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Magazine Chapter 4: Problem 10
Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. Minimize \(\quad \begin{aligned} & c=0.4 x+0.1 y \\ \text { subject to } & 30 x+20 y \geq 600 \\ & 0.1 x+0.4 y \geq 4 \\ & 0.2 x+0.3 y \geq 4.5 \\ & x \geq 0, y \geq 0 \end{aligned}\)
Short Answer
Expert verified
The feasible region for the given LP problem is unbounded with only one vertex (15, 9.5). Thus, the objective function has no optimal solution.
Step by step solution
01
Graphing linear inequalities
First, we will graph the linear inequalities to find the feasible region.
The given inequalities are:
1. \(30x + 20y \geq 600\)
2. \(0.1x + 0.4y \geq 4\)
3. \(0.2x + 0.3y \geq 4.5\)
4. \(x \geq 0\), \(y \geq 0\)
Graph all the inequalities and shade the region that satisfies all of them.
02
Find the vertices of the feasible region
Once the feasible region is graphed, identify its vertices by finding the intersections of the boundaries of the inequalities. Let's consider the pairs of inequalities for finding the points of intersection:
1. Intersection of \(30x + 20y \geq 600\) and \(0.1x + 0.4y \geq 4\):
Solve the system of equations:
\(30x + 20y = 600\) and \(0.1x + 0.4y = 4\)
This system of equations has no solution. So, there is no intersection point.
2. Intersection of \(30x + 20y \geq 600\) and \(0.2x + 0.3y \geq 4.5\):
Solve the system of equations:
\(30x + 20y = 600\) and \(0.2x + 0.3y = 4.5\)
This system of equations also has no solution. So, there is no intersection point.
3. Intersection of \(0.1x + 0.4y \geq 4\) and \(0.2x + 0.3y \geq 4.5\):
Solve the system of equations:
\(0.1x + 0.4y = 4\) and \(0.2x + 0.3y = 4.5\)
The solution is \((x, y) = (15, 9.5)\).
Now, find the points on the axis where the inequalities intersect (the intersections with x-axis and y-axis):
1. Intersection with \(x\)-axis:
- For \(30x + 20y \geq 600\), setting \(y = 0\), we get \(x = 20\).
- For \(0.1x + 0.4y \geq 4\), setting \(y = 0\), we get \(x = 40\).
- For \(0.2x + 0.3y \geq 4.5\), setting \(y = 0\), we get \(x = 22.5\).
2. Intersection with \(y\)-axis:
- For \(30x + 20y \geq 600\), setting \(x = 0\), we get \(y = 30\).
- For \(0.1x + 0.4y \geq 4\), setting \(x = 0\), we get \(y = 10\).
- For \(0.2x + 0.3y \geq 4.5\), setting \(x = 0\), we get \(y = 15\).
However, from the graph, it is clear that no intersection points on the axes belong to the feasible region. The only vertex of the feasible region is (15, 9.5).
03
Check if the feasible region is bounded or unbounded
Since the feasible region has only one vertex (15, 9.5), it must be unbounded.
04
End Result
Now, we can claim that the feasible region exists but is unbounded, and thus, the objective function has no optimal solution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Feasible Region
The feasible region in a linear programming problem is crucial as it represents all possible solutions that satisfy the given constraints. Essentially, it is the area on a graph where all the inequalities overlap.
For this problem, the inequalities provided define a space where all solutions must lie. However, in this particular scenario, the feasible region is quite unique. By graphing each inequality, you will notice exactly where solutions are possible. Each segment of overlap between these inequalities constrains the feasible region further. This bounded area encapsulates where the solution could potentially optimally reside.
In our given case, despite graphing all inequalities, we find that the region is confined to only one point that satisfies all the equations. This is quite interesting as it implies a boundary that's not completely contained, leading us to more concepts like unboundedness.
For this problem, the inequalities provided define a space where all solutions must lie. However, in this particular scenario, the feasible region is quite unique. By graphing each inequality, you will notice exactly where solutions are possible. Each segment of overlap between these inequalities constrains the feasible region further. This bounded area encapsulates where the solution could potentially optimally reside.
In our given case, despite graphing all inequalities, we find that the region is confined to only one point that satisfies all the equations. This is quite interesting as it implies a boundary that's not completely contained, leading us to more concepts like unboundedness.
Graphing Inequalities
Graphing inequalities allows us to visually interpret and understand the constraints of a linear programming problem. To graph an inequality, begin by converting it into an equation by using an equal sign instead of an inequality symbol.
For example:
This visual method simplifies identifying possible solutions and aids in understanding more complex concepts like vertices and the feasibility of regions.
For example:
- Convert \(30x + 20y \geq 600\) to \(30x + 20y = 600\).
- Plot this line on a graph. Use points where the line intersects the x-axis and y-axis for easier plotting.
- Shade above the line to represent the region where \(30x + 20y\) is greater than or equal to 600.
This visual method simplifies identifying possible solutions and aids in understanding more complex concepts like vertices and the feasibility of regions.
Objective Function
In linear programming, the objective function represents the goal that needs to be optimized. Whether maximizing profits or minimizing costs, the objective function gives purpose to the problem being solved.
In the exercise presented, the objective function is defined as \(c = 0.4x + 0.1y\). Here, you are minimizing the value of \(c\), where \(x\) and \(y\) meet the criteria set by the constraints or inequalities.
Once you have identified the feasible region, the objective function allows you to evaluate different vertex points within this region to find which one gives you the smallest possible value for \(c\). However, in scenarios where solutions lie on one vertex or the region is unbounded, the quest for an optimum solution becomes a theoretical challenge.
After verifying if the region is bounded and calculating vertices, you use the objective function to test these solutions to find the preferred outcome.
In the exercise presented, the objective function is defined as \(c = 0.4x + 0.1y\). Here, you are minimizing the value of \(c\), where \(x\) and \(y\) meet the criteria set by the constraints or inequalities.
Once you have identified the feasible region, the objective function allows you to evaluate different vertex points within this region to find which one gives you the smallest possible value for \(c\). However, in scenarios where solutions lie on one vertex or the region is unbounded, the quest for an optimum solution becomes a theoretical challenge.
After verifying if the region is bounded and calculating vertices, you use the objective function to test these solutions to find the preferred outcome.
Unbounded Solution
An unbounded solution in linear programming implies that there is no finite maximum or minimum value for the objective function within the feasible region. This occurs when the bounds of the feasible region do not close, allowing one or more dimensions of the feasible region to stretch infinitely.
For this exercise, considering the intersection of inequalities led us to a feasible region with only one vertex point. This suggests that the feasible region itself is unbounded, as no defined boundary restricted the feasible area completely.
As a consequence, it was found that we cannot achieve an optimal solution for our given objective function \(c = 0.4x + 0.1y\), as it heads towards infinity or cannot reach an optimal point within the confines of the existing region. Understanding unbounded solutions is essential as it sheds light on constraints' sufficiency and may suggest revisiting the conditions to achieve a different optimal strategy.
In real-life applications, unbounded solutions indicate the need to reassess constraints for more practical and close-ended solutions.
For this exercise, considering the intersection of inequalities led us to a feasible region with only one vertex point. This suggests that the feasible region itself is unbounded, as no defined boundary restricted the feasible area completely.
As a consequence, it was found that we cannot achieve an optimal solution for our given objective function \(c = 0.4x + 0.1y\), as it heads towards infinity or cannot reach an optimal point within the confines of the existing region. Understanding unbounded solutions is essential as it sheds light on constraints' sufficiency and may suggest revisiting the conditions to achieve a different optimal strategy.
In real-life applications, unbounded solutions indicate the need to reassess constraints for more practical and close-ended solutions.
