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An unbounded solution of linear programming problem is reflected in the simplex method, when
- a)All the ratio of ‘right hand sides’ to coefficients in key column become negative
- b)All the ratios of right hand sides to coefficients in key columns become zero
- c)All right hand sides become negative
- d)All right hand become zero
Correct answer is option 'A'. Can you explain this answer?
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An unbounded solution of linear programming problem is reflected in t...
Unbounded Solution : If the feasible region is not bounded, it is possible that the value of the objective function goes on increasing without leaving the feasible region. This is known as an unbounded solution.
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An unbounded solution of linear programming problem is reflected in t...
Explanation:
The simplex method is an iterative algorithm used to solve linear programming problems. It starts with an initial feasible solution and then iteratively improves it until an optimal solution is reached. In each iteration, the simplex method selects a pivot element to enter and leave the basis, which results in a new feasible solution.
An unbounded solution in linear programming occurs when the objective function can be made arbitrarily large or small without violating any of the constraints. This means that there is no finite optimal solution, and the objective function value can be increased or decreased indefinitely.
Simplex Method and Unbounded Solution:
In the simplex method, the selection of the pivot element is based on the ratios of the right-hand sides (RHS) to the coefficients in the key column. The key column represents the coefficients of the basic variables in the current basis.
If all the ratios of the RHS to the coefficients in the key column become negative, it indicates that the objective function can be made arbitrarily small (negative) without violating any of the constraints. This implies an unbounded solution because there is no finite optimal solution, and the objective function value can be decreased indefinitely.
The simplex method detects this unboundedness and terminates, indicating that the problem is unbounded. It means that the constraints are not sufficiently restrictive to limit the objective function value.
Other Options:
The other options mentioned in the question are not correct:
- If all the ratios of the RHS to the coefficients in the key column become zero, it means that the objective function cannot be increased or decreased. This indicates a degenerate solution, not an unbounded solution.
- If all the right-hand sides become negative, it does not necessarily indicate an unbounded solution. It means that all the constraints are violated, and there is no feasible solution.
- If all the right-hand sides become zero, it indicates a degenerate solution, not an unbounded solution.
Therefore, the correct answer is option 'A', where all the ratios of the RHS to the coefficients in the key column become negative. This reflects an unbounded solution in the simplex method.
The simplex method is an iterative algorithm used to solve linear programming problems. It starts with an initial feasible solution and then iteratively improves it until an optimal solution is reached. In each iteration, the simplex method selects a pivot element to enter and leave the basis, which results in a new feasible solution.
An unbounded solution in linear programming occurs when the objective function can be made arbitrarily large or small without violating any of the constraints. This means that there is no finite optimal solution, and the objective function value can be increased or decreased indefinitely.
Simplex Method and Unbounded Solution:
In the simplex method, the selection of the pivot element is based on the ratios of the right-hand sides (RHS) to the coefficients in the key column. The key column represents the coefficients of the basic variables in the current basis.
If all the ratios of the RHS to the coefficients in the key column become negative, it indicates that the objective function can be made arbitrarily small (negative) without violating any of the constraints. This implies an unbounded solution because there is no finite optimal solution, and the objective function value can be decreased indefinitely.
The simplex method detects this unboundedness and terminates, indicating that the problem is unbounded. It means that the constraints are not sufficiently restrictive to limit the objective function value.
Other Options:
The other options mentioned in the question are not correct:
- If all the ratios of the RHS to the coefficients in the key column become zero, it means that the objective function cannot be increased or decreased. This indicates a degenerate solution, not an unbounded solution.
- If all the right-hand sides become negative, it does not necessarily indicate an unbounded solution. It means that all the constraints are violated, and there is no feasible solution.
- If all the right-hand sides become zero, it indicates a degenerate solution, not an unbounded solution.
Therefore, the correct answer is option 'A', where all the ratios of the RHS to the coefficients in the key column become negative. This reflects an unbounded solution in the simplex method.
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An unbounded solution of linear programming problem is reflected in the simplex method, whena)All the ratio of ‘right hand sides’ to coefficients in key column become negativeb)All the ratios of right hand sides to coefficients in key columns become zeroc)All right hand sides become negatived)All right hand become zeroCorrect answer is option 'A'. Can you explain this answer?
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