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 = the number of officers beginning work at 8:00 a.m.,

x₂ = the number of officers beginning work at noon

x₃ = the number of officers beginning work at 4:00 p.m.,

x₄ = the number of officers beginning work at 8:00 p.m.

x₅ = the number of officers beginning work at mid-night,

x₆ = the number of officers beginning work at 4:00 a.m.

We need to solve the following set of equations:

Minimize x₁ + x₂ + x₃ + x₄ + x₅ + x₆

subject to constraints: x₆ + x₁ >= 5

x₁ + x₂ >= 6

  x₂ + x₃ >= 7

  x₃ + x₄ >= 7

  x₄ + x₅ >= 4

x₅ + x₆ >= 6

AND x₁ >=0; x₂ >= 0; x₃ >= 0; x₄ >= 0; x₅ >= 0;  x₆ >= 0

Using the solver add-in in MS Excel, we get the following solution:

x₁ =3; x₂ = 3; x₃ = 3; x₄ = 4; x₅ = 3; x₆ = 3

We have the following table:

Starting Time	Officers Starting
8:00 a.m.	3
Noon	3
4:00 p.m.	3
8:00 p.m.	4
Midnight	3
4:00 a.m.	3