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Chapter 5: Problem 6

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function \(\quad z=2 x+3 y\) \(\begin{aligned} \text { Constraints } & & x \geq 0, y & \geq 0 \\ & & 2 x+y & \leq 8 \\ & & & 2 x+3 y & \leq 12 \end{aligned}\)

Short Answer

Expert verified
The maximum value of the objective function \(z\) is 12, and it occurs when \(x=0\) and \(y=4\).

Step by step solution

01

Graphing inequalities

First the given inequalities should be plotted in a graph. Draw the line for \(2x + y = 8\) and shade the region where \(2x + y \leq 8\). Do the same for the line \(2x + 3y = 12\) shading the region where \(2x + 3y \leq 12\). The first quadrant of the coordinate system is the feasible region due the restrictions \(x \geq 0\) and \(y \geq 0\).
02

Identify vertices

Four corner points, or vertices, are visible: (0,0), (4,0), (0,4) and (2,2).
03

Determine the objective function value at corners

Next, the value of the objective function, \(z = 2x + 3y\), is computed at each vertex. For point (0,0), \(z = 2(0) + 3(0) = 0\). For point (4,0), \(z = 2(4) + 3(0) = 8\). For point (0,4), \(z = 2(0) + 3(4) = 12\). For point (2,2), \(z = 2(2) + 3(2) = 10\).
04

Identify the maximum value and the corresponding point

By comparing the calculated values of \(z\), we see that \(z\) reaches its maximum value, 12, at point (0,4). Hence, the objective function reaches its maximum value when \(x=0\) and \(y=4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Objective Function
The heart of a linear programming problem is the objective function, which defines the goal that you're trying to achieve. It's an equation that represents the measure of performance, like profit or cost, which you want to maximize or minimize. In the given exercise, we have the objective function

\( z = 2x + 3y \).

This formula calculates the value of the objective function 'z' for any given combination of 'x' and 'y'. Each possible pair (x, y) has a different z value, but we're looking for the combination that gives us the highest z value under the given constraints.
System of Linear Inequalities
In linear programming, constraints are expressed as a system of linear inequalities. These inequalities represent the limitations or requirements of the problem, such as resource availabilities or other restrictions. In the exercise, we are given a set of constraints:

\( x \geq 0, y \geq 0, 2x + y \leq 8, 2x + 3y \leq 12 \).

These inequalities help define the feasible region where we can search for the optimal solution to our objective function. It's essential to understand how these constraints shape the area we're working within.
Graphing Inequalities
To visually represent the problem, graphing inequalities is a critical step. This involves plotting each inequality on a coordinate plane and shading the area that satisfies the inequality. The exercise asks us to graph lines such as

\( 2x + y = 8 \)

and \( 2x + 3y = 12 \),

following which we shade the region that satisfies the corresponding inequality (\
Feasible Region
The feasible region is the overlap of the shaded areas that satisfy all the system's inequalities. This region contains all the possible solutions to the linear programming problem that adhere to the constraints. In our problem, the feasible region lies in the first quadrant of the graph because X and Y must both be non-negative. By identifying the vertices (or corner points) of this region, we isolate the candidates for the optimal solution to the objective function.
Optimization
Optimization in linear programming involves finding the maximum or minimum value of the objective function while respecting the constraints of the problem. It involves two main steps: First, evaluate the objective function at each vertex of the feasible region. Then, determine which vertex provides the best (maximum or minimum) value. From our calculations, it's clear that the maximum value of the objective function is at the vertex (0,4), making it the optimal solution for this particular problem.

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Most popular questions from this chapter

When using the addition or substitution method, how can you tell if a system of linear equations has infinitely many solutions? What is the relationship between the graphs of the two equations?

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