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Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities,
- a)optimal value must occur at a corner point (vertex) of the feasible region.
- b)optimal value must occur at the centroid of the feasible region.
- c)optimal value must occur at the midpoints of the corner points (vertices) of the feasible region
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Let R be the feasible region (convex polygon) for a linear programming...
Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities,then , optimal value must occur at a corner point (vertex) of the feasible region.
Most Upvoted Answer
Let R be the feasible region (convex polygon) for a linear programming...
Explanation:
To understand why the optimal value must occur at a corner point (vertex) of the feasible region, let's consider the following points:
1. Feasible Region:
The feasible region is the intersection of all the linear inequality constraints. It is a convex polygon formed by the overlapping region of the constraints.
2. Objective Function:
The objective function, Z = ax + by, represents the quantity that we want to maximize or minimize. It is a linear function of the variables x and y.
3. Linear Programming Problem:
A linear programming problem involves optimizing the objective function subject to the linear inequality constraints. The goal is to find the values of x and y that maximize or minimize the objective function while satisfying all the constraints.
4. Corner Points (Vertices):
The corner points (vertices) of the feasible region are the points where the boundaries of the constraints intersect. These points define the extreme values of the variables x and y within the feasible region.
5. Convexity:
The feasible region is a convex polygon, which means that any line segment connecting two points within the region lies entirely within the region. This property is important in linear programming as it allows us to conclude that the optimal value must occur at a corner point of the feasible region.
6. Optimal Value:
The optimal value of the objective function represents the maximum or minimum value that can be achieved within the feasible region. In linear programming, the optimal value must occur at a corner point (vertex) of the feasible region.
Why the Optimal Value Occurs at a Corner Point:
To understand why the optimal value occurs at a corner point, consider the following:
- If the optimal value occurs in the interior of the feasible region, it means that there exists another point that can produce a higher or lower objective function value, violating the optimality condition.
- By moving from an interior point towards a corner point, the objective function value either increases or decreases until it reaches the corner point. This is because the direction of movement is constrained by the linear inequalities.
- At a corner point, there are no neighboring points that can lead to a higher or lower objective function value, making it the extreme value within the feasible region.
Therefore, the optimal value of the objective function must occur at a corner point (vertex) of the feasible region. Option 'A' is the correct answer.
To understand why the optimal value must occur at a corner point (vertex) of the feasible region, let's consider the following points:
1. Feasible Region:
The feasible region is the intersection of all the linear inequality constraints. It is a convex polygon formed by the overlapping region of the constraints.
2. Objective Function:
The objective function, Z = ax + by, represents the quantity that we want to maximize or minimize. It is a linear function of the variables x and y.
3. Linear Programming Problem:
A linear programming problem involves optimizing the objective function subject to the linear inequality constraints. The goal is to find the values of x and y that maximize or minimize the objective function while satisfying all the constraints.
4. Corner Points (Vertices):
The corner points (vertices) of the feasible region are the points where the boundaries of the constraints intersect. These points define the extreme values of the variables x and y within the feasible region.
5. Convexity:
The feasible region is a convex polygon, which means that any line segment connecting two points within the region lies entirely within the region. This property is important in linear programming as it allows us to conclude that the optimal value must occur at a corner point of the feasible region.
6. Optimal Value:
The optimal value of the objective function represents the maximum or minimum value that can be achieved within the feasible region. In linear programming, the optimal value must occur at a corner point (vertex) of the feasible region.
Why the Optimal Value Occurs at a Corner Point:
To understand why the optimal value occurs at a corner point, consider the following:
- If the optimal value occurs in the interior of the feasible region, it means that there exists another point that can produce a higher or lower objective function value, violating the optimality condition.
- By moving from an interior point towards a corner point, the objective function value either increases or decreases until it reaches the corner point. This is because the direction of movement is constrained by the linear inequalities.
- At a corner point, there are no neighboring points that can lead to a higher or lower objective function value, making it the extreme value within the feasible region.
Therefore, the optimal value of the objective function must occur at a corner point (vertex) of the feasible region. Option 'A' is the correct answer.
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Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities,a)optimal value must occur at a corner point (vertex) of the feasible region.b)optimal value must occur at the centroid of the feasible region.c)optimal value must occur at the midpoints of the corner points (vertices) of the feasible regiond)None of theseCorrect answer is option 'A'. Can you explain this answer?
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