Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A financier plans to invest up to $$\$ 500,000$$ in two projects. Project A yields a return of \(10 \%\) on the investment whereas project \(\bar{B}\) yields a return of \(15 \%\) on the investment. Because the investment in project \(\mathrm{B}\) is riskier than the investment in project \(\mathrm{A}\), the financier has decided that the investment in project \(\mathrm{B}\) should not exceed \(40 \%\) of the total investment. How much should she invest in each project in order to maximize the return on her investment? What is the maximum return?

Step-by-Step Solution

Answer

The financier should invest $300,000 in Project A and $200,000 in Project B to maximize the return on investment. The maximum return will be $75,000.

Step 1: Define the variables

Let's define x and y as the amounts invested in projects A and B, respectively, such that: - \(x\): Investment in Project A - \(y\): Investment in Project B

Step 2: Formulate the constraints

We are given the following constraints: 1. The total investment should not exceed $500,000, so \(x + y \leq 500,000\). 2. The investment in Project B should not exceed 40% of the total investment, so \(y \leq 0.40(x + y)\).

Step 3: Formulate the objective function

We want to maximize the ROI. We know that Project A yields a 10% return, and Project B yields a 15% return. The total return on the investment can be represented by the objective function: ROI = 0.10x + 0.15y

Step 4: Solve the constraints and find the feasible region

Express \(y\) in terms of \(x\) from constraint 2: \(y \leq 0.40(x + y)\) \(y \leq 0.40x + 0.40y\) \(0.60y \leq 0.40x\) \(y \leq \frac{2}{3}x\) So the constraints are: 1. \(x + y \leq 500,000\) 2. \(y \leq \frac{2}{3}x\) These constraints create a feasible region, and the maximum ROI will lie on the vertices of this region.

Step 5: Determine the vertices of the feasible region

To find the vertices, we will have to find the intersection of our constraints: 1. \(x = 0\) 2. \(y = 0\) 3. \(x + y = 500,000\) and \(y = \frac{2}{3}x\) For case 3, we can substitute the equation of \(y\) into the constraint: \(x + \frac{2}{3}x= 500,000\) Solving this equation, we get: \(\frac{5}{3}x = 500,000\) \(x = 300,000\) When \(x=300,000\), we can plug it back into the equation for \(y\), which gives: \(y = \frac{2}{3}(300,000)\) \(y = 200,000\) Thus, there are three vertices of the feasible region: (0, 0), (0, 500,000), and (300,000, 200,000).

Step 6: Calculate ROI for each vertex

Now we will calculate the ROI at each vertex using the objective function: 1. Vertex (0, 0): ROI = 0.10(0) + 0.15(0) = $0 2. Vertex (0, 500,000): ROI = 0.10(0) + 0.15(500,000) = $75,000 3. Vertex (300,000, 200,000): ROI = 0.10(300,000) + 0.15(200,000) = \(45,000+\)30,000 =$75,000

Step 7: Analyze the results and determine the maximum ROI

From our calculations, the maximum ROI is attained at vertices (0, 500,000) and (300,000, 200,000), both yielding a ROI of $75,000. Since the financier wants to invest in both projects, the optimal investment choice is to invest \(300,000 in Project A and \)200,000 in Project B, resulting in a maximum return of $75,000.

Key Concepts

Objective Function

In linear programming, the objective function is vital as it helps you understand what you are trying to optimize. Specifically, in the context of investments, the objective function is a formula representing the total expected return. The main goal is to maximize or minimize this function, depending on what you are trying to achieve.
In the given exercise, the investor aims to maximize the return on investment (ROI). The returns are defined as percentages of investments in two projects. For Project A, the return is 10%, while for Project B, it's 15%. Therefore, the objective function can be represented by the formula:

• ROI = 0.10x + 0.15y

This equation combines the different rates of returns with the amounts invested in each project, symbolized by x (Project A) and y (Project B). The objective is to adjust x and y within given constraints to achieve the highest possible ROI.

Constraints

Constraints in linear programming refer to the limitations or rules that must be adhered to within a problem. These are conditions that narrow down the choices of x and y—the investments in Projects A and B—and help form the feasible region.
In this problem, we have two main constraints:

• The total investment cannot exceed $500,000, formulated as: \[x + y \leq 500,000\]
• Investment in Project B can't be more than 40% of the overall investment, given by: \[y \leq 0.40(x + y)\]

By rewriting the second constraint, it transforms to: \[y \leq \frac{2}{3}x\]These equations ensure the investor doesn't overcommit financially and maintains a balance in the riskier Project B. Constraints are essential since they limit the possible solutions to those that meet all specified requirements.

Feasible Region

The feasible region is where all the constraints of a linear programming problem are satisfied simultaneously. In this exercise, it refers to the area where different combinations of investments in Project A and Project B will not exceed the total investment or other conditions outlined.
After formulating the constraints:

• \(x + y \leq 500,000\)
• \(y \leq \frac{2}{3}x\)

we can graph these inequalities on a plane, using x and y axes representing investments in Project A and B, respectively. The area where both constraints overlap is the feasible region.
To find the optimal investment, observe the vertices of this region, as the solution often lies at these points. In this case, vertices are calculated as (0, 0), (0, 500,000), and (300,000, 200,000), and these points need to be evaluated for maximum ROI.

Return on Investment (ROI)

Return on investment, or ROI, measures the profitability of an investment as a percentage of the initial outlay. In this context, we calculate ROI by applying the objective function to the vertices of the feasible region.
At each vertex of the feasible region, you calculate:

• Vertex (0, 0): ROI = 0.10(0) + 0.15(0) = $0
• Vertex (0, 500,000): ROI = 0.10(0) + 0.15(500,000) = $75,000
• Vertex (300,000, 200,000): ROI = 0.10(300,000) + 0.15(200,000) = $75,000

The goal is to find the highest ROI from these possibilities. Interestingly, the maximum ROI of $75,000 is achieved at both (0, 500,000) and (300,000, 200,000), but since the investor desires diversification, the choice would be to invest $300,000 in Project A and $200,000 in Project B. ROI allows investors to quantify the success of their financial decision in clear, numerical terms.