Maths › Decision Mathematics 1 › Float, Gantt charts and scheduling
Float, Gantt charts and scheduling
Once the critical path is known, everything else has room to move. Float measures how much, a Gantt chart shows it at a glance, and a histogram turns it into a staffing question.
Builds on Critical path analysis and Algorithms, sorting and bin packing.
IN THIS TOPIC
- Calculate the total float of every activity.
- Draw a Gantt chart showing floats, and read simultaneous activities from it.
- Draw a resource histogram, find the lower bound on workers, and schedule.
WHAT YOU PROBABLY THINK
If two activities each have three days of float, both can be delayed by three days without affecting the project.
How much slack there is
The total float of an activity is its latest finish minus its earliest start minus its duration, which is the same as the difference between its latest and earliest starts. Critical activities have zero float, by definition.
The word total matters. Float is shared along a chain: if two activities on the same path each show three days, using all three on the first leaves none for the second. The opening claim treats the floats as independent, and the Gantt chart is exactly the picture that makes the sharing visible.
WORKED EXAMPLE
Floats for the project
For the seven-activity project, find the total float of each activity.
A, C, E and G are critical, so each has float 0.
B: latest start 2, earliest start 0, so float 2.
D: latest start 8, earliest start 3, so float 5. F: latest start 10, earliest start 5, float 5.
D and F lie on the same chain, so the five days are shared between them, not five each.
Workers and levelling
A resource histogram plots how many workers are needed at each moment when every activity starts as early as possible. The peak of that histogram is often higher than necessary, and shifting non-critical activities within their float, called levelling, can flatten it.
The lower bound on the number of workers is the total of all the activity durations divided by the project duration, rounded up: that many workers would have to be busy every moment. If a schedule achieves the bound it is optimal, and if it cannot be achieved the question will ask you to say why.
YOUR TURN
How many workers
The seven activities take 3, 5, 4, 2, 6, 3 and 2 days, each needing one worker, and the project lasts 15 days. Find the lower bound on the number of workers and comment on the histogram.
Show the working
Total work = 3 + 5 + 4 + 2 + 6 + 3 + 2 = 25 worker-days.
Lower bound = 25/15 = 1.67, rounded up to 2 workers.
At earliest start times the histogram peaks at 3, between days 3 and 5, when B, C and D all run.
Delaying D within its five days of float removes that peak, so two workers do suffice: levelling has saved a third worker.
THE EXAM BIT
- Compute float as latest finish minus earliest start minus duration, and show the subtraction.
- Draw the Gantt chart to scale with the critical activities in one unbroken row.
- State the lower bound on workers before attempting a schedule.
- When levelling, say which activity you moved and by how much, and check its float allows it.
CHECK YOURSELF
A project of duration 20 days has activities totalling 62 worker-days. Find the lower bound on the number of workers.
Show a hint
Total work over duration, rounded up.
Show the answer
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Total float is latest finish minus earliest start minus duration, and it is shared along a chain rather than held separately.
The lower bound on workers is total work divided by project duration, rounded up; levelling shifts non-critical activities within their float to reach it.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Calculate the total float of every activity.
- Draw a Gantt chart showing floats, and read simultaneous activities from it.
- Draw a resource histogram, find the lower bound on workers, and schedule.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.