1. Activity Networks

2. Example 1. α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 Activity
Pre-requisites
Time required α1 - 5 α2 4 α3 3 α4 α5 α6 6 α7 8 α8 9 α9
α4, α5, α7,
α10
α4, α5 10
α11
α6, α8
α12
α9, α11 2

3. The data is often illustrated in an activity network.
Example 1 (cont’d) α4
α13
dummy
activity
α10 α1 α5 α9 α2
start
end
α12 α6 α7 α3
α11 α8

4. Example 1 (cont’d) 5
dummy
activity 10 5 5 3 4
start
end 2 6 8 3 4 9 4

5. Example 2. α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 α13 Activity
Pre-requisites
Time required α1 - 8 α2 6 α3 1 α4 16 α5 13 α6
α1, α3 α7
α5, α8 2 α8 5 α9
α4, α7 7
α10
α11 10
α12 19
α13
α9, α10 4

6. α4 α9 α7 α10 α2 α5 α13 α8 α11 start end α3 α6 α1 α12
Example 2 (cont’d) α4 α9 α7
α10 α2 α5
α13
start α8
α11
end α3 α6 α1
α12

7. Forward labelling. Labelling method.
Example 2 (cont’d) 22 16 7 29 6 2 13 6 13 19 4
start 10 5 33
end 1 14 6 8 19
-Label the Start node 0.
-Locate a node where all the immediately preceding nodes are labelled. Label this node. 8
Forward labelling. Labelling method.

8. Forward labelling gives the earliest start time for each event.
More formally the vertex labels are determined by the following rule.
For a vertex v, let u1, u2,…, uk be the immediately preceding vertices. Let l(x) denote the label of x. Then if ui are already labelled for i=1,2,…,k then
This gives the earliest starting time for activity v.
The label at the end vertex gives the minimum time required for the entire project.

9. v 5 11 3 3 5 7 8 6
l(v) = max{3+5, 6+5, 7+3} = 11. 5 8

10. -Label the End node with the shortest time for the project.
Example 2 (cont’d) 22 16 7
Backward labelling. 29 6 2 13 6 13 20 4
start 10 5 33
end 1 15 6 8 19
-Label the End node with the shortest time for the project. 9
-Locate a node where all the immediately succeeding nodes are labelled. Label this node.

11. Backward labelling gives the latest start time for each event which will not make the entire project longer than its minimum time requirement.
The vertex labels are determined by the following rule.
For a vertex v, let u1, u2,…, uk be the immediately following vertices. Let l(x) denote the label of x. Then if ui are already labelled for i=1,2,…,k then
This gives the earliest starting time for activity v.

12. v 5 15 7 6 1 4 13 17 5 3 4 16 15
l(v) = min{15-5, 13-6, 17-4} = 7.

13. float times are all 0. Such a path is called a critical path.
The float time for an event is the difference between the latest start time and the earliest start time. The float times are as follows. There is always a path from start to end on which the
Example 2 (cont’d) 1
start
end 1
float times are all 0. Such a path is called a critical path. 1

14. N.B. Any extension in the duration of an activity along a critical path will lead to an extension in the duration of the entire project.

15. Both forward labelling and backward labelling can be done on the same diagram.22 16 7 22 29 6 2 29 6 13 6 13 19 4 20 10 33 5
end 1 33 14 15 8 6 19 8 9

16. Example 3. Without diagram α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 α13
Activity
Pre-requisites
Time required
Earliest Start time
Latest Start time
Earliest Finish time
LateSt FiniSh Time α1 - 8 α2 6 α3 1 α4 16 α5 13 α6
α1, α3 α7
α5, α8 2 α8 5 α9
α4, α7 7
α10
α11 10
α12 19
α13
α9, α10 4 8 6 6 7 6 22 6 19 8 14
maX of 8 and 7 19 21 14 19 22 29
maX of 22 and 21 14 27 19 29 8 27
maX of 8 and 7 29 33
maX of 29 and 27

17. Example 3. Cont’d α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 α13 - 8 6 1
Activity
Pre-requisites
Time required
Earliest Start time
Latest Start time
Earliest Finish time
LateSt FiniSh Time α1 - 8 α2 6 α3 1 α4 16 α5 13 α6
α1, α3 α7
α5, α8 2 α8 5 α9
α4, α7 7
α10
α11 10
α12 19
α13
α9, α10 4 1 8 9
min of 9 and 14 6 6
min of 8, 6 and 7 6 8 7 9
min of 9 and 14 6 6 22 22 6 7 19 20
min of 20 and 23 8 9 14 15
min of 15 and 16 19 20 21 22 14 15 19 20
min of 20 and 29 22 22 29 29 14 16 27 29 19 23 29 33 8 14 27 33 29 29 33 33

18. Example 4. Without diagram α1 α2 α3 α5 α7 α8 α9 α10 α11 α12 - 8 α4 5
Activity
Pre-requisites
Time required
Earliest Start time
Latest Start time
Earliest Finish time
LateSt FiniSh Time α1 - 8 α2 α4 5 α3 α6 12 10 α5 9 11 α7 α8
α2, α5 4 α9
α3, α8
α10
α11
α9, α10
α12 20 8 10

19. □ Some properties of activity networks.
Property 1: In an activity network there are no directed cycles.
Proof: A cycle would imply an endless sequence of activities each one preceding the other. □

20. Property 2: (a) In the forward labelling process, there is always an unlabeled node where all the immediately preceding nodes are labelled.
(b) In the backward labelling process, there is always an unlabeled node where all the immediately succeeding nodes are already labelled.
Proof of (a): Consider any unlabeled node u1. If all the immediately preceding nodes of u1 are labelled, then u1 is a node we are seeking, and we are done. Otherwise there is a node u2, immediately preceding u1 which is unlabelled. If all the immediately preceding nodes of u2 are labelled, then u2 is a node

21. we are seeking, and we are done
we are seeking, and we are done. Otherwise, there is a node u3, immediately preceding u2 which is unlabeled. In this manner we generate a sequence of unlabeled nodes u1, u2, … . Now, the nodes are all distinct. (why?) Since we only have a finite number of nodes this sequence must terminate at some node uk. Now, uk can only be a node for which every preceding node is labelled. This is a node we are seeking □
The proof of (b) is similar.

22. A vertex at which all the arcs are entering is called a sink.
Definition In an digraph (directed graph), a vertex at which all the arcs are leaving is called a source.
A vertex at which all the arcs are entering is called a sink. v w u
u and v are sources. w and x are sinks. x

23. Theorem In an digraph (directed graph) with no cycles, there is at least one source and one sink.
This can be proved in a similar fashion to the previous result.

24. Definition A path in a digraph (directed graph) is a sequence of arcs e1, e2, …, ek, such that, for any i, ei is entering the same vertex that ei+1 is leaving, and for i<j, arc ej does not enter the same vertex that ei is leaving.
Example: e1 e2 e3 e4 e5 e6 is a path. e1 e2 e6 e3 e5 e4

25. Definition The length of a path in an edge-weighted digraph is the sum of the weights of its arcs.4 3 e2 e6 4 1 6 e8 6 5 e3 e5 5 5 3 2 7 e7 e4
Example: length of path e1 e2 e3 e4 e5 e6 = = 26.
length of path e7 e4 e5 e8 = = 17

26. Theorem In an activity network, the labels obtained in the forward labelling process give the longest path lengths from the Start vertex to that vertex.
Proof Left as an exercise.

27. rajeshlakhan.weebly.com
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