Activity networks and precedence tablesEdexcel International A Level Maths: Revision notes

A project is split into activities, each with a duration. Some activities cannot start until others have finished. A precedence table (dependence table) lists each activity with its immediate predecessors: the activities that must be completed immediately before it can start. Only immediate predecessors are listed. If A must precede C and C must precede F, then F's entry is C only, because A is already implied. An activity with no predecessors is shown with a dash and starts at the start of the project. For example, if C depends on A and D depends on A and B, the table shows C: A and D: A, B.

In an activity-on-arc network each arc (an arrow) is an activity, labelled with its letter and duration. The nodes are events: points in time when activities start or finish. The conventions are:

• one start event and one finish event only;
• every arrow points forward, and events are numbered so that arrows go from lower to higher numbers;
• an activity can start only when every activity arriving at its start event has finished;
• no two activities may share the same start event and the same finish event, and no loops are allowed. Reading the network: the predecessors of an activity are all the activities that arrive at its start event (including through dummies).

1. Draw the start event. Every activity with no predecessor starts there.
2. Work down the table. An activity with a single predecessor starts at the event where that predecessor finishes.
3. An activity with several predecessors must start at an event where all of them finish, which sometimes needs dummies.
4. Activities which nothing depends on all finish at the single finish event.
5. Check every precedence relationship against the finished network, and number the events. Example: A, B no predecessors; C: A; D: A, B. A runs from 1 to 2 and B from 1 to 3. C starts at 2. D must wait for A and B, so it starts at 3 with a dummy from 2 to 3. Check that C does not depend on B: it starts at event 2, which only A reaches.

A dummy activity is shown as a dotted or dashed arrow. It has zero duration and uses no resources. It has two uses:

• To show dependence correctly: where an activity depends on two others, but one of those has a successor depending on it alone. In the example above, the dummy from 2 to 3 makes D wait for A without making C wait for B.
• To give unique descriptions: if two activities have the same start and finish events they would both be described as 'event $i$ to event $j$'. A dummy is added to one so that every activity has a different pair of events. Use as few dummies as possible: a dummy that does not change the logic is redundant and should be removed. In exam answers, always draw the dummy with the correct direction, because the arrow shows which event depends on which.

To complete a table from a network, read each activity's start event and list the activities that arrive at it. Follow dummies backwards: an activity starting at an event reached by a dummy also depends on the activities finishing at the dummy's start. Example: A from 1 to 2, B from 1 to 3, C from 2 to 4, dummy from 2 to 3, D from 3 to 4. Event 3 is reached by B and by the dummy from event 2, where A finishes, so D depends on A and B. C starts at event 2, which only A reaches, so C depends on A alone. Check that you list immediate predecessors only, and that dummies themselves do not appear in the table.