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.

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Example Precedence Table

Activity	Duration	Immediate Predecessors
A	4	-
B	6	A
C	3	A
D	5	B, C
E	2	D

Activity $A$ has no predecessors, so it can start immediately.
Activities $B$ and $C$ depend on $A$ , meaning $A$ must finish before they start.
Activity $D$ depends on $B$ and $C$ , so it can only start once both $B$ and $C$ 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

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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**

$A$ has no predecessors; draw it starting from the start node.
$B$ and $C$ both depend on $A$ ; draw edges from the node for $A$ to the nodes for $B$ and $C$ .
$D$ depends on $B$ and $C$ ; draw edges from $B$ and $C$ to $D$ .
$E$ depends on $C$ ; draw an edge from $C$ to $E$ .
$F$ depends on $D$ and $E$ ; draw edges from $D$ and $E$ to $F$ .

Step 3**: Add Start and End Nodes**

Connect all starting activities to the start node.
Connect all ending activities (in this case, $F$ ) 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

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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.
Wrong Direction of Edges: Drawing edges in the incorrect direction can misrepresent the sequence of activities.
Overlapping Dependencies: Adding unnecessary dependencies that are not specified in the precedence table.
Misinterpreting "Immediate Predecessors": Confusing indirect dependencies (e.g., $A \to B \to D$ ) with immediate ones.

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Key Formulas/Theorems

Dependency Rule: An activity $X$ can only start after all its immediate predecessors have been completed.
Graph Flow: Activity networks flow from a single start node to a single end node.
Precedence Table Construction: For each activity:

\text{Predecessors of } X = \{\text{Activities ending at the start of } X\}

Activity Networks & Precedence Tables Simplified Revision Notes for A-Level Edexcel Further Maths Decision Maths 1

11.1.1 Activity Networks & Precedence Tables

Key Concepts

Activity Network

Precedence Table

Example Precedence Table

Constructing an Activity Network from a Precedence Table

Steps

Completing a Precedence Table from an Activity Network

Worked Example

Note Summary

Common Mistakes

Key Formulas/Theorems

500K+ Students Use These Powerful Tools to Master Activity Networks & Precedence Tables For their A-Level Exams.

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