Problem 14 - Finite Mathematics and Applied Calculus

Question

Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Maximize \(\quad \begin{aligned} \quad p=3 x+2 y & \\ \text { subject to } & 0.1 x+0.1 y \geq 0.2 \\ y & \quad x \geq 0, y \geq 0 . \end{aligned}\)

Step-by-Step Solution

Answer

The feasible region is represented by the intersection of the shaded areas in the first quadrant above the line x+y=2. There is no optimal solution for this LP problem since the feasible region is unbounded, and the objective function p = 3x + 2y will keep increasing indefinitely as x and y increase within the feasible region.

Step 1: Graph the constraints

Plot the inequality constraints on the coordinate axis. The inequality 0.1x+0.1y ≥ 0.2 can be rewritten as x+y ≥ 2. Therefore, we will plot the line x+y=2 and shade the area above the line to represent the constraint. Also, the constraint y indicates that we should shade the whole positive y-axis. Lastly, x≥0, y≥0 indicates that we should only consider the first quadrant of the coordinate plane where x and y are non-negative. Here is the graph of the constraints: https://www.desmos.com/calculator/8rok2fwebl

Step 2: Find the feasible region

The feasible region is the area where all the constraints are satisfied simultaneously. In the graph, it is represented by the intersection of the shaded areas. In this case, the feasible region is the entire first quadrant of the coordinate plane above the line x+y=2.

Step 3: Check for optimal solution

In LP problems, any maximum or minimum value (if it exists) always occurs at the vertices, or corner points, of the feasible region. In our problem, the feasible region is unbounded, which means that there is no finite maximum value for the objective function. The reason behind this is because, as x and y increase while still remaining in the feasible region, the function p=3x+2y will keep increasing indefinitely.

Step 4: Conclude the result

Since the feasible region is not empty, and the objective function is unbounded in the feasible region, we can conclude that there is no optimal solution for this LP problem.

Key Concepts

Feasible Region

In linear programming, the feasible region is the set of all possible solutions that satisfy all the given constraints. It is the area on a graph where all constraints overlap and hold true. In our particular problem, this region is defined by the constraints of non-negativity of both variables and the inequality, which together form a quadrant on the graph.

To understand how it looks on a graph, consider that each line represents a constraint. The intersection of these lines, usually shaded, illustrates where solutions can potentially lie. In simple terms, any point within this region is a potential solution. The feasible region is crucial because any viable solution to the linear programming problem must fall within this area.

• It is bounded by inequalities.
• Any point inside it is a feasible solution.
• Where it is unbounded, solutions can increase indefinitely within constraints.

In the problem above, the feasible region is all points above the line \(x + y = 2\), within the first quadrant. This region is quite large, suggesting more space for possible solutions.

Optimization

Optimization in linear programming involves finding the maximum or minimum value of an objective function. Our goal in this exercise is to maximize the function \(p = 3x + 2y\). The objective is to identify values for \(x\) and \(y\) that bring the greatest possible output of \(p\), adhering to all problem constraints.

Normally, solutions are located at the vertices of the feasible region. These are the points where constraint lines meet. In bounded regions, it's straightforward to find the optimal point by evaluating the objective function at vertices. But, if the region is unbounded, as in our case, optimization becomes more complex.

• Solutions are often found at the boundary's corners.
• Optimization means adjusting values to achieve maximum output.
• Sometimes, as with unbounded regions, there is no finite optimum.

Here, since the feasible region is unbounded, the function can continually increase as \(x\) and \(y\) grow, meaning there's no singular optimal solution for maximization.

Inequality Constraints

Inequality constraints define limitations or boundaries on the variables in a linear programming problem. They shape the feasible region and guide potential solutions.

In our exercise, the key inequality is \(0.1x + 0.1y \geq 0.2\), which simplifies to \(x + y \geq 2\). This inequality forms a boundary line on the graph, with feasible solutions lying on or above this line.

• Inequalities restrict the variable values.
• The shaded area on the graph goes above or below the line, indicating feasible solutions.
• Non-negativity ((\(x \geq 0\), \(y \geq 0\))) restricts solutions to the first quadrant.

Understanding these constraints helps in identifying where feasible solutions can exist, as well as the shape and limitations of the feasible region. In our case, these constraints ensure solutions lie in a positive realm, informing the optimization and decision-making processes.