of 21
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Activity networks Example 2
Task Duration (days) Immediate
predecessors
A 2 -
B 3 -
C 5 -
D 6 A, B
E 8 C
F 2 C
G 4 D, E
The table below shows the tasks involved in a project,
with their durations and immediate predecessors.
Draw an activity network and use it to find the critical
activities and the minimum duration of the project.
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Activity networks Example 2
Begin with a start node,
labelled 1.
Activities A, B and C have no
preceding activities, so can
all begin at the start node.
A(2)
B(3)1
C(5)
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Activity networks Example 2
Activity D depends on both A
and B. Since A and B must
not start and finish at the
same node, a dummy activity
is needed to ensure unique
numbering.
The dummy activity has
zero duration. Now
activity D can be drawn
in, following on from both
A and B.
A(2)
B(3)1
D(6)
2
3
C(5)
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Activity networks Example 2
Activities E and F both
depend on activity C.
A(2)
B(3)1
D(6)
2
3
C(5)
4
E(8)
F(2)
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E(8)
Activity networks Example 2
Since G depends on both D
and E, these two activities
must both lead into the
same node.
Activity G can now be
drawn in.
A(2)
B(3)1
D(6)
2
3G(4)
5
C(5)
4
E(8)
F(2)
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Activity networks Example 2
Finally, activities F and G
must finish at the end node.
A(2)
B(3)1
D(6)
2
3G(4)
5 6
E(8)
F(2)
C(5)
4
F(2)
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Activity networks Example 2
The next step is to find the
earliest event times (EETs).
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
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Event 1 occurs at time zero.
Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
The earliest that event 2 can
occur is after A has finished, at
time 2.
0
2
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
Event 3 cannot occur until
both A and B have finished,
so the earliest time at which
event 3 can occur is 3.
3
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
Event 4 cannot occur until C
has finished, so the earliest
time at which event 5 can
occur is 5.
5
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
5The earliest that D can finish
is at time 9, and the earliest
that E can finish is at time
13, so the earliest that event
5 can occur is time 13.
13
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
5
13
The earliest that F can finish
is at time 7, and the earliest
that G can finish is at time
17, so the earliest that event
5 can occur is time 17.
17
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
5
13
17
The next step is to find the
latest event times (LETs),
starting from the finish node
and working backwards.
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
5
13
17
Event 6 must not occur later
than time 17, or the project
will be delayed .
17
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The latest G can start is at
time 13, so the latest time
for event 5 is 13.
Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
5
13
17 17
13
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
5
13
17 17
13
The latest that E can start is
at time 5, and the latest that
F can start is at time 15, so
the latest possible time for
event 4 is 5.
5
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
5
13
17 17
13
5The latest that D can start is
at time 7, so the latest
possible time for event 3 is 7.
7
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
5
13
17 17
13
5
7
The latest that the dummy
activity (with zero duration)
can start is at time 7, so the
latest possible time for event
2 is 7.
7
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
5
13
17 17
13
5
7
7
The latest that event 1 can
start is at time 0.
0
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Activity networks Example 2
A(2)
B(3)1
D(6)
2
3G(4)
C(5) E(8)
4
5 6
F(2)
0
2
3
5
13
17 17
13
5
7
7
0
The critical activities are
activities for which the float is
zero: i.e. the latest event time
for activity j the earliest
event time for activity i is equal
to the activity duration.
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Activity networks Example 2
A(2)
B(3)
D(6)
2
3G(4)
C(5) E(8)
F(2)
0
2
3
5
13
17 17
13
5
7
7
0
The critical activities are
C, E and G.
The project can be
completed in 17 days.
For analysis of the float in this
example, see Example 2 in the
Notes and Examples.
65
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http://localhost/var/www/apps/conversion/tmp/scratch_5/4.1.pdfhttp://localhost/var/www/apps/conversion/tmp/scratch_5/4.1.pdf