The Megabuck Hospital Corp. is to build a state-subsidized nursing home catering

Question

The Megabuck Hospital Corp. is to build a state-subsidized nursing home catering to homeless patients as well as high-income patients. State regulations require that every subsidized nursing home must house a minimum of 740 homeless patients and no more than 1,100 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 2,400 patients. The board of directors, under pressure from a neighborhood group, insists that the number of homeless patients should not exceed twice the number of high-income patients. Due to the state subsidy, the hospital will make an average profit of $9,500 per month for every homeless patient it houses, whereas the profit per high-income patient is estimated at $8,000 per month. How many of each type of patient should it house in order to maximize profit?

high-income patients=

homeless pateints=

profit=

Explanation / Answer

1. h740 (minimum)
2. H0 (minimum H)
3. H1100 (maximum)
4. Hh/2 (minimum ratio for H, maximum ratio for h)
5. H+h=2400 (capacity)

Z=profit = 9500h + 8000H
Z is to be evaluated at corners of the polygon.
We see that the corners of the polygon of feasibility are at
(740,1100), (1200,1100), (1400,700), (740,370).

at (740,1100), z=profit=(9500)(740)+(8000)(1100)

                                   =7030000+8800000=15830000

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