Project Management, Easy steps to critical path determination and floats calculations for projects

1. Easy Steps to Critical Path
Determination and Floats Calculations
for Projects
Project Management
Dr. Mahmoud Abbas Mahmoud
Asst. Prof.
2016

2. Easy Steps to Critical Path
Determination and Floats Calculations
for Projects
We will illustrate these
steps with the following
Example:
Dr. Mahmoud Abbas Mahmoud

3. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
Example:
For the project given in the following (AOA) network diagram:
i) Find the critical activities.
ii) Find the critical path (CP).
iii) Find project completion time.
iv) Find the free float (FF) for each activity.
v) Summarize the results in a table.
Note: all times are in (weeks)

4. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
Critical Path Calculations

5. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
Forward Pass:

6. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
Forward Pass:
Calculating the Earliest Start Time (ES):

7. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
Forward Pass:
Calculating the Earliest Start Time (ES):
Earliest Start Time (ES)j=Max [ (ES)i + Dij ]

8. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
Earliest Start Time (ES) for Start Event = (0)
NOTE: (ES) for any Start Event = (0)

9. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0

10. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
10[0+10]= 10
0

11. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15 Max [(0+7) , (10+5)] = Max [7,15]= 15

12. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
16
[10+ 6]= 16

13. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
16
24
Max [(15+9) , (16+0)] =
Max [24,16]= 24

14. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
16
24
36
Max [(16+8) , (24+12)]
= Max [24,36]= 36

15. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
16
24
36
Project completion time is: 36 weeks

16. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
16
24
36
Backward Pass:

17. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
16
24
36

18. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
16
24
36
36

19. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
16
24
36
36
Calculating the Latest Completion Time (LC):

20. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
16
24
36
36
Calculating the Latest Completion Time (LC):
Latest Completion Time (LC)i=Min [ (LC)j - Dij ]

21. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
16
24
24
36
36
36 – 12 = 24

22. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
24
16
24
24
36
36
Min [(36-8),(24-0)=
Min[28,24] = 24

23. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
15
15
24
16
24
24
36
36
24 – 9 = 15

24. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
10
10
15
15
24
16
24
24
36
36
Min [(24-6),(15-5)=
Min[18,10] = 10

25. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Min [(15-7),(10-10)=
Min[8, 0] = 0

26. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36

27. A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
Dr. Mahmoud Abbas Mahmoud
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
How to find the Critical Activities and
the Critical Path ???

28. Dr. Mahmoud Abbas Mahmoud
Remember that only a critical
activity can have a Zero Total
Float (TF)

29. Dr. Mahmoud Abbas Mahmoud
Therefore, we will calculate
the total floats for all
activities and each activity
has a ZERO total float (TF) will
be a critical activity.

30. Total Float (TF) calculations
Dr. Mahmoud Abbas Mahmoud
TFij = (LC)j – (ES)i – Dij
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36

31. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
A
B
C
D
E
F
G
Dummy

32. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
B
C
D
E
F
G
Dummy
For activity A (1, 2)
TF = 10 – 0 – 10 = 0*

33. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
B
C
D
E
F
G
Dummy

34. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
C
D
E
F
G
Dummy
For activity B (1, 3)
TF = 15 – 0 – 7 = 8

35. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
C
D
E
F
G
Dummy

36. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
D
E
F
G
Dummy
For activity C (2, 3)
TF = 15 – 10 – 5 = 0*

37. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
D
E
F
G
Dummy

38. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
E
F
G
Dummy
For activity D (3, 4)
TF = 24 – 10 – 6 = 8

39. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
E
F
G
Dummy

40. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
Yes0E
F
G
Dummy
For activity E (3, 5)
TF = 24 – 15 – 9 = 0*

41. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
Yes0E
F
G
Dummy

42. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
Yes0E
No12F
G
Dummy
For activity F (4, 6)
TF = 36 – 16 – 8 = 12

43. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
Yes0E
No12F
G
Dummy

44. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
Yes0E
No12F
Yes0G
Dummy
For activity G (5, 6)
TF = 36 – 24 – 12 = 0*

45. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
Yes0E
No12F
Yes0G
Dummy

46. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
Yes0E
No12F
Yes0G
No8Dummy
For Dummy activity (4, 5)
TF = 24 – 16 – 0 = 8

47. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
Yes0E
No12F
Yes0G
No8Dummy

48. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
CriticalTotal FloatActivity
Yes0A
No8B
Yes0C
No8D
Yes0E
No12F
Yes0G
No8Dummy
The Critical Activities are: (A, C, E, G)
Critical Path is: (A C E G)
Project completion time is: 36 weeks

49. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36

50. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Free Float (FF) Calculations
FFij = (ES)j – (ES)i – Dij

51. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Free FloatActivity
A
B
C
D
E
F
G
Dummy

52. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Free FloatActivity
0A
B
C
D
E
F
G
Dummy
For activity A (1, 2)
FF = 10 – 0 – 10 = 0

53. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Free FloatActivity
0A
8B
C
D
E
F
G
Dummy
For activity B (1, 3)
FF = 15 – 0 – 7 = 8

54. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Free FloatActivity
0A
8B
0C
D
E
F
G
Dummy
For activity C (2, 3)
FF = 15 – 10 – 5 = 0

55. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Free FloatActivity
0A
8B
0C
0D
E
F
G
Dummy
For activity D (3, 4)
FF = 16 – 10 – 6 = 0

56. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Free FloatActivity
0A
8B
0C
0D
0E
F
G
Dummy
For activity E (3, 5)
FF = 24 – 15 – 9 = 0

57. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Free FloatActivity
0A
8B
0C
0D
0E
12F
G
Dummy
For activity F (4, 6)
FF = 36 – 16 – 8 = 12

58. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Free FloatActivity
0A
8B
0C
0D
0E
12F
0G
Dummy
For activity G (5, 6)
FF = 36 – 24 – 12 = 0

59. Dr. Mahmoud Abbas Mahmoud
A= 10
B= 7
1
2
3
C= 5
D= 6
E= 9
4
5
6Dummy
F= 8
G= 12
0
0
0
10
10
15
15
24
16
24
24
36
36
Free FloatActivity
0A
8B
0C
0D
0E
12F
0G
8Dummy
For Dummy activity (4, 5)
FF = 24 – 16 – 0 = 8

60. Dr. Mahmoud Abbas Mahmoud
Free FloatActivity
0A
8B
0C
0D
0E
12F
0G
8Dummy
Think !!!!

61. 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 (D) has
zero Free Float but it is not
critical.
Free FloatActivity
0A
8B
0C
0D
0E
12F
0G
8Dummy

62. Summary
Activity
Symbol
Activity
(i, j)
Duration
Dij
ESi LCj TFij FFij
A (1, 2) 10 0 10 0* 0
B (1, 3) 7 0 15 8 8
C (2, 3) 5 10 15 0* 0
D (2, 4) 6 10 24 8 0
E (3, 5) 9 15 24 0* 0
Dummy (4, 5) 0 16 24 8 8
F (4, 6) 8 16 36 12 12
G (5, 6) 12 24 36 0* 0
Notice that total float (TF) values marked with (*) mark
are for critical activities.
Dr. Mahmoud Abbas Mahmoud

63. Dr. Mahmoud Al-Naimi