A hospital employs volunteers to staff the reception desk between 8:00 A.M. and 10:00 P.M. Each volunteer works three consecutive hours except for those starting at 8:00 P.M. who work for two hours only. The minimum need for volunteers is approximated by a step function over 2-hour intervals starting at 8:00 A.M. as 8, 6, 8, 6, 4, 6, and 5. Because most volunteers are retired individuals, they are willing to offer their services at any hour of the day (8:00 A.M. to 10:00 P.M.). However, because of the large number of charities competing for their service, the number needed must be kept as low as possible. Determine an optimal schedule (using AMPL, Solver, or TORA) for the start time of the volunteers.
The correct answer and explanation is :
To determine an optimal volunteer schedule for staffing the hospital’s reception desk between 8:00 A.M. and 10:00 P.M., we can approach this as a linear programming problem. The goal is to minimize the total number of volunteers while ensuring that the reception desk is adequately staffed according to the specified needs.
Problem Breakdown:
- Operational Hours: 8:00 A.M. to 10:00 P.M. (14 hours)
- Volunteer Shift Duration: Three consecutive hours, except for shifts starting at 8:00 P.M., which last two hours.
- Volunteer Requirements: Minimum number of volunteers needed in 2-hour intervals starting at 8:00 A.M. are 8, 6, 8, 6, 4, 6, and 5.
Decision Variables:
Let ( x_i ) be a binary decision variable representing whether a volunteer starts at time slot ( i ), where ( i ) corresponds to the following times:
- 1: 8:00 A.M.
- 2: 10:00 A.M.
- 3: 12:00 P.M.
- 4: 2:00 P.M.
- 5: 4:00 P.M.
- 6: 6:00 P.M.
- 7: 8:00 P.M.
Objective Function:
Minimize the total number of volunteers:
[ \text{Minimize} \quad Z = 8x_1 + 6x_2 + 8x_3 + 6x_4 + 4x_5 + 6x_6 + 5x_7 ]
Constraints:
- Coverage Constraints: Ensure that the reception desk is staffed according to the required number of volunteers in each 2-hour interval:
- For 8:00 A.M. to 10:00 A.M.: ( x_1 + x_2 \geq 8 )
- For 10:00 A.M. to 12:00 P.M.: ( x_2 + x_3 \geq 6 )
- For 12:00 P.M. to 2:00 P.M.: ( x_3 + x_4 \geq 8 )
- For 2:00 P.M. to 4:00 P.M.: ( x_4 + x_5 \geq 6 )
- For 4:00 P.M. to 6:00 P.M.: ( x_5 + x_6 \geq 4 )
- For 6:00 P.M. to 8:00 P.M.: ( x_6 + x_7 \geq 6 )
- For 8:00 P.M. to 10:00 P.M.: ( x_7 \geq 5 )
- Shift Duration Constraints: Each volunteer works for three consecutive hours, except for those starting at 8:00 P.M. who work for two hours only. This can be represented as:
- ( x_1 + x_2 + x_3 \geq 1 )
- ( x_2 + x_3 + x_4 \geq 1 )
- ( x_3 + x_4 + x_5 \geq 1 )
- ( x_4 + x_5 + x_6 \geq 1 )
- ( x_5 + x_6 + x_7 \geq 1 )
- ( x_6 + x_7 \geq 1 )
Solution Approach:
This linear programming problem can be solved using optimization tools such as AMPL, Solver, or TORA. By inputting the objective function and constraints into one of these tools, we can determine the optimal start times for the volunteers that minimize the total number of volunteers while ensuring that the reception desk is adequately staffed throughout the day.
Explanation:
The hospital requires volunteers to cover specific time intervals with varying staffing levels. By formulating this as a linear programming problem, we can systematically determine the minimum number of volunteers needed and their optimal start times. This approach ensures that the hospital meets its staffing requirements efficiently, respecting the constraints of volunteer shift durations and minimizing the total number of volunteers utilized.
Visual Representation:
