Maximize P = 7x + Y A Step-by-Step Linear Programming Guide

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Linear programming is a powerful mathematical technique used to optimize a linear objective function subject to a set of linear constraints. This method is widely applied in various fields, including business, economics, engineering, and operations research, to solve resource allocation problems, maximize profits, minimize costs, and improve efficiency. This article delves into the process of maximizing the objective function P = 7x + y, subject to the constraints 2x + y ≤ 150, 4x + 3y ≤ 350, x + y ≥ 80, and x, y ≥ 0. We will explore the graphical method, a fundamental approach to solving linear programming problems, and provide a step-by-step guide to finding the optimal solution.

Understanding Linear Programming

In the realm of linear programming, the primary goal is to find the best possible solution (either maximum or minimum) for a linear objective function. This objective function is subject to a set of linear constraints, which represent limitations or requirements in the problem. These constraints define a feasible region, which is the set of all possible solutions that satisfy all the constraints simultaneously. The optimal solution lies within this feasible region, and linear programming techniques help us identify the specific point that maximizes or minimizes the objective function.

Linear programming problems typically involve several key components:

  • Objective Function: This is the linear function that we aim to maximize or minimize. It represents the quantity we want to optimize, such as profit, cost, or resource utilization. In our case, the objective function is P = 7x + y.
  • Decision Variables: These are the variables that we can control to achieve the optimal solution. In our problem, the decision variables are x and y, representing the quantities of two different resources or products.
  • Constraints: These are the linear inequalities or equalities that restrict the values of the decision variables. They represent limitations on resources, production capacity, or other requirements. Our constraints are 2x + y ≤ 150, 4x + 3y ≤ 350, x + y ≥ 80, and x, y ≥ 0.
  • Feasible Region: This is the region defined by the constraints, representing all possible combinations of decision variables that satisfy the constraints. The optimal solution must lie within this feasible region.
  • Optimal Solution: This is the point within the feasible region that maximizes or minimizes the objective function. It represents the best possible solution to the problem.

Graphical Method: A Step-by-Step Approach

The graphical method is a visual approach to solving linear programming problems with two decision variables. It involves plotting the constraints on a graph, identifying the feasible region, and then finding the corner point that optimizes the objective function. This method provides a clear understanding of the problem and the solution process.

1. Graphing the Constraints

The first step in the graphical method is to graph each constraint on a coordinate plane. To do this, we treat each inequality as an equation and plot the corresponding line. Then, we determine the region that satisfies the inequality by testing a point (such as the origin (0, 0)) in the inequality. If the point satisfies the inequality, the region containing the point is the feasible region for that constraint. If not, the region on the other side of the line is the feasible region.

Let's graph our constraints:

  • 2x + y ≤ 150: To graph this inequality, we first graph the line 2x + y = 150. We can find two points on this line by setting x = 0 and solving for y, which gives us the point (0, 150), and setting y = 0 and solving for x, which gives us the point (75, 0). We then draw a line through these two points. To determine the feasible region, we test the point (0, 0) in the inequality: 2(0) + 0 ≤ 150, which is true. Therefore, the feasible region for this constraint is the region below the line.
  • 4x + 3y ≤ 350: Similarly, we graph the line 4x + 3y = 350. Setting x = 0 gives us y = 350/3 ≈ 116.67, and setting y = 0 gives us x = 350/4 = 87.5. We draw a line through the points (0, 116.67) and (87.5, 0). Testing the point (0, 0) in the inequality: 4(0) + 3(0) ≤ 350, which is true. The feasible region is below the line.
  • x + y ≥ 80: We graph the line x + y = 80. Setting x = 0 gives us y = 80, and setting y = 0 gives us x = 80. We draw a line through the points (0, 80) and (80, 0). Testing the point (0, 0) in the inequality: 0 + 0 ≥ 80, which is false. The feasible region is above the line.
  • x, y ≥ 0: These constraints simply mean that x and y must be non-negative, which restricts the feasible region to the first quadrant of the coordinate plane.

2. Identifying the Feasible Region

The feasible region is the area on the graph that satisfies all the constraints simultaneously. It is the intersection of the feasible regions for each individual constraint. In our case, the feasible region is a polygon bounded by the lines representing the constraints and the x and y axes. This region represents all possible combinations of x and y that satisfy the problem's constraints. Identifying the feasible region is a crucial step, as it narrows down the possible solutions to a manageable set.

3. Finding the Corner Points

The corner points of the feasible region are the vertices of the polygon. These points are the intersections of the lines representing the constraints. To find the coordinates of the corner points, we solve the systems of equations formed by the intersecting lines. In our problem, we need to find the intersections of the following lines:

  • 2x + y = 150 and 4x + 3y = 350
  • 2x + y = 150 and x + y = 80
  • 4x + 3y = 350 and x + y = 80
  • x = 0 and x + y = 80
  • y = 0 and 2x + y = 150

Solving these systems of equations will give us the coordinates of all the corner points of the feasible region.

4. Evaluating the Objective Function at the Corner Points

The fundamental theorem of linear programming states that the optimal solution (maximum or minimum) of a linear programming problem, if it exists, will occur at one of the corner points of the feasible region. Therefore, to find the maximum value of our objective function P = 7x + y, we need to evaluate it at each corner point of the feasible region.

We substitute the coordinates of each corner point into the objective function and calculate the corresponding value of P. The corner point that yields the highest value of P is the optimal solution that maximizes the objective function.

5. Determining the Optimal Solution

After evaluating the objective function at all corner points, we compare the values of P and identify the maximum value. The corner point corresponding to this maximum value is the optimal solution. The coordinates of this point represent the values of x and y that maximize the objective function P = 7x + y, subject to the given constraints. This final step is critical in determining the best possible outcome for the problem.

Detailed Solution for Maximize P = 7x + y

Now, let's apply the graphical method to solve our specific problem: Maximize P = 7x + y subject to the constraints 2x + y ≤ 150, 4x + 3y ≤ 350, x + y ≥ 80, and x, y ≥ 0.

1. Graphing the Constraints (Revisited)

As we discussed earlier, we graph each constraint on a coordinate plane. The lines corresponding to the inequalities are:

  • 2x + y = 150
  • 4x + 3y = 350
  • x + y = 80

And the regions satisfying the inequalities are:

  • 2x + y ≤ 150: Region below the line 2x + y = 150
  • 4x + 3y ≤ 350: Region below the line 4x + 3y = 350
  • x + y ≥ 80: Region above the line x + y = 80
  • x, y ≥ 0: First quadrant

2. Identifying the Feasible Region (Revisited)

The feasible region is the polygon formed by the intersection of these regions. By graphing the constraints, we can visually identify the feasible region. It is a quadrilateral bounded by the lines and the axes.

3. Finding the Corner Points (Detailed)

Now, we need to find the coordinates of the corner points of the feasible region. These points are the intersections of the lines:

  • Intersection of 2x + y = 150 and 4x + 3y = 350: Multiplying the first equation by 3, we get 6x + 3y = 450. Subtracting the second equation from this, we get 2x = 100, so x = 50. Substituting x = 50 into the first equation, we get 2(50) + y = 150, so y = 50. Thus, the intersection point is (50, 50).

  • Intersection of 2x + y = 150 and x + y = 80: Subtracting the second equation from the first, we get x = 70. Substituting x = 70 into the second equation, we get 70 + y = 80, so y = 10. Thus, the intersection point is (70, 10).

  • Intersection of 4x + 3y = 350 and x + y = 80: Multiplying the second equation by 3, we get 3x + 3y = 240. Subtracting this from the first equation, we get x = 110. Substituting x = 110 into the second equation, we get 110 + y = 80, so y = -30. However, this point (110, -30) is not in the feasible region since y ≥ 0. We need to find the intersection within the feasible region. Let's re-examine.

    Multiplying the equation x + y = 80 by 4, we get 4x + 4y = 320. Subtracting 4x + 3y = 350 from this, we get y = -30. Substituting y = -30 into x + y = 80 gives x = 110. This point is outside the feasible region.

    Let's solve this system again using substitution. From x + y = 80, we have y = 80 - x. Substitute this into 4x + 3y = 350: 4x + 3(80 - x) = 350 => 4x + 240 - 3x = 350 => x = 110. Then y = 80 - 110 = -30. This confirms the earlier result, and it's outside the feasible region. This indicates a potential error in the feasible region or an extraneous intersection point.

    Instead, we need to find the intersection point of 4x + 3y = 350 and the constraint y = 0. Substituting y = 0 into 4x + 3y = 350, we get 4x = 350, so x = 87.5. The point is (87.5, 0).

  • Intersection of x + y = 80 and y = 0: Substituting y = 0 into x + y = 80, we get x = 80. Thus, the intersection point is (80, 0).

  • Intersection of 2x + y = 150 and x = 0: Substituting x = 0 into 2x + y = 150, we get y = 150. The point is (0, 150).

  • Intersection of 4x + 3y = 350 and x = 0: Substituting x = 0 into 4x + 3y = 350, we get 3y = 350, so y = 350/3 ≈ 116.67. The point is (0, 350/3).

  • Intersection of x + y = 80 and x = 0: Substituting x = 0 into x + y = 80, we get y = 80. Thus, the intersection point is (0, 80).

By analyzing the graph and the feasible region, we identify the relevant corner points as (50, 50), (70, 10), (87.5, 0), and (80, 0).

4. Evaluating the Objective Function at the Corner Points (Detailed)

Now, we evaluate P = 7x + y at each corner point:

  • At (50, 50): P = 7(50) + 50 = 350 + 50 = 400
  • At (70, 10): P = 7(70) + 10 = 490 + 10 = 500
  • At (87.5, 0): P = 7(87.5) + 0 = 612.5
  • At (80, 0): P = 7(80) + 0 = 560

5. Determining the Optimal Solution (Detailed)

Comparing the values of P, we find that the maximum value is 612.5, which occurs at the corner point (87.5, 0).

Conclusion

Therefore, the maximum value of P = 7x + y subject to the given constraints is 612.5, and it occurs when x = 87.5 and y = 0. This comprehensive guide has demonstrated the step-by-step process of solving a linear programming problem using the graphical method. By understanding the concepts of objective functions, constraints, feasible regions, and corner points, you can effectively apply this powerful technique to optimize various real-world scenarios. The key to successful linear programming lies in accurate graphing and precise calculation, ensuring that the optimal solution is correctly identified.

This example highlights the importance of carefully analyzing the constraints and the feasible region to identify the correct corner points. The graphical method, while limited to problems with two variables, provides a valuable visual representation and a clear understanding of the optimization process. In more complex scenarios with multiple variables, other techniques such as the simplex method are employed to find the optimal solution. However, the fundamental principles of linear programming remain the same: defining the problem, identifying the constraints, and finding the solution that optimizes the objective function. Linear programming is a cornerstone of optimization, enabling informed decision-making in a wide array of applications.