Problem 26 - Precalculus with Limits

Question

Finding Minimum and Maximum Values, find the minimum and maximum values of the objective function and where they occur, subject to the constraints \(x \geq 0, y \geq 0, x+4 y \leq 20\) \(x+y \leq 18,\) and \(2 x+2 y \leq 21 .\) $$ z=2 x+4 y $$

Step-by-Step Solution

Answer

The maximum and minimum values of the function \(z\) occur at the vertices of the feasible region plotted from the constraints. These points are found by substituting the vertices into the objective function, and the largest and smallest values are the maximum and minimum respectively.

Step 1: Graph the inequalities

Graph all the constraints on a graph. This will form a feasible region on the graph. The points within this region satisfy all the constraints.

Step 2: Identify the vertices

Once the feasible region is plotted, identify the corner points or vertices of this region. These points would be the intersections of the lines drawn.

Step 3: Substitute vertices into the objective function

Substitute the coordinates of these vertices into the function \(z = 2x + 4y\). This will give the value of the objective function at these points.

Step 4: Find the maximum and minimum values

The maximum and minimum values of the function \(z\) are the largest and smallest of the values obtained in step 3 respectively. The points at which these occur are the solution of the problem.

Key Concepts

Objective Function

In linear programming, the objective function is crucial as it represents the mathematical expression we are trying to optimize. In this specific exercise, the objective function is given by the equation \( z = 2x + 4y \).
This equation quantifies the criterion of optimization, which can either be maximization or minimization. Here are some key points about the objective function:

• It is a linear equation, meaning it comprises variables that are raised to the power of one.
• The coefficients (in this case, 2 and 4) represent how much each unit of the variable contributes to the overall objective.

To find the optimal solution, you need to evaluate this function at specific points, which could represent potential solutions.

Constraints

Constraints are essential in determining the feasible solutions by defining the limitations under which the objective function must operate.
In the exercise, the constraints are represented by the inequalities:

• \( x \geq 0 \)
• \( y \geq 0 \)
• \( x + 4y \leq 20 \)
• \( x + y \leq 18 \)
• \( 2x + 2y \leq 21 \)

These inequalities form the boundaries of the feasible region and depict possible conditions the solution must satisfy.Constraints can originate from various requirements, such as resource availability, time limitations, or capacity restrictions. They work alongside the objective function to ensure that the solution is not only optimal but also practical.

Feasible Region

The feasible region is the set of all possible points that satisfy the constraints laid out in a linear programming problem.
This region forms a polygon on a graph, bounded by the lines of the inequalities. Each point within the feasible region represents a viable solution. Interesting details about the feasible region:

• It is generally a convex shape, meaning that if you pick any two points within it, the line segment joining them will lie entirely within the region.
• The corners, or vertices, of the feasible region are especially significant as potential optimum solutions often reside at these points.

Identifying this region involves plotting each constraint on the graph and shading the area where all constraints overlap. This guides the search for the optimal solution within known boundaries.

Graphical Method

The graphical method is a visual approach used to solve a linear programming problem with two variables. This method is particularly handy for understanding how constraints shape the feasible region and affect the objective function. Steps in the graphical method include:

• Plot each constraint as a line on a graph and determine the feasible region.
• Identify the vertices of the feasible region, as these are potential points where the objective function could attain its optimal value.
• Evaluate the objective function at each vertex to find out which provides the maximum or minimum value.

The graphical method is not only straightforward but also an excellent way to visualize linear programming problems, helping enhance comprehension and solution accuracy. While it's typically limited to two-variable problems, it lays the groundwork for understanding more complex problem-solving techniques.