PERT

© 2006 Prentice Hall, Inc.
6/18/20201
Welcome To Heartiest
Students
Dereje Jima
Construction Planning and Management
Department of Civil Engineering
ADDIS COLLEGE
Semester I
2017

6/18/20202
Stochastic Project Scheduling
Lecture
On
Stochastic Project Scheduling – Project Evaluation
and Review Technique
[PERT]

6/18/20203
Stochastic Project Scheduling [PERT]
Introduction
In some situations, estimating activity duration becomes
a difficult task due to ambiguity inherited in and the
risks associated with some work.
Thus, the duration of an activity is estimated as a
range of time values rather than being a single
value.
This section deals with the scheduling of the project

6/18/20204
[PERT]
Scheduling with Uncertain Durations
Using the probabilistic distribution of activity
durations.
Duration of a particular activity is assumed to be a
random variable that is distributed in a particular fashion.
An activity duration might be assumed to be distributed
as a normal or a beta distributed random variable as
shown in figure below.

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[PERT]
Figure shows the probability or chance of experiencing
a particular activity duration based on a probabilistic
distribution.

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[PERT]
The beta distribution is often used to characterize
activity durations, since it can have an absolute
minimum and an absolute maximum of possible
duration times.
The normal distribution is a good approximation to the
beta distribution in the center of the distribution and
is easy to work with, so it is often used as an
approximation.

6/18/20207
[PERT]
The most common formal approach to incorporate
uncertainty in the scheduling process is to apply the
critical path scheduling process and then analyze the
results from a probabilistic perspective.
This process is usually referred to as the Program
Evaluation and Review Technique [PERT] method.
The critical path [determined] is then used to analyze
the duration of the project incorporating the uncertainty
of the activity durations along the critical path.

6/18/20208
[PERT]
The expected project duration is equal to the sum of
the expected durations of the activities along
the critical path.
Assuming that activity durations are independent
random variables, the variance or variation in the
duration of this critical path is calculated as the
sum of the variances along the critical path.

6/18/20209
[PERT]
With the mean and variance of the identified critical
path known, the distribution of activity durations can also
be computed.

6/18/202010
[PERT]
Program Evaluation and Review Technique
PERT introduces uncertainty into the estimates for
activity and project durations.
PERT is well suited for those situations where there is
either insufficient background information to specify
accurately time and cost or where project activities
require research and development.

6/18/202011
[PERT]
AON diagramming can be easily used alternatively
AOA [original development approach].
The method is based on the well-known “central
limit theorem”.

6/18/202012
[PERT]
The theorem states that:
… “Where a series of sequential independent activities
lie on the critical path of a network, the sum of the
individual activity durations will be distributed in
approximately normal fashion, regardless of the
distribution of the individual activities themselves….
… The mean of the distribution of the sum of the activity
durations will be the sum of the means of the individual
activities and its variance will be the sum of the

6/18/202013
[PERT]
Primary assumptions of PERT can be summarized as
follows:
Any PERT path must have enough activities to make
central limit theorem valid.
The mean of the distribution of the path with the
greatest duration, from the initial node to a given
node, is given by the maximum mean of the duration
distribution of the paths entering the node.
PERT critical path is longer enough than any other path in

6/18/202014
Six Steps PERT & CPM
1. Define the project and prepare the work breakdown
structure
2. Develop relationships among the activities - decide
which activities must precede and which must follow
others
3. Draw the network connecting all of the activities

6/18/202015
Step Cont’d
4. Assign time and/or cost estimates to each activity
5. Compute the longest time path through the network –
this is called the critical path
6. Use the network to help plan, schedule, monitor, and
control the project

Questions PERT & CPM Can Answer
1. When will the entire project be completed?
2. What are the critical activities or tasks in the project?
3. Which are the noncritical activities?
4. What is the probability the project will be completed by
a specific date?
6/18/202016
[PERT]

5. Is the project on schedule, behind schedule, or ahead
of schedule?
6. Is the money spent equal to, less than, or greater than
the budget?
7. Are there enough resources available to finish the
project on time?
8. If the project must be finished in a shorter time, what is
the way to accomplish this at least cost?6/18/202017
[PERT]

6/18/202018
[PERT]
Variability in Activity Times
CPM assumes we know a fixed time estimate for
each activity and there is no variability in activity
times
PERT uses a probability distribution for activity
times to allow for variability

6/18/202019
[PERT]
Variability in Activity Times: Three time
estimates are required
Optimistic time (a): Estimate of the minimum time
required for an activity if exceptionally good luck is
experienced. If everything goes according to
plan.
Most–likely or modal time (m) [Most
realistic/probable estimate]: Time required if the
activity is repeated a number of times under

6/18/202020
[PERT]
Pessimistic time (b) [Assuming very
unfavorable conditions]: Estimate of the maximum
time required if unusually bad luck is experienced.

6/18/202021
[PERT]
These three time estimates become the framework on
which the probability distribution curve for the
activity is erected.
Many authors argue that beta distribution is mostly
fit construction activities.
The three times are thought to be easier for
managers to estimate subjectively.

6/18/202022
[PERT]
Using the three times estimates, the expected mean
time (te) is derived using equation below.
Then, te is used as the best available time
approximation for the activity in question.
The standard deviation (σ) is given by equation below,
The variance (ν) can be determined as ν = σ2.

6/18/202023
[PERT]
Estimate follows beta distribution
Standard Deviation
Expected time:
Variance of times:
t = (a + 4m +
b)/6
v = [(b – a)/6]2

6/18/202024
[PERT]
By adopting activity expected mean time, the critical
path calculations proceed as CPM.
Associated with each duration in PERT, however,
is its standard deviation or its variance.
The project duration is determined by summing up the
activity expected mean time along the critical path
and thus will be an expected mean duration.

6/18/202025
[PERT]
Since the activities on the critical path are
independent of each other, central limit theory gives
the variance of the project duration as the sum of
the individual variances of these critical path
activities.

6/18/202026
[PERT]
Once the expected mean time for project duration
(TX) and its standard deviation (σX) are determined,
it is possible to calculate the chance of meeting
specific project duration (TS).
Then normal probability tables are used to
determine such chance using Equation below.

6/18/202027
[PERT]
The procedure for hand probability computations using
PERT can be summarized in the following steps:
1. Make the usual forward and backward pass
computations based on a single estimate (mean) for
each activity.
2. Obtain estimates for a, m, and b for only critical
activities. If necessary, adjust the length of the critical
path as dictated by the new te values based on a, m,
and b.

6/18/202028
[PERT]
3. Compute the variance for event x (νX) by
summing the variances for the critical activities
leading to event x.
4. Compute Z and find the corresponding normal
probability.

6/18/202029
[PERT]
Estimate follows beta distribution
Expected time:
Variance of times:
t = (a + 4m + b)/6
v = [(b − a)/6]2
Probability
of 1 in 100
of > b
occurring
Probability
of 1 in 100
of
< a
occurring
Probability
Optimistic
Time (a)
Most Likely
Time (m)
Pessimistic
Time (b)
Activity
Time

6/18/202030
[PERT]
Most Expected
Optimistic Likely Pessimistic Time Variance
Activity a m b t = (a + 4m + b)/6 [(b – a)/6]2
A 1 2 3 2 .11
B 2 3 4 3 .11
C 1 2 3 2 .11
D 2 4 6 4 .44
E 1 4 7 4 1.00
F 1 2 9 3 1.78
G 3 4 11 5 1.78
H 1 2 3 2 .11
Computing Variance

6/18/202031
[PERT]
Project variance is computed by summing the
variances of critical activities
s2 = Project variance
= (variances of activities
on critical path)
p
Probability of Project Completion

6/18/202032
[PERT]
Project variance
s2 = .11 + .11 + 1.00 + 1.78 + .11 = 3.11
Project standard deviation
sp = Project variance
= 3.11 = 1.76 weeks
p

6/18/202033
[PERT]
Standard deviation = 1.76 weeks
15 Weeks
(Expected Completion Time)

6/18/202034
[PERT]
What is the probability this project can be completed
on or before the 16 week deadline?
Z = – /sp
= (16 wks – 15 wks)/1.76
= 0.57
due expected date
date of completion
Where Z is the number of
standard deviations the due
date lies from the mean

6/18/202035
[PERT]
Z= − /sp
= (16 wks − 15 wks)/1.76
= 0.57
due expected date
date of completion
Where Z is the number of
standard deviations the due
date lies from the mean
.00 .01 .07 .08
.1 .50000 .50399 .52790 .53188
.2 .53983 .54380 .56749 .57142
.5 .69146 .69497 .71566 .71904
.6 .72575 .72907 .74857 .75175
From Appendix I

6/18/202036
[PERT]
Time
Probability
(T ≤ 16 weeks)
is 71.57%
0.57 Standard deviations
15 16
Weeks Weeks

6/18/202037
[PERT]
Probability
of 0.01
Z
From Appendix I
Probability
of 0.99
2.33
Standard
deviations0 2.33
Determining Project Completion Time

6/18/202039
[PERT]
[PERT]

PERT

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