Plan operations with critical path analysis
Read activity networks, calculate minimum duration and floats, and use critical path analysis as a planning aid with limits.
Cambridge International AS & A Level Business 9609, 2026–2028 syllabus and 2029 syllabus, section 9.3.5. Operations examples and project estimates depend on the stated industry, site and period. Written practice is self-checked. Specialist review has not been completed for this lesson; these exercises do not award an official mark or establish exam readiness.
Before you begin
- Apply the operational concept to the stated process, product or project and identify the assumptions behind any comparison or calculation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does a dummy activity represent in a network?
What you will learn
- Identify activities, nodes and dummy activities
- Calculate project duration, critical path, total float and free float
- Interpret network results and evaluate critical path analysis
A network shows task dependencies
A network diagram represents activities, dependencies and project timing. In an activity-on-arrow network, nodes represent events and arrows represent activities with durations. A dummy activity has zero duration and uses no resources; it shows a dependency that would otherwise be unclear.
A forward pass finds earliest event times by adding activity durations and taking the largest value at merge points. A backward pass finds latest event times by subtracting durations and taking the smallest value at split points. The critical path is the longest-duration path through the network; its activities have zero total float in a correctly calculated network.
Float indicates scheduling flexibility
For activity i to j with duration d, total float = latest time at j − earliest time at i − d. Free float = earliest time at j − earliest time at i − d. Total float is the delay available without extending the project; free float is delay available without delaying the earliest start of following activities.
Critical path analysis (CPA) can identify minimum project duration and activities needing close attention. It depends on complete dependencies and duration estimates and may omit resource limits, uncertainty, quality or changing priorities. Use it with judgement and update it when actual progress changes.
Pause and explain
What is the critical path?
Worked example
Activities are A: 1→2 (3 days), B: 1→3 (4), C: 2→4 (4), D: 3→4 (2) and E: 4→5 (2). Find the earliest project duration, critical path and total float of B.
Show the worked solution
- Forward pass: event 2=3, event 3=4, event 4=max(3+4,4+2)=7, event 5=9.
- Backward pass: event 5=9, event 4=7, event 2=3, event 3=5 and event 1=0.
- Total float of B = latest at node 3 − earliest at node 1 − duration = 5−0−4=1 day.
Answer Minimum project duration is 9 days. The critical path is A–C–E (3+4+2=9 days). Activity B has one day of total float; D also has one day. A delay beyond available total float can extend the project unless the schedule is revised.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Using the example network, calculate total float for activity D (3→4, 2 days) and explain whether one day's delay changes the earliest project finish.
Give me a hint
Use latest time at node 4, earliest at node 3 and duration.
Compare my reasoning
- Latest at node 4 is 7 and earliest at node 3 is 4.
- Total float = 7−4−2 = 1 day.
- A delay of one day uses the float but does not extend the 9-day minimum duration.
Activity D has one day of total float. A delay of one day uses that float and can still meet the 9-day minimum finish if no other delay uses the same schedule allowance; a longer delay can extend the project or require rescheduling.
Look for these in your work
- Float is calculated correctly
- Relationship to project finish is interpreted
- Shared schedule risk is recognised
Explain why free float can be smaller than total float and how a project manager might use each measure.
Give me a hint
Free float protects the next activity's early start; total float protects project completion.
Compare my reasoning
- Free float compares with the earliest next-event time.
- Total float uses the latest allowable next-event time.
- Use free float to protect immediate sequencing and total float to manage overall deadline.
Free float can be smaller because it measures delay without affecting the earliest start of the next activity, while total float allows delay up to the latest time that does not extend the whole project. Managers can use free float for local handovers and total float to prioritise schedule recovery across the project.
Look for these in your work
- Definitions are distinguished
- Relative size is explained
- Both have a management use
A project is behind schedule after a supplier changes a delivery date. Explain why the manager should update the network and name two limits of relying on CPA alone.
Give me a hint
Recalculate affected dependencies and combine time analysis with resources and uncertainty.
Compare my reasoning
- Update activity duration or dependency data and recalculate forward and backward passes.
- CPA may assume fixed durations and complete dependencies.
- It may not show staff, cash, quality or supplier capacity constraints.
The manager should update the supplier activity and downstream dependencies, recalculate earliest and latest times and review the critical path and floats. CPA relies on the accuracy of the network and duration estimates and may omit resource capacity, cash, quality or uncertainty. It should inform, not replace, project judgement and stakeholder communication.
Look for these in your work
- Schedule data are updated
- Two limitations are identified
- The tool is used with judgement
Common mistakes
- Taking the smaller value at a forward-pass merge
- Confusing total float with free float
- Assuming CPA includes resource constraints or uncertain durations automatically
What can you explain now?
What does a dummy activity represent in a network?
Compare with the explanation
A dependency with zero duration and no resource use
It clarifies precedence when needed in the network.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do forward and backward passes reveal project duration and float, and what real-world constraints does a network leave out?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.