Activity networks Example 2

1
Activity networks Example 2
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.
2
Activity networks Example 2
A(2)
1
B(3)
C(5)
Begin with a start node, labelled 1.
Activities A, B and C have no preceding
activities, so can all begin at the start node.
3
Activity networks Example 2
2
A(2)
D(6)
1
3
B(3)
C(5)
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.
4
Activity networks Example 2
2
A(2)
D(6)
1
3
B(3)
C(5)
E(8)
4
Activities E and F both depend on activity C.
F(2)
5
Activity networks Example 2
2
A(2)
D(6)
G(4)
1
3
5
B(3)
C(5)
E(8)
E(8)
4
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.
F(2)
6
Activity networks Example 2
2
A(2)
D(6)
G(4)
1
3
5
6
B(3)
C(5)
E(8)
F(2)
4
Finally, activities F and G must finish at the
end node.
F(2)
7
Activity networks Example 2
2
A(2)
D(6)
G(4)
1
3
5
6
B(3)
C(5)
E(8)
F(2)
4
The next step is to find the earliest event times
(EETs).
8
Activity networks Example 2
2
2
A(2)
D(6)
G(4)
1
3
5
6
B(3)
0
C(5)
E(8)
F(2)
4
Event 1 occurs at time zero.
The earliest that event 2 can occur is after A
has finished, at time 2.
9
Activity networks Example 2
2
2
A(2)
D(6)
G(4)
1
3
5
6
B(3)
0
3
C(5)
E(8)
F(2)
4
Event 3 cannot occur until both A and B have
finished, so the earliest time at which event 3
can occur is 3.
10
Activity networks Example 2
2
2
A(2)
D(6)
G(4)
1
3
5
6
B(3)
0
3
C(5)
E(8)
F(2)
4
Event 4 cannot occur until C has finished, so the
earliest time at which event 5 can occur is 5.
5
11
Activity networks Example 2
2
2
A(2)
13
D(6)
G(4)
1
3
5
6
B(3)
0
3
C(5)
E(8)
F(2)
4
5
The 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.
12
Activity networks Example 2
2
2
A(2)
13
D(6)
G(4)
1
3
5
6
B(3)
0
3
17
C(5)
E(8)
F(2)
4
5
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.
13
Activity networks Example 2
2
2
A(2)
13
D(6)
G(4)
1
3
5
6
B(3)
0
3
17
C(5)
E(8)
F(2)
4
5
The next step is to find the latest event times
(LETs), starting from the finish node and working
backwards.
14
Activity networks Example 2
2
2
A(2)
13
D(6)
G(4)
1
3
5
6
B(3)
0
3
17
17
C(5)
E(8)
F(2)
4
5
Event 6 must not occur later than time 17, or the
project will be delayed .
15
Activity networks Example 2
2
2
A(2)
13
13
D(6)
G(4)
1
3
5
6
B(3)
0
3
17
17
C(5)
E(8)
F(2)
4
The latest G can start is at time 13, so the
latest time for event 5 is 13.
5
16
Activity networks Example 2
2
2
A(2)
13
13
D(6)
G(4)
1
3
5
6
B(3)
0
3
17
17
C(5)
E(8)
F(2)
4
5
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
17
Activity networks Example 2
2
2
A(2)
13
13
D(6)
G(4)
1
3
5
6
B(3)
0
3
17
17
7
C(5)
E(8)
F(2)
4
5
5
The latest that D can start is at time 7, so the
latest possible time for event 3 is 7.
18
Activity networks Example 2
2
7
2
A(2)
13
13
D(6)
G(4)
1
3
5
6
B(3)
0
3
17
17
7
C(5)
E(8)
F(2)
4
5
5
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.
19
Activity networks Example 2
2
7
2
A(2)
13
13
D(6)
G(4)
1
3
5
6
B(3)
0
3
17
17
7
0
C(5)
E(8)
F(2)
4
5
5
The latest that event 1 can start is at time 0.
20
Activity networks Example 2
2
7
2
A(2)
13
13
D(6)
G(4)
1
3
5
6
B(3)
0
3
17
17
7
0
C(5)
E(8)
F(2)
4
5
5
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.
21
Activity networks Example 2
2
7
2
A(2)
13
13
D(6)
G(4)
3
6
5
1
B(3)
0
3
17
17
7
0
C(5)
E(8)
F(2)
4
5
5
The critical activities are C,
For analysis of the float in this example, see
Example 2 in the Notes and Examples.
E
and G.
The project can be completed in 17 days.