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Contents · 0% complete
  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn7367Paper 3 · Discrete optionCritical path analysisActivity networks and critical paths
Critical path analysis · Subtopic 1 of 1

Critical Path Analysis

A project's activities and their dependencies form a network. A forward pass gives earliest times, a backward pass latest times. Activities with zero total float form the critical path, whose length is the minimum project duration. Total float == latest finish −- earliest start −- duration. A lower bound for workers is ⌈total activity timeduration⌉\left\lceil\frac{\text{total activity time}}{\text{duration}}\right\rceil.

Edexcel YMA01 WDM11Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 50 minRevision 1 · updated 2026-10-27
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Section 1 of 14

Learning objectives

  • Draw activity networks from precedence tables (with dummies where needed).
  • Find early and late times, the critical path and floats.
  • Draw Gantt charts and resource histograms; find a lower bound for workers.
Pause and think

To make breakfast you boil the kettle (3 min), make toast (4 min) and butter it after it's toasted (1 min). What's the shortest time?

Section 2 of 14

Core idea

Start hereBuilding a house

Foundations before walls, walls before roof; but plumbing and electrics can run at the same time. Some tasks can slip without delaying the move-in date; others can't. Critical path analysis finds which.

The big idea

Forward pass (max) gives earliest event times; backward pass (min) gives latest. Activities with zero total float form the critical path, and its length is the project's minimum duration.

1. Early and late times

Activities: A(3), B(4), C(2) after A, D(5) after A and B, E(3) after C and D.

Worked exampleForward and backward passes

Find the duration, critical activities and floats.

Goal
Duration and floats.
Strategy
Max forwards, min backwards.
  1. D: max⁡(3,4)=4→9D:\ \max(3,4)=4\to9
    D waits for both A and B.
  2. E: max⁡(5,9)=9→12E:\ \max(5,9)=9\to12
    E waits for C and D.
  3. C: latest finish 9⇒float 9−3−2=4C:\ \text{latest finish }9\Rightarrow\text{float }9-3-2=4
    C only needs to finish by E's start.
  4. A: latest finish min⁡(7,4)=4⇒float 1A:\ \text{latest finish }\min(7,4)=4\Rightarrow\text{float }1
    A must be done for D at 4.
AnswerDuration 1212; critical path B–D–E; floats A 11, C 44.
✓ CheckCritical durations 4+5+3=124+5+3=12 ✓.
total float=latest finish−earliest start−duration\text{total float}=\text{latest finish}-\text{earliest start}-\text{duration}
where
float
how long the activity can be delayed without delaying the project.

2. Resources

Lower bound for workers
  1. total work=∑d=17\text{total work}=\sum d=17
    Each activity needs one worker for its duration.
  2. workers×T≥17\text{workers}\times T\ge17
    ww workers over T=12T=12 days provide at most 12w12w worker-days.
  3. w≥⌈1712⌉=2w\ge\left\lceil\tfrac{17}{12}\right\rceil=2
    Round up.
CPA workflow

From data to schedule.

Common misconceptionMax and min Forward: max of incoming. Backward: min of outgoing. Swapping them is the commonest error.
Key takeawayDummies Dummy activities (duration 0, dashed) show dependence without work, e.g. when D needs A and B but C needs only A.
Pause and think

If C takes 6 days instead of 2, what changes?

Summary

  • Forward: max; backward: min.
  • Critical: zero float.
  • Workers ≥⌈∑d/T⌉\ge\lceil\sum d/T\rceil.
Key words
Critical path
Longest chain of activities; sets the duration.
Total float
Possible delay without delaying the project.
Gantt chart
Bar chart of activity timings.
HistoryPolaris and DuPont. CPA and PERT were developed in 1957–58 for DuPont chemical plants and the US Navy's Polaris missile, cutting project times dramatically.
Fun factPit stops. An F1 pit stop of about 2 seconds is a CPA masterpiece: around 20 people working in parallel on a planned critical path.
Section 3 of 14

Key formulae

Forward passej=max⁡(ei+dij)e_j=\max(e_i+d_{ij})
Backward passli=min⁡(lj−dij)l_i=\min(l_j-d_{ij})
Float and workersfloat=lj−ei−d;workers≥⌈∑dT⌉\text{float}=l_j-e_i-d;\quad\text{workers}\ge\left\lceil\tfrac{\sum d}{T}\right\rceil
TrickCheck the critical path. On a critical activity, early and late times at both ends match and the duration fills the gap exactly.
Section 4 of 14

Worked examples

Example 1 · Critical path0 / 3 steps
A(3), B(4), C(2) after A, D(5) after A and B, E(3) after C and D. Find the duration and critical path.
    Example 2 · Float0 / 2 steps
    Find the floats of C and A.
      Section 5 of 14

      Explore the graph

      Resource histogram (all activities start early)
      Loading the interactive graph…
      Try this
      1. Workers needed if everything starts as early as possible: 2 until time 5, then 1.
      2. The line at 1712\frac{17}{12} is the average workload: at least 2 workers are needed.

      Total activity time 1717 over 1212 days.

      In the real worldConstruction. Building firms track critical paths daily: a delay to a critical activity delays the whole building; float lets other tasks slip safely.
      Section 6 of 14

      Common mistakes

      IncorrectForward pass uses min
      CorrectUse max
      An event can't happen until all preceding activities finish.
      IncorrectBackward pass uses max
      CorrectUse min
      Must be early enough for every following activity.
      IncorrectLower bound rounded down
      CorrectRound up
      You can't have part of a worker.

      Activity with float 0

      Me with no time to spare all week.

      Section 7 of 14

      Exam tip

      Exam tipDummies. Use a dummy when two activities share some but not all predecessors, or to stop two activities having the same start and end events.
      Section 8 of 14

      Guided practice

      SA-DM-CP-G012 marksL2 StandardNon-calculatorCritical path
      Activities: A(3), B(4), C(2) after A, D(5) after A and B, E(3) after C and D. Find the minimum project duration.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 9 of 14

      Practice questions

      SA-DM-CP-0012 marksL2 StandardNon-calculatorCritical path
      Find the total float of activity C.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-DM-CP-0022 marksL3 ExamNon-calculatorCritical path
      Find the total float of activity A.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-DM-CP-0032 marksL2 StandardNon-calculatorScheduling
      Each activity needs one worker. Find a lower bound for the number of workers needed to finish in 1212 days.

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      Salam Academy Original Practice · Proposed mark scheme

      More questions on every skill in this subtopic are in the question bank under Practise.

      Section 10 of 14

      Challenge

      SA-DM-CP-0042 marksL3 ExamNon-calculatorCritical path
      Activity B is delayed by 22 days. Find the new minimum duration.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 11 of 14

      Summary

      • Forward pass: max.
      • Backward pass: min.
      • Critical: zero float.
      • Workers ≥⌈∑dT⌉\ge\lceil\frac{\sum d}T\rceil.
      Section 12 of 14

      Quick quiz

      Q1 · SA-DM-CP-Q011 markL1 FoundationNon-calculatorCritical path
      A critical activity is one that
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Section 13 of 14

      Same topic, different board

      Specification
      7367 A-level Further Mathematics
      Where
      Discrete · Critical path analysis
      Expected
      Activity networks (activity on node), earliest and latest times, critical paths, float, Gantt charts, resource levelling.
      Calculator
      Permitted on all papers.
      Typical style
      AQA mixes short multiple-choice questions with longer structured ones.
      Open in
      7367 · Paper 3
      Section 14 of 14

      Past paper questions

      No past paper questions are tagged to this subtopic yet. We add papers every week.

      Browse past papers

      Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7367, First teaching 2017 (official specification).