Critical Path Analysis (Edexcel IAL Maths D1)

Edexcel International A-Level Mathematics· D1 §4.1 to §4.4 (Issue 3 2018 specification)· 25 min read

1. Activity-on-Arc Networks from Precedence Tables★★☆☆☆⏱ 6 min

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📘 Definition

Activity-on-Arc (AOA) Network

A directed graph where arcs represent project activities, nodes represent events (the start or end of one or more activities), and dashed 0-duration dummy arcs show logical precedence where no real activity occurs.

To draw an AOA network, start with a single start node, add arcs for each activity using the precedence table, and use dummies to resolve cases where two activities share some but not all predecessors. Ensure the network ends with a single end node.

📐 Worked Example

Draw the AOA network for the following precedence table: Activity A has no predecessors, Activity B has no predecessors, Activity C has predecessor A, Activity D has predecessors A and B.

1
1. Label the start event as Node 1. Add arcs A (Node 1 → Node 2) and B (Node 1 → Node 3) as they have no predecessors.
2
1. Activity C only follows A, so add arc C from Node 2 → Node 4 (the end event for this activity).
3
1. Activity D follows both A and B, so add a dashed dummy arc (0 duration) from Node 2 → Node 3 to show A is a predecessor for D. Add arc D from Node 3 → Node 4 (final end event).

Exam tip:

Always label dummies as dashed lines with 0 duration: you will lose presentation marks if you omit this formatting.

2. Deriving Precedence Tables from AOA Networks★★☆☆☆⏱ 5 min

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To build a precedence table from an AOA network, list all non-dummy activities, then trace back all immediate predecessors for each activity. Account for logical links from dummies, but do not list dummies as activities themselves.

📐 Worked Example

Produce the precedence table for the AOA network drawn in the previous section.

1
1. List all real activities: A, B, C, D.
2
1. Activity A starts at Node 1 (no incoming arcs), so it has no immediate predecessors.
3
1. Activity B starts at Node 1 (no incoming arcs), so it has no immediate predecessors.
4
1. Activity C starts at Node 2, which only receives arc A, so its only predecessor is A.
5
1. Activity D starts at Node 3, which receives arc B and a dummy arc from Node 2 (fed by A), so its predecessors are A and B.
6
1. Tabulate the activities and their immediate predecessors for the final table.

Exam tip:

Never include dummy activities in your precedence table, only real project activities are assessed.

3. Critical Path Algorithm: Earliest & Latest Event Times★★★☆☆⏱ 7 min

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📘 Definition

Critical Path

The longest sequence of activities in a project, equal to the minimum possible total project duration. Any delay to an activity on the critical path will delay the entire project.

First complete a forward pass to calculate earliest event times ( $e_{i}$ ): start with $e_{1} = 0$ for the first node, then for each node, $e_{i}$ is the maximum value of $e_{p r e v} + t_{p r e v, i}$ for all incoming arcs. Next complete a backward pass to calculate latest event times ( $l_{j}$ ): start with $l_{f ina l} = e_{f ina l}$ for the end node, then for each node, $l_{j}$ is the minimum value of $l_{n e x t} - t_{j, n e x t}$ for all outgoing arcs. Total float is calculated using the formula below:

F(i,j) = l_j - e_i - t_{i,j}

📐 Worked Example

For the earlier network, activity durations are: A=2, B=3, C=4, D=5. Calculate event times, total float, and identify the critical path.

1
1. Forward pass: $e_{1} = 0$ , $e_{2} = 0 + 2 = 2$ , $e_{3} = ma x (0 + 3, 2 + 0) = 3$ , $e_{4} = ma x (2 + 4, 3 + 5) = 8$ . Project duration = 8 units.
2
1. Backward pass: $l_{4} = 8$ , $l_{3} = 8 - 5 = 3$ , $l_{2} = min (8 - 4, 3 - 0) = 3$ , $l_{1} = min (3 - 2, 3 - 3) = 0$ .
3
1. Calculate float: A: $3 - 0 - 2 = 1$ , B: $3 - 0 - 3 = 0$ , C: $8 - 2 - 4 = 2$ , D: $8 - 3 - 5 = 0$ .
4
1. Critical activities are B and D, so critical path is 1→3→4, total duration 8.

Exam tip:

Show all working for forward and backward passes: you will get method marks even if your final critical path is incorrect.

4. Total Float, Gantt Charts & Basic Scheduling★★★☆☆⏱ 7 min

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📘 Definition

Gantt (Cascade) Chart

A bar chart where each bar represents an activity, plotted against time. The bar runs from the earliest start time to the latest finish time of the activity, with a shaded section for the activity duration starting at earliest start. The unshaded section equals total float.

For basic scheduling, assign critical activities to workers first to avoid delaying the project, then use the float of non-critical activities to fill gaps in worker schedules, respecting all precedence rules.

📐 Worked Example

Using the earlier activity data, schedule the project with 2 workers to complete in the minimum 8 unit duration.

1
1. Calculate timings for each activity: A (ES=0, EF=2, LS=1, LF=3), B (ES=0, EF=3, LS=0, LF=3), C (ES=2, EF=6, LS=4, LF=8), D (ES=3, EF=8, LS=3, LF=8).
2
1. Assign critical activities first: Worker 1 completes B from 0-3, then D from 3-8.
3
1. Fill gaps with non-critical activities: Worker 2 completes A from 0-2, then C from 2-6.
4
1. Verify all precedence rules are followed, and total project duration remains 8 units.

Exam tip:

When drawing Gantt charts, label axes clearly and shade activity durations to distinguish them from available float.

5. Common Pitfalls

Wrong move:

Using activity-on-node networks instead of activity-on-arc

Why:

Edexcel D1 exclusively assesses activity-on-arc, so activity-on-node submissions get zero marks

Correct move:

Always draw arcs for activities, nodes for events, per Edexcel specification

Wrong move:

Forgetting to add dummies for overlapping precedence rules

Why:

Missing dummies breaks logical precedence, leading to incorrect event time and float calculations

Correct move:

Add dashed 0-duration dummy arcs to show required logical links between events

Wrong move:

Completing backward pass before forward pass

Why:

Latest event time for the end node equals its earliest event time, which is only known after the forward pass

Correct move:

Finish full forward pass first to get project duration, then complete backward pass

Wrong move:

Including dummy activities in precedence tables or float calculations

Why:

Dummies are only logical links with 0 duration, not real project activities

Correct move:

Only include non-dummy activities in tables, calculations, and Gantt charts

Wrong move:

Scheduling non-critical activities before critical activities

Why:

Delays to critical activities increase total project duration, even if non-critical activities are finished early

Correct move:

Assign all critical activities to workers first, then fill gaps with non-critical activities using their float

6. Quick Reference Cheatsheet

Concept	Rule / Formula	Key Note
AOA Network	Arcs = activities, Nodes = events	Dummies = dashed, 0 duration, show precedence logic
Forward Pass	$e_{i} = ma x (e_{p r e v} + t_{p r e v, i})$	Start at first node, $e_{1} = 0$
Backward Pass	$l_{j} = min (l_{n e x t} - t_{j, n e x t})$	Start at final node, $l_{f ina l} = e_{f ina l}$
Total Float	$F (i, j) = l_{j} - e_{i} - t_{i, j}$	Critical activities have $F = 0$
Gantt Chart	Bar from ES to LF, shaded for activity duration	Unshaded bar length = total float
Scheduling	Assign critical activities first	Use float of non-critical activities to fill gaps

7. Frequently Asked

Can I use activity-on-node networks for Edexcel D1 critical path questions?

No. Edexcel exclusively assesses activity-on-arc networks for this topic, and you will receive zero marks for activity-on-node submissions.

How do I identify critical activities?

Critical activities have a total float of 0, meaning any delay to the activity will directly delay the total project duration. They lie on the critical path.

Do dummy activities count towards project duration?

No. Dummy activities have a duration of 0 and only exist to show logical precedence links between events. They are never included in float calculations or Gantt charts.

Going deeper

official_specEdexcel IAL Maths D1 SpecificationRefer to page 34 for critical path analysis requirements

What's Next

You have now mastered all critical path analysis content required for Edexcel IAL D1. This topic typically appears as an 8-12 mark question on every D1 paper, so practice full timed past paper questions to build speed and accuracy, particularly for drawing clear labeled AOA networks. Next, you can review other core D1 network topics including the route inspection problem and linear programming to complete your D1 preparation. You should also practice converting between precedence tables and networks, as this is a common opening section for critical path exam questions.