In the name of Allah the Most Gracious the Most
Merciful
Construction Management
and Planning
2
Float

Total Float is Calculated by Subtracting Early
Start and Duration from the Activity’s Late
Finish Time

An Activity that has Zero Total Float is a Critical
Activity

An Activity’s Start may be Critical Even Though
an Activity itself may not be Critical

Start Float may be Calculated by Subtracting
an Activity’s Early Start Time from Its Late Start
Time
Float

Finish Float is Calculated by Subtracting an
Activity Early Finish Time from its Late Finish
Time
Project Scheduling
Probabilistic PERT
History of PERT
• Project Evaluation and Review Technique
(PERT)
– U S Navy (1958) for the POLARIS
missile program
– Multiple task time estimates
(probabilistic nature)
– Activity-on-arrow network construction
– Non-repetitive jobs (R & D work)
6
PERT Probability Approach to
Project Scheduling
• Activity completion times are seldom known with certainty.
• Completion time estimates can be estimated using the
Three Time Estimate approach. In this approach, three
time estimates are required for each activity:
a = an optimistic time to perform the activity
m = the most likely time to perform the activity
b = a pessimistic time to perform the activity
PERT
– pessimistic time (a) - the time the
activity would take if things did not go
well
– most likely time ( m) - the consensus
best estimate of the activity’s duration
– optimistic time (b) - the time the activity
would take if things did go well
8
3-Time Estimate Approach
Probability Distribution
• With three time estimates, the activity completion time can
be approximated by a Beta distribution.
• Beta distributions can come in a variety of shapes:
a m
b
a
m
b
a
mb
Mean and Standard Deviation for
Activity Completion Times
• The best estimate for the mean is a weighted average
of the three time estimates with weights 1/6, 4/6, and
1/6 respectively on a, m, and b.
• Since most of the area is with the range from a to b (ba), and since most of the area lies 3 standard
deviations on either side of the mean (6 standard
deviations total), then the standard deviation is
approximated by Range/6.
 = the mean completion
time =
a + 4m + b
6
 = the standard deviation
=
b-a
6
The Project Completion Time
Distribution
The three assumptions imply that the overall
project completion time is normally distributed,
with:
 = Sum of the ’s on the critical path
2 = Sum of the 2 ’s on the critical path
PERT analysis
• Draw the network.
• Analyze the paths through the network and find the
critical path.
• The length of the critical path is the mean of the project
duration probability distribution which is assumed to be
normal
• The standard deviation of the project duration probability
distribution is computed by adding the variances of the
critical activities (all of the activities that make up the
critical path) and taking the square root of that sum
• Probability computations can now be made using the
normal distribution table.
12
Probability computation
• Determine probability that project is
completed within specified time
Z=
x-

where  = tp = project mean time
 = standard deviation
x = (proposed ) specified time
13
Normal Distribution of Project Time
Probability
Z
 = tp
x
Time
14
Immed. Optimistic Most Likely Pessimistic
Activity Predec. Time (Hr.) Time (Hr.) Time (Hr.)
A
-4
6
8
B
-1
4.5
5
C
A
3
3
3
D
A
4
5
6
E
A
0.5
1
1.5
F
B,C
3
4
5
G
B,C
1
1.5
5
H
E,F
5
6
7
I
E,F
2
5
8
J
D,H
2.5
2.75
4.5
K
G,I
3
5
7
15
16
PERT Example
Activity
A
B
C
D
E
F
G
H
I
J
K
Expected Time (μ)
(a + 4m + b)/6
6
4
3
5
1
4
2
6
5
3
5
Variance (σ2)
[(b – a)/6]2
4/9
4/9
0
1/9
1/36
1/9
4/9
1/9
1
1/9
4/9
17
18
σ2path = σ2A + σ2C + σ2F + σ2I + σ2K
= 4/9 + 0 + 1/9 + 1 + 4/9
= 2
σpath = 1.414
Mean project completion time = 23
Proposed completion time = 24
z = (24 - 23)/(24-23)/1.414 = .71
From the Standard Normal Distribution
table:
P(z < .71) = .5 + .2612 = .7612
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Using the PERT-CPM Template for
Probabilistic Models
• Instead of calculating µ and  by hand, the
Excel template may be used.
• Instead of entering data in the µ and 
columns, input the estimates for a, m, and
b into columns C, D, and E.
– The template does all the required
calculations
– After the problem has been solved, probability
analyses may be performed.
Enter a, m, b instead of 
Call Solver
Click Solve
Go to PERT OUTPUT worksheet
Call Solver
Click Solve
To get a cumulative
probability, enter
a number here
P(Project is completed in less than 180 days