Civil Engineering Reference
In-Depth Information
Duration
Activity
Optimistic
(
To
)
Most
Likely (
Tm
)
Pessimistic
(
Tp
)
Expected
Duration (
Te
)
Standard
Deviation (
σ
e
)
Variance
(
V
e
=σ
e
)
A
4
5
6
5.00
0.33
0.11
C
13
9
16
9.83
1.50
2.25
F
4
7
13
7.50
1.50
2.25
TE
=
22.33
VE
=
4.61
(
V
e
=σ
e
)
The
Z
value associated with a probability of 95% is 1.645, and
T
S
=σ
E
⋅
Z
+
T
E
=
2
.
15
1
.
645
+
22
.
33
=
25
.
87
≈
26 days
⋅
Note
: The preceding calculations were done using an Excel spreadsheet.
The answers were rounded for simplicity.
Determining the Probability of a Certain Project Finish Date (Multiple Paths
Considered)
In statistics and probability, the probability that two
independent
events will happen
simultaneously is equal to the product of the two probabilities:
Pr(
A
∩
B
)=Pr(
A
)
⋅
Pr(
B
)
(11.9)
Equation 11.9 holds true only if events A and B are independent. If they are
dependent, the probability that both events will happen at the same time will depend
on the correlation between the two events and cannot be defined in one simple
equation. For example, in the network of Figure 11.10, paths BDG and CFI may
be, but not necessarily, independent. We cannot be sure until we know everything
about them. Even though they are not linked by a logical relationship, they could be
dependent, for example, if activities on both paths are performed by the same crew
or use materials that will be delivered in the same shipment.
Theoretically, if we need to calculate the probability that the project in
Figure 11.10 will finish by a certain date, we need to consider all paths. In our small
example, we have five paths: ABDG, ABEHI, ABDHI, ACEHI, and ACFI. Since they
all share at least one activity, A, they have dependencies. Calculating the probability
that the project will finish by a certain date requires not only complicated calculations
but also correlation factors between activities that may be difficult to obtain or
estimate. In real-life projects, we may have tens, hundreds, or even thousands of
paths in a CPM network. For practical reasons, we focus on one path at a time, as we
did in the last example.
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