Vertices Of The Feasible Region A Step By Step Guide

In the realm of mathematical optimization, we often encounter scenarios where we aim to maximize or minimize an objective function, subject to a set of constraints. These constraints, typically expressed as inequalities, define a feasible region within which the optimal solution must lie. Understanding the boundaries and corners of this feasible region is crucial for identifying potential solutions. This article delves into the process of determining the vertices of a feasible region, using a specific example as our guide.

Defining the Feasible Region

At the heart of optimization problems lie constraints, which act as limitations or restrictions on the variables involved. These constraints can represent a variety of real-world scenarios, such as resource limitations, production capacities, or contractual obligations. Mathematically, constraints are often expressed as inequalities, defining a range of values that the variables can take. The feasible region is the area where all these constraints are simultaneously satisfied. It's like a playground where our solution can roam freely, but only within the defined boundaries.

Understanding the Constraints

Let's consider a specific set of constraints:

$ \begin{array}{l} 4 x+3 y \leq 12 \ 2 x+6 y \leq 15 \ x \geq 0 \ y \geq 0 \end{array} $

These constraints define a feasible region in the xy-plane. Each inequality represents a boundary line, and the feasible region lies on one side of the line. The inequalities $x \geq 0$ and $y \geq 0$ restrict the feasible region to the first quadrant, where both x and y are non-negative. This is a common constraint in real-world problems, as quantities like production levels or resource allocation cannot be negative.

The inequalities $4x + 3y \leq 12$ and $2x + 6y \leq 15$ represent linear constraints, which define straight lines as boundaries. To visualize these constraints, we can rewrite them as equations:

$4x + 3y = 12$ and $2x + 6y = 15$

These equations represent the boundary lines of the feasible region. The inequality signs ($\leq$) indicate that the feasible region lies on the side of the line where the inequality holds true. For example, for the constraint $4x + 3y \leq 12$, we can test a point, such as (0, 0), to see if it satisfies the inequality. Plugging in x = 0 and y = 0, we get:

$4(0) + 3(0) \leq 12$

$0 \leq 12$

Since this is true, the feasible region for this constraint lies on the same side of the line as the origin (0, 0).

Graphing the Feasible Region

To fully visualize the feasible region, it's helpful to graph the boundary lines and identify the area that satisfies all the constraints. Let's start by graphing the line $4x + 3y = 12$. We can find two points on this line by setting x = 0 and y = 0:

• When x = 0, $3y = 12$, so y = 4. This gives us the point (0, 4).
• When y = 0, $4x = 12$, so x = 3. This gives us the point (3, 0).

Plotting these points and drawing a line through them gives us the boundary line for the first constraint. Similarly, we can graph the line $2x + 6y = 15$:

• When x = 0, $6y = 15$, so y = 2.5. This gives us the point (0, 2.5).
• When y = 0, $2x = 15$, so x = 7.5. This gives us the point (7.5, 0).

Plotting these points and drawing a line through them gives us the boundary line for the second constraint. Remember, since we have inequalities ($\leq$), we draw solid lines to indicate that the points on the line are included in the feasible region.

Now, we need to shade the area that satisfies both inequalities. As we determined earlier, the feasible region for $4x + 3y \leq 12$ lies on the same side of the line as the origin. Similarly, the feasible region for $2x + 6y \leq 15$ also lies on the same side of the line as the origin. The constraints $x \geq 0$ and $y \geq 0$ restrict the feasible region to the first quadrant. The feasible region is the intersection of all these areas, forming a polygon in the first quadrant.

Identifying the Vertices

The vertices of the feasible region are the corner points where the boundary lines intersect. These points are crucial because the optimal solution to a linear programming problem (maximizing or minimizing an objective function) will always occur at one of the vertices of the feasible region. This is a fundamental concept in linear programming, and it simplifies the process of finding the optimal solution. Instead of searching the entire feasible region, we only need to evaluate the objective function at the vertices.

Finding the Intersection Points

To find the vertices, we need to determine the points where the boundary lines intersect. We can do this by solving the equations of the lines simultaneously. In our example, we have four boundary lines:

1. $4x + 3y = 12$
2. $2x + 6y = 15$
3. $x = 0$
4. $y = 0$

We need to find the intersection points of these lines, taken two at a time. Let's start with the intersections with the axes:

• Intersection of line 1 ($4x + 3y = 12$) and the y-axis ($x = 0$): Substituting $x = 0$ into the equation $4x + 3y = 12$, we get $3y = 12$, so $y = 4$. This gives us the vertex (0, 4).
• Intersection of line 1 ($4x + 3y = 12$) and the x-axis ($y = 0$): Substituting $y = 0$ into the equation $4x + 3y = 12$, we get $4x = 12$, so $x = 3$. This gives us the vertex (3, 0).
• Intersection of line 2 ($2x + 6y = 15$) and the y-axis ($x = 0$): Substituting $x = 0$ into the equation $2x + 6y = 15$, we get $6y = 15$, so $y = 2.5$. This gives us the vertex (0, 2.5).
• Intersection of line 2 ($2x + 6y = 15$) and the x-axis ($y = 0$): Substituting $y = 0$ into the equation $2x + 6y = 15$, we get $2x = 15$, so $x = 7.5$. This gives us the vertex (7.5, 0).
• Intersection of the x-axis ($y = 0$) and the y-axis ($x = 0$): This is simply the origin (0, 0).

Now, we need to find the intersection of the two linear constraints:

• Intersection of line 1 ($4x + 3y = 12$) and line 2 ($2x + 6y = 15$): We can solve this system of equations using substitution or elimination. Let's use elimination. Multiply the first equation by -2 and the second equation by 4:

  $-8x - 6y = -24$ $8x + 24y = 60$

  Adding the two equations, we get:

  $18y = 36$

  So, $y = 2$. Substituting this value back into the first equation ($4x + 3y = 12$):

  $4x + 3(2) = 12$ $4x + 6 = 12$ $4x = 6$ $x = 1.5$

  This gives us the vertex (1.5, 2).

Verifying the Vertices

We have found five potential vertices: (0, 4), (3, 0), (0, 2.5), (7.5, 0), (0, 0), and (1.5, 2). However, not all of these points may be vertices of the feasible region. We need to check if each point satisfies all the constraints.

• (0, 4):
  • $4(0) + 3(4) = 12 \leq 12$ (True)
  • $2(0) + 6(4) = 24 \leq 15$ (False) This point does not satisfy the second constraint, so it's not a vertex.
• (3, 0):
  • $4(3) + 3(0) = 12 \leq 12$ (True)
  • $2(3) + 6(0) = 6 \leq 15$ (True)
  • $3 \geq 0$ (True)
  • $0 \geq 0$ (True) This point satisfies all the constraints, so it's a vertex.
• (0, 2.5):
  • $4(0) + 3(2.5) = 7.5 \leq 12$ (True)
  • $2(0) + 6(2.5) = 15 \leq 15$ (True)
  • $0 \geq 0$ (True)
  • $2.5 \geq 0$ (True) This point satisfies all the constraints, so it's a vertex.
• (7.5, 0):
  • $4(7.5) + 3(0) = 30 \leq 12$ (False) This point does not satisfy the first constraint, so it's not a vertex.
• (0, 0):
  • $4(0) + 3(0) = 0 \leq 12$ (True)
  • $2(0) + 6(0) = 0 \leq 15$ (True)
  • $0 \geq 0$ (True)
  • $0 \geq 0$ (True) This point satisfies all the constraints, so it's a vertex.
• (1.5, 2):
  • $4(1.5) + 3(2) = 6 + 6 = 12 \leq 12$ (True)
  • $2(1.5) + 6(2) = 3 + 12 = 15 \leq 15$ (True)
  • $1.5 \geq 0$ (True)
  • $2 \geq 0$ (True) This point satisfies all the constraints, so it's a vertex.

Therefore, the vertices of the feasible region are (0, 0), (3, 0), (0, 2.5), and (1.5, 2). These are the corner points of the polygon that represents the feasible region. Any optimal solution to a linear programming problem with these constraints will occur at one of these vertices.

Significance of Vertices in Optimization

The vertices of the feasible region hold paramount importance in optimization problems, particularly in the realm of linear programming. Linear programming deals with optimizing a linear objective function (maximizing profit or minimizing cost, for example) subject to linear constraints. The fundamental theorem of linear programming states that if an optimal solution exists, it will always occur at a vertex of the feasible region. This theorem dramatically simplifies the process of finding optimal solutions.

Instead of searching the entire feasible region, which could be an infinite set of points, we only need to evaluate the objective function at the vertices. This reduces the problem to a finite set of calculations. By comparing the values of the objective function at each vertex, we can easily identify the optimal solution (or solutions, in case of multiple optima).

This principle is the cornerstone of various optimization algorithms, such as the simplex method, which efficiently explores the vertices of the feasible region to find the optimal solution. Understanding the vertices and their role in optimization is crucial for anyone working with linear programming and related fields. The ability to identify and analyze the vertices of a feasible region is a powerful tool for decision-making and problem-solving in a wide range of applications.

Conclusion

Determining the vertices of the feasible region is a fundamental step in solving optimization problems with constraints. By understanding the constraints, graphing the feasible region, and finding the intersection points of the boundary lines, we can identify the vertices. These vertices represent the corner points of the feasible region and play a crucial role in finding optimal solutions. The vertices are the key to unlocking the optimal solution in linear programming problems. The feasible region represents the set of possible solutions that satisfy all constraints, and the vertices are the extreme points of this region. By evaluating the objective function at these vertices, we can efficiently determine the optimal solution that maximizes or minimizes the objective within the given constraints. This process is not only essential for solving mathematical problems but also has practical applications in various fields, including economics, engineering, and operations research.

In conclusion, the ability to identify the vertices of a feasible region is a powerful tool in mathematical optimization. It allows us to narrow down the search for optimal solutions to a finite set of points, making the problem much more manageable. This concept is fundamental to linear programming and has widespread applications in various fields. By mastering the techniques of graphing constraints, finding intersection points, and verifying vertices, we can effectively solve a wide range of optimization problems and make informed decisions in complex situations.