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384 CHAPTER 9 PROJECT SCHEDULING: PERT/CPM

Table 9.4 Optimistic, Most Probable and Pessimistic Activity Time Estimates (In Weeks) for the
Porta-Vac Project
Activity Optimistic (a) Most Probable (m) Pessimistic (b)

A 4 5 12
B 1 1.5 5
C 2 3 4
D 3 4 11
E 2 3 4
F 1.5 2 2.5
G 1.5 3 4.5
H 2.5 3.5 7.5
I 1.5 2 2.5
J 1 2 3

a þ 4m þ b
t ¼ (9:4)
6
For activity A we have an average or expected time of:

4 þ 4ð5Þþ 12 36
t A ¼ ¼ ¼ 6 weeks
6 6
With uncertain activity times, we can use the variance to describe the dispersion
or variation in the activity time values. The variance of the activity time is given by
1
the formula :
2
b a
2
¼ (9:5)
6
The difference between the pessimistic (b) and optimistic (a) time estimates greatly
affects the value of the variance. Large differences in these two values reflect a high
degree of uncertainty in the activity time. Using Equation (9.5), we obtain the
2
measure of uncertainty – that is, the variance – of activity A, denoted :
A
12 4 2 2
8
2
¼ ¼ ¼ 1:78
A
6 6
Equations (9.4) and (9.5) are based on the assumption that the activity time
2
distribution can be described by a beta probability distribution. With this assump-
tion, the probability distribution for the time to complete activity A is as shown in
Figure 9.11. Using Equations (9.4) and (9.5) and the data in Table 9.4, we calculated
the expected times and variances for all Porta-Vac activities; the results are sum-
marized in Table 9.5.
1
1 The variance equation is based on the notion that a standard deviation is approximately / 6 of the difference
between the extreme values of the distribution: (b a)/6. The variance is the square of the standard deviation.
2 2
The equations for t and s require additional assumptions about the parameters of the beta probability
distribution. However, even when these additional assumptions are not made, the equations still provide
good approximations of t and s . 2

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