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Activity Networks & Precedence Tables (Edexcel A Level Further Maths: Decision Maths 1)

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Paul

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Paul

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Activity Networks & Precedence Tables

What is an activity network?

  • An activity network is a graph that shows the activities needed - and in what order - to complete a project
    • e.g.  the project could be 'building a house' with activities such as 'foundations', 'walls' and 'roof'
  • Some activities will depend on others being completed first
    • e.g. the activity 'foundations' would need to be completed before the 'walls' are built
  • Some activities can occur at the same time
    • e.g. 'windows' and 'doors' can be fitted at the same time
  • The arcs (edges) of the graph represent the activities
    • this may be referred to as an activity-on-arc network
  • The nodes (vertices) of the graph represent events within the project
    • events can be thought of as 'stepping stones'
    • the project cannot progress beyond an event until all the activities leading to that event are completed
      • 'you can't jump off the stepping stone until all activities leading to that stone are completed'

What does an activity network look like?

  • Events (Nodes) are labelled with numbers, generally increasing in the direction of the project
    • The event at the start of the project is called the source node
      • it is labelled with 0 or S
    • The event at the end of the project is called the sink node
      • it will be the highest numbered node or labelled with T
  • Arcs are labelled with their activities, with the duration given in brackets
    • activities are denoted by capital letters - A, B, C, D, etc
    • arrows are drawn on the arcs to show the order in which the project progresses
      • so strictly speaking, an activity network is a directed graph
      • in broad terms, this is generally from left to right across the activity network

non-dummy-activity-network

What is a precedence table?

  • A precedence table shows a list of the activities for a project
    • For each activity, the table includes a list of the activities that must already have been completed
      • but only the immediately preceding activities are listed, not all of them
    • Activities that do not have any precedents are indicated by '-'
      • these activities can begin at the start of the project
      • they will be attached to the source node

What does a precedence table look like?

  • As well as a list of activities a precedence table may also show
    • the duration of each activity
    • the number of workers required to complete that activity

precedence-example

Drawing an Activity Network

How do I draw an activity network?

  • An activity network can be drawn from a precedence table
  • Starting with the source node
    • add an arc for each activity, one at a time, considering its immediately preceding activities
    • an event (node) will be needed prior to each activity commencing
      • more than one activity can commence from the same event
      • more than one activity can finish at the same event
  • A crucial feature of an activity network is that each activity has a unique pair of start and end nodes
  • Any activities that do not precede another will go to the sink node at the end of the project
  • In general, activity networks
    • use straight, arrowed lines for arcs
    • numbered circles for events/nodes

Examiner Tip

  • A rough, curly-edged activity network often helps to start off with
    • This will give you a mental picture of what the network looks like
    • You can easily make changes, scribble bits out, etc with a rough diagram
    • When you are happy with it, you can redraw it neatly with straight edges
      • Remember the arrows, event/node numbers and activities with their durations!

Worked example

Draw an activity network for the precedence table given below.

Activity Preceding activities Duration
A - 4
B A 5
C A 3
D B 6
E C 4
F D, E 6

Starting with the source node, node 0, it is only activity A that can begin
So we will have one arc starting at the source node
Label the arc with an arrow, the activity name (A) and its duration (4)

drawact-we-ans-1

Activities B and C both depend on A, so add event/node 1 with arcs for B and C attached
Leave plenty of room (between B and C) in case anything later needs to go in between them

drawact-we-ans-2

Activity D follows from B only, and activity E follows C only
Looking ahead though, activity F has D and E as immediate predecessors, so D and E need to meet at an event
So use event 2 to start activity D, event 3 to start activity E, and event 4 where they meet, ready for activity F

drawact-we-ans-3

Activity F is the last activity of the project so goes to the sink node, event 5

drawact-we-ans-4

Check that all activities have a unique start and end node
For example, activity B starts at event/node 1 and ends at event/node 2 (this may be written as an ordered pair, (1, 2))
No other activity starts at 1 AND ends at 2 (C is (1, 3))
Checking everything else, the final answer is

drawact-we-final-ans

Completing a Precedence Table

How do I complete a precedence table?

  • A precedence table can be constructed from an activity network
  • A basic table listing the activities and their duration can be constructed from the labels on the activity network
  • To complete the preceding activities column in the table
    • start at the source node
      • any activities starting at the source node do not have preceding activities so use '-' in the table
    • for all other activities look at the event/node the activity starts at
      • any activities ending at this event/node are the immediately preceding activities
  • The numbers of workers (for each activity) are not shown on an activity network so this column in the table would not be required

Worked example

Construct a precedence table for the activity network shown below.

complprec-we-qu

The activities are A, B, C, D, E and F, with their durations given in brackets, so two columns of the precedence table can be completed immediately

Activity Preceding activities Duration
A   4
B   5
C   3
D   6
E   4
F   6

Starting at the source node 0, only activity A has no preceding activities, so this can be completed with a '-'
Work through each other activity considering the activities that go to its start event/node

  • activity B starts at event/node 1
    • activity A ends at event/node 1
    • B has immediate predecessor A
  • C starts at 1
    • A ends at 1 
    • C has immediate predecessor A
  • D starts at 2
    • B ends at 2
    • D has immediate predecessor B
  • E starts at 3
    • C ends at 3
    • E has immediate predecessor C
  • F starts at 4
    • D and E end at 4
    • F has immediate predecessors D and E 

F is the last activity (it ends at the sink node, event 5) so the precedence table can be completed

Activity Preceding activities Duration
A - 4
B A 5
C A 3
D B 6
E C 4
F D, E 6

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    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.