Step by step network construction and (cpm) calculations

Step by Step
Network Construction and (CPM)
Calculations
Project Management
Dr. Mahmoud Abbas Mahmoud
Asst. Prof.
2016

Step by Step
Network Construction and (CPM)
Calculations
Duration
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Example*
Draw the (AOA) network diagram for
the activities given in the table, find
the critical path (CP), and give all
(CPM) calculations (complete network
analysis table).
*This Example is adapted from “Operation Research - An
Introduction - 8th Edition - H.A. Taha (2007)
These steps will be illustrated
with an Example.

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0
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Note:
Practically, nodes numbering
can be start by (0) or (1).

A
0
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A
0
1
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B
A
0
1
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B
A
0
2
1
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B
A
0
2
1
C
Duration
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B
A
0
2
1
C
D
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B
A
0
2
1
C
D
E
Duration
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B
A
0
2
1
C 3
D
E
Duration
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B
A
0
2
1
C 3
D
E
G
F
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B
A
0
2
1
C 3
D
E 4
G
H
F
Duration
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???????

B
A
0
2
1
C 3
D
E 4
G
H
F
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Use Dummy !

B
A
0
2
1
C 3
D
E 4
G
H
F
Duration
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3-B
2AC
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5C,D,EI
6H,FJ

B
A
0
2
1
C 3
D
E 4
I
G
H
F
Duration
(week)
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2AC
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B
A
0
2
1
C 3
D
E 4
5
I
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H
F
Duration
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6H,FJ

B
A
0
2
1
C 3
D
E 4
5
I
J
G
H
F
Duration
(week)
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Activity
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6H,FJ

B
A
0
2
1
C 3
D
E 4
5 6
I
J
G
H
F
Duration
(week)
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7C,D,EH
5C,D,EI
6H,FJ

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
(AOA) Network Drawing is Completed
Duration
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
Critical Path Calculations
Duration
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
Forward Pass:
Duration
(week)
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
Forward Pass:
Calculating the Earliest Start Time (ES):
Duration
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
Forward Pass:
Calculating the Earliest Start Time (ES):
Earliest Start Time (ES)j=Max [ (ES)i + Dij ]
Duration
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
Duration
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NOTE: (ES) for Start Event = (0) Is NOT RELATED
with the Number Given to Start Event (in Circle).
Earliest Start Time (ES) for Start Event = (0)

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
Duration
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
2
0
[2+0]= 2
Duration
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
Duration
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[0+3]= 3

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6 Max [(3+3) , (2+2)] = max [6,4]= 6
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
max[(3+2) , (6+0)] = max [5,6] = 6
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
13
Max [(6+3) ,(6+7)] =max [9,13] =13
Duration
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
19
13
Max [(6+2),(13+6),(6+5]=
Max [8,19,9] = 19
Duration
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
13
Backward Pass:
Duration
(week)
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19

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
13
Backward Pass:
Calculating the Latest Completion Time (LC):
Duration
(week)
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19

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
13
Backward Pass:
Calculating the Latest Completion Time (LC):
Latest Completion Time (LC)i=Min [ (LC)j - Dij ]
Duration
(week)
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Activity
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19

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
19
13
Duration
(week)
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Activity
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
19
19
13
Duration
(week)
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Activity
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
19
19
13
13
19 – 6 =13
Duration
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
6
19
19
13
13
Min [(19-5),(13-7)=
Min[14,6] = 6
Duration
(week)
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
2
0
6
6
6
6
19
19
13
13
Min [(19-2),(13-3),(6-0)] = Min [17,10,6] = 6
Duration
(week)
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
3
2
0
6
6
6
6
19
19
13
13
Min [(6-2),(6-3)] =
Min [4, 3] = 3
Duration
(week)
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
3
3
4
2
0
6
6
6
6
19
19
13
13
6-2 = 4
Duration
(week)
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
Min [(3-3),(4-2)] =
Min [0, 2] = 0
Duration
(week)
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B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
Duration
(week)
Preceding
Activity
Activity
2-A
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2AC
3BD
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3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
Finding the Critical Activities and the Critical Path

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
To Identify the Critical Activities
Remember the Three Conditions:
For a critical activity
i) (LC)i = (ES)i
ii) (LC)j = (ES)j
iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
Critical Activity Conditions:
i) (LC)i = (ES)i
ii) (LC)j = (ES)j
iii)(ES)j - (ES)i = (LC)j - (LC)i = Dij
Duration
(week)
Preceding
Activity
Activity
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3BD
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2C,DG
7C,D,EH
5C,D,EI
6H,FJ

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
The Critical Activities are: (0, 2), (2, 3), (3, 4), (4, 5) and (5, 6)
The Critical Path is: (0 – 2 – 3 – 4 – 5 – 6)
Project Completion Time is: 19 weeks

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
Total Float (TF) and Free Float (FF)
TFij = (LC)j – (ES)i – Dij
FFij = (ES)j – (ES)i – Dij

Notice that only a critical activity
can have a Zero Total Float (TF)
B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
For activity A (0, 1)
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0
For activity B (0, 2)
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0
For activity C (1, 3)
TF = 6 – 2 – 2 = 2
FF = 6 – 2 – 2 = 2 Dr. Mahmoud Abbas Mahmoud

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0
TF = 6 – 2 – 2 = 2
FF = 6 – 2 – 2 = 2
For activity D (2, 3)
TF = 6 – 3 – 3 = 0 *
FF = 6 – 3 – 3 = 0

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0
TF = 6 – 2 – 2 = 2
FF = 6 – 2 – 2 = 2
TF = 6 – 3 – 3 = 0 *
FF = 6 – 3 – 3 = 0
For activity E (2, 4)
TF = 6 – 3 – 2 = 1
FF = 6 – 3 – 2 = 1

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0
TF = 6 – 2 – 2 = 2
FF = 6 – 2 – 2 = 2
TF = 6 – 3 – 3 = 0 *
FF = 6 – 3 – 3 = 0
TF = 6 – 3 – 2 = 1
FF = 6 – 3 – 2 = 1
For Dummy activity (3, 4)
TF = 6 – 6 – 0 = 0 *
FF = 6 – 6 – 0 = 0

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0
TF = 6 – 2 – 2 = 2
FF = 6 – 2 – 2 = 2
TF = 6 – 3 – 3 = 0 *
FF = 6 – 3 – 3 = 0
TF = 6 – 3 – 2 = 1
FF = 6 – 3 – 2 = 1
TF = 6 – 6 – 0 = 0 *
FF = 6 – 6 – 0 = 0
For activity F (3, 5)
TF = 13 – 6 – 3 = 4
FF = 13 – 6 – 3 = 4

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0
TF = 6 – 2 – 2 = 2
FF = 6 – 2 – 2 = 2
TF = 6 – 3 – 3 = 0 *
FF = 6 – 3 – 3 = 0
TF = 6 – 3 – 2 = 1
FF = 6 – 3 – 2 = 1
TF = 6 – 6 – 0 = 0 *
FF = 6 – 6 – 0 = 0
TF = 13 – 6 – 3 = 4
FF = 13 – 6 – 3 = 4
For activity G (3, 6)
TF = 19 – 6 – 2 = 11
FF = 19 – 6 – 2 = 11

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0
TF = 6 – 2 – 2 = 2
FF = 6 – 2 – 2 = 2
TF = 6 – 3 – 3 = 0 *
FF = 6 – 3 – 3 = 0
TF = 6 – 3 – 2 = 1
FF = 6 – 3 – 2 = 1
TF = 6 – 6 – 0 = 0 *
FF = 6 – 6 – 0 = 0
TF = 13 – 6 – 3 = 4
FF = 13 – 6 – 3 = 4
TF = 19 – 6 – 2 = 11
FF = 19 – 6 – 2 = 11
For activity H (4, 5)
TF = 13 – 6 – 7 = 0 *
FF = 13 – 6 – 7 = 0

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0
TF = 6 – 2 – 2 = 2
FF = 6 – 2 – 2 = 2
TF = 6 – 3 – 3 = 0 *
FF = 6 – 3 – 3 = 0
TF = 6 – 3 – 2 = 1
FF = 6 – 3 – 2 = 1
TF = 6 – 6 – 0 = 0 *
FF = 6 – 6 – 0 = 0
TF = 13 – 6 – 3 = 4
FF = 13 – 6 – 3 = 4
TF = 19 – 6 – 2 = 11
FF = 19 – 6 – 2 = 11
TF = 13 – 6 – 7 = 0 *
FF = 13 – 6 – 7 = 0
For activity I (4, 6)
TF = 19 – 6 – 5 = 8
FF = 19 – 6 – 5 = 8

B=3
A=2
0
2
1
C=2 3
D=3
E=2 4
5 6
I=5
J=6
G=2
H=7
F=3
0
0
3
3
4
2
0
6
6
6
6
19
19
13
13
TF = 4 – 0 – 2 = 2
FF = 2 – 0 – 2 = 0
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0
TF = 6 – 2 – 2 = 2
FF = 6 – 2 – 2 = 2
TF = 6 – 3 – 3 = 0 *
FF = 6 – 3 – 3 = 0
TF = 6 – 3 – 2 = 1
FF = 6 – 3 – 2 = 1
TF = 6 – 6 – 0 = 0 *
FF = 6 – 6 – 0 = 0
TF = 13 – 6 – 3 = 4
FF = 13 – 6 – 3 = 4
TF = 19 – 6 – 2 = 11
FF = 19 – 6 – 2 = 11
TF = 13 – 6 – 7 = 0 *
FF = 13 – 6 – 7 = 0
For activity I (4, 6)
TF = 19 – 6 – 5 = 8
FF = 19 – 6 – 5 = 8
For activity J (5, 6)
TF = 19 – 13 – 6 = 0 *
FF = 19 – 13 – 6 = 0

We can notice from these results that all
critical activities must have zero free
floats but not each activity that has zero
free float must be critical.
For example, activity A (0, 1) has a zero
Free Float but it is not critical.

Earliest Completion (EC) and Latest Start (LS) Times
Activity
Symbol
Activity
(i, j)
Duration
Dij
(ES)i
ECij
= (ES)i + Dij
LSij
= (LC)j - Dij
(LC)j
A (0, 1) 2 0 2 2 4
B (0, 2) 3 0 3 0 3
C (1, 3) 2 2 4 4 6
D (2, 3) 3 3 6 3 6
E (2, 4) 2 3 5 4 6
Dummy (3, 4) 0 6 6 6 6
F (3, 5) 3 6 9 10 13
G (3, 6) 2 6 8 17 19
H (4, 5) 7 6 13 6 13
I (4, 6) 5 6 11 14 19
J (5, 6) 6 13 19 13 19

Summary of (CPM) Calculations
Activity
Symbol
Activity
(i, j)
Duration
Dij
(Es)i ECij LSij (LC)j TFij FFij
A (0, 1) 2 0 2 2 4 2 0
B (0, 2) 3 0 3 0 3 0* 0
C (1, 3) 2 2 4 4 6 2 2
D (2, 3) 3 3 6 3 6 0* 0
E (2, 4) 2 3 5 4 6 1 1
Dummy (3, 4) 0 6 6 6 6 0* 0
F (3, 5) 3 6 9 10 13 4 4
G (3, 6) 2 6 8 17 19 11 11
H (4, 5) 7 6 13 6 13 0* 0
I (4, 6) 5 6 11 14 19 8 8
J (5, 6) 6 13 19 13 19 0* 0
Notice that only a critical activity can have a Zero total float (TF) values are marked with (*)
mark to indicate the critical activities.

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