Critical paths and scheduling
| English | 中文 | Pinyin |
|---|---|---|
| float/fləʊt/ | 浮动时间 | fú dòng shí jiān |
Must every activity wait for every other?
- A school event has activities that can run in parallel. Adding every duration overestimates the minimum completion time.
- This lesson studies float 浮动时间: The time an activity can be delayed without delaying completion under the network model.
Choose the mathematical structure
- For an activity network, calculate earliest event times forward and latest event times backward. Total float for activity i to j is L_j-E_i-duration. A critical activity has zero total float; more than one critical path may exist.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines float?
The time an activity can be delayed without delaying completion under the network model.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Activities A=3 and B=5 start together. C=4 follows A, and D=2 follows both B and C. The earliest completion of C is 7; D must wait until max(5,7)=7 and finishes at 9. Path A-C-D is critical; B has float 2.
Critical paths and scheduling
For an activity network, calculate earliest event times forward and latest event times backward
Compare the model with the worked case and explain one change.
For A=3,C=4 after A,D=2 after C, find total path duration.
The path A-C-D takes 3+4+2=9 time units.
Test a tempting shortcut
- Do not add durations of independent parallel activities. A zero-float activity belongs to a critical path, but there may be several such paths. Resource limits can require a schedule longer than the network's theoretical minimum.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The minimum project duration is always the sum of every activity duration. This claim is false. Explain which definition or assumption it violates.
B=5 must finish before D starts at time 7. Find B's float.
B normally finishes at 5 but can finish at 7: total float=7-5=2.
The minimum project duration is always the sum of every activity duration.
Do not add durations of independent parallel activities. A zero-float activity belongs to a critical path, but there may be several such paths. Resource limits can require a schedule longer than the network's theoretical minimum.
Interpret a new situation
- Draw precedence relationships before assigning times. Distinguish activity duration from event time. Explain what a delay does to the completion date, and record any resource assumptions.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
A project starts at time 0 and finishes at 9. Find its minimum duration.
Project duration=end time-start time=9-0=9.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL further mathematics; official unit D1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The time an activity can be delayed without delaying completion under the network model. Choose the relationship, show the method, check its assumptions and interpret the result.