Scheduling Activities (Edexcel A-Level Further Mathematics): Revision Notes

Example 1: Scheduling with Simple Dependencies

Step 1: Given Information

Step 2: Create a Gantt Chart

Visualise the timeline:

Day:       0   1   2   3   4   5   6   7   8   9   10  11
Activity A ███
Activity B         ████
Activity C         ██
Activity D                 █████

Step 3: Worker Assignment

• Assign workers so that simultaneous tasks use different workers:
• Worker 1: Activity $A, B, D$
• Worker 2: Activity $C$

Worker Timeline:

Day:       0   1   2   3   4   5   6   7   8   9   10  11
Worker 1:  AAAAAAAA BBBBBBBBB DDDDD
Worker 2:  CCCCCC

Step 4: Verify the Schedule

• Total workers required: 2.
• All dependencies are respected.

Example 2: Minimising Workers with Float

Step 1: Given Information

Step 2: Analyse Float and Adjust Start Times

• $B$: Float = 2, so it can start anytime between week 2 and week 4.

• $C$: Float = 1, so it can start anytime between week 2 and week 3. Adjust $B$ and $C$ to reduce worker overlap:

• Start $B$ at week 4 (latest possible).

• Start $C$ at week 2 (earliest possible).

Step 3: Worker Assignment

Visualise the timeline:

Week:      0   1   2   3   4   5   6   7   8   9   10
Activity A ██
Activity C         ██
Activity B                 ███
Activity D                         ████

Assign workers:

• Worker 1: Activity $A, B, D$
• Worker 2: Activity $C$

Worker Timeline:

Week:      0   1   2   3   4   5   6   7   8   9   10
Worker 1:  AA      BBBBBBBBB DDDDDDDD
Worker 2:      CC

Step 4: Verify the Schedule

• Total workers required: 2.
• All dependencies and float are respected.

Example 3: Complex Dependencies and Worker Minimization

Step 1: Given Information

Step 2: Initial Gantt Chart

Visualise activities without adjustment:

Day:       0   1   2   3   4   5   6   7   8   9   10
Activity A ███
Activity B         ████
Activity C         ██
Activity D                 ███
Activity E                     ██

Step 3: Adjust Non-Critical Activities

• $B$: Has 1 day of float, can be delayed to start at day 4.
• $C$: Has 1 day of float, can be delayed to start at day 4.

Revised schedule:

Day:       0   1   2   3   4   5   6   7   8   9   10
Activity A ███
Activity B             ████
Activity C             ██
Activity D                     ███
Activity E                         ██

Step 4: Worker Assignment

Assign workers to minimise overlap:

• Worker 1: Activity $A, B, D, E$
• Worker 2: Activity $C$

Common Mistakes

1. Ignoring float: Not using float to reduce worker overlap and minimise total workers.
2. Violating dependencies: Starting dependent activities too early, breaking logical order.
3. Misaligning durations: Confusing start times with durations, leading to incorrect overlaps.
4. Overloading workers: Failing to account for the number of workers already allocated to overlapping tasks.
5. Omitting critical path: Changing start times of critical activities, delaying project completion.

Key Formulas

1. Total Float:

2. Worker Assignment:

• Assign tasks sequentially based on start and finish times.
• Use float to delay non-critical activities if it reduces worker overlap.

3. Worker Utilization:

• Total workers = Maximum number of overlapping activities at any time.

4. Dependencies:

• Start times must respect the precedence order of activities.