Question

formulate but do not solve the following exercise as a linear - programming problem. madison finance has a total of $20 million earmarked for homeowner loans and auto loans, where x is homeowner loans in millions of dollars and y is auto loans in millions of dollars. on the average, homeowner loans have a 8% annual rate of return, whereas auto loans yield a 12% annual rate of return. madison has also stipulated that the total amount of homeowner loans should be greater than or equal to 4 times the total amount of automobile loans. determine the total amount of loans of each type madison should extend to each category to maximize its returns p in millions of dollars. maximize p = subject to the constraints total loans distribution of loans x > 0 y > 0 resources book

Explanation:

Step1: Define the objective function

The return on homeowner - loans is 8% or 0.08 of $x$ (in millions of dollars) and on auto - loans is 12% or 0.12 of $y$ (in millions of dollars). The objective is to maximize the return $P$, so $P = 0.08x+0.12y$.

Step2: Set up the total loans constraint

The total amount of money earmarked for loans is $80$ million. So, $x + y=80$.

Step3: Set up the distribution of loans constraint

The total amount of homeowner loans should be greater than or equal to 4 times the total amount of auto - loans. So, $x\geq4y$.

Answer:

$P = 0.08x + 0.12y$
Subject to the constraints:
$x + y=80$
$x\geq4y$
$x\geq0$
$y\geq0$