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.