In a cascade diagram, where do the bars of non-critical activities start?

At their earliest start time, with the float drawn after the bar.

Formula for the float of an activity?

Latest finish - earliest start - duration.

What is a resource histogram?

A bar chart of the number of workers needed in each time period.

How do you read the peak demand from a histogram?

It is the tallest bar: the most workers needed at any one time.

How do you find total worker-days?

Sum of (duration \times workers) over all activities, which is the area under the histogram.

What is resource levelling?

Delaying non-critical activities within their floats to reduce peaks in demand without delaying the project.

Why is levelling done with a heuristic?

There is no method guaranteed to give the best schedule, so a sensible rule is used and checked.

Which activities must you not delay when levelling?

Critical activities, because delaying them delays the whole project.

Lower bound for the number of workers (one worker per activity)?

\frac{total of durations}{minimum completion time}, rounded up.

Total duration 27 days, minimum time 16 days. Lower bound?

\frac{27}{16}=1.69\ldots, so at least 2 workers.

What must you do after finding the lower bound?

Give a schedule showing that this number of workers is sufficient, respecting all precedences.

What happens if a resource limit cannot be met without moving critical activities?

The project completion time must increase.

Gantt charts, resource histograms and levellingAQA A-Level Further Maths: Flashcards

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What does a cascade (Gantt) diagram show?

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What does a cascade (Gantt) diagram show?: Each activity as a horizontal bar against time, with its float, so you can see which activities run at any moment.
In a cascade diagram, where do the bars of non-critical activities start?: At their earliest start time, with the float drawn after the bar.
Formula for the float of an activity?: Latest finish $-$ earliest start $-$ duration.
What is a resource histogram?: A bar chart of the number of workers needed in each time period.
How do you read the peak demand from a histogram?: It is the tallest bar: the most workers needed at any one time.
How do you find total worker-days?: Sum of (duration $\times$ workers) over all activities, which is the area under the histogram.
What is resource levelling?: Delaying non-critical activities within their floats to reduce peaks in demand without delaying the project.
Why is levelling done with a heuristic?: There is no method guaranteed to give the best schedule, so a sensible rule is used and checked.
Which activities must you not delay when levelling?: Critical activities, because delaying them delays the whole project.
Lower bound for the number of workers (one worker per activity)?: $\frac{\text{total of durations}}{\text{minimum completion time}}$ , rounded up.
Total duration 27 days, minimum time 16 days. Lower bound?: $\frac{27}{16}=1.69\ldots$ , so at least 2 workers.
What must you do after finding the lower bound?: Give a schedule showing that this number of workers is sufficient, respecting all precedences.
What happens if a resource limit cannot be met without moving critical activities?: The project completion time must increase.

Exam questions on Gantt charts, resource histograms and levelling

A project has seven activities. Activity A takes 4 days and has no predecessor. B takes 3 days and has no predecessor. C takes 5 days and follows A. D takes 2 days and follows both A and B. E takes 6 days and follows C. F takes 3 days and follows D. G takes 2 days and follows both E and F.
A cascade (Gantt) diagram is drawn with every activity starting as early as possible. For activity F, state the day on which its bar ends and the length of the float drawn after it.2 marks
A project is planned with every activity starting at its earliest time. P runs from day 0 to day 3 and needs 2 workers. Q runs from day 0 to day 4 and needs 3 workers. R runs from day 3 to day 5 and needs 2 workers. S runs from day 4 to day 7 and needs 4 workers. T runs from day 5 to day 7 and needs 2 workers. Each worker is paid for every day on which they are needed.
The firm employs only 5 workers. State the period during which the schedule needs more than 5 workers, and by how many it is short.2 marks
A project consists of seven activities, each needing one worker. Durations are in days. A takes 5 days and has no predecessor. B takes 3 days and has no predecessor. C takes 4 days and follows A. D takes 6 days and follows A. E takes 2 days and follows B. F takes 3 days and follows both C and E. G takes 4 days and follows both D and F.
Find the minimum completion time and list the critical activities.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).