1. Activity on Arrow
Project Management

2. Elements of an AoA (Activity-on-Arrow) diagram
Activity (arrow)
Work element or task
Can be real or not real
Name or identification of the tasks (label) must be added
Event (node)
The start and/or finish of one or more activities
Tail (preceding) and head (succeeding) nodes

3. Conventions Time flows from left to right
Arrows’ direction
Labels’ order
Head nodes always have a number (or label) higher that of the tail node. This is the same with the arrow labels (alphabetic order).
Activity labels are placed below the arrow (despite the pictures in the textbook), duration of activity is based above the arrow
A network has only one starting and only one ending event.
These conventions are not universal. There are many other to choose from.

4. Graphical representation
Arrows, nodes, bending
Identification of activities
Representation of time
Representation of deadlines (external constraints)

5. Dependency rule b and c are independent from each other:
b depends on a (b is a successor of a): 1 2 3 12 13 a b
b and c are independent from each other: 3 4 2 1 b c a 13 12 8

6. Consequences of the dependency rule
An event cannot be realised until all activities leading to it are complete.
No activity can start until its tail event is realised.

7. Merge and burst nodes Merge nodes: Burst nodes:
Events into which a number of activities enter and one (or several) leave.
Burst nodes:
Events that have one (or more) entering activities generating a number of emerging activities.

8. Two typical errors in logic
Looping: underlying logic must be at fault
Dangling: an activity is undertaken with no result 5 6 7 e f g 2 4
5 end a c d
1 start 3 b

9. Interfacing
When an event is common to two or more subnetworks it is said to be an ‘interface’ event between those subnetworks and is represented by a pair of concentric circles. 11 13 aa ac ab 21 ba 24 bc bb bd
12 22

10. Milestones
Events which have been identified as being of particular importance in the progress of the project.
Identified by an inverted triangle over the event node (occasionally with an imposed time for the event)
1/1/2014 1 2 3 a b

11. Multiple starts and finishes
Only used in computer programs
All starting activities can occur at the start and all finish activities will occur at the end of the project.

12. Hammock activities
Artificial activities created for the representation of the overhead cost with the aim of cost control.
Embrace activities belong to the same cost centre
Zero duration time (not taking part in the time analysis)
Overhead cost rate is assumed to be constant over the life of the hammock.

13. Hammock activity 1 2 1 3 4 12 2 a b c
h (hammock)

14. Dummy activities
Activities that do not require resources but may in some cases take time.
They are drawn as broken arrows.
They are always subject to the basic dependency rule.
Thre occassions to use dummies:
Identity dummies
Logic dummies
Transit time dummies

15. Identity dummies
When two or more parallel activities have the same tail and head nodes. 1 3 2 a b 4 3

16. Logic dummies
When two chains of activities have a common node yet they are at least partly independent of each other. Hint: examine ANY crossroads.
Example:
Activitiy c depends on activity a
Activity d depends on activities a and b
Solution:
separate c from b with a dummy activity

17. Logic dummy example: 2 6 4 a b 3 7 8 1 g f e d c h 5

18. Logic dummy example:

19. Transit time dummies
If a delay must occur after the competition of an activity before the successor activity can start. 2 4 a b 3 1 d c 5 2

20. Overlapping activities
If the activities are not fully discrete
The second activity can start before the first is completed but not before it is at least partly completed. 1 2 3 10 15 a b b 1 2 3 a1 a2 5 7 15

21. Standard layout for recording data
Network Analysis (Computation)
Occurrence times of Events = Early and late timings of event occurrence = Early and late event times
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Earliest
Event
Time
Latest
Label
Activity
Tail
Head
Standard layout for recording data

22. Early Event Time (EET = E =TE)
Early Event Time (Earliest occurrence time for event) is the earliest time at which an event can occur, considering the duration of precedent activities.
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Forward Pass for Computing EET
Each activity starts as soon as possible, i.e., as soon as all of its predecessor activities are completed.
Direction: Left to right, from the beginning to the end of the project
Set: EET of the initial node = 0
Add: EETj = EETi + Dij
Take the maximum
The estimated project duration = EET of the last node. j i
EETj
EETi
Activity
Dij

23. Early Event Times (EET = E =TE)
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24. Early Event Times (TE) 12 40 80 4 5 9 K L M 50 15 70 24 4
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25. Early Event Times (TE) 20 10 30 50 40 60 70 2 3 4 1 5 7
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26. Early Event Times (TE) 20 10 30 50 40 3 60 70 2 4 1 5 7 9 16 8
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27. Late Event Time (LET = L =TL)
Late Event Time (Latest occurrence time of event) is the latest time at which an event can occur, if the project is to be completed on schedule.
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Backward Pass for Computing LET
Direction: Right to left, from the end to the beginning of the project
Set: LET of the last (terminal) node = EET for it
Subtract: LETi = LETj - Dij
Take the minimum j i
LETi
EETj
LETj
EETi
Activity
Dij

28. Late Event Times (TL) 8 13 50 16 9 3 7 40 60
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29. Late Event Times (TL) 2 20 8 50 11/9/2018 1:21 PM 10 4 16 30 70 5 9 74 30 50 3 60 70 1 5 7 9 16 8
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30. Late Event Times (TL) 2 20 8 50 11/9/2018 1:21 PM 13 10 4 16 30 70 5 94 30 50 40 3 60 70 1 5 7 9 16 8 13
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31. Network Analysis (Computation)
Activity Times (Schedule)
Early Start (ES): The earliest time at which an activity can be started.
ESij = EETi
Early Finish (EF): The earliest time at which an activity can be completed.
EFij = ESij + Dij
Late Finish (LF): The latest time at which an activity can be completed without delaying project completion.
LFij = LETj
Late Start (LS): The latest time at which an activity can be started.
LSij = LFij  Dij
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32. Example: Activity Times2 20 10 4 30 50 40 3 60 70 1 5 7 9 16 8 13
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ES20-50 = EET20 = 2
EF20-50 = ES + D = = 5
LF20-50 = LET50 = 13
LS20-50 = LF – D = 13 – 3 = 10

33. Network Analysis (Computation)
Activity Floats
Total Float (TF)
Total float or path float is the amount of time that an activity’s completion may be delayed without extending project completion time.
Total float or path float is the amount of time that an activity’s completion may be delayed without affecting the earliest start of any activity on the network critical path.
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34. Network Analysis (Computation)
Activity Floats
Total Float (TF)
Total path float time for activity (i-j) is the total float associated with a path.
For arbitrary activity (ij), the total float can be written as:
Path Float =Total Float (TFij)
= LSij  ESij
= LFij  EFij
= LETj – EETi  Dij
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35. Example: Total Float Times2 20 10 4 30 50 40 3 60 70 1 5 7 9 16 8 13
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TF20-50 = LS ES20-50
TF20-50 = 10 – 2 = 8
TF20-50 = LF EF20-50
TF20-50 = 13 – 5 = 8
TF20-50 = LET50 – EET20 - D20-50
TF20-50 = 13 – = 8

36. Network Analysis (Computation)
Activity Floats
Free Float (FF)
Free float or activity float is the amount of time that an activity’s completion time may be delayed without affecting the earliest start of succeeding activity.
Activity float is “owned” by an individual activity, whereas path or total float is shared by all activities along a slack path.
Total float equals or exceeds free float (TF ≥ FF).
For arbitrary activity (ij), the free float can be written as:
Activity Float = Free Float (FFij)
= ESjk  EFij
= EETj – EETi  Dij
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37. Example: Free Float Times2 20 10 4 30 50 40 3 60 70 1 5 7 9 16 8 13
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FF20-50 = ES EF20-50
FF20-50 = 8 – 5 = 3
FF20-50 = EET50 – EET20 - D20-50
FF20-50 = 8 – = 3

38. Network Analysis (Computation)
Activity Floats
Interfering Float (ITF)
Interfering float is the difference between TF and FF.
If ITF of an activity is used, the start of some succeeding activities will be delayed beyond its ES.
In other words, if the activity uses its ITF, it “interferes” by this amount with the early times for the down path activity.
For arbitrary activity (ij), the Interfering float can be written as:
Interfering Float (ITFij)
= TFij  FFij
= LETj  EETj
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39. Example: Interfering Float Times2 20 10 4 30 50 40 3 60 70 1 5 7 9 16 8 13
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ITF20-50 = TF FF20-50
IFF20-50 = 8 – 3 = 5
ITF20-50 = LET50 – EET50
ITF20-50 = 13 – 8 = 5

40. Network Analysis (Computation)
Activity Floats
Independent Float (IDF)
It is the amount of float which an activity will always possess no matter how early or late it or its predecessors and successors are.
The activity has this float “independent” of any slippage of predecessors and any allowable start time of successors. Assuming all predecessors end as late as possible and successors start as early as possible.
IDF is “owned” by one activity.
In all cases, independent float is always less than or equal to free float (IDF ≤ FF).
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41. Network Analysis (Computation)
Activity Floats
Independent Float (IDF)
For arbitrary activity (ij), the Independent Float can be written as:
Independent Float (IDFij)
= Max (0, EETj  LETi – Dij)
= Max (0, Min (ESjk) - Max (LFli)  Dij)
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42. Example: Independent Float Times2 20 10 4 30 50 40 3 60 70 1 5 7 9 16 8 13
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IDF20-50 = Max. (0, [EET50 – LET20 - D20-50])
IDF20-50 = Max. (0, [8 – 10 – 3]) = 0

43. Network Analysis (Computation)
Critical Path
Critical path is the path with the least total float = The longest path through the network.
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Subcritical Paths
Subcritical paths have varying degree of path float and hence depart from criticality by varying amounts.
Subcritical paths can be found in the following way:
Sort the activities in the network by their path float, placing those activities with a common path float in the same group.
Order the activities within a group by early start time.
Order the groups according to the magnitude of their path float, small values first.

44. Example Activity Preceding Activity Time (days) A None 5 B 7 C 4 D 8 E
Draw an arrow diagram to represent the following project. Calculate occurrence times of events, activity times, and activity floats. Also determine the critical path and the degree of criticality of other float paths.
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Activity
Preceding Activity
Time (days) A
None 5 B 7 C 4 D 8 E 6 F G
E, C, F 3 H I

45. Example Activity on arrow network and occurrence times of events
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[7]
[5]
[8]
[4]
[3]
[6] A B C D E F G H 10 5 12 15 28 24 21 13 6 I

46. Example Activity times and activity floats 5 12 15 8 3 9 21 17 13 18
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Activity ES EF LF LS TF FF
ITF
IDF A 5 B 12 15 8 3 C 9 21 17 D 13 E 18 F G 24 H 28 30 26 2 I

47. Third most critical path
Example
Critical path and subcritical paths
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Activity ES EF LF LS TF
Criticality A 5
Critical Path D 13 F 21 G 24 I 30 H 28 26 2
a “near critical” path B 12 15 8 3
Third most critical path E 18 C 9 17
Path having most float

48. Case Study Installation of a new machine and training the operator 100
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Activity Code
Activity Description
Depends on
Level
Duration
(day)
100
Inspect the machine after installation
300 4 1
200
Hire the operator
None 25
Install the new machine
500, 400 3 2
400
Inspect and store the machine after delivery
700
500
Hire labor to install the new machine 20
600
Train the operator
200, 300
Order and deliver the new machine 30

49. Case Study: Installation of a New Machine and Training the Operator30 1 2 20 3 25 36 33 31 10 60 50 40
Hire the operator
Hire labor to install the new machine
Order & deliver the machine
Inspect the machine after delivery
Install the m.
Inspect the m.
Train the operator
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50. Activity i-j number ES EF LS LF TF FF ITF IDF
Case Study: Installation of a New Machine and Training the Operator
Activity times and activity floats
Activity
i-j number ES EF LS LF TF FF
ITF
IDF
Hire the operator
10-50 25 8 33
Hire labor to install the machine
10-30 20 11 31
Order and deliver the machine
10-20 30
Inspect the m. after delivery
20-30
Install the machine
30-40
Inspect the machine
40-60 34 35 36 2
Train the operator
50-60
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Critical path: 10-20, 20-30, 30-40,
Near critical path: 40-60
Third most critical path: 10-50
Path having most float: 10-30