# Thread: Linear programming solved using graphical method question.

1. ## Linear programming solved using graphical method question.

I need help formulating this linear programming problem:

Annabelle Sizemore has cashed in some treasury bonds and a life insurance policy that her parents had accumulated over the years for her. She has also saved some money in certificates of deposit and savings bonds during the 10 years since she graduated from college. As a result, she has $120,000 available to invest. Given the recent rise in the stock market, she feels that she should invest all of this amount there. She has researched the market and has decided that she wants to invest in an index fund tied to S&P stocks and in an Internet stock fund. However, she is very concerned about the volatility of Internet stocks. Therefore, she wants to balance her risk to some degree. She has decided to select an index fund from Shield Securities and an Internet stock fund from Madison Funds, Inc. She has also decided that the proportion of the dollar amount she invests in the index fund relative to the Internet fund should be at least one-third but that she should not invest more that twice the amount in the Internet fund that she invests in the index fund. The price per share of the index fund is$175, whereas the price per share of the internet fund is \$208. The average annual return during the last 3 years for the index fund has been 17%, and for the Internet stock fund it has been 28%. She anticipates that both mutual funds will realize the same average returns for the coming year that they have in the recent past; however, at the end of the year she is likely to reevaluate her investment strategy anyway. Thus, she wants to develop an investment strategy that will maximize her return for the coming year.

Q1 ) Formulate a linear programming model for Annabelle that will indicate how much money she should invest in each fund and solve this model by using the graphical method.

I understand the first constraint is 175x + 208y =< 120000
But I don't know how to formulate the other constraints and how to solve them graphically. Thanks.

2. ## Re: Linear programming solved using graphical method question.

If $x$ is the number of shares she should purchase of the internet fund and $y$ is the number of shares she should purchase of the index fund, then $175x$ is the amount she should invest in the internet fund and $208y$ is the amount she should invest in the index fund.

So, you were given proportions of those amounts. You were told that "the dollar amount she invests in the index fund relative to the Internet fund should be at least one-third" which means that she wants to invest at least one third as much in the internet fund that she is investing in the index fund: $175x \ge \dfrac{1}{3}(208y)$.

Then "she should not invest more that twice the amount in the Internet fund that she invests in the index fund" means the amount she spends on the internet fund should be less than or equal to twice the amount she spends on the index fund: $175x \le 2(208y)$.

So, now you have the three constraints:

$175x + 208y \le 120,000$

$\dfrac{208}{3}y \le 175 x \le 416y$ (I combined two of the constraints into one)

I would recommend looking in your book to see how to solve this system graphically.