Problem 15 - Precalculus with Limits

Question

Solving a Linear Programming Problem, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints. $$ \begin{array}{c}{\text { Objective function: }} \\ {z=4 x+5 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+y \geq 8} \\ {3 x+5 y \geq 30}\end{array} $$

Step-by-Step Solution

Answer

The maximum value of the objective function occurs at the corner points that make \(z = 4x + 5y\) as large as possible, and the minimum value will occur at the corner points that make \(z = 4x + 5y\) as small as possible. The exact points will depend on the graph of the feasible region.

Step 1: Plot the inequalities

We'll start by plotting the constraints on a graph: \(x \geq 0\), \(y \geq 0\), \(x+y \geq 8\), and \(3x + 5y \geq 30\). This will give us the feasible region.

Step 2: Identify the corner points of the feasible region

Next, we have to identify the corner points of the feasible region. These will be the points of intersection of the boundary lines of the feasible region.

Step 3: Evaluate the objective function

We then evaluate the objective function \(z = 4x + 5y\) at each corner point to find the maximum and minimum values.

Step 4: Determine where the maximum and minimum values occur

Finally, based on the results, we can determine where the maximum and minimum values of the objective function occur.

Key Concepts

Objective Function

In linear programming, the objective function is the mathematical expression that defines what you want to maximize or minimize. Think of it as the goal of your problem-solving. In this exercise, the objective function is given as \( z = 4x + 5y \). This function tells us that for every increment in \( x \), the value of \( z \) increases by 4, and for every increment in \( y \), \( z \) increases by 5.
To understand its significance, you must evaluate how different combinations of \( x \) and \( y \) affect \( z \).

Some points to remember about an objective function in linear programming:

• Objective functions are always linear.
• The coefficients show how much each variable contributes to the total outcome.
• Our aim is to find the set of \( x \) and \( y \) that maximizes or minimizes \( z \).

Hence, by substituting corner points from the feasible region into the objective function, one can identify the points that yield maximum and minimum values.

Feasible Region

The feasible region is a critical concept in linear programming, representing the set of all possible points that satisfy the given constraints. Imagine it as the playground where the solutions can "play" and where the best solution lies.
In this exercise, we define the feasible region by plotting the constraints: \( x \geq 0 \), \( y \geq 0 \), \( x+y \geq 8 \), and \( 3x + 5y \geq 30 \).
The overlap between these constraints on a graph shows our feasible region.

Key characteristics of the feasible region include:

• It is typically a polygon, often in the form of a triangle or quadrilateral.
• All solutions must lie within this boundary to be valid.
• It is defined by the intersection of all the inequalities.

The corners or vertices of this polygon are essential because they are potential solutions which may yield the desired maximum or minimum values of the objective function.

Constraints

Constraints in linear programming act as the rules or limitations within which the objective function must be optimized. They restrict the values that the variables can take. For this problem, the constraints are:

• \( x \geq 0 \): \( x \) cannot be negative, ensuring the solution remains in the first quadrant.
• \( y \geq 0 \): \( y \) must also be non-negative.
• \( x+y \geq 8 \): Ensures that the total of \( x \) and \( y \) exceeds or equals 8.
• \( 3x + 5y \geq 30 \): Adds another dimension to the limitation, creating an additional boundary.

These constraints must all be satisfied concurrently. They define the scope of allowable solutions, ensuring that any solution to the objective function falls within the feasible region. Constraints transform a solution space that could be infinitely large into a manageable set of possibilities.