A man has decided to invest at most $90,000 in securities in the form of corporate stocks. He has three options: blue chip stocks he assumes will yield a 6% return, growth stocks @ 12%, and speculative stocks @ 20%. No more than 25% of his investment should be in growth and speculative stocks and at least 15,000 of his investments should be in growth stocks. How much should he invest in each group in hope of maximizing return?

Question

A man has decided to invest at most $90,000 in securities in the form of corporate stocks. He has three options: blue chip stocks he assumes will yield a 6% return, growth stocks @ 12%, and speculative stocks @ 20%. No more than 25% of his investment should be in growth and speculative stocks and at least 15,000 of his investments should be in growth stocks. How much should he invest in each group in hope of maximizing return?

I just need to know how to set up the constraints and objective function. This is a case and a big part of my grade! THANKS!!

Answer 1

You have:

Max (.06*B + .12*G + .20*S) subject to:
B+G+S = 90,000
G+S <= 22,500
G >= 15,000

You could set this up as a linear programming problem. However, getting to the optimal solution is obvious.
G=15,000, S=7500, B=the remainder

Answer 2

To set up the constraints and objective function for this problem, we first need to define the decision variables.

Let:
- X1 = investment in blue chip stocks (in dollars)
- X2 = investment in growth stocks (in dollars)
- X3 = investment in speculative stocks (in dollars)

Now, let's set up the constraints:

1. Total investment constraint: The man has decided to invest at most $90,000. So, the total investment should not exceed $90,000.
X1 + X2 + X3 ≤ 90,000

2. Percentage constraint: The investment in growth and speculative stocks should not exceed 25% of the total investment.
(X2 + X3) ≤ 0.25 * (X1 + X2 + X3)

3. Minimum investment constraint: At least $15,000 should be invested in growth stocks.
X2 ≥ 15,000

4. Non-negativity constraint: The investment cannot be negative.
X1, X2, X3 ≥ 0

Now, let's define the objective function:

The objective is to maximize the return on investment. The returns from each type of stock investment can be calculated by multiplying the investment amount by the respective returns:

Total Return = 0.06 * X1 + 0.12 * X2 + 0.20 * X3

Therefore, the objective function is:
Maximize Z = 0.06 * X1 + 0.12 * X2 + 0.20 * X3

By formulating the problem with these constraints and the objective function, you can use linear programming techniques to find the optimal solution that maximizes the return.