Project Scheduling Basics: The document introduces the basics of project scheduling, including network-based scheduling methods like Gantt charts and Critical Path Method (CPM). Scheduling Techniques: It explains different scheduling techniques such as bar charts for planning and control, and Gantt charts for precise activity sequencing. Critical Path Method (CPM): The document details the CPM, which helps identify the critical set of activities, calculate early and late start times, and determine the minimum project duration. Float Calculation: It covers the concept of float in project scheduling, which is the amount of time an activity can be delayed without affecting the overall project timeline. Different types of float are discussed, including total, free, and interfering float.

1. Chapter 10
Project Scheduling Using Critical Path
Method (CPM)
Dr. Yantao YU
(with credit to Prof. Jack C.P. CHENG)
Objectives:
• Students will understand the basics of project scheduling.
• Students will be able to model project schedules in networks
(AON, AOA).
• Students will be able to analyze project schedules using CPM.
1

2. Warm-up
2
• Morning routine scheduling
• You need to arrive in the
classroom by 9:00 am.
• What time should you wake up?
Shower & Get dressed
(30 min)
Walking to Canteen
(10 min)
Breakfast
(20 min)
Reading &
Checking social media
(20 min)
Walking to classroom
(10 min)
https://www.mentimeter.com/app/presentation/alu7ic58ytw21ei7sqhb8vexu52gnxpq/6erwem281ok2/edi t

3. • 1. Project scheduling
• 2. Network-based scheduling
• 3. Critical Path Method (CPM)
3
Chapter 10
Project Scheduling Using Critical Path Method
(CPM)

4. 1 Project Scheduling
• Project scheduling addresses issues in time planning and
management.
• Common scheduling methods: bar charts or Gantt charts
• During the past 40 years: network-based scheduling methods
• Project breakdown, identification of the relationships / logic
among activities, and activity sequencing are needed.
• Scheduling networks can be analyzed using critical path
method (CPM) and PERT (Program Evaluation and Review
Technique).
4

5. 1.1 Bar Chart
• In a bar chart, work activities are represented as time scaled bars
• The length of the bar has two meanings:
– The length indicates the planned duration.
– Proportionally scaled baseline.
5

6. 1.1 Bar Chart
• Bar chart acts as both a planning-scheduling model and a
reporting-control model.
– Planning-scheduling model ( ____focus): it indicates the
planned start, execution and completion of work activities.
– Reporting-control model (____ focus): it indicates actual
performance toward completion of work activities.
6

7. • Bar chart progress models: (a) bar chart schedule (plan focus) and
(b) bar chart updating (control focus)
8
• Using different shading patterns, bar chart can indicate periodic
progress toward physical completion of work activities.

8. 1.1 Bar Chart
• Bar chart is developed by breaking down the project into a number
of components.
• The breakdown usually focuses on physical components of the
project.
• The relative positioning of work activities indicates their planned
schedule and sequence.
9
1.2 Gantt Chart
• Traditional bar charts lack precision in establishing the exact
sequence between activities.
• A Gantt chart overcomes this disadvantage.
• A Gantt chart is a type of bar chart, developed by Henry Gantt in
the 1910s, that illustrates a project schedule and emphasizes the
sequencing / dependency of activities by using arrow connectors.

9. • Gantt chart example for a gas station.
10

10. Warm-up
11
• Morning routine scheduling
Activity Duration 1 2 3 4 5 6 7 8 9
1. Shower & Get dressed 30 min
2. Walking to Canteen 10 min
3. Breakfast 20 min
4. Social media 20 min
5. Walking to classroom 10 min
Sequences between activities:
1->2->3->5
Time unit: 10 mins

11. • Gantt chart examples using commercial software.
12

12. • Review questions
– This a Gantt chart or traditional bar chart? Why?
– What do the purple bars represent?
– What do the red, yellow, green bars represent?
13

13. 1.3 Scheduling Logic
• Logical sequence relating activities to one another must be
developed.
• Example:
14
(b) Exploded view of pier
(a) Schematic view of pier

14. 1.3 Scheduling Logic
• Conceptual model of pier components
15
• Logical modeling rationales:
(a)
(b) adjacency of contact modeling (c) physical construction order modeling

15. 1.3 Scheduling Logic
16
• The physical makeup of the pier?
adjacency of contact modeling

16. 1.3 Scheduling Logic
• Pile-driving sequences (the order in which
the piles will actually appear on the site)
18
physical construction
order modeling

17. • 1. Project scheduling
• 2. Network-based scheduling
• 3. Critical Path Method (CPM)
19
Chapter 10
Project Scheduling Using Critical Path Method
(CPM)

18. 2 Network Based Scheduling
• A scheduling network consists of nodes and directional
links/arrows.
• The node may represent an activity or an event in time
depending on the notation.
• The link may indicate the logical sequence between activities
or represent an activity.
20
(b) Node to represent an event
(Activity on Arrow, AOA)
(a) Node to represent an activity
(Activity on Node, AON)

19. 2 Network Based Scheduling
21
Node to represent an activity
(Activity on Node, AON)
adjacency of contact modeling
physical construction order modeling

20. • Example of AON and AOA
22
Activity on Node (AON)
(or Activity Network in
Precedence Network)
- Nodes: activities
- Arrows: sequence or
relationship of activities.
Activity on Arrow (AOA)
(or Activity Network in
Arrow Notation)
- Nodes: events in time
- Arrows: activities.
(a)
(b)

21. • Dummy activity in AOA
23
WRONG!
What is wrong?
AON
AOA

22. • Conversion between Precedence Network or Activity on Node
(AON) and Arrow Network or Activity on Arrow (AOA)
24
AOA
(Arrow Network)
AON
(Precedence
Network)

23. • Conversion between Precedence Network (Activity on Node) and
Arrow Network (Activity on Arrow)
25
AOA
(Arrow Network)
AON
(Precedence
Network)

24. • Conversion between Precedence Network (Activity on Node) and
Arrow Network (Activity on Arrow)
27
AOA
(Arrow Network)
AON
(Precedence
Network)

25. • Conversion between Precedence Network (Activity on Node) and
Arrow Network (Activity on Arrow)
29
AON
(Precedence Network)
- Activity B and Activity C cannot begin until Activity A is completed;
- Activity B and Activity C must both be completed before Activity D can start.

26. • Conversion between Precedence Network (Activity on Node) and Arrow
Network (Activity on Arrow)
30
WRONG
CORRECT
- Activity B and Activity C cannot begin until Activity A is completed;
- Activity B and Activity C must both be completed before Activity D can start.
AON
(Precedence Network)

27. • Conversion between Precedence Network (Activity on Node) and
Arrow Network (Activity on Arrow)
31
AOA
(Arrow
Network)
AON
(Precedence
Network)

28. • Conversion between Precedence Network (Activity on Node) and
Arrow Network (Activity on Arrow)
32
AOA
(Arrow
Network)
AON
(Precedence
Network)
Alternative AOA representation?
WRONG
CORRECT

29. • Conversion between Precedence Network (Activity on Node) and
Arrow Network (Activity on Arrow)
33
AOA
(Arrow
Network)
AON
(Precedence
Network)
What is the AOA representation?
Is dummy arrow needed?
A
B C
D
E

30. • Conversion between Precedence Network (Activity on Node) and
Arrow Network (Activity on Arrow)
34
AOA
(Arrow
Network)
AON
(Precedence
Network)
What is the AOA representation?
Is dummy arrow needed?
A
B C
D
E
0 1
2
5
4
3
A
B
C
D E
0 1
2
4
3
A
B C
D E
1
2
OR

31. 2.1 Network Based Scheduling Example
• Example: Construction of Concrete Footings
• Activities involved:
– A. Lay out of foundation
– B. Dig foundation
– C. Place formwork
– D. Place concrete
– E. Obtain steel reinforcement
– F. Cut and bend steel reinforcement
– G. Place steel reinforcement
– H. Obtain concrete
35

32. 2.1 Network Based Scheduling Example
• Example: Construction of Concrete Footings
1. Foundation Chain
A. Lay out of foundation
B. Dig foundation
C. Place formwork
G. Place steel reinforcement
D. Place concrete
36
2. Steel Chain
E. Obtain steel reinforcement
F. Cut and bend steel reinforcement
G. Place steel reinforcement
D. Place concrete
3. Concrete Chain
H. Obtain concrete
D. Place concrete

33. 2.1 Network Based Scheduling Example
• Example: Construction of Concrete Footings
• Activities involved:
– A. Lay out of foundation
– B. Dig foundation
– C. Place formwork
– D. Place concrete
– E. Obtain steel reinforcement
– F. Cut and bend steel reinforcement
– G. Place steel reinforcement
– H. Obtain concrete
37
Foundation Chain
A, B, C, G, D
Steel Chain
E, F, G, D
Concrete Chain
H, D

34. 2.1 Network Based Scheduling Example
• Example: Construction of Concrete Footings
• Precedence Network (AON):
38
Initial sketch
First draft

35. 2.1 Network Based Scheduling Example
• Example: Construction of Concrete Footings
• Arrow Network (AOA):
39
Initial sketch
First draft

36. • 1. Project scheduling
• 2. Network-based scheduling
• 3. Critical Path Method (CPM)
40
Chapter 10
Project Scheduling Using Critical Path Method
(CPM)

37. 3 Critical Path Method (CPM)
The objectives of analysing a project schedule network:
1. Find the critical set of activities that establishes the longest
path and defines the minimum duration of the project.
2. Calculate the early start times for each activity.
3. Calculate the late start times for each activity.
4. Calculate the float, which means the time available for delay
for each activity.
41
3.1 Network Schedule Analysis – AON

38. 3.1 Network Schedule Analysis – AON
Critical Activities:
1. Cannot be delayed without extending the project duration.
2. Float associated with a critical activity is zero.
3. Critical activities lie along the longest path through the
network (i.e. critical path).
Critical Path:
• The longest path, consisting of the critical set of activities
42

39. 3.2 Identifying Critical Path
• Notation:
43
• Which are the critical activities?
• What is the minimum project duration?
• When should each activity start?
EST(I): Early start time of activity I
EFT(I): Early finish time of activity I
LST(I): Late start time of activity I
LFT(I): Late finish time of activity I
DUR(I): Duration of activity I

40. • How about this schedule?
– Critical activities? Minimum project duration? Starting time?
44

41. 3.2 Identifying Critical Path
Forward pass algorithm
• To calculate the earliest event times for each activity.
• To calculate the minimum duration of the project.
Backward pass algorithm
• To calculate the latest event times for each activity.
Identify Critical Activities
• Identify activities for which the earliest and latest start times are the
same.
45

42. 3.2.1 Forward Pass Algorithm
• Notation:
• To calculate the early start time (EST) and early finish time (EFT):
• Example:
46
EST(I): Early start time of activity I
EFT(I): Early finish time of activity I
LST(I): Late start time of activity I
LFT(I): Late finish time of activity I
DUR(I): Duration of activity I
EFT(I) = EST(I) + DUR(I)
EST(J) = max [EFT(I)], for all I precedes activity J
EST(start node) = 0
I2
3
I3
6
I1
4
J
8
EST=16, EFT=20
EST=15, EFT=18
EST=16, EFT=22
EST=22, EFT=30

43. 3.2.1 Forward Pass Algorithm
• Example:
47

44. 3.2.1 Forward Pass Algorithm
• Example:
48
The forward pass can tell the minimum duration of the entire project.

45. 3.2.2 Backward Pass Algorithm
• Notation:
• To calculate the late start time (LST) and late finish time (LFT):
• Example:
49
EST(I): Early start time of activity I
EFT(I): Early finish time of activity I
LST(I): Late start time of activity I
LFT(I): Late finish time of activity I
DUR(I): Duration of activity I
LST(J) = LFT(J) – DUR(J)
LFT(I) = min [LST(J)], for all J follows activity I
J2
4
J3
7
J1
6
I
5
LST=34, LFT=40
LST=32, LFT=36
LST=28, LFT=35
EST=23, LFT=28
min {34, 32, 28}

46. • Example:
50
3.2.2 Backward Pass Algorithm

47. 3.2.2 Backward Pass Algorithm
• Example:
51
Critical Path Activities:
LST(I) = EST(I)
LFT(I) = EFT(I)
If the calculations are performed
correctly, the EST and LST of the
initial activity (A) should be zero (0).
LST(J) = LFT(J) – DUR(J)
LFT(I) = min [LST(J)],
for all J follows activity I

48. 3.2 Identifying Critical Path – Summary
Review Question:Consider the following project scheduling network.
(a) Is the network AON or AOA? (b) What is the minimum project
duration? (c) What is the LST of Activity B? (d) Is Activity C a critical
activity?
52
A
6
B
5
E
4
C
6
D
7
F
5
G
3
H
6
1. (a) AOA; (b) 24; (c) 9; (d) yes
2. (a) AON; (b) 27; (c) 9; (d) no
3. (a) AON; (b) 24; (c) 6; (d) no
4. (a) AOA; (b) 25; (c) 9; (d) no
5. (a) AON; (b) 27; (c) 6; (d) yes
6. (a) AON; (b) 27; (c) 9; (d) yes

49. 3.3 Float Calculation
• Float is the amount of time by which an activity can be delayed
without delaying the total project.
• Critical activities:
– the earliest and latest start times are the same;
– cannot be delayed without delaying the completion of the project.
– The float of a critical activity = 0
• Activities having positive float are not critical.
• Five types of float:
– Total Float
– Free Float
– Interfering Float
– Safety Float
– Independent Float 57

50. 3.3.1 Total Float
• Total Float: The total, or maximum, number of time units that an
activity can be delayed without increasing the total project duration.
58
TF(I) = LFT(I) – EFT(I)
TF(E) = LFT(E) – EFT(E)
= 12 – 9 = 3
TF(C) = LFT(C) – EFT(C)
= __________________
A critical activity has zero
float.
TF(D) = LFT(D) – EFT(D)
= 12 – 12 = 0
TF(B) = LFT(B) – EFT(B)
= __________________

51. 3.3.2 Free Float
• Use of the total float available to an activity may reduce the float
available to activities that follow it in sequence.
– If Activity C delays by 2 days, EFT(C) = 9
 EST of E = 9  E can delay only by 1 days
• Free Float: The amount of time an activity can be delayed without
impacting activities that follow it.
59
FF(I) = min[EST(J)] – EFT(I), for all J follows Activity I

52. 3.3.2 Free Float
• Free Float: The amount of time an activity can be delayed without
impacting activities that follow it.
60
FF(I) = min[EST(J)] – EFT(I), for all J follows Activity I
FF(C) = min[EST(E)] – EFT(C)
= 7 – 7 = 0
FF(B) = min[EST(D),EST(E)] –
EFT(B)
= min(6, 7) – 6 = 6 – 6 = 0
FF(E) = min[EST(F)] – EFT(E)
= 12 – 9 = 3

53. 3.3.3 Interfering Float
• Interfering Float: The amount of the total float utilized that
interferes with the following activities.
61
IF(I) = TF(I) – FF(I)

54. 3.3.3 Interfering Float
• Interfering Float: The amount of the total float utilized that
interferes with the following activities.
62
IF(I) = TF(I) – FF(I)
IF(C) = TF(C) – FF(C) = 3 – 0 = 3
IF(E) = TF(E) – FF(E) = 3 – 3 = 0
Although 3 days of delay can
occur on activity C without
impacting total project duration,
each day of IF used in
conjunction with activity C will
“interfere” with the float available
for following activities.

55. 3.3.4 Safety Float
• Safety Float: The maximum amount of time of an activity that can
be delayed without causing an increase in the total project
duration, assuming that preceding activities have been completed
as late as possible.
63
SF(J) = LST(J) – max[LFT(I)], for all I precedes Activity J
SF(C) = LST(C) – max[LFT(A)]
= 5 – 2 = 3
SF(E) = LST(E) –
max[LFT(B),LFT(C)]
= .

56. 3.3.5 Independent Float
• Independent Float: The maximum amount of time an activity can
be delayed without delaying the early start of the succeeding
activities and without being affected by the allowable delays of the
preceding activities.
65
IndF(I) = min[EST(J)] – DUR(I) – max[LFT(H)]
, for all J follows Activity I, H precedes Activity I

57. 3.3.5 Independent Float
66
• Positive independent float means that float exists even if preceding activities use
all of their float and no float is taken from the following activities.
• Independent float could be negative. In this case. we set the value to zero.
• Independent float is always less than or equal to free float. Yes or No?
IndF(C)
= min[EST(E)] – DUR(C) – max[LFT(A)]
= 7 - 5 - 2 = 0
IndF(B)
= min[EST(D),EST(E)] – DUR(B) – max[LFT(A)]
= 6 - 4 - 2 = 0
FF(I) = min[EST(J)] – EFT(I)
IndF(I) = min[EST(J)] – DUR(I) – max[LFT(H)]

58. 67
3.3 Float Calculation – Summary
LST(I)
SF(I)
TF(I) = LFT(I) – EFT(I)
FF(I) = min[EST(J)] – EFT(I)
IF(I) = TF(I) – FF(I)
SF(I) = LST(I) – max[LFT(H)]
IndF(I) = min[EST(J)]
– DUR(I) – max[LFT(H)]

59. Activity Duration Critical? TF FF IF SF IndF
A 2
B 4
C 5
D 6
E 2
F 8
• Exercise:
3.3 Float Calculation – Summary

60. Activity Duration Critical? TF FF IF SF IndF
A 2 Yes 0 0 0 0 0
B 4 Yes 0 0 0 0 0
C 5 No 3 0 3 3 0
D 6 Yes 0 0 0 0 0
E 2 No 3 3 0 0 0
F 8 Yes 0 0 0 0 0
69
• Exercise:

61. 3.4 Critical Path Method Example – Gas Station
• Consider a small gas station
• Plans for the gas station:
70

62. 71

63. • Duration of activities for the small gas station.
72
Predecessor
Activities
3,4
5
1 5,6,7
1 8
2,3 8
3 8
3 8
4,5,6,7 9
8 10,11,12
9 13,15,16,17
9 13,15,16,17
9 14
10, 11 21
12 15,16,17
10,11,14 18,19,20
10,11,14 18,19,20
10,11,14 21
15,16 21
15,16 21
15,16 21
13,17,18,19,20 22
21
Processor
Activities

64. Review questions
• Calculation of EST and EFT
– Forward path algorithm
• Calculation of LFT and LST
– Backward path algorithm
• Critical path
– EST(I) = LST(I) or EFT(I) = LFT(I)
• Total float, free float, interfering float
73
TF(I) = LFT(I) – EFT(I)
FF(I) = min[EST(J)] – EFT(I)
IF(I) = TF(I) – FF(I)

65. Calculation of EST and EFT
74

66. Calculation of LFT and LST
75

67. • Results of EST, EFT, LST and LFT.
• Critical path?  EST(I) = LST(I) OR EFT(I) = LFT(I)
76

68. Calculation of Floats
77

69. Summary
• Project Scheduling
– For (1) planning, (2) control
– Representations: (1) Bar charts, (2) Gantt charts
– Follows scheduling logic
• Network Based Scheduling
– (1) Activity on Node (AON)/Precedence Network, (2) Activity on Arrow
(AOA)/Arrow Notation
• Conversion between each other; dummy arrows in AOA
• Critical Path Method (CPM)
– Early start time, early finish time, late start time, late finish time
– Forward pass algorithm (EST, EFT), backward pass algorithm (LST, LFT)
– Critical activities; Minimum project duration
– Total float, free float, interfering float, safety float, independent float
– Critical activities: total float is 0.
78

70. References
• Halpin, D. W. (2012). Construction Management, 4th edition, John Wiley
& Sons, Inc.
79