How do I formulate this question to be solved in the POM-QM software?
Deborah inherited a large sum of money; she wants to use a portion of this money to set up a trust fund for her 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 fund and 10% for the stock fund. Whatever portion of the inheritance she finally decides to commit to the trust fund, she wants to invest at least 30% of the amount in the bond fund. In addition, she wants to select a mix that will enable her to obtain a total return of at least 7.5%.
To formulate this question to be solved in the POM-QM software, you can frame it as a linear programming problem. Here are the steps to formulate the problem:
Step 1: Define the decision variables:
Let:
x = amount invested in the bond fund (in percentage of the inheritance)
y = amount invested in the stock fund (in percentage of the inheritance)
Step 2: Define the objective function:
The objective is to maximize the total return on the investment. The total return is calculated by multiplying the amount invested in each fund by its respective projected return rate. The objective function is:
Maximize: 0.06x + 0.10y
Step 3: Define the constraints:
a) The total amount invested in both funds should be less than or equal to the entire inheritance.
x + y <= 100
b) At least 30% of the total investment should be in the bond fund.
x >= 0.30 * (x + y)
c) The total return on the investment should be at least 7.5%.
0.06x + 0.10y >= 0.075 * (x + y)
Step 4: Define the non-negativity constraints:
x, y >= 0
Once you have formulated the problem with these steps, you can input the objective function, constraints, and decision variables into the POM-QM software to solve for the optimal values of x and y that maximize the total return while satisfying the given constraints.