PROJECT CONTROL WITH PERT/CPM
CPM (Critical Path Method) determines the longest path and the critical activities
along this path in a project network; the project completion time can't be shorter than the
duration of this longest path unless such techniques as time crashing are used. PERT
(Program Evaluation and Review Technique) incorporates probabilistic elements into the
computation of activity durations and hence the project completion time; PERT uses
optimistic (o), most probable (m), and pessimistic (p) activity times to estimate the expected
activity times. Expected time is given by (o + 4m + p) / 6 and variance is given by ((p o)/6)2 . The reason for dividing by 6 is due to the fact that the area under the normal curve
between -3 and 3 accounts for more than 99% of the total probability of 100%. In case of
standard normal curve,  = 1; hence -3 = -3(1) = -3 and 3() = 3(1) = 3 and the difference
between 3 and -3 is 6. An example will illustrate the PERT/CPM technique.
Example:
The optimistic, most probable, and pessimistic times (in days) for completion of
activities for a certain project are as follows:
ACTIVITY
IMMEDIATE
PREDECESSOR
OPTIMISTIC
TIME (o)
A
-
4
MOST
PROBABLE
TIME (m)
5
B
-
6
8
10
C
A
6
6
6
D
B
3
4
5
E
B
2
3
4
F
C,D
8
10
12
G
E
6
7
8
H
C,D
12
13
20
I
F,G
10
12
14
a) Find the critical path.
b) Find the probability that all critical activities will be
completed in 35 days or less.
PESSIMISTIC
TIME (p)
6
Solution:
a) The expected times are obtained by using the formula
(o + 4m + p) / 6 and the
2
variances are obtained by using the formula ((p-o)/6) and are summarized in the following
table:
EXPECTED
TIME
5
8
6
4
3
10
7
14
12
ACTIVITY
A
B
C
D
E
F
G
H
I
VARIANCE
.11
.44
0
.11
.11
.44
.11
1.78
.44
The project network diagram based on the expected values in the table above is depicted
below:
C
ES EF
6
14
7
6,12
20,34
4
0,5
12,26
H
4
2
5
5,11
F
12
22,34 22,34
1,6
12,22
12,22
A
8,12
LS LF
8,12
10
1
B
6
0,8
0,8
7
11,18
G
15,22
D
8
3
E
8,11
12,15
3
5
I
The numbers in brackets above the arrows show the earliest start (ES) and earliest
finish (EF) for each activity respectively whereas the numbers in brackets below the arrows
show the latest start (LS) and latest finish (LF) times respectively. EF = ES + activity
duration and LF = LS + activity duration Activity slack = LS - ES = LF - EF. Activity slack
shows how much an activity can be delayed without affecting the project completion time.
For example, in our problem slack for activity E = 12 - 8 = 4 ; hence we can delay activity E
for 4 days and still finish the project in the expected completion time of 34 days.
On the other hand, when the activity slack is zero, that activity can be delayed zero
days, meaning that it can't be delayed; hence an activity is critical whenever its slack is zero.
Such an activity is called critical because any delay in the completion of that activity would
delay the whole project. In our example, activities B, D, F and I have zero slack; hence they
are critical activities. All other activities in our example are non critical. Knowing which
activities are critical and which are non critical may be quite useful; whenever there is an
unexpected delay in critical activities we can shift resources like capital, manpower, etc.
from non critical activities that can be delayed to the critical activities which can't be
delayed.
In our problem, the critical path can be depicted as follows:
Critical path: B D F I or 1  3 4  6  7
Expected completion time,  = 34 days.
Variance along the critical path:
2 = 2B + 2D + 2F + 2I = .44 + .11 + .44 + .44 = 1.43
 = 1.43 = 1.2
b) P(x ≤ 35) = P(z ≤ (35 - 34)/1.2) = P(z ≤ 0 .83) = 0.7967 ≤ 0 .8 = 80%
NOTE: z = (x-) /  for standard normal distribution. In our problem x = 35,  = 34 and
 = 1.2 and that is how we get P(z ≤ (35 - 34)/1.2) above.
PERT/CPM also gives managers a pretty good idea about the probability of
completing the projects so that they can plan ahead to expedite certain activities if necessary.
Also, these probability computations help managers to select and accept those projects that
have higher chances of completion within a specified time frame.