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In case of solution of linear programming problem using graphical method, if the constraint line of one of the non-redundant constraints is parallel to the objective function line, then it indicates
- a)An infeasible solution
- b)A degenerate solution
- c)An unbound solution
- d)A multiple number of optimal solutions
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
In case of solution of linear programming problem using graphical meth...
All points on the line is a solution. So there are infinite no of optimal solutions.
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In case of solution of linear programming problem using graphical meth...
Explanation:
When we solve a linear programming problem using graphical method, we draw the feasible region which is the intersection of all non-redundant constraints. We then find the optimal solution by maximizing or minimizing the objective function over this feasible region. In some cases, the constraint line of one of the non-redundant constraints may be parallel to the objective function line. This situation indicates that there are multiple number of optimal solutions to the problem. Let's see why.
Reasoning:
When the constraint line and the objective function line are parallel, it means that the objective function is not affected by the constraint represented by this line. Therefore, any point on this line is as good as any other point on the same line, in terms of optimizing the objective function. This means that we have multiple number of points that can be optimal solutions.
Example:
Consider the following linear programming problem:
Maximize Z = 3x + 4y
Subject to:
2x + 3y ≤ 12
4x + 6y ≤ 24
x ≥ 0, y ≥ 0
If we draw the feasible region, we get a rectangle with vertices (0,0), (0,4), (2,3), and (4,0). The objective function line Z = 3x + 4y has a slope of -3/4. If we draw a line with this slope passing through the feasible region, we get a line that intersects the feasible region at point (2,3). The constraint line 2x + 3y = 12 is also passing through this point and is parallel to the objective function line. This indicates that there are multiple number of optimal solutions to this problem, which lie on the line 2x + 3y = 12.
Conclusion:
When the constraint line of one of the non-redundant constraints is parallel to the objective function line, it indicates that there are multiple number of optimal solutions to the problem. This happens because the objective function is not affected by the constraint represented by this line.
When we solve a linear programming problem using graphical method, we draw the feasible region which is the intersection of all non-redundant constraints. We then find the optimal solution by maximizing or minimizing the objective function over this feasible region. In some cases, the constraint line of one of the non-redundant constraints may be parallel to the objective function line. This situation indicates that there are multiple number of optimal solutions to the problem. Let's see why.
Reasoning:
When the constraint line and the objective function line are parallel, it means that the objective function is not affected by the constraint represented by this line. Therefore, any point on this line is as good as any other point on the same line, in terms of optimizing the objective function. This means that we have multiple number of points that can be optimal solutions.
Example:
Consider the following linear programming problem:
Maximize Z = 3x + 4y
Subject to:
2x + 3y ≤ 12
4x + 6y ≤ 24
x ≥ 0, y ≥ 0
If we draw the feasible region, we get a rectangle with vertices (0,0), (0,4), (2,3), and (4,0). The objective function line Z = 3x + 4y has a slope of -3/4. If we draw a line with this slope passing through the feasible region, we get a line that intersects the feasible region at point (2,3). The constraint line 2x + 3y = 12 is also passing through this point and is parallel to the objective function line. This indicates that there are multiple number of optimal solutions to this problem, which lie on the line 2x + 3y = 12.
Conclusion:
When the constraint line of one of the non-redundant constraints is parallel to the objective function line, it indicates that there are multiple number of optimal solutions to the problem. This happens because the objective function is not affected by the constraint represented by this line.
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In case of solution of linear programming problem using graphical method, if the constraint line of one of the non-redundant constraints is parallel to the objective function line, then it indicatesa)An infeasible solutionb)A degenerate solution c)An unbound solutiond)A multiple number of optimal solutionsCorrect answer is option 'D'. Can you explain this answer?
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