Problem 49 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Patricia has at most $$\$ 30,000$$ to invest in securities in the form of corporate stocks. She has narrowed her choices to two groups of stocks: growth stocks that she assumes will yield a \(15 \%\) return (dividends and capital appreciation) within a year and speculative stocks that she assumes will yield a \(25 \%\) return (mainly in capital appreciation) within a year. Determine how much she should invest in each group of stocks in order to maximize the return on her investments within a year if she has decided to invest at least 3 times as much in growth stocks as in speculative stocks.

Step-by-Step Solution

Answer

Patricia should invest $$22,500$$ in growth stocks and $$7,500$$ in speculative stocks to maximize her return on investment, which will be $$5,625$$ within a year.

Step 1: Define the variables

Let G be the amount of money invested in growth stocks and S be the amount of money invested in speculative stocks.

Step 2: Write the constraints

The first constraint is the budget constraint: \(G + S \leq 30,000\) The second constraint is related to the at least 3 times as much invested in growth stocks: \(G \geq 3S\)

Step 3: Write the objective function

We want to maximize the total return: \(R = 0.15G + 0.25S\)

Step 4: Graph the Constraints

Graph the constraint lines \(G + S = 30,000\) and \(G = 3S\) on the same plane. Identify the feasible region formed by the region that satisfies both constrains.

Step 5: Find the vertices of the feasible region

The feasible region bounded by the constraints should be a polygon. Identify the vertices of this polygon. These will be the candidates for the solution to the optimization problem. In this case, we have: 1. Intersection of \(G+S=30,000\) and \(G=3S\): Substitute G in the first equation with \(3S\): \(3S+S=30,000\) Solving for S: \(S = 7,500\) Plugging the value of S back into G: \(G = 3(7,500) = 22,500\) 2. If Patricia invests all $30,000 in growth stocks: \(G = 30,000, S = 0\) 3. If Patricia invests all $30,000 in speculative stocks: This is _not_ a feasible solution because it violates the constraint \(G \geq 3S\).

Step 6: Plug the vertices into the objective function

Calculate the value of R for each vertex: 1. R(22,500,7,500) = 0.15(22,500) + 0.25(7,500) = 5,625 2. R(30,000,0) = 0.15(30,000) + 0.25(0) = 4,500

Step 7: Determine the optimal solution

The highest return on investment is obtained when Patricia invests \(22,500 in growth stocks and \)7,500 in speculative stocks, with a return of $5,625 within a year.

Key Concepts

Linear Programming

Linear programming (LP) is a mathematical method for determining a way to achieve the best outcome in a given mathematical model. The model is represented as linear relationships. In the context of investment optimization, LP can be applied to maximize an investment return based on various constraints. The task is to allocate resources, in this case, money, across different investment options to receive the highest possible return.

In our example, Patricia is using LP to decide how much to invest in growth and speculative stocks. She models her decision based on the assumption of linear returns for simplicity—15% for growth stocks and 25% for speculative stocks. The linear constraints here include not exceeding the investment budget and maintaining a 3:1 investment ratio between growth and speculative stocks.

Why is LP suitable for this problem?

• It helps handle multiple constraints efficiently.
• LP problems can be solved using graphical methods for two variables, which is the case with Patricia's problem.
• Solutions to LP problems often focus on corner points of the feasible region, simplifying the search for the best outcome.

Optimization Constraints

Optimization constraints are restrictions or conditions that the variables in an optimization problem must satisfy. Constraints define the feasible solution space and limit the values that the decision variables can take.

In Patricia’s investment scenario, there are two key constraints:

• Budget Constraint: Patricia can invest at most $30,000, forming a linear equation (G + S ≤ 30,000).
• Investment Ratio Constraint: Patricia wants to invest at least three times as much in growth stocks as in speculative stocks. This requirement translates into another linear equation, (G ≥ 3S).

These constraints guide the creation of the feasible region, which visually represents all possible solutions that satisfy the problem's conditions. By determining this feasible region through the graphical method, Patricia can visualize exactly where she should consider investing her money while adhering to her constraints.

Objective Function

An objective function is a formula that provides the value that you want to optimize—it's the target of the optimization problem. In linear programming, this function is linear and involves our decision variables. It embodies the goal of the problem, whether it's to maximize profit, minimize cost, or in Patricia's case, maximize the return on her investment portfolio.

Patricia's objective function for the return on her investment is expressed as:

R = 0.15G + 0.25S

This function represents the total monetary return from investments, taking into account the 15% return from growth stocks (G) and the 25% return from speculative stocks (S). Patricia's goal is to find the values of G and S that will maximize her return R while conforming to the constraints of her investment capacity and strategy.

To identify this optimal point, Patricia calculates the objective function at the vertices of the feasible region and compares the values. The vertex yielding the highest function value indicates the best investment allocation according to her linear model, achieving the most favourable outcome given her strategy and constraints.