Finding Vertices Of A Feasible Region A Step-by-Step Guide

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In the realm of mathematical optimization, identifying the feasible region and its vertices is a fundamental step in solving linear programming problems. The feasible region represents the set of all possible solutions that satisfy a given set of constraints. The vertices of this region, also known as corner points, are the points where the boundary lines of the constraints intersect. These vertices are crucial because they represent the potential optimal solutions to the problem. This article delves into the process of determining the vertices of a feasible region, using a specific example to illustrate the methodology. We will explore the underlying concepts, step-by-step procedures, and practical applications of this essential technique in optimization.

Understanding the Feasible Region

Before diving into the specifics of vertex identification, it's important to grasp the concept of a feasible region. In simple terms, the feasible region is the area on a graph that satisfies all the constraints of a problem simultaneously. These constraints are typically expressed as linear inequalities, which define lines that bound the region. The feasible region can be a polygon, a line segment, a point, or even an unbounded area. The shape and size of the feasible region are determined by the nature and combination of the constraints. Understanding the feasible region is critical because it narrows down the search for the optimal solution to only those points within this region.

To visualize the feasible region, we typically graph each constraint as a line on a coordinate plane. For inequalities involving "less than or equal to" (≤) or "greater than or equal to" (≥), the region lies on one side of the line, while for strict inequalities (< or >), the region does not include the line itself. The intersection of all these regions forms the feasible region. Identifying the feasible region often involves shading the areas that satisfy each constraint and then finding the area where all shaded regions overlap.

Key characteristics of a feasible region include:

  • It is defined by a set of linear constraints.
  • It represents all possible solutions to the problem.
  • It can be bounded or unbounded.
  • Its vertices are potential optimal solutions.

The Significance of Vertices

Vertices, also known as corner points, hold a special significance in the context of linear programming. The Fundamental Theorem of Linear Programming states that if a linear objective function has an optimal solution within a bounded feasible region, that solution must occur at one of the vertices. This theorem dramatically simplifies the optimization process because it means we only need to evaluate the objective function at the vertices rather than at every point within the feasible region.

Geometrically, vertices are the points where the boundary lines of the constraints intersect. They represent the extreme points of the feasible region, and it's at these extremes where the objective function is most likely to reach its maximum or minimum value. The vertices are determined by solving systems of linear equations formed by the intersecting constraint lines. Each vertex corresponds to a solution that simultaneously satisfies the equations of the intersecting lines.

Identifying vertices is crucial for:

  • Finding optimal solutions in linear programming problems.
  • Reducing the search space for optimization.
  • Understanding the boundaries of the feasible region.
  • Providing a clear picture of the solution space.

Step-by-Step Guide to Identifying Vertices

To effectively identify the vertices of a feasible region, a systematic approach is required. Here's a step-by-step guide to help you navigate the process:

Step 1: Graphing the Constraints

The first step is to graph each constraint as a line on a coordinate plane. This involves converting the inequalities into equations and plotting the lines. For example, the constraint x + y ≤ 7 can be graphed by first plotting the line x + y = 7. To do this, find two points on the line (e.g., (0, 7) and (7, 0)) and draw a line through them. The same process is repeated for each constraint.

When graphing inequalities, it's important to determine which side of the line represents the solution set. For inequalities involving ≤ or ≥, the line is included in the solution set and is drawn as a solid line. For strict inequalities (< or >), the line is not included and is drawn as a dashed line. To determine which side of the line to shade, you can test a point (e.g., (0, 0)) in the original inequality. If the point satisfies the inequality, shade the side of the line that contains the point; otherwise, shade the opposite side.

Step 2: Identifying the Feasible Region

After graphing all the constraints, the feasible region is the area where all the shaded regions overlap. This is the region that satisfies all the constraints simultaneously. The feasible region can be bounded, meaning it is enclosed on all sides, or unbounded, meaning it extends infinitely in one or more directions. Accurately identifying the feasible region is crucial as it forms the basis for finding the vertices.

Step 3: Finding the Intersection Points

The vertices of the feasible region are the points where the constraint lines intersect. To find these points, solve the systems of linear equations formed by the intersecting lines. Each pair of intersecting lines represents a system of two equations with two variables, which can be solved using various methods, such as substitution, elimination, or matrix operations. The solutions to these systems of equations are the coordinates of the vertices.

Step 4: Verifying the Vertices

Once you've found the intersection points, it's important to verify that they are indeed vertices of the feasible region. This involves checking whether each point satisfies all the constraints. If a point does not satisfy all the constraints, it is not a vertex of the feasible region and should be discarded. This verification step ensures that you only consider valid vertices for further analysis.

Applying the Guide to the Example Problem

Let's apply the step-by-step guide to the given problem:

Constraints:

  • x + y ≤ 7
  • x - 2y ≤ -2
  • x ≥ 0
  • y ≥ 0

Step 1: Graphing the Constraints

  1. Graph the line x + y = 7. Points (0, 7) and (7, 0) lie on this line. Since the inequality is x + y ≤ 7, shade the region below the line.
  2. Graph the line x - 2y = -2. Points (0, 1) and (2, 2) lie on this line. Since the inequality is x - 2y ≤ -2, shade the region above the line.
  3. The constraint x ≥ 0 represents the region to the right of the y-axis.
  4. The constraint y ≥ 0 represents the region above the x-axis.

Step 2: Identifying the Feasible Region

The feasible region is the area where all shaded regions overlap. This region is a quadrilateral bounded by the x-axis, y-axis, and the two constraint lines.

Step 3: Finding the Intersection Points

  1. Intersection of x + y = 7 and x - 2y = -2:
    • Solve the system of equations:
      • x + y = 7
      • x - 2y = -2
    • Subtract the second equation from the first:
      • 3y = 9
      • y = 3
    • Substitute y = 3 into x + y = 7:
      • x + 3 = 7
      • x = 4
    • The intersection point is (4, 3).
  2. Intersection of x + y = 7 and y = 0 (x-axis):
    • Substitute y = 0 into x + y = 7:
      • x + 0 = 7
      • x = 7
    • The intersection point is (7, 0).
  3. Intersection of x - 2y = -2 and x = 0 (y-axis):
    • Substitute x = 0 into x - 2y = -2:
      • 0 - 2y = -2
      • y = 1
    • The intersection point is (0, 1).
  4. Intersection of x = 0 (y-axis) and y = 0 (x-axis):
    • The intersection point is (0, 0).

Step 4: Verifying the Vertices

  1. (4, 3):
    • 4 + 3 ≤ 7 (True)
    • 4 - 2(3) ≤ -2 (True)
    • 4 ≥ 0 (True)
    • 3 ≥ 0 (True)
  2. (7, 0):
    • 7 + 0 ≤ 7 (True)
    • 7 - 2(0) ≤ -2 (False)
    • 7 ≥ 0 (True)
    • 0 ≥ 0 (True) Point (7,0) does not satisfy the second inequality.
  3. (0, 1):
    • 0 + 1 ≤ 7 (True)
    • 0 - 2(1) ≤ -2 (True)
    • 0 ≥ 0 (True)
    • 1 ≥ 0 (True)
  4. (0, 0):
    • 0 + 0 ≤ 7 (True)
    • 0 - 2(0) ≤ -2 (False)
    • 0 ≥ 0 (True)
    • 0 ≥ 0 (True) Point (0,0) does not satisfy the second inequality.

Based on the calculations and verification, vertices of the feasible region are (0, 1) and (4, 3).

Analyzing the Answer Choices

Given the answer choices:

A. (0,0), (0,1), (4,3), (7,0) B. (0,1), (4,3), (7,0)

From our calculations, we found two valid vertices are (0, 1) and (4, 3). But both choices also contain points (0,0) and (7,0). We already tested that (7,0) and (0,0) are not a solution since they do not satisfy all constraints. Based on our verification, (7, 0) does not satisfy the constraint x - 2y ≤ -2, and (0,0) does not satisfy the constraint x - 2y ≤ -2.

Conclusion

In conclusion, identifying the vertices of a feasible region is a crucial step in solving linear programming problems. By graphing the constraints, identifying the feasible region, finding the intersection points, and verifying the vertices, we can effectively determine the potential optimal solutions. In the given example, we systematically identified the vertices of the feasible region defined by the constraints. This process highlights the importance of understanding the underlying concepts and applying a structured approach to optimization problems. Mastering the technique of vertex identification not only enhances problem-solving skills but also provides a solid foundation for more advanced topics in mathematical optimization.

Therefore, the correct vertices of the feasible region, based on our step-by-step analysis and verification, are (0, 1) and (4, 3).