Resource Histograms (Edexcel A-Level Further Mathematics): Revision Notes
📚 Revision Notes
11.1.5 Resource Histograms
Introduction
A resource histogram is a bar chart used to visually represent the number of workers (or resources) required over the duration of a project. This tool helps to assess the workforce demands and ensures that the necessary resources are available at the right times.
In A-Level Further Maths, constructing resource histograms involves:
- Allocating workers to specific activities.
- Using the project timeline (e.g., derived from a Gantt chart) to determine the number of workers required at each time interval.
Steps to Construct a Resource Histogram
- Understand the Problem
- Collect Key Information
- Build the Histogram
1. Understand the Problem
- Each activity has a start time, duration, and number of workers required.
- Use the timeline of activities (e.g., from a Gantt chart) to calculate the total number of workers needed at each time step.
2. Collect Key Information
For each activity, note:
- Start and finish times.
- Number of workers required.
- Duration of the activity.
3. Build the Histogram
- Divide the timeline into units (e.g., days, weeks).
- For each unit, sum the workers needed across all activities in progress.
- Draw a bar for each time interval representing the total workforce.
Worked Example
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Example 1: Construct a Resource Histogram
Step 1: Given Information
| Activity | Start Time | Duration (days) | Workers Required |
|---|---|---|---|
| A | 0 | 3 | 2 |
| B | 2 | 4 | 3 |
| C | 4 | 2 | 4 |
| D | 5 | 3 | 1 |
Step 2: Analyse Overlaps
- Activity A runs from day 0 to day 2.
- Activity B starts at day 2 and overlaps with Activity A for 1 day.
- Activity C starts at day 4, overlaps with Activity B for 1 day, and then continues independently.
- Activity D starts at day 5, overlaps with Activity C for 2 days.
Step 3: Calculate Workers at Each Day
| Day | Workers from A | Workers from B | Workers from C | Workers from D | Total Workers |
|---|---|---|---|---|---|
| 0 | 2 | 0 | 0 | 0 | 2 |
| 1 | 2 | 0 | 0 | 0 | 2 |
| 2 | 2 | 3 | 0 | 0 | 5 |
| 3 | 0 | 3 | 0 | 0 | 3 |
| 4 | 0 | 3 | 4 | 0 | 7 |
| 5 | 0 | 0 | 4 | 1 | 5 |
| 6 | 0 | 0 | 4 | 1 | 5 |
| 7 | 0 | 0 | 0 | 1 | 1 |
Step 4: Draw the Resource Histogram
The resource histogram can be represented as follows:
Day: 0 1 2 3 4 5 6 7
Workers: ██ ██ █████ ███ ███████ █████ █████ █
Key:
- The height of each bar represents the total workers needed for that day.
- Activities contributing to each total are determined from the table.
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Example 2: Histogram with Non-Uniform Durations
Step 1: Given Information
| Activity | Start Time | Duration (weeks) | Workers Required |
|---|---|---|---|
| A | 0 | 2 | 5 |
| B | 1 | 3 | 4 |
| C | 3 | 2 | 6 |
Step 2: Analyse Overlaps
- Activity A runs from week 0 to week 1.
- Activity B starts at week 1, overlaps with Activity A in week 1, and continues until week 3.
- Activity C starts at week 3, overlaps with Activity B in week 3, and then continues independently.
Step 3: Calculate Workers at Each Week
| Week | Workers from A | Workers from B | Workers from C | Total Workers |
|---|---|---|---|---|
| 0 | 5 | 0 | 0 | 5 |
| 1 | 5 | 4 | 0 | 9 |
| 2 | 0 | 4 | 0 | 4 |
| 3 | 0 | 4 | 6 | 10 |
| 4 | 0 | 0 | 6 | 6 |
Step 4: Draw the Resource Histogram
Week: 0 1 2 3 4
Workers: █████ █████████ ████ ██████████ ██████
Key:
- Bars clearly show workforce demand by week.
- Overlaps of activities result in taller bars.
Note Summary
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Common Mistakes
- Forgetting overlaps: Not summing workers from activities that occur simultaneously.
- Incorrect time intervals: Misaligning activities with the timeline.
- Misinterpreting durations: Confusing start and finish times with durations.
- Omitting zero contributions: Failing to account for periods when no workers are required.
- Poor visualization: Drawing inconsistent or unclear bars in the histogram.
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Key Formulas
- Resource calculation: Total workers = sum of workers for overlapping activities.
- Start and finish times:
- Finish time = start time + duration.
- Histogram representation:
- Height of the bar = total workers required at each time step.
- Overlaps:
- Add contributions from all activities running during the same time interval.