Question

The given linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the maximum value of the objective function and where it occurs. Objective function: \(z=2.5 x+y\) Constraints: \(\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x+5 y & \leq 15 \\ 5 x+2 y & \leq 10 \end{aligned}\)

Short answer

The maximum value of the objective function \(z=2.5x+y\) is 5 and it occurs at the point (2,0). The unusual characteristic of this problem is the maximum value occurs at a point that touches only two of the constraints.

Step-by-step solution

Graph the Constraints

Plot the constraints on a graph. Here, the constraints form a quadrilateral shape in the first quadrant because all constraints are in the form of 'greater than or equal to':\(1. x \geq 0\) \\(2. y \geq 0\) \\(3. 3x + 5y \leq 15\) transitions into \(y \leq -0.6x + 3\)\\(4. 5x + 2y \leq 10\) transitions into \(y \leq -2.5x + 5\)

Find the Vertices of the Solution Region

The vertices of the solution region are the intersection points of the constraints. Here, we have four vertices at the points: \( (0,0), (0, 3), (2,0) and (1,2) )

Substitute Vertices in the Objective Function and find the Maximum Value

Substitute every set of vertices coordinates into the objective function \(z=2.5x+y\) and find the maximum z-value:For \( (0,0) , z = 2.5(0) + 0 = 0\)\For \( (0,3) , z = 2.5(0) + 3 = 3\) \For \( (2,0) , z = 2.5(2) + 0 = 5\) \For \( (1,2) , z = 2.5(1) + 2 = 4.5\)Hence, the maximum value of the objective function is 5 and it occurs at the point (2,0).

Identify the Unusual Characteristic

This problem displays an unusual characteristic of having the maximum value of the objective function at a point that touches only two of the constraints. Typically, in linear programming problems, the maximum or minimum value often touches three or more constraints.

Key concepts

Objective Function

The objective function in linear programming is essentially the heart of the problem. It represents the goal that needs to be optimized, whether that is maximizing profits, minimizing costs, or achieving the best possible outcome of any quantitative measure. In our exercise, the objective function is defined as \(z=2.5x+y\).

When you're tasked with finding the maximum or minimum value of this function, it involves substituting the coordinates of the vertices of the solution region into the objective function to see which vertex yields the best value. Understanding the objective function's role is crucial because it's the driving force behind decision-making in linear programming scenarios.

Constraints in Linear Programming

Constraints are the limitations or requirements that must be met in a linear programming problem. They define the feasible region within which an optimal solution must lie. In the given exercise, the constraints include non-negativity restrictions \(x \geq 0\) and \(y \geq 0\), as well as limitations based on resource availability or other restrictions \(3x+5y \leq 15\) and \(5x+2y \leq 10\).

These inequalities ensure that any solution not only optimizes the objective function but also adheres to these real-world limits. When solving linear programming problems, not violating these constraints is as important as optimizing the objective function.

Graphing Linear Inequalities

Graphing linear inequalities is a visual approach to identifying the feasible region where an optimal solution can be found. In this exercise, the inequalities are graphed on a coordinate system, creating boundaries beyond which solutions are not viable. For example, the constraint \(3x+5y \leq 15\) would be expressed as a line on the graph with all the possible solutions falling on or below this line.

Graphing each inequality helps in visualizing the region where all constraints overlap – this is where the possible solutions to our linear programming problem reside. It’s a straightforward yet powerful step in solving these problems, as it transitions abstract concepts into tangible visual elements.

Solution Region

The solution region or feasible region is the area on the graph where all the constraints of a linear programming problem are satisfied simultaneously. This is where the acceptable solutions that adhere to the limitations imposed by the problem live. In our problem, the solution region is a quadrilateral in the first quadrant, created by the intersection of the constraints.

Any point within this region is a viable candidate for the optimal solution. Identifying this region correctly is crucial since the objective function's optimal value must come from this specified area. When plotting the region, accuracy is key to ensure all possible solutions are considered.

Vertices of the Solution Region

The vertices of the solution region are the corner points where the linear inequalities intersect. It's a critical aspect of linear programming because, due to the nature of linear equations, the optimal solution will always occur at one of these vertices. In the given exercise, the vertices are at the coordinates \((0,0)\), \((0, 3)\), \((2,0)\) and \((1,2)\).

By testing these points in the objective function, we can identify which vertex provides the maximum or minimum value required for the problem. This method, known as 'corner-point' method, significantly reduces the complexity of the problem because instead of considering an infinite number of points within the solution region, we only need to compare a finite set of vertices.