Problem:
A personnel manager must schedule the police force in such a way as to satisfy the staffing requirements shown in the table below:
|
TIME
|
Minimum number of officers required
|
|
Midnight-4 A.M.
|
5
|
|
4 A.M.-8 A.M.
|
7
|
|
8 A.M.- Noon
|
15
|
|
Noon-4 P.M.
|
7
|
|
4 P.M.-8 P.M.
|
12
|
|
8 P.M.-Midnight
|
9
|
|
Shift
|
Starting Time
|
Ending Time
|
|
1
|
Midnight
|
8:00 A.M.
|
|
2
|
4:00 A.M.
|
Noon
|
|
3
|
8:00 A.M.
|
4:00 P.M.
|
|
4
|
Noon
|
8:00 P.M.
|
|
5
|
4:00 P.M.
|
Midnight
|
|
6
|
8:00 P.M.
|
4:00 A.M.
|
The officers work 8-hour shifts. There are 6 such shifts each day. The starting and ending times for each shift are given in the table above. The personnel manager wants to determine how many officers should work each shift in order to minimize the total number of officers employed, while still satisfying the staffing requirements.
Required:
Formulate the problem as an integer linear programming model. Write the Objective function and the Constraints
To formulate the Objective function, note that the total number of officers is the sum of the number of officers assigned to each shift. The personnel manager wants to minimize this sum.
To formulate the Constraints note that officers are on duty for 2 time intervals. For example, shift 1 is on duty during the first time interval (Midnight to 4:00 A.M.). and the second time interval (4:00 A.M.-8:00 A.M.). Each time interval has two shifts of officers.
Solve the given numerical problem and illustrate step by step calculation.