Question

The Clark County Sheriff’s Department schedules police officers for 8-hour shifts. The beginning times for the shifts are 8:00 a.m., noon, 4:00 p.m., 8:00 p.m., midnight, and 4:00 a.m. An officer beginning a shift at one of these times works for the next 8 hours. During normal weekday operations, the number of officers needed varies depending on the time of day. The department staffing guidelines require the following minimum number of officers on duty:

Time of Day Minimum No. of Officers on Duty

8:00 a.m.–noon 5

Noon–4:00 p.m.    6

4:00 p.m.–8:00 p.m.    7

8:00 p.m.–midnight 7

Midnight–4:00 a.m. 4

4:00 a.m.–8:00 a.m. 6

Determine the number of police officers that should be scheduled to begin the 8-hour shifts at each of the six times to minimize the total number of officers required. (Hint: Let x1 = the number of officers beginning work at 8:00 a.m., x2 = the number of officers beginning work at noon, and so on.) If your answer is zero, enter “0”.

Starting Time Officers Starting

8:00 a.m. _____________

Noon    _______________

4:00 p.m. _______________

8:00 p.m.    ________________

Midnight    _________________

4:00 a.m. _________________

Answer 1

Let

x1= No. of police officers required during shift 1

x2= No. of police officers required during shift 2

x3= No. of police officers required during shift 3

x4= No. of police officers required during shift 4

x5= No. of police officers required during shift 5

x6= No. of police officers required during shift 6

Objective is : Min Z= x1+x2+x3+x4+x5+x6

Constraints:

• x1 5
• x26
• x37
• x47
• x54
• x66

Along with this, The Non-Negativity Constraint is : x1,x2,x3,x4,x5,x6 0

Use Spreadheet:

Click Data, open Solver, insert the data & solve as specified below.

The Results Provided will be as below:

Ans: The No. of police officers required will be:

6,0,7,0,6,0 Respectively.