All revision notes topics

Resource histograms and schedulingEdexcel A-Level Further Maths: Revision notes

Section 1

Resource histograms

A resource histogram is a chart showing the number of workers needed at each time during a project. Time is on the horizontal axis and the number of workers is on the vertical axis. In this specification, each activity has a given number of workers per activity, which may be more than one. Each activity is drawn as a block whose width is its duration and whose height is the number of workers it needs, and blocks that run at the same time are stacked. The area of a block is its worker-days: workers multiplied by days. The total area of the histogram is the total worker-days of the project.

Key termsresource histogramworker-days
Common mistake

Treating every activity as needing one worker. Use the number given for each activity.

Section 2

Constructing a histogram

  1. Find the earliest start time of each activity from a forward pass.
  2. Place each activity at its earliest start, with height equal to its number of workers.
  3. For each time interval, add the workers of every activity in progress.
  4. Draw the total as a stepped chart. Worked example: C (3 workers) runs 4 to 7, D (2) runs 5 to 7 and F (2) runs 5 to 8. Between 5 and 7, C, D and F are all in progress, so 3 + 2 + 2 = 7 workers are needed. Watch the boundaries: an activity ending at time 5 is not in progress after 5, and one starting at 5 is in progress after 5.
Key termstime interval
Exam tip

List the interval boundaries first: every time at which an activity starts or ends.

Section 3

Resource levelling

A histogram with earliest start times often has high peaks. Resource levelling reduces the peak, and so the greatest number of workers needed, by moving activities.

  • Only non-critical activities can be moved, and only within their float: a critical activity cannot move without delaying the project.
  • Delay an activity so that it does not overlap with others that need many workers.
  • If an activity is delayed by no more than its float, the project time does not change. Example: F has float 3 and 2 workers. At its earliest start it overlaps with C and D, giving a peak of 7. Starting F at 7 or 8 reduces the peak to 5 and the project still takes 13 days.
Key termsresource levelling
Common mistake

Moving a critical activity, or delaying an activity by more than its float.

Section 4

Scheduling with the least number of workers

Scheduling assigns each activity to a worker at specific times so that every activity is done in order. With one worker per activity, the lower bound for the number of workers is (sum of durations) ÷ (project time), rounded up. To schedule with the least number of workers:

  1. Give the critical activities to one worker, as they leave no gaps.
  2. Fit the remaining activities onto the other workers, starting each after its predecessors and finishing before its successors need it, using the floats.
  3. Check each worker does one activity at a time and that all precedences hold. Example: with durations totalling 27 and a project time of 17, at least 2 workers are needed. Worker 1 does A, C, E, G; worker 2 does B, D, F. Any valid schedule is accepted. If the lower bound cannot be met, more workers or more time is needed.
Key termsschedulinglower bound for workers
Exam tip

Show each worker's activities with start and finish times and check each precedence in turn.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Resource histograms and scheduling

  1. A project has six activities. Each activity, its duration in days, the number of workers it needs and its immediate predecessors are: A 3 days, 2 workers (none); B 4 days, 1 worker (none); C 2 days, 3 workers (A); D 3 days, 2 workers (B); E 4 days, 2 workers (A); F 2 days, 4 workers (C, D, E). Every activity starts as soon as it can, and the project takes 9 days.
    State which activities are in progress when the greatest number of workers is needed, and the time interval for which this happens.2 marks
  2. A project has six activities. Each activity, its duration in days, the number of workers it needs and its immediate predecessors are: A 3 days, 2 workers (none); B 4 days, 3 workers (none); C 2 days, 2 workers (A); D 5 days, 1 worker (A); E 3 days, 2 workers (B, C); F 2 days, 3 workers (D, E). Every activity starts as soon as it can, and the project takes 10 days.
    State the greatest number of workers needed at any one time, and the time interval during which it occurs.2 marks
  3. A project has seven activities, each needing one worker. Each activity, its duration in days and its immediate predecessors are: A 3 (none); B 5 (none); C 4 (A); D 2 (A); E 6 (B, C); F 3 (D); G 4 (E, F). The minimum completion time of the project is 17 days, and the critical activities are A, C, E and G.
    Show that at least 2 workers are needed to complete the project in 17 days.3 marks
See the full worksheet

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).