G. Johnson recently inherited a large sum o money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: 1) a bond fund and 2) a stock fund. The projected returns over the life of the investments are 6% for the bond and 10% for the stock fund. WHatever protion of the inheritance he finally decides to commit to the trust fund, he wants to invest at least 30% of that amount in the bond fund. In addition he wants to select a mix that will enable him to obtain a total return of at least 7.5%..Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives.
Let the total amount of the trust fund be T. The amount in the bond fund, B, and the amount in the stock fund, S, will equal the amount of the fund: T = S + B. Also B >= .3T
The return is .06B + .1(T - B) and that is to be >=.075T
.06B + .1T - .1B >= .075T
.025T - .04B >= 0
.025T >= .04B
25T >= 40B
5T >= 8B
B <= .625T
The amount invested in the bond fund should be no more than 62.5% of the total amount, nor less than 30% of the total.
To formulate a linear programming model for this scenario, we need to define decision variables, objective function, and constraints.
Let's define the decision variables:
Let x be the percentage of the inheritance committed to the bond fund.
Let y be the percentage of the inheritance committed to the stock fund.
Now, let's define the objective function:
Our objective is to maximize the total return on the investment. Since we want to obtain a total return of at least 7.5%, we can maximize the return as follows:
Maximize: 0.06x + 0.10y
Next, let's define the constraints:
1) The total percentage invested should be equal to 100%:
x + y = 100
2) At least 30% of the total investment amount should be allocated to the bond fund:
x >= 0.30(x + y)
3) The total return on the investment should be at least 7.5%:
0.06x + 0.10y >= 0.075(x + y)
In summary, the linear programming model can be formulated as follows:
Maximize: 0.06x + 0.10y
Subject to:
x + y = 100 (Total percentage invested should be 100%)
x >= 0.30(x + y) (At least 30% should be allocated to the bond fund)
0.06x + 0.10y >= 0.075(x + y) (Total return should be at least 7.5%)
This model can be solved using linear programming techniques, such as the Simplex method, to determine the percentage that should be allocated to each investment alternative.