1. Analysing the AoA network
Project Management

2. Total Project Time
The minimum time in which the project can be completed.
Calculation: forward pass
Forward pass: calculating the earliest event times (EETs) and the earliest start times (ESTs) of all activities.
Earliest Finishing Time = EST + Duration

3. Critical path
Path: continuous series of project activities connected by logical relationships as designated in the project schedule network diagram.
Critical path: sequence of activities that has no float time, and that determines the duration of the project. It is the longest path. Activities on the critical path (or paths) are the critical activities.
The critical path can be identified by a backward pass, calculating the Latest Event Times (LETs) and the Latest Finishing Times (LFTs).
Latest Starting Time = Latest Finishing Time - Duration

4. Activity times & event times
EET = EST of all emerging activities
LET = LFT of all entering activities
Deadline
Activity identifier
Duration 1 2
EET
LET

5. TPT
EST 0
EFT 14 14 14 1 2 a 14
LST 0
LFT 14
TPT = 14

6. Float 1 2 Float on activity ‘a’: Float: 6 a 20 EST 0 EFT 14 14 14 6 2014 14 1 2 a 6 20
LST 6
LFT 20
Float: 6

7. 4 6 5 Calculate the… EET of event 6 LETs of events 4 and 522 ? 5 24 6 34 d e 8 10
Calculate the…
EET of event 6
LETs of events 4 and 5
ESTs and EFTs of activities ‘d’ and ‘e’
LSTs and LFTs of activities ‘d’ and ‘e’

8. 4 6 5 EST and EFT of ‘d’: 22 and 30 EST and EFT of ‘e’: 24 and 348 4 30 d 26 26 34 34 6 34 34 24 24 5 10 34 e 24 24
EST and EFT of ‘d’: 22 and 30
EST and EFT of ‘e’: 24 and 34
LST and LFT of ‘d’: 26 and 34
LST and LFT of ‘e’: 24 and 34

9. 15 ? 30 10 a 10 ? 20 ? 35 12 b 25 ? 8 c

10. 15 20 30 10 a 10 17 20 22 35 12 b 25 18 8 c

11. Calculate all event and activity times and the float-times, find the critical path

12. Calculate all event and activity times and the float-times, find the critical path16 16 16 31 8 24 24 24 39 20 30 29 39 20 20 20 35 35 35 38 38 38 50 51 20 20 20 35 35 36 39 39 39 51 51 35 51 35 51 30 21 51

13. Activity times and float for the previous diagram
EST
LST
EFT
LFT
Float A B C D E G H J K

14. Activity times and float for the previous diagram
EST
LST
EFT
LFT
Float A 8 16 24 B 20 C 21 30 51 D 35 E 29 39 9 G 36 38 1 H J 31 K 50

15. Four characteristics of the critical path
It starts at the first node
It is continuous
It ends at the last node
It has no float

16. Floats in the AoA network

17. Activity and event times
Activity times:
EST
LST
EFT
LFT
Event times:
EET
LET

18. Float
Time available for an activity or path in addition to its duration.
It can be positive or negative
It is a property of activities (and available only at given activities)
In case of more succeeding activities, the minimum have to be taken.
Types of float:
Total float
Free float

19. Total float 2 1 4 3 The total float possessed by an activity.
Calculation for activity j:
Total floatj = LFTj – EFTj
Identify the activity/activities possessing float below, and calculate the total float(s). 2 ?
Total float =
= 35 – 22 = 13
Total float =
= 25 – 12 = 13 12 12 35 12 10 12 25 25 1 ? 25 4 ? 22 c a 13 23 35 23 12 15 3 ? 8 35 35 15 15 b 15 d 27 27
Total float =
= 27 – 15 = 12 27
Total float =
= 35 – 23 = 12
Critical path = ?
TFT = ?

20. Total float 2 1 4 3 The total float possessed by an activity.
Calculation for activity j:
Total floatj = LFTj – EFTj = LFTj – (ESTj + Dj)
Identify the activities possessing float below, and calculate the total float for each.
Total float =
= 11 – (0+5) = 6 2 5
Total float = 23 – (5+12) = 6 12 5 11 c a 1 4 23 15 8 23 b 3 15 d 15

21. Calculating free float
The FREE FLOAT is the float possessed by an activity which, if used, will not change the float in later activities.
Free floatj = ESTj+1 – EFTj = EEThead – EETtail - D
Free float =
= 5 – 0 – 5 = 0 2 5
Free float = 23 – 5 – 12 = 6 12 5 11 c a 1 4 23 15 8 23 b 3 15 d 15

22. Positive float on the critical path
If the target time (deadline) for the project (or for a part of the project) is grater than TPT, than a float will appear on the critical path(s), too (and also on other paths).
The float will be the minimal on the critical path(s).

23. Negative float
Negative float = the time by which activities on the path or paths concerned must be reduced if the TPT is to be met. Negative float is a type of the Total Float.
If it appears on both critical and non-critial paths, than the critical path has the greatest negative float (in absolute value).
Negative float apperas when the TPT is longer than the project’s target time. 1 ? 2 a 10 3 c 15

24. Negative float
Negative float = the time by which activities on the path or paths concerned must be reduced if the TPT is to be met. Negative float is a type of the Total Float.
If it appears on both critical and non-critial paths, than the critical path has the greatest negative float (in absolute value).
Negative float apperas when the TPT is longer than the project’s target time. 1 -5 2 10 5 a 3 20 15 c
Total float: -5

25. Total float = 11 – 0 – 5 = 6 Free float = 5 – 0 – 5 = 0
Slack
It is the ’float’ measured at events, and not at activities.
Slack = LET - EET
Slack: 6
Total float = 23 – 5 – 12 = 6 Free float = 23 – 5 – 12 = 6
Total float = 11 – 0 – 5 = 6 Free float = 5 – 0 – 5 = 0 2 5 12 5 11 c a 1 4 23
TF = 0; FF = 0
TF = 0; FF = 0 15 8 23 b 3 15 d
Slack: 0
Slack: 0 15
Slack: 0

26. Readings
Lockyer – Gordon (2005) Chapter 13

27. Analysing the AoN network
Project Management

28. Data on the activity node
Activity label & description
Total Float
EST
EFT
LST
Duration
LFT

29. Total Project Time
The shortest time in which the project can be completed. Determined by the critical path.
Calculation: forward pass
Forward pass: The earliest start times (EST) of all activities are calculated.
Trom these the earliest finishing times (EFT) are also calculated

30. Critical path
sequence of activities that has no float time (or has the maximum negative float time in absolute value), and that determines the duration of the project. It is the longest path. Activities on the critical path are the critical activities.
The critical path can be identified by a backward pass, calculating the Latest Finishing Times (LFT), and from these the Latest Starting Times (LST).

31. Floats in AoN
Total float: the time by which an activity can be delayed or extended without affecting the TPT. It can be used to delay the start of an activity or to increase its duration. TF = LST - EST
Free float: the time by which an activity can be delayed or extended without affecting the start of any succeding activity. FF = ESTj+1 - EFTj

32. Example: organising a conference
Objectives: to organise a 3 days long open scientific conference with participants, lectures, buffet reception, a conference book of the best studies and TV and radio interviews with some of the most known lecturers.
Create the WBS chart and create the task list with estimated durations and precedence relations (in a table form)
Plot both the AoA and the AoN diagram
Calculate the TPT, identify the critical path, the total, and the free float times.

33. Organising participants Invitation and marketing Organising interviews
Example: WBS
Project
Book
Event management
Marketing
Editing
Publishing
Infra-structure
Organising participants
Arranging event
Invitation and marketing
Organising interviews
Collecting articles
Peer reviewing
Facilities
Staff
Materials

34. Task list with precedence relations
Activity label
Task description
Duration (weeks)
Immediate predecessors a
Invitation 2 – b
Organising participants 4 c
Facilities 3 d
Staffing e
Materials f
Collecting articles 6 g
Peer reviewing h
Organising interviews 1
c, d, e i
Publishing 5 j
Arranging event
h, i
Activity label
Task description
Duration (weeks)
Immediate predecessors a
Invitation 2 b
Organising participants 4 c
Facilities 3 d
Staffing e
Materials h
Collecting articles 6 j
Peer reviewing k
Publishing 5 l
Organising interviews 1 m
Arranging event
Activity label
Task description
Duration
Immediate predecessors a
Invitation b
Organising participants c
Facilities d
Staffing e
Materials h
Collecting articles j
Peer reviewing k
Publishing l
Organising interviews m
Arranging event

35. AoA 4 9 19 3 c 5 10 19
4 d 1 2a 2 3 6 6 10 19 4b 9 20 3e 1h 10 21 1j 6 5 f
TPT = 21 i 7 12 8 15 3g
CP: a-b-f-g-i-j

36. AoN TPT: 21 CP: a-b-f-g-i-j c a b d h j e i f g 1 6 9 10 16 19 3 9 2 29 a 2 b 2 6 4 d 6 10 15 19 4 h 10 11 9 19 20 1 j 20 21 1 1 e 6 9 10 16 19 3 i 15 20 5 f 6 12 g 12 15 3
TPT: 21
CP: a-b-f-g-i-j

37. Activity times for the previous diagram (finalize individually)
Duration
EST
LST
EFT
LFT
Total float
Free float a 2 b 4 6 c 3 9 16 19 10 1 d e f g h i j

38. Example 2 (for individual practice)
a) Draw the AoA and AoN diagram with the data below:
Activity label
Duration (weeks)
Immediate predecessors a 1 – b 2 c 5 d 3 e f g 4
f, c, d
b) Determine the TPT and the critical path and activity floats.
c) Compute the EETs, LETs and slack for every node in the AoA & ESTs, LSTs, EFTs, LFTs in the AoN diagram.

39. Solution: AoA 3 5 1 2 6 7 4 2 b 2 e 2 f 1 7 1 a 5 c 4 g 7 3 d TPT: 11
Float: 0 3 5
Float: 0 2 b 2 e 2 f
Float: 0
Slack: 0
Slack: 0 1 2 1 6 7 7 11 1 a 5 c 4 g
T. float: 1
F. float: 1
Float: 0
Float: 0
Slack: 0
Slack: 0
Slack: 0
Slack: 0 4 7 3 d
T. float: 3
F. float: 3
Slack: 3
TPT: 11
CP: a-b-e-f-g

40. Solution: AoN TPT: 11 CP: a-b-e-f-g b e f g a c d 1 3 2 3 5 2 5 7 2 1b 1 3 2 e 3 5 2 f 5 7 2 1 g 7 11 4 a 1 c 1 6 2 7 5 3 d 1 4 3 7
TPT: 11
CP: a-b-e-f-g

41. Example 3 3 2 b 5 3 e 5 1 2 2 a 2 5 c 5 7 8 1 f 14 2 8 14 4 2 4 g 6 3 d 12 2 h 5 12
Calculate the EETs and LETs.
Create a precedence table (with task, duration, immediate predecessor, total and free floats).

42. Solution Activity Duration Total float Free float a 2 b c 5 1 d 3 e fb c 5 1 d 3 e f g 4 h

43. ‘Crashing’ – reducing task durations by increased costs

44. Definition of crashing
Obtaining reduction in time at an increased cost (increasing the employed resources).
Cost-slope: the cost of reducing duration time by a unit of time.
Let’s see the following example: a 4 b 2 c e d 5 f 3 c 6 8 1 7 9 2 e 8 10 1 9 11 2 a 4 b 4 6 2 f 11 14 3 d 6 11 5

45. Procedure for crashing
Crash one time unit at a time
Only the crashing of critical activities has any effect on TPT
Crash that activity first that is the cheapest to reduce in time
Be aware of multiple critical paths
Stop crashing when:
the crash-time is reached at every ‘crashable’ activity,
benefits of possible crashing are lower than crashing costs.

46. Crashing table
If the costs to reduce times are known, then a table can be set up showing the relative costs for the reduction in time of each activity by a constant amount.
Crash-time is the minimum duration of an activity. It is given by technical factors.
Activity
(label)
Duration
(day)
Float (day)
Crash time
Cost-slope (€/day) a 4 2
100 b
150 c 1
110 d 5 3
200 e
160 f
500
Crash time is the minimum duration of an activity (one cannot crash the duration of an activity below the crash time).
Benefit of reducing TPT by one day:
400 €/day

47. Solution method step: identify the critical activities
step: find the critical activity with cheapest crash cost, and if its cost slope is lower than the daily benefit from crashing, reduce its duration with one day. If there is no activity to crash, or it is too costly, stop crashing and go to step 4.
step: reidentify the critical path, and go back to step two.
step: identify the final critical path(s), TPT and the total net benefit of crashing.

48. Path / activity crashed Path / activity crached
Solution
Path durations
Path / activity crashed
normal
step 1
step 2
step 3
step 4
step 5 – a d
d, c
none
a-b-c-e-f 13 12 11 10
a-b-d-f 14
Cost:
100
200
310
Cumulated net benefit:
300
600
800
890
Path durations
Path / activity crached
normal
step 1
step 2
step 3
step 4
step 5
Cost:
Cumulated net benefit:
After crashing:
there are two critical paths
TPT is 10 days
total benefit of crashing is €890

49. Example 2 (for individual work)b 2 d 2 e 5 a 3 g 3 7 c 3 f 3
Identify the critical path and the TPT.

50. Example 2 (for individual work)b 3 5 2 d 5 7 2 e 7 12 5 a 3 g 12 15 3 7 c 3 6 1 4 7 f 6 9 3 12
Critcal: a-b-d-e-g TPT: 15
Using tbe table on the next slide, calculate the optimal TPT with crashing.

51. What is the total profit on crashing? 10 days €3000
Activity
(label)
Normal duration (day)
Float (day)
Crash time
Cost-slope (€/day) a 3 1
500 b 2
550 c
150 d 5
900 e 4
400 f
100 g
200
Activity
(label)
Normal duration (day)
Float (day)
Crash time
Cost-slope (€/day) a 3 1
500 b 2
550 c
150 d 5
900 e 4
400 f
100 g
200
Benefit of reducing TPT by one day:
1200 €/day
What is the new TPT?
What is the total profit on crashing?
10 days
€3000

52. Reading
Lockyer – Gordon (2005) Chapter 8 pp & Chapter 14

53. Thanks for the attention!