Activity Networks Example 2

7/29/2019 Activity Networks Example 2

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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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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 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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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)


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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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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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.


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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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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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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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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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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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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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.


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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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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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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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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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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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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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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Activity Networks Example 2

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