Activity networks and precedence tablesEdexcel A-Level Further Maths: Revision notes

A critical path analysis models a project as an activity network. In the activity on arc form:

• each activity is an arc, with its duration written on it;
• each event is a node, marking the moment an activity starts or finishes;
• the network has one start event and one finish event;
• an event is reached only when all the activities ending there are complete, and only then can activities starting there begin. Events are numbered, usually in increasing order from start to finish. Each activity is identified by its letter, and the network must contain no cycles.

A precedence table lists each activity with its immediate predecessors: the activities that must be finished before it can start, and which have no other activity between them. Tables only show immediate predecessors. Example: if C depends on B, and B depends on A, then C's immediate predecessor is B only, not A. A is implied because B cannot start until A is finished. Activities with no predecessors start at the start event. An activity that is not a predecessor of any other activity ends at the finish event.

A dummy activity is an arc of zero duration (drawn as a dotted arrow) that uses no resources. It has two uses.

1. Logic: to show a dependency without making another activity depend on it. Example: S depends on P and Q, but R depends only on P. Then a dummy runs from the end of P to the start of S, which is where Q ends.
2. Unique identification: if two activities have the same start and end events, one is given its own end event and joined to the shared event with a dummy, so each activity has a unique pair of events. A dummy shows only that one event must be reached before another. Do not add dummies that are not needed.

1. Start with the activities that have no predecessors, drawn from the start event.
2. An activity with a single predecessor starts at the event where that predecessor ends.
3. An activity with several predecessors starts at one event where all of them end, using dummies where needed.
4. Where two activities share all their predecessors, they start at the same event.
5. Finish with all the activities that have no successors joined to one finish event, then number the events. Example: A none, B none, C is A and B, D is B. B ends at the start of D, A ends at the start of C, and a dummy runs from the end of B to the start of C. Without the dummy D would wrongly depend on A.

To find the immediate predecessors of an activity, look at its start event. The predecessors are the activities that end at that event, plus, for each dummy ending at that event, the activities that end at the start of the dummy. Example: D starts at event 3, where B ends and a dummy from event 2 arrives. Event 2 is where A ends, so D's immediate predecessors are A and B. Dummies are not listed as predecessors. If the dummy comes from an event where several activities end, all of them are predecessors.