The following tablesummarizes the 24-hour operation of police officers in a policestation. The minimum requirements will vary accordingly to the timeof the day as shown in the table:

Shift (i)	Time	Min no. of officersrequired
1	02:00am to 06:00am	25
2	06:00am to 10:00am	50
3	10:00am to 02:00pm	88
4	02:00pm to 06:00pm	105
5	06:00pm to 10:00pm	45
6	10:00pm to 02:00am	32

All officers(full-time and part-time) will report for duty at the start of oneof the shifts shown in the table. A full-time officer will remainon duty for 12 consecutive hours whereas a part-time officer willremain on duty for only 4 consecutive hours. For each shift, theno. of part-time officers cannot exceed 30% of the total no. ofofficers on duty.

Let Xi and Yi be theno. of full-time and part-time police officers respectively whowill start work in shift i where i = 1 to 6.

Formulate an LPproblem to minimize the total no. of police officers to be hired tomeet the needs throughout the day. Solve this LP to find theoptimal solution.

Show your output report and comment onthe results/key findings. (20 marks)

For each shift, theexcess headcount is defined as the no. of officers (full-time andpart-time) working in that shift minus the stated minimum requiredno. of officers. Derive the new objective function for minimizingthe total excess headcounts for the day. Solve the LP to find thenew optimal solution.

Show your outputreport and briefly comment on the results/key findings and comparewith the earlier results/key findings. (10 marks)