Network Analysis
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NetWork(start/end node)
A. Draw network?
B. Determine the critical path
Activity
(I,j)
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Activity
Duration
(1,2)
A
5
(1,4)
B
4
(1,5)
C
3
(2,3)
D
1
(2,5)
E
2
(2,6)
F
3
(3,4)
G
7
(3,6)
I
9
(4,6)
J
1
(4,7)
K
10
(5,6)
L
4
(5,7)
M
5
(6,7)
N
2
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NetWork(Precedence list)
A.
B.
Draw network?
Determine the critical path
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Activities
Predecessor
A
-
B
-
C
A
D
B
E
C,D
F
E
Output
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Program Evaluation and review Technique PERT
Example: Consider the following project that has activities, where the a,m and b time
are given in the following table:
 Find Dij, V(Dij) ?
 Find the critical path and its expected and variance
Of total duration time of project?
activity
Solution by QM and Tora
A
(I,j)
(a,m,b)
(1,2)
(3,5,7)
B
(1,3)
(4,6,8)
C
(2,4)
(1,3,5)
D
(3,4)
(5,8,11)
E
(4,5)
(1,2,3)
F
(5,6)
(9,11,13)
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Critical path method (CPM)
Critical path: It is a sequence of critical
activities.
The critical path calculation involve
• Forward pass: determine the earliest
occurrence times of the events
.
• Backward pass : determine the latest
occurrence time of the events( latest
completion time) .
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Where
j  Max
i  Dij
 j  i
The project starts at time 0
and
i  Min 
j  Dij
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 j  i
Construction of the time schedule
• Construction of preliminary schedule for the
project.
Activities
Noncritical activities
Critical activities
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Critical Activities
• An activity (i,j) will be critical if
•
i=
j
•
i=
j
•
ji= ji=Dij
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Solution by Tora
Open tora select from menu project
planning CPM critical method
and click enter.
Enter activities and durations
as the following table:
Activity
(I,j)
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Activity
Duration
(1,2)
A
5
(1,4)
B
2
(1,5)
C
2
(2,3)
D
7
(2,5)
E
5
(2,6)
F
4
(3,4)
G
3
(3,6)
I
10
(4,6)
J
5
(4,7)
K
9
(5,6)
L
4
(5,7)
M
3
(6,7)
N
3
Click on solve problem and select from options CPM bar chart and CPM
calculations then we get :
• Figure 1:
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Determination of the floats
Floats : are the slack times available within •
the
allotted span of the uncritical activity.
TFij  
jiD 
1- Total floats
ij
2- Free floats
FFij  
j-
Where
i  Dij

FFij  TFij
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Figure 2:
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Red –Flagging Rule:
The implication of the rule is that a noncritical
activity will be red flagged if its.
FFij  TFij
From previous example Red-Flagging are B and C.
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