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Chapter 4: Problem 8

Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Maximize \(\quad \begin{array}{ll}\text { subject to } & p=x+2 y \\ & 30 x+20 y \leq 600 \\ & 0.1 x+0.4 y \leq 4 \\ & 0.2 x+0.3 y \leq 4.5 \\ & x \geq 0, y \geq 0\end{array}\)

Short Answer

Expert verified
The short answer is: The feasible region is given by the overlapping shaded areas found after graphing the constraint inequalities. The vertices of the feasible region are found by solving the intersection of the line equations or their intersections with the axes. The vertices are: (0, 0), (0, 15), (6, 12), (20, 0), and (10, 0). To find the optimal solution, evaluate the objective function, \(p=x+2y\), at each vertex: (0, 0): \(p=0\) (0, 15): \(p=30\) (6, 12): \(p=30\) (10, 0): \(p=10\) (20, 0): \(p=20\) The optimal solution occurs at vertices (0, 15) and (6, 12), with an optimal value of \(p=30\).

Step by step solution

01

Graph the feasible region

First, plot the constraint inequalities on a graph to find the feasible region. Rewrite the inequalities as equalities to obtain their line equations and plot the lines. Then, determine the valid region based on the inequality constraints: 1. \(30x + 20y \leq 600\). During the x-intercept, y=0, \(x=\frac{600}{30}\), therefore x-intercept is 20. For the y-intercept, x=0, \(y=\frac{600}{20}\), therefore y-intercept is 30. Plot the line \(30x+20y=600\) and shade the region below the line. 2. \(0.1x+0.4y\leq4\). At the x-intercept, y=0, \(x=\frac{4}{0.1}\), therefore x-intercept is 40. For the y-intercept, x=0, \(y=\frac{4}{0.4}\), therefore y-intercept is 10. Plot the line \(0.1x+0.4y=4\) and shade the region below the line. 3. \(0.2x+0.3y \leq 4.5\). At the x-intercept, y=0, \(x=\frac{4.5}{0.2}\), therefore x-intercept is 22.5. For the y-intercept, x=0, \(y=\frac{4.5}{0.3}\), therefore y-intercept is 15. Plot the line \(0.2x+0.3y=4.5\) and shade the region below the line. 4. \(x\geq0\), \(y\geq0\). Select only the region in the first quadrant. The feasible region is defined by the overlapping shaded regions.
02

Find the vertices

To find the vertices of the feasible region, look at the intersections between the lines or where the lines meet the x and y axes: 1. Intersection between \(30x+20y=600\) and \(0.1x+0.4y=4\). Solve the system of equations. 2. Intersection between \(0.1x+0.4y=4\) and \(0.2x+0.3y=4.5\). Solve the system of equations. 3. Intersection between \(0.2x+0.3y=4.5\) and \(30x+20y=600\). Solve the system of equations. 4. Intersection between \(30x+20y=600\) and the y-axis. 5. Intersection between \(0.2x+0.3y=4.5\) and the x-axis. The vertices are found by solving these systems of equations or evaluating the coordinates at the axis intersections.
03

Evaluate the objective function at the vertices

Using the objective function \(p=x+2y\), evaluate p at each vertex found in step 2. The maximum p value is the optimal solution. Check if there is no optimal solution or whether the feasible region is empty or the objective function is unbounded. If none of the conditions are true, provide the optimal solution based on the maximum p value from the vertex evaluations.

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Key Concepts

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

Feasible Region
Understanding the feasible region is crucial to solving linear programming (LP) problems. In essence, it is the graphical representation of all possible solutions to the given constraint inequalities in an LP problem. To visualize the feasible region, you need to plot each constraint inequality on a coordinate system.

For example, if we have inequalities like \(30x + 20y \leq 600\) or \(0.1x + 0.4y \leq 4\), we turn them into equalities to plot the lines: \(30x + 20y = 600\) and \(0.1x + 0.4y = 4\). The feasible region is typically a polygon or a polyhedron in multidimensional cases, delineated by these lines. It consists of all the points that satisfy all the constraints simultaneously, including not only the points on the boundary lines but also those within the enclosed area. It's where you'll find the set of potential solutions for the objective function to analyze further.
Objective Function
The objective function in a linear programming problem is a formula that summarizes the goal of the LP problem, such as maximizing profits or minimizing costs. It is usually given in the form of a linear equation dependent on the decision variables of the problem, such as \( p = x + 2y \), where \( p \) represents the quantity to be optimized.

Once the feasible region is established, the objective function is evaluated at all the vertices of this region to find the optimal solution. This method, known as the 'corner-point principle', states that the maximum or minimum value of the objective function, if it exists, will be located at one of the vertices of the feasible region.
Constraint Inequalities
Constraint inequalities are mathematical expressions that set boundaries on the values that decision variables in a linear programming problem can take. They describe limitations or requirements like resource constraints or policy rules that must be adhered to.

For instance, in the example exercise, constraints like \(30x + 20y \leq 600\), \(0.1x + 0.4y \leq 4\), and \(0.2x + 0.3y \leq 4.5\) are inequalities each representing a specific boundary condition. To solve an LP problem effectively, these constraint inequalities need to be carefully expressed and solved as a system, ensuring that all of the problem's limitations are captured within the feasible region.
System of Equations
In the context of linear programming, a system of equations arises when we look for the points of intersection among the lines representing constraint inequalities. To identify the feasible region's vertices, which are potential candidates for optimization, we solve these systems of equations.

Typically, each pair of lines will intersect at a point unless they are parallel. By finding the intersection points, as in the steps provided in the exercise, you determine the corners of the feasible region. These vertices can then be checked against the objective function, enabling you to ascertain which one will give the desired maximum or minimum value as the solution to the LP problem. Solving these equations is key to narrowing down potential solutions and plays a vital role in the optimization process of linear programming.

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Most popular questions from this chapter

We suggest the use of technology. Round all answers to two decimal places. \(\begin{array}{ll}\operatorname{Maximize} & p=2.2 x+2 y+1.1 z+2 w \\ \text { subject to } & x+1.5 y+1.5 z+\quad w \leq 50.5 \\ 2 x+1.5 y-\quad z-\quad w \geq 10 \\ & x+1.5 y+\quad z+1.5 w \geq 21 \\ x & \geq 0, y \geq 0, z \geq 0, w \geq 0\end{array}\)

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Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. \(\vee\) Minimize \(c=-x+2 y\) subject to \(\begin{aligned} y & \leq \frac{2 x}{3} \\\ x & \leq 3 y \\ y & \geq 4 \\ x & \geq 6 \\ x+y & \leq 16 . \end{aligned}\)
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