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 900 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 1,800 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? HINT [See Example 3.] (If an answer does not exist, enter DNE.)

Answer 1

Let the number of homeless patients be x

Let the number of high income patients be y

Maximize, Z(Profit) = 9500*x + 8000*y

Constraints:

x >= 740 [number of homeless patients must be 740]

y <= 900 [number of high-income patients must be less than 900]

x <= 2* y [number of homeless patients should not exceed more than two times high-income patients]

x + y <= 1800 [overall capacity]

So, the maxima will occur when we will have 1200 homeless patients and 600 high-income patients [The answer is pretty straight forward since we are getting high amount for homeless patients and it will be only bounded by constraint, then x <= 2*y]

Number of homeless patients = 1200

Number of high-income patients = 600

Maximum Profit = 9500 * 1200 + 8000 * 600 = 16200000$

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