Activity Networks & Precedence Tables (Edexcel A-Level Further Mathematics): Revision Notes
11.1.1 Activity Networks & Precedence Tables
Activity networks and precedence tables are tools used to model and manage projects efficiently. They show the relationships between activities, highlighting dependencies and the order in which tasks must be completed.
Key Concepts
Activity Network
An activity network is a directed graph where:
- Nodes represent events (the start or completion of activities).
- Edges represent activities, labelled with their duration.
- The direction of edges indicates the sequence of activities.
Precedence Table
A precedence table lists activities and their immediate predecessors. It specifies the order in which activities must occur, based only on their direct dependencies.
Example Precedence Table
| Activity | Duration | Immediate Predecessors |
|---|---|---|
| A | 4 | - |
| B | 6 | A |
| C | 3 | A |
| D | 5 | B, C |
| E | 2 | D |
- Activity has no predecessors, so it can start immediately.
- Activities and depend on , meaning must finish before they start.
- Activity depends on and , so it can only start once both and are completed.
Constructing an Activity Network from a Precedence Table
Steps
- Identify Nodes and Activities:
- Assign a start node and label each activity with its duration.
- Draw Dependencies:
- For each activity, draw an edge from its predecessors' node to its own node.
- Connect Events:
- Ensure all activities flow in the correct order from a single start node to a single end node.
Completing a Precedence Table from an Activity Network
To derive a precedence table from an activity network:
- List All Activities:
- Identify each activity (edges in the network) and their durations.
- Identify Predecessors:
- For each activity, determine which nodes (events) must be completed before it can begin.
- Create the Table:
- Populate the table with the activity, its duration, and its immediate predecessors.
Worked Example
Question Construct an activity network from the following precedence table:
| Activity | Duration | Immediate Predecessors |
|---|---|---|
| A | 3 | - |
| B | 2 | A |
| C | 4 | A |
| D | 5 | B, C |
| E | 3 | C |
| F | 6 | D, E |
Step 1**: Identify Nodes and Activities**
- Start with a single start node.
- Label each activity with its duration.
Step 2**: Draw Dependencies**
- has no predecessors; draw it starting from the start node.
- and both depend on ; draw edges from the node for to the nodes for and .
- depends on and ; draw edges from and to .
- depends on ; draw an edge from to .
- depends on and ; draw edges from and to .
Step 3**: Add Start and End Nodes**
- Connect all starting activities to the start node.
- Connect all ending activities (in this case, ) to a single end node.
Final Activity Network
The network has a directed flow from the start node to the end node, with all dependencies and durations represented.
Note Summary
Common Mistakes
-
Incorrect Dependencies: Forgetting that precedence tables only list immediate predecessors, not all preceding activities.
-
Omitting Start or End Nodes: Failing to include a single start and end node in the activity network leads to an incomplete graph.
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Wrong Direction of Edges: Drawing edges in the incorrect direction can misrepresent the sequence of activities.
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Overlapping Dependencies: Adding unnecessary dependencies that are not specified in the precedence table.
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Misinterpreting "Immediate Predecessors": Confusing indirect dependencies (e.g., ) with immediate ones.
Key Formulas/Theorems
-
Dependency Rule: An activity can only start after all its immediate predecessors have been completed.
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Graph Flow: Activity networks flow from a single start node to a single end node.
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Precedence Table Construction: For each activity: