Filbeck Investments manages the money invested by several wealthy clients. The company tailors its investment strategy based on the client’s risk tolerance and desire for the return on investment. Sanath Kumar, an investor, has authorized Filbeck Investments to invest $1 million in stocks and bonds. Each unit of stock fund costs $60 and yields an annual rate of return of 9%. Each unit of bond fund costs $90 and yields an annual rate of return of 3%. Sanath Kumar wants to minimize the risk associated with his investments, but he wants his investments to generate at least $50,000 in annual income.

Based on the risk analysis conducted by the company, stocks have a risk index of 7, and bonds have a risk index of 2. The higher the risk index is, the riskier the investment. Sanath Kumar has also instructed Filbeck Investments that at least $400,000 be invested in bonds.

1. Formulate this problem as a linear programming model with the objective of minimizing the total risk index for the portfolio.

2. Solve the problem to determine the units to be invested in each fund and the annual income generated by the optimal investment strategy.

3. What would be the optimum portfolio if Sanath Kumar’s objective is to maximize annual return on investments?