Problem 36 - 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 \(\mathrm{B}\) 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 \(B\) 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 all \( \$ 500,000\) in project A and none in project B to maximize the return on her investment, yielding a maximum return of \( \$ 50,000\).

Step 1: Define Objective Function

We want to maximize the return on investment. The return on investment for project A is 10%, and for project B, it is 15%. So, the total return will be given by the following equation: Objective Function (Z): Z = 0.10a + 0.15b We need to maximize Z.

Step 2: Set Up Constraints

First, let's list the constraints given by the problem: 1) The total investment is up to $ \$ 500,000: a + b ≤ 500,000 2) The investment in project B should not exceed 40% of the total investment: b ≤ 0.40(a + b) Now, let's rewrite the inequality b ≤ 0.40(a + b) to be more simplified: b ≤ 0.40a + 0.40b 0.60b ≤ 0.40a b ≤ (2/3)a So, we have the following constraints: a + b ≤ 500,000 b ≤ (2/3)a Additionally, we need to consider non-negativity constraints: a ≥ 0 b ≥ 0

Step 3: Determine Feasible Region and Corner Points

Now we need to graph the constraints and identify the feasible region and corner points. This can be done either by plotting the constraints on a graph paper or using a graphing calculator or computer software. Plotting the constraints: - a + b ≤ 500,000: This will result in a region below and to the left of the line a + b = 500,000 - b ≤ (2/3)a: This will result in a region below and to the right of the line b = (2/3)a The corner points of the feasible region are given by the intersections of these constraints. We find them by solving the corresponding equations: Intersection of a + b = 500,000 and b = (2/3)a yields (300,000, 200,000). Other corner points are (500,000, 0) and (0, 0), which are intersections with the non-negativity constraints.

Step 4: Evaluate Objective Function at Corner Points and Find Optimal Solution

Now we can evaluate the objective function Z = 0.10a + 0.15b at the corner points to find the optimal solution: 1) (a, b) = (300,000, 200,000): Z = 0.10(300,000) + 0.15(200,000) = 45,000 2) (a, b) = (500,000, 0): Z = 0.10(500,000) + 0.15(0) = 50,000 3) (a, b) = (0, 0): Z = 0.10(0) + 0.15(0) = 0 The maximum value of Z is 50,000, which corresponds to investing \( \$ 500,000\) in project A and \( \$ 0\) in project B.

Step 5: Conclusion and Maximum Return

The financier should invest all \( \$ 500,000\) in project A and none in project B to maximize the return on her investment. The maximum return will be \( \$ 50,000\).

Key Concepts

Objective Function

An objective function in mathematics, specifically within the realm of optimization problems, is a formula that defines the quantity to be maximized or minimized. Think of it as the 'target' of the problem. In the context of our financier's dilemma, the objective function is the total return on the investment in both projects, A and B. Expressed mathematically, it is represented as \(Z = 0.10a + 0.15b\), where \(Z\) marks the total return, \(a\) is the amount invested in project A, and \(b\) is the amount invested in project B.

The objective function is the centerpiece of an optimization problem. It's where we condense our goals into mathematical form, shaping the direction of our efforts. In the case of our financier, by finding the optimal values of \(a\) and \(b\) that maximizes \(Z\), we effectively determine the best investment strategy.

Constraints in Optimization

The real world is full of limitations, and these are mimicked in optimization problems by constraints. Constraints in optimization are the mathematical expressions of these limitations that define the boundaries of the feasible region. They can represent budget ceilings, resource capacities, or other restrictions relevant to the problem at hand. In our investment example, the constraints include a budget limit of \(a + b \leq 500,000\) and the relative risk mitigation measure that the investment in project B, \(b\), cannot be more than 40% of the total investment, leading to the inequality \(b \leq (2/3)a\).

Moreover, there are non-negativity constraints, which reflect the fact that the financier cannot invest a negative amount: both \(a \geq 0\) and \(b \geq 0\). These constraints act as 'rules' the solutions must obey, carving out the feasible region wherein our optimal solution resides.

Feasible Region

The feasible region is the 'playground' of possible solutions that satisfy all the constraints of an optimization problem. It is here where we search for the optimal solution. In graphical terms, for our example, the feasible region is the area on a graph where all the constraints overlap. This region contains all the potential investment strategies that abide by the financier's restrictions.

By plotting the constraints on a graph, we can visually examine this space. In this scenario, our feasible region is delineated by the lines representing the constraints: \(a + b \leq 500,000\) and \(b \leq (2/3)a\), including the axes since we cannot invest negative amounts. Identifying corner points of the feasible region is crucial because, according to linear programming theory, the optimal solution will reside at one of these points. For the financier, these corner points, determined algebraically or graphically, can pinpoint the exact investment strategy needed to maximize the return.

Return on Investment

Return on investment (ROI) is a measure of the profitability of an investment. It's the percentage of money gained or lost on an investment relative to the amount invested. Our financier's goal is to maximize this ROI, which in the context of the given problem, translates to optimizing her investments in projects A and B for the highest return.

In the solution process, examining the returns at the corner points of the feasible region allows us to choose the strategy that offers the best ROI. The concept of ROI in the discipline of optimization underlines the intention behind setting up objectives and adhering to constraints. For the financier, the maximum return is obtained by investing entirely in project A, which offers a lower, but secure ROI of 10%, largely because project B's risk was regulated through constraints. This strategic allocation of resources, guided by the principles of optimization, ensures the most efficient use of capital to achieve financial goals.