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. management 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 $p = 0.08x + 0.12y$ subject to the constraints total loans $x + y = 80$ × distribution of loans $xgeq4y$ √ $xgeq0$ $ygeq0$ resources ebook

Explanation:

Step1: Identify total loan amount constraint

The total amount of money earmarked for loans is $20$ million. So the sum of homeowner loans $x$ and auto - loans $y$ should be $x + y=20$, not $x + y = 80$ as in the incorrect formulation.

Step2: Correct the linear - programming model

The objective function to maximize the return $P$ is $P=0.08x + 0.12y$ (since homeowner loans have an $8\%$ or $0.08$ annual rate of return and auto - loans have a $12\%$ or $0.12$ annual rate of return). The constraints are:

1. Total loan amount: $x + y=20$
2. Distribution of loans: $x\geq4y$
3. Non - negativity: $x\geq0,y\geq0$

Answer:

Maximize $P = 0.08x+0.12y$
subject to the constraints:
$x + y=20$
$x\geq4y$
$x\geq0$
$y\geq0$