Final answer:
To maximize the profit with an investment of up to $100,000, the investment fund manager should use linear programming techniques, factoring in returns of 10% for shares and 16% for gold, and respecting the constraints provided. The objective function is to maximize P = 0.10x + 0.16y with constraints that manage the relationship between the investments in shares and gold.
Step-by-step explanation:
The investment fund manager needs to decide how much to invest in shares (x) and gold (y) under certain constraints to maximize profit, using a maximum investment of $100,000 in either asset. The constraints given are that the value of the shares (x) must be less than or equal to 3 times the value of gold (y), and the value of the shares must be greater than or equal to the value of the gold. We also know that historically, shares have returned 10% annually, while gold has returned 16% per year. To maximize the profit using linear programming, we need to establish objective and constraint functions:
Objective Function
Maximize P = 0.10x + 0.16y
Constraints
- x ≥ y (shares greater than or equal to gold)
- x + y ≤ 100,000 (total investment does not exceed $100,000)
- x ≥ 0, y ≥ 0 (cannot invest negative amounts)
By solving this linear programming problem, usually with graphical method or simplex method, the fund manager can determine the optimal amount of shares and gold purchased to maximize profit.