Activity networks – Example 1

Task Duration (hours) Immediate predecessors

A 3 -

B 4 -

C 6 -

D 5 A

E 1 B

F 6 B

G 7 C, 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.

Activity networks – Example 1

Activities A, B and C do not depend on any other activity, so they all begin at node 1.

A(3)

B(4)1

C(6)

First draw a start node labelled 1.

Activity networks – Example 1

Activity D depends on A, so add event node 2 at the end of A.

A(3)

B(4)1

C(6)

D(5)

Now add activity D.

2

Activity networks – Example 1

Activities E and F both depend on B, so add event node 3 at the end of B.

A(3)

B(4)1

C(6)

Now add activities E and F.

E(1)

D(5)

F(6)

3

2

C(6)

B(4) E(1)

F(6)

3

Activity networks – Example 1

Activity G depends on C, D and E, so all these three events need to end at the same node.

A(3)

This is easiest if you redraw the network so that C is between A and B.

D(5)2

B(4)

C(6)

E(1)

F(6)

31

Activity networks – Example 1

A(3)

D(5)2

C(6)

B(4) E(1)

F(6)

3

Now add node 4, with C, D and E leading into it.

Now add activity G.

1 4

D(5)

G(7)

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

F(6)

3

A finish node is now needed. Any activities not leading into a node must end at the finish node.

1 4

D(5)

G(7)

5

F(6)

G(7)

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

The next step is to find the early event times (EETs). An EET is the earliest time that an event (denoted by the numbered nodes) can occur. The event cannot occur until all activities leading into the event node have finished.

1 4

D(5)

5

F(6)

G(7)

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

Event 1 occurs at time zero.

1 4

D(5)

5

F(6)

G(7)0

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

Event 2 cannot occur until A is finished. The earliest time for this is 3.

1 4

D(5)

5

F(6)

G(7)0

3

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

Event 3 cannot occur until B is finished. The earliest time for this is 4.

1 4

D(5)

5

F(6)

G(7)0

3

4

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

Event 4 cannot occur until C, D and E are all finished.

1 4

D(5)

5

F(6)

G(7)0

3

4

The earliest C can finish is 6. The earliest D can finish is 3 + 5 = 8. The earliest E can finish is 4 + 1 = 5.

So the earliest time for event 4 is 8.

8

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

Event 5 cannot occur until F and G are both finished.

1 4

D(5)

5

F(6)

G(7)0

3

4

The earliest F can finish is 4 + 6 = 10. The earliest G can finish is 8 + 7 = 15.

So the earliest time for event 5 is 15.

8

15

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

The next step is to find the late event times (LETs), working backwards through the network. A LET is the latest time that an event can occur without delaying the project. The LET is found by finding the latest time that each activity leading out of the event can begin – the LET is the earliest of these.

1 4

D(5)

5

F(6)

G(7)0

3

4

8

15

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

Event 5 must occur by time 15, or the project will not finish in the minimum possible time.

1 4

D(5)

5

F(6)

G(7)0

3

4

8

15 15

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

The only activity leading from event 4 is G, which must start by time 8 if the project is not to be delayed. So event 4 must occur by time 8.

1 4

D(5)

5

F(6)

G(7)0

3

4

8

15 15

8

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

The activities leading from event 3 are E (which must start by time 7) and F (which must start by time 9). So event 3 must occur by time 7.

1 4

D(5)

5

F(6)

G(7)0

3

4

8

15 15

8

7

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

The only activity leading from event 2 is D, which must start by time 3. So event 2 must occur by time 3.

1 4

D(5)

5

F(6)

G(7)0

3

4

8

15 15

8

7

3

Activity networks – Example 1

A(3)

2

C(6)

B(4) E(1)

3

Finally, event 1 must occur by time zero.

1 4

D(5)

5

F(6)

G(7)0

3

4

8

15 15

8

7

3

0

Activity networks – Example 1

The completed network shows that the project can be completed in 15 hours.

The critical activities are the activities (i, j) for which the LET for j – the EET for i is equal to the activity duration.

The critical activities are A, D and G.

For analysis of the float in this example, see the Notes and Examples.

A(3)

C(6)

B(4) E(1)

3

D(5)

F(6)

G(7)0

3

4

8

15 15

8

7

3

0

1

2

4

5