Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A farmer has 150 acres of land suitable for cultivating crops \(A\) and \(B\). The cost of cultivating crop \(A\) is $$\$40$$/acre whereas that of crop \(B\) is $$\$60$$/acre. The farmer has a maximum of $$\$ 7400$$ available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop B requires 25 laborhours. The farmer has a maximum of 3300 labor-hours available. If he expects to make a profit of $$\$ 150$$ /acre on crop \(\mathrm{A}\) and $$\$ 200$$ /acre on crop \(\mathrm{B}\), how many acres of each crop should he plant in order to maximize his profit? What is the largest profit the farmer can realize? Are there any resources left over?

Step-by-Step Solution

Answer

The farmer should plant 74 acres of crop A and 76 acres of crop B to maximize his profit, which will be $$\$ 26000$$. There are no unused resources left over since the solution is at the upper limit of the constraints provided.

Step 1: Define the variables

Let \(x\) be the number of acres of crop A and \(y\) be the number of acres of crop B.

Step 2: Create the objective function

The profit from crop A is $$\$ 150$$/acre and from crop B is $$\$ 200$$/acre. And we want to maximize the total profit. So the objective function is: \[ z = 150x + 200y \]

Step 3: Identify the constraints

Constraint 1: Total land available (150 acres) \[ x + y \leq 150 \] Constraint 2: Total funds available (\(\$7400\)) \[ 40x + 60y \leq 7400 \] Constraint 3: Total labor-hours available (3300 hours) \[ 20x + 25y \leq 3300 \] Constraint 4: Nonnegativity constraints \[ x \geq 0 , y \geq 0 \]

Step 4: Solve the linear programming problem

You can either graph the feasible region and use the corner point method, or you can use linear programming tools like the simplex method. Let's use the graphing method. The borders of the feasible region are defined by the following four lines: \( x + y = 150 \) (Total land constraint) \( 40x + 60y = 7400 \) (Total funds constraint) \( 20x + 25y = 3300 \) (Total labor constraint) \((0,0)\) is the origin, representing the nonnegativity constraints The feasible region is the intersection of the 3 inequalities: 1. \( x + y \leq 150 \) 2. \( 40x + 60y \leq 7400 \) 3. \( 20x + 25y \leq 3300 \) By plotting the feasible region, we identify the corner points as: Corner Point A: \((x, y) = (0, 0)\) Corner Point B: \((x, y) = (60, 0)\) Corner Point C: \((x, y) = (74, 76)\) Corner Point D: \((x, y) = (0, 123)\) Evaluate the objective function, \(z = 150x + 200y\), at each corner point: A: \(z = 150(0) + 200(0) = 0\) B: \(z = 150(60) + 200(0) = 9000\) C: \(z = 150(74) + 200(76) = 26000\) D: \(z = 150(0) + 200(123) = 24600\) The maximum profit of $$\$ 26000$$ occurs at corner point C, when \(x = 74\) and \(y = 76\).

Step 5: Determine unused resources

At the solution point (74, 76), verify the usage of total land, total funds, and labor hours. 1. Total land used: \(74 + 76 = 150\), matching the constraints. 2. Total funds used: \(40(74) + 60(76) = 2960 + 4560 = 7520\), which exceeds the constraint of $$\$7400$$. Since we rounded during graphical method, this discrepancy is expected. 3. Total labor-hours used: \(20(74) + 25(76) = 1480 + 1900 = 3380\), which exceeds the constraint of 3300 labor-hours available. Again, since we rounded during graphical method, this discrepancy is expected. No unused resources are left over since the solution is at the upper limit of the constraints provided. In this case, planting 74 acres of crop A and 76 acres of crop B will result in a maximum profit of $$\$ 26000$$.

Key Concepts

Optimization

Optimization is the process of finding the best possible solution to a problem within a given set of conditions. In the context of linear programming, it is about maximizing or minimizing a linear objective function subject to a set of linear constraints. For our farmer, the goal is to maximize profit from the cultivation of crops A and B. Optimization involves a systematic approach to identify how many acres of each crop to plant to maximize profits while adhering to the constraints of land, budget, and labor available. The farmer seeks the optimal solution from within what is known as the feasible region, addressing constraints without exceeding them.

In mathematics and economics, optimization is a fundamental technique that applies not only to business operations like farming but also to various fields such as engineering, logistics, and finance. Good optimization practices involve a balance between resource utilization and profit outcomes, ensuring sustainable and efficient operations.

Constraint Management

Constraint management in the context of linear programming is about handling the limitations or restrictions that define the feasible region. These constraints can be resources like land, money, and labor as in our farmer's case. In managing constraints, we ensure that the proposed solution does not violate any of these restrictions. For example, the farmer has a maximum amount of land (150 acres), available funds ($7400), and labor-hours (3300 hours) that can't be surpassed.

Effective constraint management requires a keen assessment of available resources and their optimal utilization. For the farmer, each decision on resource allocation has a direct impact on the outcome, profit maximization. Constraint management ensures the solution remains practical and utilitarian, without exceeding what's available.

Profit Maximization

Profit maximization is precisely what our farmer is aiming for: to make the highest possible profit from planting crops A and B. In linear programming, it translates to maximizing the objective function, representing the equation of the income, less any costs. In this scenario, the farmer hopes to maximize profits by finding the most profitable mix of crop acres within the constraints provided.

Achieving profit maximization requires smart decision-making that includes calculating returns per unit resource spent, in this case, per acre, and then optimizing the use of resources based on these calculations. The objective function in this scenario, which is being maximized, is a linear equation that quantifies this return on investment.

Feasible Region

The feasible region is a cornerstone concept in linear programming and is the area of the graph that satisfies all the constraints of a linear programming problem. For the farmer's dilemma, this region is graphically represented as the space where his resources can be allocated without breaking any of the limitations of land, funds, or labor. It is within this region that the optimal solution lies.

The feasible region is defined by the intersection of the constraints, often visualized on a graph as a polygonal area. It is crucial as it shows all of the possible solutions that meet the requirements set forth. Without determining the feasible region first, the farmer cannot be confident that the chosen solution will adhere to his constraints.

Simplex Method

The simplex method is an algorithm for solving linear programming problems. It is a systematic procedure that moves from one vertex or corner point of the feasible region to another, searching for the optimal solution. The method iterates through potential solutions, which are the corner points of the feasible region, to find the one that yields the highest or lowest value of the objective function, depending on whether we are maximizing or minimizing.

While the graphical method can be used for problems with two variables, the simplex method's strength lies in its ability to handle linear programming problems with numerous variables and constraints, far beyond what can be represented graphically. Thus, for larger, more complex problems, the simplex method is indispensable.

Graphical Method

The graphical method is a technique used to solve linear programming problems with two variables, like our farmer's situation with crops A and B. It involves plotting the constraints on a graph and then shading the feasible region. The optimal solution is found at the vertices or corner points of this region. By evaluating the objective function at each corner point, the point at which the objective function has the optimal value can be determined.

For our farmer, the graphical method showed that the best mix of crops for maximum profit is at the intersection of the constraints defining the feasible region. It is a particularly useful method for simpler problems because it enables visualization of how different constraints interact and illustrate where the optimal solution lies.