A local bank wants to build a bond portfolio to maximize the expected return from a set of 5 bonds with a $1 million available for investment. The expected annual return, the work-case annual return on each bond, and the "duration" of each bond are given in the following table. (The duration of a bond is a measure of the bond's sensitivity to interest rates).

The average worst-case return for the portfolio must be at least 7.2 percent. The average duration of the portfolio must be at most  6. Because of diversification requirements, at most 40 percent of the total amount invested can be inveted in a single bond.

1. Write down the mathematical formulation of the optimization problem using a linear programming model, specifying the decision variables, objectives function, and constraints.

                            Expected Rate of Return   Worst-Case Rate of Return    Duration

Bond 1                           15%                                    8%                                   8

Bond 2                           11.5%                                 7.5%                                7

Bond 3                           10.5%                                 6.8%                                 6

Bond 4                            9.5%                                  7%                                   5

Bond 5                            8.5%                                 7.4%                                3