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If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that
[1996]
- a)Right hand side of the primal constraint can altered without affecting the optimum solution
- b)Changing the right hand side of the primal constraint will disturb the optimum solution
- c)The objectives function is unbounded
- d)The problem is degenerate
Correct answer is option 'C'. Can you explain this answer?
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If at the optimum in a linear programming problem, a dual variable cor...
The Dual Variable and Primal Constraint in Linear Programming
In linear programming, the dual variable is associated with each primal constraint. The dual variable represents the rate of change of the objective function with respect to a unit change in the right-hand side of the primal constraint.
Understanding the Optimum Solution
The optimum solution in a linear programming problem is the combination of decision variables that maximizes or minimizes the objective function while satisfying all the given constraints. The values of the dual variables are determined based on this optimum solution.
Interpreting a Dual Variable of Zero
If the dual variable corresponding to a particular primal constraint is zero at the optimum solution, it means that a unit change in the right-hand side of that constraint does not affect the objective function. In other words, the objective function is unbounded in the direction of that constraint.
To understand this further, let's consider a simple example. Suppose a linear programming problem has a primal constraint of the form "ax + by ≤ c", where "a", "b", and "c" are constants and "x" and "y" are decision variables. If the dual variable associated with this constraint is zero, it implies that a unit increase in the right-hand side of the constraint, i.e. "c", does not affect the objective function.
Significance of an Unbounded Objective Function
If the objective function is unbounded, it means that there is no finite optimal solution for the problem. This can occur when the feasible region is unbounded or when the objective function has a non-zero slope in the direction of the constraint with a zero dual variable.
In practical terms, an unbounded objective function may indicate that the problem has multiple optimal solutions or that the problem is infeasible. It is important to analyze the problem further to determine the significance of an unbounded objective function and its implications for decision-making.
Conclusion
In summary, if the dual variable corresponding to a primal constraint is zero at the optimum solution of a linear programming problem, it indicates that the objective function is unbounded in the direction of that constraint. This implies that a unit change in the right-hand side of the constraint does not affect the objective function. Understanding the significance of a zero dual variable helps in analyzing the problem and making appropriate decisions.
In linear programming, the dual variable is associated with each primal constraint. The dual variable represents the rate of change of the objective function with respect to a unit change in the right-hand side of the primal constraint.
Understanding the Optimum Solution
The optimum solution in a linear programming problem is the combination of decision variables that maximizes or minimizes the objective function while satisfying all the given constraints. The values of the dual variables are determined based on this optimum solution.
Interpreting a Dual Variable of Zero
If the dual variable corresponding to a particular primal constraint is zero at the optimum solution, it means that a unit change in the right-hand side of that constraint does not affect the objective function. In other words, the objective function is unbounded in the direction of that constraint.
To understand this further, let's consider a simple example. Suppose a linear programming problem has a primal constraint of the form "ax + by ≤ c", where "a", "b", and "c" are constants and "x" and "y" are decision variables. If the dual variable associated with this constraint is zero, it implies that a unit increase in the right-hand side of the constraint, i.e. "c", does not affect the objective function.
Significance of an Unbounded Objective Function
If the objective function is unbounded, it means that there is no finite optimal solution for the problem. This can occur when the feasible region is unbounded or when the objective function has a non-zero slope in the direction of the constraint with a zero dual variable.
In practical terms, an unbounded objective function may indicate that the problem has multiple optimal solutions or that the problem is infeasible. It is important to analyze the problem further to determine the significance of an unbounded objective function and its implications for decision-making.
Conclusion
In summary, if the dual variable corresponding to a primal constraint is zero at the optimum solution of a linear programming problem, it indicates that the objective function is unbounded in the direction of that constraint. This implies that a unit change in the right-hand side of the constraint does not affect the objective function. Understanding the significance of a zero dual variable helps in analyzing the problem and making appropriate decisions.
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If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that[1996]a)Right hand side of the primal constraint can altered without affecting the optimum solutionb)Changing the right hand side of the primal constraint will disturb the optimum solutionc)The objectives function is unboundedd)The problem is degenerateCorrect answer is option 'C'. Can you explain this answer?
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If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that[1996]a)Right hand side of the primal constraint can altered without affecting the optimum solutionb)Changing the right hand side of the primal constraint will disturb the optimum solutionc)The objectives function is unboundedd)The problem is degenerateCorrect answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that[1996]a)Right hand side of the primal constraint can altered without affecting the optimum solutionb)Changing the right hand side of the primal constraint will disturb the optimum solutionc)The objectives function is unboundedd)The problem is degenerateCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that[1996]a)Right hand side of the primal constraint can altered without affecting the optimum solutionb)Changing the right hand side of the primal constraint will disturb the optimum solutionc)The objectives function is unboundedd)The problem is degenerateCorrect answer is option 'C'. Can you explain this answer?.
If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that[1996]a)Right hand side of the primal constraint can altered without affecting the optimum solutionb)Changing the right hand side of the primal constraint will disturb the optimum solutionc)The objectives function is unboundedd)The problem is degenerateCorrect answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that[1996]a)Right hand side of the primal constraint can altered without affecting the optimum solutionb)Changing the right hand side of the primal constraint will disturb the optimum solutionc)The objectives function is unboundedd)The problem is degenerateCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If at the optimum in a linear programming problem, a dual variable corresponding to a particular primal constraint is zero, then it means that[1996]a)Right hand side of the primal constraint can altered without affecting the optimum solutionb)Changing the right hand side of the primal constraint will disturb the optimum solutionc)The objectives function is unboundedd)The problem is degenerateCorrect answer is option 'C'. Can you explain this answer?.
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