YÖNEYLEM ARAŞTIRMASI
Known as the "Project Evaluation and Review Technique", PERT ANALYSIS was
first applied to the US Navy's Special Project Office Polaris Project in 1958, and two years
were gained in the completion of the project. It is seen that this technique is used in the
investments of large-scale projects such as the Keban Dam and the Istanbul Bosphorus
Bridge in Turkey .
PERT is a method that minimizes delays, hang-ups and conflicts in production,
coordinates and synchronizes various parts of the whole job , and accelerates the completion
of projects. 2 PERT achieves the desired goal.
It is an approach to the planning, coordination, and control of business efforts necessary to
achieve
In the CPM method, the time periods of the activities are known precisely. However,
in practice, it is not possible to know the exact duration of the activities. If the durations of the
activities are not known precisely, there may be probabilistic time periods.
In such cases, PERT analysis is used. While developing the first PERT analysis, the activity times that
are not known with certainty were accepted as a probability distribution and thus
the critical path obtained was called the probabilistic critical path . Also, in practice it would be
unrealistic to ask the person or the organization the probability distribution of the completion
times of specific activities; instead, three separate periods are determined for
each activity. These periods are;
The most optimistic period'. It is the time when the activity will be completed as quickly as possible
when everything goes as desired . Its symbol is . a is such a period that 99% of this activity cannot
be completed in less than a. e i i i time: It is the time to finish the activity in the worst
cases. The most pessimistic period
is such a duration that this activity, like the most
optimistic period, cannot last longer than 99%. In a sense, it is the maximum completion time of
the activity. The most probable i e duration (possible activity duration): This is the duration of
the activity to be completed under expected conditions based on past experience. Its
symbol is .
The probability distribution for each activity is the beta distribution.
↑
*
µ = I x and variance = σ2 =Is2
To summarize, the steps to follow to implement the PERT method are:
For each activity, estimates of a, b, and m should be obtained. These estimates should be made by the
project manager or obtained from people involved in similar projects .
the mean and variance formulas, the average duration and variance of each activity are found.
Based on the average activity times, the critical path is determined.
After the critical activities are determined, the averages and variances of the critical activities are added to
find the variance and averages of the total project duration .
variance determined in step 4 and the total project duration are normally distributed.
Consider a project with eight events and eleven activities. Table 9.10 below gives
us the PERT forecast times, expressed in days .
1
1
2
2
3
4
-5
5
6
6
7
-
-
-
-
-
T
3
2
3
4
5
7
6
6
7
7
8
8
3
3
5
3
2
4
2
5
4
6
3
3
2
5
2
3
6
②3
6
6
7
4
5
3
7
4
4
eDrawing of the network diagram of the project
-
S Finding the average expected time and variance of each activity
Finding the critical path of the project
Calculation of the standard deviation of the distribution showing the total
expected time of the project
The probability that the project will be completed in 19 days or less
When it is desired to be 95% sure that the project will be completed in the time
available for the completion of the project , it is not necessary to find out what the
project's completion time should be .
#-6
E
1
>
>
a
9
↓
·
↑
-
SOLUTION
a) The network diagram of the project is shown in figure 9.36. Event 1 indicates the start of
the project and event 8 indicates the completion of the project.
-
~
S
-
~
-
-
↑
x=6
Figure 9.36
b) To find the expected time and variance of each activity
D
O
formulas are used.
Accordingly,
and v2 values are given by calculating as in Table 9.11 .
↓
Activities
1---> 2
1 ---> 3
2----> 4
2- ---> 5
3- --> 7
4---> 6
5- --> 6
5---> 7
6---> 7
6---> 8
7 ---> 8
4.17
2
4.83
4.17
6
2.83
3
2
5
2.17
3
2
2
0.25
0.11
0.25
0.25
0.11
0.25
0.44
0.11
0.44
0.25
0.11
The
of the expected times ( ) in the network diagram for the
relevant activities will help us find the critical path.
cr"t"cal path
~
c) 1 — >2— » 4—>6—>7—>8 , since the path that takes the most expected times
-
S
-
-
-
( ) from start to finish is called the “critical path,” and the time is = 19.83 days. The
gap time value of each event on the critical path is positive. In addition, when we
look at the variance calculations , there are no activities on the "critical path" with the
largest variance value ( 2 ). By analyzing large variance- valued activities, the project can
be accelerated to finish earlier.
&
d) Total expected time of the project
+ 4.83 + 2.83 + 5 + 3
~µµ == 4.17
19.83 days.
~Variance =
= 0.25 + 0.25 + 0.25 + 0.44 + 0.11
= 1.3 days
and the standard deviation is
=
= 1.14 days. The total time of the project is
approximately normally distributed, with a mean µ = 19.83 and variance
= 1.14.
-
-
e) Analytical tools provide the possibility of finishing the project at the agreed
time. While it is important to know the probability of a single event to be
completed in the stipulated time, it is more important to complete the entire
project in the desired time (y).
z icon represents the probability value in the normal distribution table.
=
-soruda
ver"ge d
& y =19days
µ= 19.83 days
L
= 1.14
=
z = -0.73 is located in the area to the left of the mean under the normal curve. The value of 0.73
in the area under the normal probability curve is 0.7673 . Probability of realization of the project
&
at the desired time Pr = 1 - 0.7673 = 0.2327
-
-
The probability of completing the project in 19 days or less is 23%.
f)
-
=>
S
The following formula may be useful to find the desired predictive value for y
.
>
y = µ+ (z)
↓ &
y = 19.83 + (1.14) (1.645)
y = 21.71 days