Formulate the following problems into linear programming models. Clearly define your decision variables, objective function, and constraints. For each constraint, explain, in complete sentences, how it works. If your model uses both parameters and decision variables, make sure that you indicate which is which. Finally, explain why the model that you propose is a linear optimization problem.

1. Alice works in an investment company on Wall Street. She is considering investing in four bounds, and $1 million dollar is available for the investment. The expected annual return, the worst-case annual return on each bond, and the "duration" of each bond are given in Table 1. The duration of a bond is a measure of the bond's sensitivity to interest rates. Alice wants to maximize the expected return subject to the following three constraints. (0) Constraint 1: The worst-case return of the bond portfolio must be at least 8%. Constraint 2: The average duration of the portfolio must be at most 6. For example, a portfolio that invested $600,000 in bond 1 and $400,000 in bond 4 have an average duration of (600,000*3+400,000*9)/1,000,000 = 5.4. (iii) Constraint 3: Because of diversification requirements, at most 40% of the total amount invested can be invested in a single bond. Bond Table 1 Expected return (%) 13 Worst-case return (%) Duration 1 6 3 2 8 8 4 3 12 10 7 4 14 9 9 1. Alice works in an investment company on Wall Street. She is considering investing in four bounds, and $1 million dollar is available for the investment. The expected annual return, the worst-case annual return on each bond, and the "duration" of each bond are given in Table 1. The duration of a bond is a measure of the bond's sensitivity to interest rates. Alice wants to maximize the expected return subject to the following three constraints. (0) Constraint 1: The worst-case return of the bond portfolio must be at least 8%. Constraint 2: The average duration of the portfolio must be at most 6. For example, a portfolio that invested $600,000 in bond 1 and $400,000 in bond 4 have an average duration of (600,000*3+400,000*9)/1,000,000 = 5.4. (iii) Constraint 3: Because of diversification requirements, at most 40% of the total amount invested can be invested in a single bond. Bond Table 1 Expected return (%) 13 Worst-case return (%) Duration 1 6 3 2 8 8 4 3 12 10 7 4 14 9 9