Tom won the lottery and wants to invest his winning of $$320,000.$ Tom considers investing this $$320,000$ on these four possible investment options: a biotech stock, a healthcare stock, a saving certificate deposit, and a mutual fund. According to the financial specialist$$s estimation, the return rates on the biotech stock to be $23\%$ and the healthcare stock to be $15\%.$ The saving certificate deposit has a return rate of $7\%$ and the mutual fund has a $19\%$ return rate.

To diversify the investment, Tom would like the following setups for the investment:

$1)$ At least $$100,000$ must be invested on the biotech stock.

$2)$ The combined investment on the two stocks $($biotech and healthcare$)$ cannot be more than $$225,000.$

$3)$ No more than $$85,000$ be invested in the saving certificate deposit.

$4)$ The investment on the mutual fund should be no more than half of the investment on the biotech stock

Like most investors, Tom wants to maximize the return of the investment.

Set up a linear program to help Tom with the investment decision. Clearly define your decision variables and list your objective function and the constraints. You do NOT need to solve your linear program.