P E R T

1. PERTPERT

2. PERTPERT
In 1957 the Critical Path Method (CPM) was In 1957 the Critical Path Method (CPM) was
developed as a network model for project developed as a network model for project
management. management.
It is a deterministic method that uses a fixed It is a deterministic method that uses a fixed
time estimate for each activity. time estimate for each activity.
While CPM is easy to understand and use, but While CPM is easy to understand and use, but
does not consider uncertainty in activity time does not consider uncertainty in activity time
estimation.estimation.
Uncertainty such as weather, equipment failure, Uncertainty such as weather, equipment failure,
absenteeism can have a great impact on the absenteeism can have a great impact on the
completion time of a complex project. completion time of a complex project.

3. PERTPERT
The The Program Evaluation and Review Program Evaluation and Review
TechniqueTechnique (PERT) is a network model that (PERT) is a network model that
allows for randomness in activity allows for randomness in activity
completion times. completion times.
Generally used when there is a risk of time Generally used when there is a risk of time
associated with project.associated with project.
–R & D projects where correct time R & D projects where correct time
determinations cannot be made.determinations cannot be made.
–Example : project launching the spacecraft.Example : project launching the spacecraft.

4. PERTPERT
PERT was developed in the late 1950's for PERT was developed in the late 1950's for
the U.S. Navy's Polaris ballistic missile the U.S. Navy's Polaris ballistic missile
system project having thousands of system project having thousands of
contractors. contractors.
This project was notable in that it finished
18 months ahead of schedule and within
budget.
It has the potential to reduce both the time It has the potential to reduce both the time
and cost required to complete a project. and cost required to complete a project.

5. PERTPERT
This method uses statistical tools for This method uses statistical tools for
Implication of uncertainties on project time Implication of uncertainties on project time
Or Or
Stochastic Modeling of NetworkStochastic Modeling of Network
A distinguishing feature of PERT is its A distinguishing feature of PERT is its
ability to deal with uncertainty in activity ability to deal with uncertainty in activity
completion times. For each activity, the completion times. For each activity, the
model usually includes three time model usually includes three time
estimates:estimates:

6. Three Time EstimatesThree Time Estimates
1
2
3
4
5
2-5-12
4-7-16
1-6-23
3-7-20
2-5-10

7. TimesTimes
Optimistic timeOptimistic time – Shortest possible time in which an – Shortest possible time in which an
activity can be completed under ideal conditions. activity can be completed under ideal conditions.
This is denoted by tThis is denoted by t
oo
Pessimistic timePessimistic time - the longest time that an activity - the longest time that an activity
might require. If everything went wrong and might require. If everything went wrong and
abnormal situation prevails.however, it doesn't”t abnormal situation prevails.however, it doesn't”t
include highly unusual catastrophies such as include highly unusual catastrophies such as
earthquake, floods, fires. It is denoted by tearthquake, floods, fires. It is denoted by t
pp
Most likely timeMost likely time (Most Frequent-Mode)- the (Most Frequent-Mode)- the
completion time having the highest probability. completion time having the highest probability.
Normal condition prevails. It is denoted by tNormal condition prevails. It is denoted by t
LL

8. 1
2
3
4
5
2-5-12
4-7-16
1-6-23
3-7-20
2-5-10
t
0
t
p
t
m

9. Problem: 54 trenches of same dimensions by different partiesProblem: 54 trenches of same dimensions by different parties

Find :Optimistic, Pessimistic & Most Likely TimesFind :Optimistic, Pessimistic & Most Likely Times

10. TimesTimes

11. Most Likely TimeMost Likely Time
Tallest peak of the curve- Most Likely time or Mode

12. Expected Time & Standard Expected Time & Standard
Deviation: Beta DistributionDeviation: Beta Distribution
Expected time = ( Optimistic + 4 x Expected time = ( Optimistic + 4 x
Most likely + Pessimistic ) / 6Most likely + Pessimistic ) / 6
Expected time : Time corresponding 50 Expected time : Time corresponding 50
% probability of performance% probability of performance
SD: How tightly a set of
values is clustered around
the mean.
Standard Deviation:
Sigma: measure of
uncertainty = (b-a)/6

13. Calculate Expected Time & Calculate Expected Time &
Standard Deviation:Standard Deviation:
Write down their significanceWrite down their significance

14. Expected Time & Standard Expected Time & Standard
DeviationDeviation
Activity Activity tt
oo tt
mm tt
pp
11 44 77 1616
22 11 66 2323
Comment on Standard Deviation: Second case measure of dispersion
is higher

15. A systematic and scientific method of finding critical path lies in the
calculation
of event time which is described by
i) The Earliest Expected Occurance Time (T
E
)
ii) The Latest Allowable Occurance Time (T
L
)
The Earliest Expected Time (T
E
) is the time when an event can be expected to
occur earliest. The calculation of TE of an event is same as calculation of
EOT of CPM network
If more than one activity are directed to the event, maximum of the sum of
T
E
's along various path will give the expected mean time of the event.
Expected mean time of the initial event is taken as zero and process is
repeated for each succeeding event and ultimately to the final event. The
method is usually called the forward pass.
(T
E
)j = Max [(T
E
)i + t
ij
]
The Latest Allowable Occurence Time (T
L
) :
The latest time by which an event must occur to keep the project on schedule
is called the latest allowable occurence time (T
L
). The calculation of T
L
of an
event is same as that LOT of CPM network by the method known as
Backward Pass. ; (T
L
)i = min ((T
L
)j – t
ij
)

16. Scheduled Completion Time (T
s
)
Whenever a PERT network is taken in hand decision is made regarding the
completion time of the project and the accepted figure is called the
Scheduled Completion Time (T
s
). Naturally. T
s
refers to the latest allowable
occurence time (T
L
) of the last event of the project, i.e. (Ts= T
L
)·
SLACK .
Time box having two compartments is made at each event. the value in the
left compartment indicating the value of T
E
and that of in the right
compartment indicating T
L
of that event. And the slack of the event is given
by,
Slack (S) = (T
L
– T
E
)
Thus the slack is difference between event times denoting the range within
which an event time can vary. Thus, slack gives the idea of "time to spare".
Slack means more time to work and less to worry about. It also spots which
are potential trouble areas.
Slack may be positive, zero or negative depending upon the value of T
E
and
T
L
of that event.

17. POSITIVE SLACK
When T
L
is more than T
E
. positive slack is obtained. It indicates the
project is ahead of schedule meaning thereby the excess resources.
ZERO SLACK
When T
L
is equal to T
E
zero slack' is obtained. It indicates that the
project is going on schedule meaning thereby adequate resources.
NEGATIVE SLACK
When the scheduled completion time Ts (and hence T
L
) is less than T
E

negative slack is obtained. It indicates the project is behind schedule
meaning thereby the lack of resources.
CRITICAL EVENT
The event having the least slack value is known as a critical event
CRITICAL PATH
The path joining the critical events is called a critical path of the PERT
network. The critical path may be one or more than one. Time wise. the
critical path is the longest path connecting the initial event to the final
event. A critical path is distinctly marked in the network. usually by a
thick line or double lines.

18. Determine the Expected time for Determine the Expected time for
Each Path & Find the critical PathEach Path & Find the critical Path

19. Critical PathCritical Path

20. Probability of Meeting The Probability of Meeting The
Schedule DateSchedule Date

21. Normal Distribution FunctionNormal Distribution Function
Sum of all expected time of all activities along Sum of all expected time of all activities along
critical path is equal to the expected time of last critical path is equal to the expected time of last
event= 50 % time of completion of projectevent= 50 % time of completion of project
Though individual activities assume Though individual activities assume
random( beta distribution) but Trandom( beta distribution) but T
EE of the project as of the project as
a whole assume normal distributiona whole assume normal distribution

22. Normal Distribution FunctionNormal Distribution Function

23. Normal Distribution FunctionNormal Distribution Function

24. Normal Deviate Normal Deviate
(x): Distance from (x): Distance from
the mean the mean
expressed in expressed in
terms of sigmaterms of sigma
1. Normal Deviate = 0, it is
the expected time, probability
of completion = 50 %
2. Normal Deviate = 1,
probability of completion = 84
%.
3. Normal Deviate = -1,
probability of completion = 16
%

25. Normal DeviateNormal Deviate
If Ts is the schedules time of completionIf Ts is the schedules time of completion
& Te is the expected time of completion& Te is the expected time of completion
Z = Ts-Te/sigmaZ = Ts-Te/sigma
Sigma = (Sum of variances along critical path)Sigma = (Sum of variances along critical path)
0.50.5

Variance = (tp-to/6)Variance = (tp-to/6)
22

29. Exp. For the given PERT network, determine
a) Expected time, Standard deviation and variance of the PROJECT and
show the critical path also.
b) Probability of completion of project in 35 days.
c) Time duration that will provide 90% probability of its completion in
time.
The three time estimates of each activity. are mentioned on the network.

30. Expected mean time of activity
t
e
= (t
a
+ 4t
m
+ t
b
)/6
Standard deviation of activity d
t
= (t
b
- t
a
)/6
Variance of activity vt = (standard deviation)
2
.
Earliest Expected Mean Time (T
E
) and Latest allowable occurrence
time (T
L
) are marked in time box at each event. Slack (S) = (T
L
- T
E
) is
also mentioned on the network. Since scheduled completion time of
project is not mentioned, for the last event (8), T
L
= T
E
has been taken.

33. Least slack value = 0
:: All the events having zero slack are critical.
CRITICAL PATH-I = 1- 2- 3 - 6-7 - 8
CRITICAL PATH-II = 1- 2-4 - 6-7 – 8
Expected Mean Time of Project (m
T
) = 31 days.
Variance of project along critical path I
(VT I) = 1 + 7.1 + 5.44 +1.78 + 0.44 = 15.76
Variance along critical path II (VrII ) = 1 + 4 + 1 + 1.78 + 0.44 = 2.86
:. Variance of the project (V
T
) = 15.76
Standard Deviation of the project (d
T ) = sqrt(15.76) = 3.97
b) Probability factor (z) corresponding to x = 35 days
z = (x- m
T
)/ d
t
= (35-31)/3.97 = 1.007 = 1.0
probability % corresponding to z = 1.0 (from table)
pr= 84.13%
c) for 90% probability, the value of z = 1.32 (from table )
1.32 = (x- 31 )/3.97
So x = 36.24 days.

34. Four activities to be undertaken in series for the completion
of II project are as follows,
Estimate the time required at
(i)95% probability, and
(ii)5% probability to complete the work.
(iii)Also which of the above four activities has the most reliable time
estimates?

36. Problem:Problem:
Expected Project Length is 50 weeksExpected Project Length is 50 weeks
Variance 16Variance 16
How many weeks required to complete the How many weeks required to complete the
project to complete withproject to complete with
– 95 % Probability95 % Probability
–75 % probability75 % probability
–40 % Probability40 % Probability
57 weeks
53 weeks
49 weeks

37. Find The probability of completion Find The probability of completion
within 35 dayswithin 35 days
10
9
9
7
11
5
8
Critical path 1-2-4-5, Te= 30Variance 1-2= (18-6/6)
2
=4, + 9 + 9 = 22SD= 4.69

38. Benefits of PERTBenefits of PERT
PERT is useful because it provides the following PERT is useful because it provides the following
information:information:
–Expected project completion time.Expected project completion time.
–Probability of completion before a specified date.Probability of completion before a specified date.
–The critical path activities that directly impact the The critical path activities that directly impact the
completion time.completion time.
–The activities that have slack time and that can lend The activities that have slack time and that can lend
resources to critical path activities.resources to critical path activities.
–Activity start and end dates.Activity start and end dates.

39. LimitationsLimitations
The activity time estimates are somewhat subjective and The activity time estimates are somewhat subjective and
depend on judgement. In cases where there is little depend on judgement. In cases where there is little
experience in performing an activity, the numbers may experience in performing an activity, the numbers may
be only a guess. be only a guess.
Even if the activity times are well-estimated, PERT Even if the activity times are well-estimated, PERT
assumes a beta distribution for these time estimates, but assumes a beta distribution for these time estimates, but
the actual distribution may be different.the actual distribution may be different.
Even if the beta distribution assumption holds, PERT Even if the beta distribution assumption holds, PERT
assumes that the probability distribution of the project assumes that the probability distribution of the project
completion time is the same as the that of the critical completion time is the same as the that of the critical
path. Because other paths can become the critical path if path. Because other paths can become the critical path if
their associated activities are delayed, PERT consistently their associated activities are delayed, PERT consistently
underestimates the expected project completion time.underestimates the expected project completion time.