Linear Programming-Investment Strategy

Invest at most $90,000 in securities in the form of corporate stocks. Options into three groups of stocks: blue-chip stocks will yield 3% return (dividends and capital appreciation) within a year, growth stocks will yield 10% return within a year, and speculative stocks will yield 16% return (mainly due to capital appreciation) within a year. Because of the relative risks involved in his investments, he has further decided that no more than 25% of his investment should be in growth and speculative stocks combined, and at least $18,000 of his investments should be in growth stocks. Obviously he wants to maximize his return on investment (ROI) and he wants his ROI to be at least $4,400 in the year.

I am completely lost and cannot figure out how to set this up. Can someone please help?

To set up this problem as a Linear Programming (LP) problem, we need to define the decision variables, objective function, and the constraints.

Decision Variables:
Let:
x1 = amount invested in blue-chip stocks (in dollars)
x2 = amount invested in growth stocks (in dollars)
x3 = amount invested in speculative stocks (in dollars)

Objective Function:
We want to maximize the return on investment (ROI), which is given by the sum of the returns on each type of stock investment:

Maximize: 0.03x1 + 0.1x2 + 0.16x3

Constraints:
1. Total investment must be at most $90,000:
x1 + x2 + x3 <= 90,000

2. No more than 25% of the investment should be in growth and speculative stocks combined:
x2 + x3 <= 0.25(x1 + x2 + x3)

3. At least $18,000 of the investments should be in growth stocks:
x2 >= 18,000

4. The ROI should be at least $4,400 in a year:
0.03x1 + 0.1x2 + 0.16x3 >= 4,400

Therefore, the complete LP problem is:

Maximize: 0.03x1 + 0.1x2 + 0.16x3
subject to:
x1 + x2 + x3 <= 90,000
x2 + x3 <= 0.25(x1 + x2 + x3)
x2 >= 18,000
0.03x1 + 0.1x2 + 0.16x3 >= 4,400

To solve this LP problem and find the optimal investment strategy, you can use various optimization software or algorithms such as the Simplex Method or the interior-point algorithm. These methods will find the values of the decision variables (x1, x2, x3) that maximize the ROI while satisfying all the given constraints.