Problem 8 - 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 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 \(\bar{B}\) 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?

Step-by-Step Solution

Answer

In order to maximize the return on investment, the financier should invest $$x=300,000$$ in Project A and $$y=200,000$$ in Project B. This results in a total return of $$Z = 0.1(300,000) + 0.15(200,000) = 60,000$$ dollars.

Step 1: Define the objective function

Let's define our objective function (the total return) as $$Z = 0.1x + 0.15y$$ which we'd like to maximize.

Step 2: Identify the constraints

We have two constraints: 1. Total investment should not exceed $$\$500,000$$: $$x + y \leq 500,000$$ 2. Investment in project B must not exceed $$40\%$$ of total investment: $$y \leq 0.4(x+y)$$

Step 3: Write the constraints as linear inequalities

We already have the constraints as linear inequalities: 1. $$x + y \leq 500,000$$ 2. $$y \leq 0.4(x+y)$$ or equivalently, $$0.6y \leq 0.4x$$. Additionally, the investment amounts cannot be negative, so we have two more constraints: 3. $$x \geq 0$$ 4. $$y \geq 0$$

Step 4: Plot the feasible region

Plot each constraint on a graph and shade the feasible region, which is the area satisfying all constraints simultaneously.

Step 5: Find the corner points of the feasible region

The corner points of the feasible region are the points where the constraints intersect. In this case, these points are: 1. Intersection of $$x + y = 500,000$$ and $$0.6y = 0.4x$$ 2. Intersection of $$x + y = 500,000$$ and $$y = 0$$ 3. Intersection of $$0.6y = 0.4x$$ and $$x = 0$$

Step 6: Determine the optimal solution

Evaluate the objective function $$Z = 0.1x + 0.15y$$ at each of the corner points and choose the one that provides the maximum value: 1. Intersection of $$x + y = 500,000$$ and $$0.6y = 0.4x$$: Solve the system of equations to find $$x$$ and $$y$$, plug them into the objective function to get the value of $$Z$$. 2. Intersection of $$x + y = 500,000$$ and $$y = 0$$: This point corresponds to $$x = 500,000$$ and $$y = 0$$, so $$Z = 0.1(500,000) + 0.15(0) = 50,000$$. 3. Intersection of $$0.6y = 0.4x$$ and $$x = 0$$: This point corresponds to $$x = 0$$ and $$y = 0$$, so $$Z = 0.1(0) + 0.15(0) = 0$$. Compare the values of $$Z$$ at each corner point and choose the investment amounts corresponding to the highest value of $$Z$$. This will be the optimal investment strategy to maximize the return on the financier's investment.

Key Concepts

Optimization with Constraints

In financial decision-making, optimization with constraints plays a pivotal role as it determines how to best allocate resources to maximize returns or minimize costs under given limitations. Let's take an example from a financier needing to invest in two projects with an overall cap of \(500,000. We use linear programming, a mathematical method, to optimize an objective function—in our case, the total return from investments—subject to a set of constraints.

In our scenario, the constraints are twofold: the total investment should not surpass \)500,000, and the investment in the riskier project B should not be more than 40% of the total. Additional constraints are that investments cannot be negative, which are common sense rules in finance. Through linear inequalities, we can express these conditions mathematically, creating a structure within which we seek the best possible outcome.

Feasible Region in Linear Programming

The feasible region is a cornerstone concept of linear programming. It represents all the possible points that satisfy the constraints of a given problem. To visualize it, you plot the constraints on a graph, and where they all intersect creates a shape—the feasible region. This is the area we're interested in because any point outside of it violates at least one constraint and is thus not viable.

For the financier’s problem, the feasible region is a polygon on the graph where the investment in project A and B are both within the acceptable limits. By plotting the constraints, we determine this polygon, and the potential investment options are at the vertices. In the context of our financier, these are the possible combinations of investments in both projects that she could make without breaking any rules.

Objective Function Maximization

In linear programming, the objective function maximization is about finding the highest possible value of a function within the feasible region. This function represents what we're trying to achieve, such as maximizing profit or minimizing cost. For our financier, the objective function is her total expected return from the investments, expressed as \(Z = 0.1x + 0.15y\).

The process involves evaluating the objective function at the corner points—where the constraints' boundaries intersect—of the feasible region. These points often contain the maximum or minimum value of the function, owing to the nature of linear systems. By comparing the values calculated at each corner point, the financier can determine the optimal investment amounts to yield the highest return. Ultimately, it ensures an informed decision-making process, guided by clear, mathematical rationale.

Investment Strategy Planning

In the finance world, investment strategy planning is key to achieving financial goals. This involves analyzing various investment options, assessing risks and returns, and aligning investment decisions with financial constraints and objectives. Linear programming aids in this process through a structured, strategic approach considering all factors involved.

With regards to our example, the financier uses linear programming to plan her strategy, as it helps translate her financial goals into an objective function and constraints. By maximizing this function, she identifies the most beneficial investment plan without exceeding her risk tolerance for project B and her budget ceiling. This systematic approach leads to a solid investment strategy that aims to maximize returns while adhering to all investment guidelines and risk considerations.