What do arcs and nodes represent in an activity network?

**Arcs (edges)** represent **activities**; **nodes (vertices)** represent **events**. This is an activity-on-arc network. Events are 'stepping stones' — the project cannot pass an event until all activities leading to it are complete.

What are the source and sink nodes?

The **source node** is the start of the project, labelled 0 or S. The **sink node** is the end, the highest numbered node or labelled T.

What does a precedence table show?

A precedence table lists each activity and its **immediately preceding** activities (those that must be completed before it can start). It may also show duration. Activities with no predecessors are marked '-'.

Why must each activity have a unique (start, end) node pair?

So each activity is uniquely identified in the network. If two activities shared the same pair, they would be indistinguishable. Dummies are used when the structure would otherwise create duplicate pairs.

How do you find the immediate predecessors from an activity network?

Look at the **event (node)** where the activity starts. All activities that **end** at that event are the immediate predecessors of the activity.

Where do activities with no predecessors start?

Activities with no predecessors (marked '-' in the precedence table) start at the **source node** (node 0).

Where do activities with no successors go?

Activities that do not precede any other activity go to the **sink node** at the end of the project.

What is an activity-on-arc network?

An activity-on-arc network is one where **activities** are represented by **arcs** (edges) and **events** by **nodes** (vertices). The alternative is activity-on-node, where activities are on nodes.

Can more than one activity start from the same event?

Yes. Multiple activities can **commence** from the same event (e.g. B and C both start after A finishes). Multiple activities can also **finish** at the same event.

When drawing a network from a precedence table, what order should you add activities?

Add arcs **one at a time**, considering each activity's immediate predecessors. You need an event before each activity starts. Look ahead to see where activities will need to meet (e.g. for a common successor).

Activity Networks & Precedence Tables | Pearson Edexcel International AS Maths

Critical Path Analysis

Activity Networks & Precedence Tables

Summary

Activity networks are directed graphs that represent project activities as arcs and events as nodes. They show the order of activities and their dependencies. Precedence tables list activities with their immediately preceding activities and durations. Networks can be drawn from precedence tables and vice versa. Each activity must have a unique (start, end) node pair.

1. What is an Activity Network?

An activity network is a graph showing the activities needed to complete a project and the order in which they should be undertaken.

Arcs (edges) represent activities (activity-on-arc network). Nodes (vertices) represent events — 'stepping stones' — the project cannot progress past an event until all activities leading to it are complete.

Some activities depend on others (e.g. walls after foundations). Some can run in parallel (e.g. windows and doors).

The network is a directed graph; arrows show the direction of progress. Projects typically flow left to right.

2. Structure of an Activity Network

Source node: the start of the project, labelled 0 or S. Sink node: the end, the highest numbered node or T.

Events are numbered, generally increasing in the direction of the project.

Arcs are labelled with activity name (A, B, C, ...) and duration in brackets. Arrows show direction.

Conventions: straight arrowed lines for arcs; numbered circles for events.

3. What is a Precedence Table?

A precedence table lists activities and, for each, the immediately preceding activities that must be completed first.

Only immediate predecessors are listed. Activities with no predecessors are marked '-' and start at the source node.

The table may also include the duration of each activity.

Example: Activity F with preceding D, E means F cannot start until both D and E are finished.

4. Drawing an Activity Network from a Precedence Table

Start with the source node. Add arcs one at a time, considering each activity's immediate predecessors.

An event (node) is needed before each activity starts. Multiple activities can start from the same event or finish at the same event.

Crucial rule: each activity must have a unique (start, end) node pair. Activities with no successors go to the sink node.

Tip: sketch a rough diagram first, then redraw neatly with straight edges.

5. Completing a Precedence Table from an Activity Network

6. Worked Example

Activity Networks & Precedence Tables

Summary

1. What is an Activity Network?

An activity network is a graph showing the activities needed to complete a project and the order in which they should be undertaken.

Some activities depend on others (e.g. walls after foundations). Some can run in parallel (e.g. windows and doors).

The network is a directed graph; arrows show the direction of progress. Projects typically flow left to right.

2. Structure of an Activity Network

Source node: the start of the project, labelled 0 or S. Sink node: the end, the highest numbered node or T.

Events are numbered, generally increasing in the direction of the project.

Arcs are labelled with activity name (A, B, C, ...) and duration in brackets. Arrows show direction.

Conventions: straight arrowed lines for arcs; numbered circles for events.

3. What is a Precedence Table?

A precedence table lists activities and, for each, the immediately preceding activities that must be completed first.

Only immediate predecessors are listed. Activities with no predecessors are marked '-' and start at the source node.

The table may also include the duration of each activity.

Example: Activity F with preceding D, E means F cannot start until both D and E are finished.

4. Drawing an Activity Network from a Precedence Table

Start with the source node. Add arcs one at a time, considering each activity's immediate predecessors.

An event (node) is needed before each activity starts. Multiple activities can start from the same event or finish at the same event.

Crucial rule: each activity must have a unique (start, end) node pair. Activities with no successors go to the sink node.

Tip: sketch a rough diagram first, then redraw neatly with straight edges.

5. Completing a Precedence Table from an Activity Network

List activities and durations from the network labels.

For preceding activities: start at the source. Activities starting there have '-'.

For other activities: look at the event the activity starts at. Activities ending at that event are the immediate predecessors.

Work through each activity systematically from source toward sink.

6. Worked Example

Given: A(-,4), B(A,5), C(A,3), D(B,6), E(C,4), F(D,E,6). A starts at source. B and C follow A and diverge from event 1. D follows B (event 2), E follows C (event 3). D and E meet at event 4 for F. F goes to sink (event 5).

Check: B is (1,2), C is (1,3), D is (2,4), E is (3,4), F is (4,5). All unique.