Question

The Bloomington Police Department needs at least the number of officers listed in the table below during each 4-hour period of the day. Police officers can be hired to work 8 or 12 consecutive hours. Police officers receive $ 12 an hour for each of the first 8 hours they work, and collect $ 15 an hour for each of the next 4 hours they work. Formulate a PL that is used with the daily needs of police officers in Bloomington.

Periodo	policias
0:00  4:00	32
4:00 a 8:00	40
8:00 a 12:00	26
12:00 a 16:00	18
16:00 a 20:00	22
20:00 a 0:00	40

What is the minimum amount Bloomington must pay daily?

Answer 1

LP is formulated as under:

Let Xt be the number of police officers hired to work 8 hours joining duty at t hours, where t=0,4,8,12,16,20

Yt be the number of police officers hired to work 12 hours joining duty at t hours, where t=0,4,8,12,16,20

Min 96(X0+X4+X8+X12+X16+X20+Y0+Y4+Y8+Y12+Y16+Y20) (total daily cost, Daily cost of officer working 8-hours = 12*8 = $ 96. Daily cost of officer working 12-hours = 12*8+15*4 = $ 156)

s.t.

X0+X20+Y0+Y16+Y20 >= 32 (number of police officers on duty from 0:00 to 4:00 hours)

X0+X4+Y0+Y4+Y20 >= 40 (number of police officers on duty from 4:00 to 8:00 hours)

X4+X8+Y0+Y4+Y8 >= 26 (number of police officers on duty from 8:00 to 12:00 hours)

X8+X12+Y4+Y8+Y12 >= 18 (number of police officers on duty from 12:00 to 16:00 hours)

X12+X16+Y8+Y12+Y16 >= 22 (number of police officers on duty from 16:00 to 20:00 hours)

X16+X20+Y12+Y16+Y20 >= 40 (number of police officers on duty from 20:00 to 0:00 hours)

Xt, Yt >= 0

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Create Excel model as follows:

Enter Solver Parameters:

Click Solve to generate the solution.

After that, values appear automatically in yellow cells

Click OK

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The minimum amount Bloomington must pay daily = $ 8,760