Problem 15 - Finite Mathematics and Applied Calculus

Question

We suggest the use of technology. Round all answers to two decimal places. $$ \begin{array}{lc} \text { Maximize } & p=2.5 x+4.2 y+2 z \\ \text { subject to } & 0.1 x+y-2.2 z \leq 4.5 \\ & 2.1 x+y+z \leq 8 \\ & x+2.2 y & \leq 5 \\ & x \geq 0, y \geq 0, z \geq 0 & \end{array} $$

Step-by-Step Solution

Answer

The optimal solution will be the values of the decision variables \((x, y, z)\) at the vertex of the feasible region that maximizes the profit function, \(p = 2.5x + 4.2y + 2z\). Round all values to two decimal places. Use graphing software or calculator to plot the constraints and evaluate the profit function at the vertices.

Step 1: Graph the Constraints

Use graphing software or calculator to plot the constraint inequalities in 3D space. The feasible region is given by the area where all constraints are satisfied simultaneously. Inequalities for plotting: 1. \(0.1x + y - 2.2z \leq 4.5\) 2. \(2.1x + y + z \leq 8\) 3. \(x + 2.2y \leq 5\) 4. \(x \geq 0\), \(y \geq 0\), \(z \geq 0\)

Step 2: Identify Vertices and Feasible Region

The feasible region is a polyhedron, which is a 3D polytope. Look for the vertices of this polyhedron, where the constraints intersect each other. These vertices are possible optimal solutions.

Step 3: Evaluate Profit Function at Vertices

Now, evaluate the profit function \(p = 2.5x + 4.2y + 2z\) for all identified vertices within the feasible region.

Step 4: Find the Maximum Profit

Determine which vertex has the highest profit value (maximum). This vertex will represent the optimal solution. Round the values of \(x, y, z\) to two decimal places as required.

Step 5: Present the Optimal Solution

Report the optimal solution as the values of \(x, y, z\) that resulted in the maximum profit value.

Key Concepts

Objective Function

The objective function is at the heart of any linear programming problem. It represents the goal you are trying to achieve, often either maximizing or minimizing a particular value. In this exercise, the objective function is given by \( p = 2.5x + 4.2y + 2z \).This tells us that our goal is to maximize the value of \( p \), which is a function of three variables: \( x \), \( y \), and \( z \). Each of these variables has a coefficient, indicating how much each contributes to the total value of \( p \). Think of it as a profit equation where each term represents earnings from different sources.

Maximizing \( p \) means finding the optimum values for \( x \), \( y \), and \( z \) that yield the highest total value of \( p \). In the context of a business, this could represent maximizing profit from sales contributed by different products, with each having a different contribution to the total profit.

Constraints

Constraints are the restrictions or limits placed on the decision variables (here, \( x \), \( y \), and \( z \)). They define the conditions that must be met in any potential solution. In our problem, the constraints are expressed through inequalities:

• \( 0.1x + y - 2.2z \leq 4.5 \)
• \( 2.1x + y + z \leq 8 \)
• \( x + 2.2y \leq 5 \)
• \( x \geq 0 \), \( y \geq 0 \), \( z \geq 0 \)

Each constraint represents a limitation that could be due to resource availability, budget limits, or other practical considerations in a real-world scenario. The non-negativity constraints \( x \geq 0 \), \( y \geq 0 \), and \( z \geq 0 \) suggest that negative values of these variables are not feasible, which makes sense since negative quantities would not be practical in real-world situations.

Understanding and plotting these constraints is crucial as they help shape the feasible region where all solutions must lie.

Feasible Region

The feasible region is a geometric space where all the constraints are satisfied simultaneously. In linear programming, it is the intersection of all the half-spaces defined by each inequality. In our 3D problem, the feasible region is a polyhedron—a 3D shape that is bounded by flat surfaces representing the constraints.

This region contains all the potential solutions to our optimization problem. However, not every point in this region will satisfy the objective of maximizing \( p \). Only certain points within the feasible region, specifically at the vertices of the polyhedron, need to be evaluated to find the optimal solution.
Finding the feasible region involves graphing each of the constraints and determining where they overlap. This overlapping portion—the feasible region—defines all possible combinations of values that \( x \), \( y \), and \( z \) can take while still meeting every constraint.

Vertex Evaluation

Vertex evaluation is a critical step in solving a linear programming problem because the optimal solution (maximum or minimum value of the objective function) often lies at one of the vertices of the feasible region.

In our problem, after determining the feasible region as a polyhedron, the next task is evaluating the objective function at each vertex of this polyhedron. Each vertex is a potential candidate for the optimal solution because it represents a combination of values for \( x \), \( y \), and \( z \) that satisfies all the constraints.

To perform vertex evaluation, we substitute the coordinates of each vertex into the objective function \( p = 2.5x + 4.2y + 2z \) and calculate the values.
By comparing these values, we can identify which vertex yields the highest value of \( p \), helping us determine which combination of \( x \), \( y \), and \( z \) is optimal for our objective.