Linear programming

The Monroe 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 eight hours.  During normal weekday operations, the number of officers needed varies depending on the time of the day.  The department staffing guidelines require the following minimum number of officers on duty:

Time of Day                           Minimum Officers on Duty

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8:00 a.m. – noon                                          5

Noon – 4:00 p.m.                                         6

4:00 p.m. – 8:00 p.m.                                 10

8:00 p.m. – midnight                                   7

Midnight – 4:00 a.m.                                   4

4:00 a.m. – 8:00 a.m.                                   6

How many police officers should be scheduled to begin the 8-hour shifts at each of the six times in order to minimize the total number of officers required?  Structure this linear programming problem (decision variables, objective function, and constraints).  You do not need to solve this problem, just structure it.

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