What is the minimum number of people required to finish all the activities in the minimum possible time?

Steps to Solve

1. Identify Activities and Durations
   List the activities along with their time durations.

• Activity A: 2 hours
• Activity B: 1 hour
• Activity C: 4 hours (requires half of A's time before starting)
• Activity D: 2 hours (requires E's completion)
• Activity E: 5 hours (requires half of A's time before starting)
• Activity F: 4 hours (requires B's completion)

2. Establish Precedence Constraints
   Identify the dependencies:

• C needs A (1 hour, i.e., half of A's duration).
• E also needs A (1 hour).
• D needs E (5 hours).
• F needs B (1 hour).

3. Calculate Total Times for Dependencies
   Consider the total times based on dependencies:

• A: 2 hours
• B: 1 hour
• For C and E (each depending on half of A), they can start once A is partially completed (1 hour).
• The total completion time for C and E after starting A will be:
  • A (1 hour) + C (4 hours) = 5 hours for C (total 5, but starts after 1).
  • A (1 hour) + E (5 hours) = 6 hours for E.

4. Estimate Completion Times
   Now calculate when D and F can start:

• D can start after E. Since E takes 6 hours, D (2 hours) starts next, finishing at hour 8.
• F can start after B (1 hour). If we have B complete at hour 1, F starts next, finishing at hour 5.

5. Determine the Overall Timeline
   The schedule based on dependencies looks like this:

• 0: A (2 hours)
• 1: B (1 hour)
• 1: C (4 hours) begins when A is 1 hour done.
• 1: E (5 hours) begins when A is 1 hour done.
• 6: D (2 hours) starts after E.
• 5: F (4 hours) starts after B.

6. Calculate Minimum Workers Needed
   To complete all tasks in parallel, we check the overlaps in tasks:

• At hour 1: A, B, C, E (4 tasks, need 4 workers).
• At hour 5: B and F (2 tasks, need 2).
• At hour 6: D starts (1 task, need 1).
  At most, we only need a maximum of 4 workers at a time.

7. Final Adjustment for Workers
   Since we need the workers optimally, and we determined 4 workers are needed, adjust the numbers to find the minimum required since workers can shift from completed tasks.