The document discusses the Program Evaluation and Review Technique (PERT), a network model used for project management that accounts for uncertainty in activity times. PERT was developed in the late 1950s for large, complex projects like the Polaris missile system. It uses three time estimates - optimistic, pessimistic, and most likely - to calculate the expected time, variance, and probability of completing a project by a certain date using statistical methods like the beta distribution and normal distribution. PERT identifies the critical path that determines the project completion time and allows project managers to focus on critical activities.

1. PERTPERT

2. PERTPERT
In 1957 the Critical Path Method (CPM) wasIn 1957 the Critical Path Method (CPM) was
developed as a network model for projectdeveloped as a network model for project
management.management.
It is a deterministic method that uses a fixedIt 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, butWhile CPM is easy to understand and use, but
does not consider uncertainty in activity timedoes 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 theabsenteeism can have a great impact on the
completion time of a complex project.completion time of a complex project.

3. PERTPERT
TheThe Program Evaluation and ReviewProgram Evaluation and Review
TechniqueTechnique (PERT) is a network model that(PERT) is a network model that
allows for randomness in activityallows for randomness in activity
completion times.completion times.
Generally used when there is a risk of timeGenerally used when there is a risk of time
associated with project.associated with project.
– R & D projects where correct timeR & 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 forPERT was developed in the late 1950's for
the U.S. Navy's Polaris ballistic missilethe U.S. Navy's Polaris ballistic missile
system project having thousands ofsystem 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 timeIt 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 forThis method uses statistical tools for
Implication of uncertainties on project timeImplication of uncertainties on project time
OrOr
Stochastic Modeling of NetworkStochastic Modeling of Network
A distinguishing feature of PERT is itsA distinguishing feature of PERT is its
ability to deal with uncertainty in activityability to deal with uncertainty in activity
completion times. For each activity, thecompletion times. For each activity, the
model usually includes three timemodel 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 too
Pessimistic timePessimistic time - the longest time that an activity- the longest time that an activity
might require. If everything went wrong andmight require. If everything went wrong and
abnormal situation prevails.however, it doesn't”tabnormal situation prevails.however, it doesn't”t
include highly unusual catastrophies such asinclude highly unusual catastrophies such as
earthquake, floods, fires. It is denoted by tearthquake, floods, fires. It is denoted by tpp
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 tLL

8. 1
2
3
4
5
2-5-12
4-7-16
1-6-23
3-7-20
2-5-10
t0 tptm

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 & StandardExpected Time & Standard
Deviation: Beta DistributionDeviation: Beta Distribution
Expected time = ( Optimistic + 4 xExpected time = ( Optimistic + 4 x
Most likely + Pessimistic ) / 6Most likely + Pessimistic ) / 6
Expected time : Time corresponding 50Expected 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 & StandardExpected Time & Standard
DeviationDeviation
ActivityActivity ttoo ttmm ttpp
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 (TE)
ii) The Latest Allowable Occurance Time (TL)
The Earliest Expected Time (TE) 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
TE'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.
(TE)j = Max [(TE)i + tij]
The Latest Allowable Occurence Time (TL) :
The latest time by which an event must occur to keep the project on schedule
is called the latest allowable occurence time (TL). The calculation of TL of an
event is same as that LOT of CPM network by the method known as
Backward Pass. ; (TL)i = min ((TL)j – tij)

16. Scheduled Completion Time (Ts)
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 (Ts). Naturally. Ts refers to the latest allowable
occurence time (TL) of the last event of the project, i.e. (Ts= TL)·
SLACK .
Time box having two compartments is made at each event. the value in the
left compartment indicating the value of TE and that of in the right
compartment indicating TL of that event. And the slack of the event is given
by,
Slack (S) = (TL – TE )
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 TE and
TL of that event.

17. POSITIVE SLACK
When TL is more than TE. positive slack is obtained. It indicates the
project is ahead of schedule meaning thereby the excess resources.
ZERO SLACK
When TL is equal to TE 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 TL ) is less than TE
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

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

19. Critical PathCritical Path

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

21. Normal Distribution FunctionNormal Distribution Function
Sum of all expected time of all activities alongSum of all expected time of all activities along
critical path is equal to the expected time of lastcritical 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 assumeThough individual activities assume
random( beta distribution) but Trandom( beta distribution) but TEE of the project asof 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 DeviateNormal Deviate
(x): Distance from(x): Distance from
the meanthe mean
expressed inexpressed 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
te = (ta + 4tm + tb )/6
Standard deviation of activity δt = (tb - ta)/6
Variance of activity vt = (standard deviation)2
.
Earliest Expected Mean Time (TE ) and Latest allowable occurrence
time (TL ) are marked in time box at each event. Slack (S) = (TL - TE ) is
also mentioned on the network. Since scheduled completion time of
project is not mentioned, for the last event (8), TL = TE 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 (µT) = 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 (VT) = 15.76
Standard Deviation of the project (δT ) = sqrt(15.76) = 3.97
b) Probability factor (z) corresponding to x = 35 days
z = (x- µT )/ δt = (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 theHow 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 completionFind The probability of completion
within 35 dayswithin 35 days
10
9
9
7
11
5
8
Critical path 1-2-4-5, Te= 30 Variance 1-2= (18-6/6)2
=4, + 9 + 9 = 22 SD= 4.69

38. Benefits of PERTBenefits of PERT
PERT is useful because it provides the followingPERT 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 theThe critical path activities that directly impact the
completion time.completion time.
– The activities that have slack time and that can lendThe 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 andThe activity time estimates are somewhat subjective and
depend on judgement. In cases where there is littledepend on judgement. In cases where there is little
experience in performing an activity, the numbers mayexperience in performing an activity, the numbers may
be only a guess.be only a guess.
Even if the activity times are well-estimated, PERTEven if the activity times are well-estimated, PERT
assumes a beta distribution for these time estimates, butassumes 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, PERTEven if the beta distribution assumption holds, PERT
assumes that the probability distribution of the projectassumes that the probability distribution of the project
completion time is the same as the that of the criticalcompletion time is the same as the that of the critical
path. Because other paths can become the critical path ifpath. Because other paths can become the critical path if
their associated activities are delayed, PERT consistentlytheir associated activities are delayed, PERT consistently
underestimates the expected project completion time.underestimates the expected project completion time.