 We arrive at a time estimate ( te) as the weighted average of
these estimates as follows:

27. Since we have taken 6 units ( 1 for P t , 4 for M t
and 1 for O t ), we divide the sum by 6.
With this time estimate, we can determine the
project completion time as better applicable as
compared to CPM.
Since PERT involves the average of three
estimates of time for each activity, this method is
very practical and the results from PERT will be have
a reasonable amount of reliability.

28. Measure of Certainty
The 3 estimates of time are such that
to < tm < tp
Therefore the range for the time estimate is tp - to.
The time taken by an activity in a project network follows a
distribution with a standard deviation of one sixth of the
range, approximately.
The certainty of the time estimate of an activity can be
analyzed with the help of the variance.
The greater the variance, the more uncertainty in the
time estimate of an activity.

29.  Example
Two experts A and B examined an activity and arrived
at the following time estimates.
Experts Time Estimate
to tm tp
A 4 6 8
B 4 7 10
Determine which expert is more certain about his
estimates of time:

33. Using the single time estimates of the activities, we get the
following network diagram for the project.
There are four such paths for the given project.They are as
follows:
Path I, A B D F H=Time for the path: 4+6+12+21+5 = 48 days.
Path II, A B D G I=Time for the path: 4+6+12+ 6+3 = 31 days.
Path III,A C E F H= Time for the path: 4+10+9+ 21+5 = 49 days.
Path IV, A C E G I= Time for the path: 4+10+9+ 6+3 = 32 days.
1 2
3
4
5
6
7
8
A, 4

34.  Compare the times for the four paths.
Maximum of {48, 31, 49, 32} = 49.
We see that Path III has the maximum time.
Therefore the critical path is Path III. i.e., 1  2 4
5  6  8
The critical activities are A, C, E, F and H.
The non-critical activities are B, D, G and I.
Project time (Also called project length) = 49 days.

36. 1
6
7
8
2
3
4
5
Path I, A C F I= Time for the path: 6+12+11+4 = 33 weeks.
Path II, A D G I= Time for the path: 6+5+7+ 4= 22 weeks.
Path III, B E H=Time for the path: 5+9+15 = 29 weeks

37. Compare the times for the three paths.
Maximum of {33, 22, 29} = 33.
It is noticed that Path I has the maximum time.
Therefore the critical path is Path I. i.e., 1  2  3
 5  8
The critical activities are A, C, F and I.
The non-critical activities are B, D, G and H.
Project time = 33 weeks.
Calculation of Standard Deviation and
Variance for the Critical Activities:

38. Variance of project time (Also calledVariance of project
length) = Sum of the variances for the critical activities =
1+4+1+ 4/9 = 58/9Weeks.
Standard deviation of project time = √Variance = √58/9 =
2.54 weeks.

39. The following are the suggested applications when
PERT or CPM is found useful.
The construction of a building or of a highway.
Planning and launching a new product.
Scheduling maintenance for a project.
The manufacture and assembly of a large machine tool.
To conduct a music or drama festival.
Preparation of budget for a company.