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386 CHAPTER 9 PROJECT SCHEDULING: PERT/CPM

Figure 9.12 Porta-Vac Project Network with Latest Start and Latest Finish Times

C 6 9 F 9 11
3 10 13 2 13 15

A 0 6 D 6 11 G 11 14 J 15 17
Finish
6 0 6 5 7 12 3 12 15 2 15 17

E 6 9 H 9 13 I 13 15
Start
3 6 9 4 9 13 2 13 15

B 0 2
2 7 9

The activity schedule for the Porta-Vac project is shown in Table 9.6. Note that
the slack time (LS – ES) is also shown for each activity. The activities with zero slack
(A, E, H, I and J) form the critical path for the Porta-Vac project network.

Variability in Project Completion Time
We know that for the Porta-Vac project the critical path of A–E–H–I–J resulted in
an expected total project completion time of 17 weeks. However, variation in critical
activities can cause variation in the project completion time. Variation in noncritical
activities ordinarily has no effect on the project completion time because of the slack
time associated with these activities. However, if a noncritical activity is delayed long
enough to expend its slack time, it becomes part of a new critical path and may affect
the project completion time. Variability leading to a longer-than-expected total time
for the critical activities will always extend the project completion time, and con-
versely, variability that results in a shorter-than-expected total time for the critical
activities will reduce the project completion time, unless other activities become
critical. Let us now use the variance in the critical activities to determine the
variance in the project completion time.
Let T denote the total time required to complete the project. The expected value
of T, which is the sum of the expected times for the critical activities is:

EðTÞ¼ t A þ t E þ t H þ t I þ t J
¼ 6 þ 3 þ 4 þ 2 þ 2 ¼ 17 weeks
The variance in the project completion time is the sum of the variances of the critical
path activities. Thus, the variance for the Porta-Vac project completion time is:

2
2
2
2
2
¼ þ þ þ þ 2 J
H
A
I
E
¼ 1:78 þ 0:11 þ 0:69 þ 0:03 þ 0:11 ¼ 2:72
2
2
2
2
2
Problem 10 involves a where ; ; ; and are the variances of the critical activities.
A E H I J
2
project with uncertain The formula for s is based on the assumption that the activity times are
activity times and asks
you to calculate the independent. If two or more activities are dependent, the formula provides only
expected completion an approximation of the variance of the project completion time. The closer the
time and the variance for activities are to being independent, the better the approximation.
the project. Knowing that the standard deviation is the square root of the variance, we
compute the standard deviation s for the Porta-Vac project completion time as:
p ffiffiffiffiffi p ffiffiffiffiffiffiffiffiffi
¼ ¼ 2:72 ¼ 1:65
2
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