Project Management, Step by step network construction and (cpm) calculations

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

2. Step by Step
Network Construction and (CPM)
Calculations
Duration
(week)
Preceding
Activity
Activity
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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.
Dr. Mahmoud Abbas Mahmoud

3. Duration
(week)
Preceding
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2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

4. 0
Duration
(week)
Preceding
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Activity
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2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Note:
Practically, nodes numbering
can be start by (0) or (1).
Dr. Mahmoud Abbas Mahmoud

5. A
0
Duration
(week)
Preceding
Activity
Activity
2-A
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2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

6. A
0
1
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

7. B
A
0
1
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

8. B
A
0
2
1
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

9. B
A
0
2
1
C
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

10. B
A
0
2
1
C
D
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

11. B
A
0
2
1
C
D
E
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

12. B
A
0
2
1
C 3
D
E
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

13. B
A
0
2
1
C 3
D
E
G
F
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

14. B
A
0
2
1
C 3
D
E 4
G
H
F
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud
???????

15. B
A
0
2
1
C 3
D
E 4
G
H
F
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud
Use Dummy !

16. B
A
0
2
1
C 3
D
E 4
G
H
F
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

17. B
A
0
2
1
C 3
D
E 4
I
G
H
F
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

18. B
A
0
2
1
C 3
D
E 4
5
I
G
H
F
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

19. B
A
0
2
1
C 3
D
E 4
5
I
J
G
H
F
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

20. B
A
0
2
1
C 3
D
E 4
5 6
I
J
G
H
F
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

21. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

22. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

23. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

24. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

25. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

26. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud
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)

27. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

28. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

29. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud
[0+3]= 3

30. 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
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

31. 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
Duration
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

32. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

33. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

34. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
19
Dr. Mahmoud Abbas Mahmoud

35. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
19
Dr. Mahmoud Abbas Mahmoud

36. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
19
Dr. Mahmoud Abbas Mahmoud

37. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

38. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

39. 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
(week)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

40. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

41. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

42. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

43. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

44. 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)
Preceding
Activity
Activity
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

45. 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
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

46. 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
Dr. Mahmoud Abbas Mahmoud

47. 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
Dr. Mahmoud Abbas Mahmoud

48. 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
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

49. 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
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

50. 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
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

51. 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
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

52. 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
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

53. 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
2-A
3-B
2AC
3BD
2BE
3C,DF
2C,DG
7C,D,EH
5C,D,EI
6H,FJ
Dr. Mahmoud Abbas Mahmoud

54. 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
Dr. Mahmoud Abbas Mahmoud

55. 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)
Dr. Mahmoud Abbas Mahmoud
TFij = (LC)j – (ES)i – Dij
FFij = (ES)j – (ES)i – Dij

56. 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
Dr. Mahmoud Abbas Mahmoud

57. 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
Dr. Mahmoud Abbas Mahmoud

58. 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
For activity B (0, 2)
TF = 3 – 0 – 3 = 0 *
FF = 3 – 0 – 3 = 0
Dr. Mahmoud Abbas Mahmoud

59. 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
For activity B (0, 2)
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

60. 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
For activity B (0, 2)
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
For activity D (2, 3)
TF = 6 – 3 – 3 = 0 *
FF = 6 – 3 – 3 = 0
Dr. Mahmoud Abbas Mahmoud

61. 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
For activity B (0, 2)
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
For activity D (2, 3)
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
Dr. Mahmoud Abbas Mahmoud

62. 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
For activity B (0, 2)
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
For activity D (2, 3)
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
For Dummy activity (3, 4)
TF = 6 – 6 – 0 = 0 *
FF = 6 – 6 – 0 = 0
Dr. Mahmoud Abbas Mahmoud

63. 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
For activity B (0, 2)
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
For activity D (2, 3)
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
For Dummy activity (3, 4)
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
Dr. Mahmoud Abbas Mahmoud

64. 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
For activity B (0, 2)
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
For activity D (2, 3)
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
For Dummy activity (3, 4)
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
For activity G (3, 6)
TF = 19 – 6 – 2 = 11
FF = 19 – 6 – 2 = 11
Dr. Mahmoud Abbas Mahmoud

65. 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
For activity B (0, 2)
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
For activity D (2, 3)
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
For Dummy activity (3, 4)
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
For activity G (3, 6)
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
Dr. Mahmoud Abbas Mahmoud

66. 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
For activity B (0, 2)
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
For activity D (2, 3)
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
For Dummy activity (3, 4)
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
For activity G (3, 6)
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
For activity I (4, 6)
TF = 19 – 6 – 5 = 8
FF = 19 – 6 – 5 = 8
Dr. Mahmoud Abbas Mahmoud

67. 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
For activity B (0, 2)
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
For activity D (2, 3)
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
For Dummy activity (3, 4)
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
For activity G (3, 6)
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
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
Dr. Mahmoud Abbas Mahmoud

68. Dr. Mahmoud Abbas Mahmoud
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.

69. 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
Dr. Mahmoud Abbas Mahmoud

70. 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.
Dr. Mahmoud Abbas Mahmoud

71. Dr. Mahmoud Al-Naimi