The document discusses the Critical Path Method (CPM) for project scheduling, including how to develop a work breakdown structure, identify different types of activities, perform forward and backward pass calculations to determine early and late start/finish dates, and identify the critical path and float for activities. It also covers how to incorporate lags, determine event times in arrow networks, and the effects of an imposed finish date on the schedule.

1. 1
Lecture - 5

2. 2
The Critical Path Method (CPM)

3. 3
Who develops the WBS?
A WBS is developed by the A/E at the end of the design phase.
and/or by the bidders during the proposal (procurement phase).

4. 4
TYPES OF ACTIVITIES
1. Production Activities
Those that can be taken directly from plans and specifications
2. Procurement Activities
Procurement of material and equipment
3. Management Decision Activities
Activities that can be created by management to avoid certain situations
Delay Concrete
Company Vacation

5. 5
Example
Draw the logic network and perform the CPM calculations for the schedule shown next.
Duration
IPA
Activity
5
-
A
8
A
B
6
A
C
9
B
D
6
B,C
E
3
C
F
1
D,E,F
G

6. 6
In mathematical terms, the ES for activity j is as follows :
ESj =max( EFi )
where (EFi) represents the EF for all preceding activities.
Likewise, the EF time for activity j is as follows :
EF j= ESj + Dur j
where Dur j is the duration of activity j
Forward pass: The process of navigating through a network from start to end and calculating the
completion date for the project and the early dates for each activity
Forward pass calculations

7. 7
A
5
G
1
C
6
D
9
B
8
E
6
F
3
22,23
5,11
5,13 13,22
13,19
11,14
0,5
Solution :
Duration
IPA
Activity
5
-
A
8
A
B
6
A
C
9
B
D
6
B,C
E
3
C
F
1
D,E,F
G

8. 8
In mathematical terms, the late finish LF for activity j is as follows :
(
LFj =min(LSk
where (LSk) represents the late start date for all succeeding activities.
Likewise, the LS time for activity j (LS j) is as follows :
LS j= LFj - Dur j
where Dur j is the duration of activity
Backward pass: The process of navigating through a network from end to start and calculating the late dates for
each activity. The late dates (along with the early dates) determine the critical activities, the critical path, and the
amount of float each activity has.
Backward pass calculations

9. 9
Solution :
A
5
G
1
C
6
D
9
B
8
E
6
F
3
22,23
5,11
5,13 13,22
13,19
11,14
0,5
22,23
13,22
19,22
16,22
5,13
10,16
0,5
CPM ( ES = LS , EF = LF , TF = FF = 0)

10. 10
Types Of Floats
There are several types of float. The simplest and most important type of float is
Total Float (TF)
 Total float (TF): The maximum amount of time an activity can be delayed from its
early start without delaying the entire project.
TF = LS – ES
or
TF = LF - EF
or
TF = LF - Dur - ES

11. 11
 Free Float: may be defined as the maximum amount of time an activity can be delayed
without delaying the early start of the succeeding activities
FFi = min(ESi+1) - EFi
where min (ESi+1) means the least (i.e., earliest) of the early start dates of succeeding activities

12. 12
In the previous example we can find the free float and total float for each activity as the following :
Activity C’s free float, FF = 11 - 11 = 0 days
And
Activity C’s total float, TF =16 - 11= 5 days …… and so on.
FF
TF
LF
LS
EF
ES
Duration
Activity
0
0
5
0
5
0
5
A
0
0
13
5
13
5
8
B
0
5
16
10
11
5
6
C
0
0
22
13
22
13
9
D
3
3
22
16
19
13
6
E
8
8
22
19
14
11
3
F
0
0
23
22
23
22
1
G
 Critical activity
 Note : We must always realize that FF ≤ TF

13. 13
Node Format
Activity Name
Activity ID
Duration
ES EF
LS LF
TF FF

14. 14
Example
Activity Duration IPA
A 5 -
B 6 A
C 5 A
D 3 A
E 3 B
F 4 B, C
G 4 D
H 7 B,C,G
I 8 E, F
J 2 F
K 3 H, J
L 2 I, J

15. 15
A
5
D
3
B
6
C
5
G
4
E
3
F
4
H
7
I
8
J
2
K
3
L
2
PF

16. 16
A
5
D
3
B
6
C
5
G
4
E
3
F
4
H
7
I
8
J
2
K
3
L
2
PF
0, 5
0, 5
5, 11
5, 10
5, 8
11, 15
11, 14
8, 12
15, 17
15, 23
12, 19
23, 25
19, 22
25
25
23, 25
22, 25
15, 23
20, 22
15, 22
12, 15
11, 15
11, 15
5, 11
6, 11
8, 11
1
3
1
3
5
3 3
Graphic Solution

17. 17
Activity ES EF LS LF TF FF
A 0 5 0 5 0 0
B 5 11 5 11 0 0
C 5 10 6 11 1 1
D 5 8 8 11 3 0
E 11 14 12 15 1 1
F 11 15 11 15 0 0
G 8 12 11 15 3 0
H 12 19 15 22 3 0
I 15 23 15 23 0 0
J 15 17 20 22 5 2
K 19 22 22 25 3 3
L 23 25 23 25 0 0
Tabular Solution

18. 18
Comments on the Solution
 Near-critical activities may be as important as critical activities
 It is a good practice for the project manager not to give subordinates two sets of
dates
 PM has to choose one set of dates within the range of Early – Late
 Management may reserve a number of days as “management float” or “time contingency”

19. 19
Float Discussion
 Total float –in general- belongs to a path rather than the activity itself
 If an activity uses “its” float, successors may lose some or all of “their” float
 Total float versus free float

20. 20
 The question is: “who owns the float”?
 Float distribution attempts
 Review the contract – what if it is not mentioned?
 Float with resource leveling
 Shifting activities within their float may affect:
 Start / finish dates of succeeding activities
 Resource usage: Labor and equipment (crews)
 Materials: delivery, storage
 Cash flow

21. 21
Lags in Node Networks
 A lag is a minimum compulsory waiting period between the start/finish of an
activity and the start/finish of the successor
 A lead is a negative lag
 The lag is added in the CPM’s forward pass calculations and subtracted in the
backward pass

22. 22
Examples:
 Concrete curing (before formwork stripping or reshoring)
 Asphalt curing (before striping)
 Waiting for a permit to be issued
 Waiting for the delivery of a custom material or equipment

23. 23
Example
Activity Duration IPA Lag
A 5 -
B 3 -
C 6 -
D 7 A
E 7 A
B 4
F 4 A,B,C
G 5 E
F 3
H 6 D
G 2
I 3 D,G

24. 24
C
6
B
3
A
5
G
5
F
4
E
7
D
7
H
6
I
3
4
3
2
PS PF
27
0
0 27
0, 5 5, 12
0, 3
0, 6
7, 14
6, 10
14, 19
21, 27
19, 22
21, 27
24, 27
14, 19
14, 21
7, 14
7, 11
2, 7
0, 3
1, 7
Graphic Solution

25. 25
Activity ES EF LS LF TF FF
A 0 5 2 7 2 0
B 0 3 0 3 0 0
C 0 6 1 7 1 0
D 5 12 14 21 9 7
E 7 14 7 14 0 0
F 6 10 7 11 1 1
G 14 19 14 19 0 0
H 21 27 21 27 0 0
I 19 22 24 27 5 5
Tabular Solution

26. 26
Effect of Imposed Finish Date
 Imposed Finish Date is the project’s completion date, as specified in the contract or
stipulated by the owner
 When compared to the calculated finish date:
 Calculated finish date < imposed finish date
• You are in good shape
• What happens if you enter the imposed date?
 Calculated finish date > imposed finish date
• Negative float appears when you enter the imposed date
• You need to accelerate / crash the schedule

27. 27
Examples with Imposed Finish Dates
Repeat Example 3 with imposed finish date of 28 days
Repeat Example 3 with imposed finish date of 22 days

28. 28
A
5
D
3
B
6
C
5
G
4
E
3
F
4
H
7
I
8
J
2
K
3
L
2
PF
0, 5
0, 5
5, 11
5, 10
5, 8
11, 15
11, 14
8, 12
15, 17
15, 23
12, 19
23, 25
19, 22
25
25
23, 25
22, 25
15, 23
20, 22
15, 22
12, 15
11, 15
11, 15
5, 11
6, 11
8, 11
1
3
1
3
5
3 3
With out Imposed finished date

29. 29
Imposed Finish Date > Calculated Finish Date
A
5
D
3
B
6
C
5
G
4
E
3
F
4
H
7
I
8
J
2
K
3
L
2
PF
0, 5
3, 8
5, 11
5, 10
5, 8
11, 15
11, 14
8, 12
15, 17
15, 23
12, 19
23, 25
19, 22
25
28
26, 28
25, 28
18, 26
23, 25
18, 25
15, 18
14, 18
14, 18
8, 14
9, 14
11, 14
3 4
6 6 6 6
8
4
3 3
3
3

30. 30
A
5
D
3
B
6
C
5
G
4
E
3
F
4
H
7
I
8
J
2
K
3
L
2
PF
0, 5
-3, 2
5, 11
5, 10
5, 8
11, 15
11, 14
8, 12
15, 17
15, 23
12, 19
23, 25
19, 22
25
22
20, 22
19, 22
12, 20
17, 19
12, 19
9, 12
8, 12
8, 12
2, 8
3, 8
5, 8
-3
-3
-3
-2 -3
-3
-2 2
0 0 0 0
Imposed Finish Date < Calculated Finish Date

31. 31
 Negative float is a situation that occurs when performing an activity even
on its early dates, fails to meet the project’s imposed finish date or other
constraint
 It may occur in one of two cases:
Before construction starts
During construction (after normal start)
Negative Float

32. 32
Event Times in Arrow Networks
 The early event time, TE, is the largest (latest) date obtained to reach an event (going from
start to finish).
 The late event time, TL, is the smallest (earliest) date obtained to reach an event (going from
finish to start).
Example
Perform the CPM calculations, including the event times, for the arrow network shown below.

33. 33
10 30
40
20 60
C
E
B
50
D
F
70
A
G
H
10
5
7
8
9
4
5
8
d1
d2
Arrow network for example

34. 34
The preceding logic is similar to that of the forward and backward passes: When you are going forward, pick
the largest number. When you are going backward, pick the smallest number.
i j
Act. Name
Dur.
TEi
TLi
TEj
TLj
CPM

35. 35
10 30
40
20 60
C
E
B
50
D
F
70
A
G
H
10
5
7
8
9
4
5
8
d1
d2
10
10
7
0
0
15
10
10
19
19
24
27
27
27
(0,10)
(0,10)
(5,10)
(0,5)
(0,7)
(8,15)
(10,18)
(11,19)
(10,19)
(10,19)
(7,11)
(15,19)
(19,24)
(22,27)
(19,27)
(19,27)

36. 36
Definitions
Activity, or task: A basic unit of work as part of the total project that is easily measured and
controlled. It is time- and resource consuming.
Backward pass: The process of navigating through a network from end to start and calculating the
late dates for each activity. The late dates (along with the early dates) determine the critical
activities, the critical path, and the amount of float each activity has.
Critical activity: An activity on the critical path. Any delay in the start or finish of a critical activity will
result in a delay in the entire project.
Critical path: The longest path in a network, from start to finish, including lags and constraints.
.

37. 37
Early dates: The early start date and early finish date of an activity.
Early finish (EF): The earliest date on which an activity can finish within project constraints.
Early start (ES): The earliest date on which an activity can start within project constraints.
Event: A point in time marking a start or an end of an activity. In contrast to an activity, an event does not consume time or
resources.
Forward pass: The process of navigating through a network from start to end and calculating the completion date for the
project and the early dates for each activity.
Late dates: The late start date and late finish date of an activity.
Late finish (LF): The latest date on which an activity can finish without extending the project duration.
Late start (LS): The latest date on which an activity can start without extending the project duration.

38. 38
A
B
C
Improper Proper
A
B
C
PF
(a) Do not end a network with more than one node