A bank has $$672,000$ in assets to allocate among investments in bonds, home mortgages, car loans, and personal loans. Bonds are expected to produce a return of $11\%,$ mortgages $9\mathrm{.}5\%,$ car loans

$10\mathrm{.}5\%,$ and personal loans $13\mathrm{.}5\%.$ To make sure the portfolio is not too risky, the bank wants to restrict personal loans to no more than the $25\%$ of the total portfolio. The bank also wants to ensure

that at least as much money is invested in mortgages as is invested in personal loans. The bank also wants to invest at least as much in bonds as they do in personal loans. $($Let $x_1,x_2,x_3,$ and $x_4$ be

the amount $($in dollars$)$ invested in bonds, mortgages, car loans, and personal loans, respectively.$)$

$($a$)$ Formulate an LP model for this problem with the objective of maximizing the expected return $($in dollars$)$ on the portfolio.

MAX:

Subject to:

total amount spent

amount for personal loans

mortgages and personal loans

bonds and personal loans

$x_1,x_2,x_3,x_{4}0$

$($b$)$ Implement your model in a spreadsheet and solve it$.$ What is the optimal solution?