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Magazine Chapter 4: Problem 2
Maximize \(\quad p=x\) subject to \(\begin{aligned} x &-y \leq 4 \\\\-x &+3 y \leq 4 \\ & x \geq 0, y \geq 0 . \end{aligned}\)
Short Answer
Expert verified
The maximum value of the objective function p(x) = x subject to the given constraints is 4, which is attained at the vertex (4, 0).
Step by step solution
01
Graph the inequalities
Begin by graphing the inequalities on the xy-plane to find the feasible region. The easiest way to do this is to graph their corresponding equalities:
1. x - y = 4
2. -x + 3y = 4
Remember to additionally include the constraints x 鈮� 0 and y 鈮� 0.
From there, determine which side of each line corresponds to the inequality:
1. For x - y 鈮� 4, choose a test point not on the line, such as (0, 0). Since 0 - 0 鈮� 4 is true, shade the region that includes (0, 0).
2. For -x + 3y 鈮� 4, use (0, 0) again as the test point. Since -0 + 3(0) 鈮� 4 is also true, shade the region that includes (0, 0) for this inequality as well.
Also, as x 鈮� 0 and y 鈮� 0, stay in the first quadrant of the xy-plane.
02
Find the vertices
The feasible region is the intersection of all shaded regions from Step 1. You'll need to identify the vertices of this region:
1. (X-Intercept) When y = 0, x - 0 鈮� 4. Thus, x = 4
2. (Y-Intercept) When x = 0, -0 + 3y 鈮� 4. Thus, y = 4/3
3. (Intersection) To find the intersection point between the lines x - y = 4 and -x + 3y = 4, solve them simultaneously using substitution or elimination method. This would lead to x = 2 and y = -2.
To sum up, the vertices of the feasible region are (0, 4/3), (4, 0), and (2, -2). However, since y 鈮� 0, the third vertex is outside the feasible region.
03
Evaluate the objective function
Now, evaluate the objective function p(x) = x for each vertex of the feasible region:
1. p(0, 4/3) = 0
2. p(4, 0) = 4
04
Determine the maximum
Compare the values of p(x) computed in Step 3. The objective function attains its maximum value p(x)=4 at the vertex (4, 0). Therefore, the maximum value of p(x) subject to given constraints is 4.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Feasible Region
In linear programming, the feasible region is an essential concept representing the set of all possible points that satisfy a system of inequalities. The feasible region is found by graphing the inequalities on a coordinate plane. Begin by plotting each inequality as if it were an equation. By considering the straight lines where each inequality becomes an equality, you can determine which side of the line contains the solutions.
For instance, in the given exercise:
For instance, in the given exercise:
- The line for the inequality \(x-y \leq 4\) separates the plane into two halves. Using a test point like (0,0), if the inequality holds true at that point, you shade in the corresponding side.
- Similarly, for \(-x + 3y \leq 4\), the inequality can be tested using (0,0), shading the correct side.
- The conditions \(x \geq 0\) and \(y \geq 0\) bind the region to the first quadrant by ensuring only positive values for \(x\) and \(y\).
Objective Function
An objective function in linear programming is the function that you aim to optimize, either by maximizing or minimizing it. It's essentially a formula that represents the goal of the problem. In the exercise, the objective function is given by \(p = x\). This means the aim is to maximize the value of \(x\).
Here's how you might approach it:
Here's how you might approach it:
- After determining the feasible region, evaluate the objective function at each vertex of the feasible region.
- The vertices are crucial because, in linear programming, if a maximum or minimum value exists, it will occur at one of the vertices due to the linearity of the objective function and constraints.
- Calculate \(p\) at each vertex and compare the values to determine which gives the highest value for maximization problems.
Inequalities
Inequalities define the constraints in a linear programming problem. Each inequality represents a boundary on the coordinate plane and divides it into two parts: one that satisfies the inequality and one that does not. Inequalities are central because they form the edges of the feasible region where the objective should be evaluated.
Consider the exercise inequations:
Consider the exercise inequations:
- \(x - y \leq 4\) and \(-x + 3y \leq 4\) each limit the possible values for \(x\) and \(y\).
- \(x \geq 0\) and \(y \geq 0\) ensure that solutions remain in the positive quadrant, rejecting any solutions where either \(x\) or \(y\) would be negative.
Vertices
Vertices are the corner points where the feasible region's boundaries intersect, caused by the graphing of inequalities. In linear programming, vertices are critical because they provide the potential locations for the maximum or minimum values of the objective function.
Here's how to work with them:
Here's how to work with them:
- Identify where the lines formed by the equalities intersect. These points become vertices of the feasible region.
- Assess each vertex. In our example, calculate the values of the objective function \(p = x\) at these points.
- Due to the constraints \(x \geq 0\) and \(y \geq 0\), exclude any vertices that lie outside the feasible region, as it happened when \(y\) turned negative in one solution.
