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CE 332 CONSTRUCTION ENGINEERING AND MANAGEMENT
PLANNING
EXPLANATION ON FAQ
(1) AoA and AoN Network Solutions
Early Start Time of the activity
Late Finish Time of the activity
EETj | LETj
(C) (D)
EETi | LETi
(A) (B)
Activity
Duration (d)
Predecessor
Corresponds to Late Finish Time of its predecessor

Successor
Corresponds to Early Start Time of its successor
From an AoA network solution you can read only the Early Start and Late Finish times of
activities directly.

For Early Finish and Late Start times you should make calculations by adding or subtracting
the durations of the activities.
Early Start Time = A
Early Finish Time = A + duration
Late Finish Time = D
Late Start Time = D – duration

From an AoN network solution you can read all Early Start, Early Finish, Late Start, and
Late Finish times directly from the node of the activity.
ES Duration EF
Activity Name
TF
LS
LF
(2) Float Calculations and Critical Path
Affects Late Finish Time of its predecessor
Affects Early Start Time of its successor
EETj | LETj
(C) (D)
EETi | LETi
(A) (B)
Predecessor
Activity
Duration (d)
A
Predecessor
B

Successor
dur
Activity
C
Successor
D
Total Float: Does not constrain durations at the activities level. So the activity may take
from its predecessor's or its successor's time (when predecessor can not finish late and
successor can not start early).
TF = Late Finish Time of Activity – Early Start Time of Activity – Activity Duration
TF = D – A – duration

Free Float: Should not affect early start time of its successor. So the activity may take from
its predecessor's time (when predecessor can not finish late and successor can start early).
FF = Early Start Time of Successor – Early Start Time of Activity – Activity Duration
FF = C – A – duration

Independent Float: Should not affect any other activity. So the activity can not take any
duration from its successor's or predecessor's time (when predecesssor can finish late and
successor can start early).
IF = Early Start Time of Successor – Late Finish Time of Predecessor – Activity Duration
IF = C – B – duration

Critical Path: Zero float activities indicate critical path candidates but the only paths that
are equal to total project duration are critical paths.
PLANNING EXAMPLES
Example 1. For the given AoA network draw the AoN network.
C
E
H
F
G
A
I
G
D
B
Solution:
Extract activities and their predecessors from the given network.
Activities
A
B
C
D
E
F
G
H
I
Predecessors
A
A,B
B,C
D,E
D
F
G,H
Redraw the network according to the predecessors.
A
C
E
F
START
H
I
B
D
G
END
Example 2. For the given AoN network draw the AoA network.
A
F
B
H
START
J
C
E
D
G
END
I
Solution:
Extract activities and their predecessors from the given network.
Activities
A
B
C
D
E
F
G
H
I
J
Predecessors
A
A,C
D
B,E
E
F,G
G
F,I
Redraw the network according to the predecessors.
B
F
J
A
H
I
C
D
E
G
Example 3. According to the given activities and predecessors draw the AoA network.
Activities
A
B
C
D
E
F
G
H
I
J
K
Predecessors
A,B
B,C
D,C
D
E,F
G
G
H,I,J
Solution:
D
G
I
J
A
B
K
E
C
H
F
Example 4. How many things are wrong the network diagram shown below?
F
G
C
H
A
T
B
B
D
R
E
Solution:
1
Activity D has no arrow head
2
Event # 5 is duplicated
3
The dummy between Event #2 and Event # 5 is unnecessary
4
Activity ID B is duplicated
5
Activity G has no ending event
6
Activity H has two arrow heads
7
Activities T, R, B form a loop
Example 5. Write True (T) or False (F) for the following statements;
-------- Network is a diagram to represent the relationship of activities to complete the
project. The network may be drawn as either an “arrow diagram” or a “precedence
diagram”.
-------- Activity on Node diagrams require the use of dummy activities.
-------- Total Float is the amount of time an activity may be delayed without delaying the
early start time of the immediately following activity.
-------- In CPM calculations, Backward Pass calculations cannot be performed without
ending the Forward Pass Calculations.
Answer:
T
Network is a diagram to represent the relationship of activities to complete the
project. The network may be drawn as either an “arrow diagram” or a “precedence
diagram”.
F
Activity on Node diagrams require the use of dummy activities.
F
Total Float is the amount of time an activity may be delayed without delaying the
early start time of the immediately following activity.
T
In CPM calculations, Backward Pass calculations cannot be performed without
ending the Forward Pass Calculations.
Example 6. What is the difference between identity dummy and logic dummy?
Answer: Identity dummy is used in order to identify activities when two or more parallel
independent activities have the same start and finish events. On the other hand, logic dummy
is used in order to prevent the error in logic arising from chains of wholly or partly
independent activities having a common event.
Example 7. By using the Legend given in the Table 1 and information given in the Table 2, solve
the given problem below.
Table 1: Legend for the Critical Path Calculations (CPM)
Early Start
Total Float
Early Finish
Activity Name
Late Start
Duration
Late Finish
Table 2: Information about the schedule
Activity Name
B
E
X
Z
T
A
C
Y
D
K
Duration (days)
10
15
20
25
10
15
15
20
30
35
Dependence
B
B
B
E, X
X, Z
E, X, Z
A, K
T, C
B
a) Construct the precedence diagram (Note: All type of dependencies are finish to start).
b) By using Activity on Node (AoN) methods, compute the total project duration.
c) Identify the critical path.
d) Find the free and independent floats of the activities C, E and K.
Solution:
a) b) c)
10 10 25
E
20 15 35
30 10 40
T
40 10 50
10 5 30
X
15 20 35
35 0 50
C
35 15 50
10 0 35
Z
10 25 35
35 10 50
A
45 15 60
50 0 80
D
50 30 80
80 0 80
FINISH
80 0 80
0
0 10
B
0 10 10
50 0 70
Y
60 20 80
10 15 45
K
25 35 60
Duration: 80 days
Critical Path: B-Z-C-D
d)
Free Float C = 50-35-15 = 0 days
Independent Float C = 50-35-15 = 0 days
Free Float E = 30-10-15 = 5 days
Independent Float E = 30-10-15 = 5 days
Free Float K = 50-10-35 = 5 days
Independent Float K = 50-10-35 = 5 days
Example 8. Draw the AoN network of the given activities and make necessary calculations.
Activities
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Duration (days)
6
3
4
9
12
8
3
3
4
4
3
1
6
7
4
3
Predecessors
A
A
A
B,D
C,D
C,D
E
E
E
G,H
H
N,I,J
K,L
Solution:
0 8 8
F
21 21 29
ES Duration EF
Activity
LS
TF
LF
0 0 0
START
0 0 0
6 3 9
B
17 11 20
15 3 18
G
20 5 23
18 6 24
M
23 5 29
6 9 15
D
6 0 15
15 3 18
H
15 0 18
18 7 25
N
18 0 25
6 4 10
C
11 5 15
15 4 19
I
21 6 25
0 6 6
A
0 0 6
12 4 16
J
21 9 25
0 12 12
E
9 9 21
12 3 15
K
23 11 26
12 1 13
L
25 13 26
Duration: 29 days
Critical Path: A-D-H-N-O
15 3 18
P
26 11 29
25 4 29
O
25 0 29
29 0 29
FINISH
29 0 29