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In the solution of a linear programming problem by Simplex method, if during iteration, all ratios of right-hand side bi to the coefficients of entering variable a are found to be negative, it implies that the problem has
- a)Infinite number of solutions
- b)Infeasible solution
- c)Degeneracy
- d)Unbound solution
Correct answer is option 'D'. Can you explain this answer?
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In the solution of a linear programming problem by Simplex method, if ...
Understanding the Simplex Method
The Simplex method is a widely-used algorithm for solving linear programming problems. In this method, we iterate through possible solutions while optimizing a specific objective function, usually maximizing or minimizing a linear function subject to certain constraints.
Negative Ratios in Iteration
During the iteration process, we calculate the ratios of the right-hand side (bi) to the coefficients of the entering variable (a). These ratios help determine which basic variable will leave the basis to maintain feasibility.
Implication of All Negative Ratios
If all these ratios are negative, it signifies that:
- Direction of Improvement: The entering variable is increasing, but all constraints are pushing in the opposite direction, meaning that increasing this variable will lead to a negative or infeasible solution for all existing constraints.
- Unbounded Solution: This scenario indicates that there is no upper limit to the objective function value. The feasible region extends infinitely in the direction of the entering variable, allowing the objective function to increase indefinitely without breaching any constraints.
Conclusion
Therefore, when all ratios of bi/a are negative, it strongly implies that the linear programming problem is unbounded. This situation can occur in cases where the constraints are not restrictive enough to limit the value of the objective function, leading to potential infinite solutions in one direction of the feasible region.
Understanding this concept is crucial for accurately interpreting the results of the Simplex method and for taking corrective actions in linear programming formulations.
The Simplex method is a widely-used algorithm for solving linear programming problems. In this method, we iterate through possible solutions while optimizing a specific objective function, usually maximizing or minimizing a linear function subject to certain constraints.
Negative Ratios in Iteration
During the iteration process, we calculate the ratios of the right-hand side (bi) to the coefficients of the entering variable (a). These ratios help determine which basic variable will leave the basis to maintain feasibility.
Implication of All Negative Ratios
If all these ratios are negative, it signifies that:
- Direction of Improvement: The entering variable is increasing, but all constraints are pushing in the opposite direction, meaning that increasing this variable will lead to a negative or infeasible solution for all existing constraints.
- Unbounded Solution: This scenario indicates that there is no upper limit to the objective function value. The feasible region extends infinitely in the direction of the entering variable, allowing the objective function to increase indefinitely without breaching any constraints.
Conclusion
Therefore, when all ratios of bi/a are negative, it strongly implies that the linear programming problem is unbounded. This situation can occur in cases where the constraints are not restrictive enough to limit the value of the objective function, leading to potential infinite solutions in one direction of the feasible region.
Understanding this concept is crucial for accurately interpreting the results of the Simplex method and for taking corrective actions in linear programming formulations.
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In the solution of a linear programming problem by Simplex method, if during iteration, all ratios of right-hand side bi to the coefficients of entering variable a are found to be negative, it implies that the problem hasa)Infinite number of solutionsb)Infeasible solution c)Degeneracy d)Unbound solutionCorrect answer is option 'D'. Can you explain this answer?
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In the solution of a linear programming problem by Simplex method, if during iteration, all ratios of right-hand side bi to the coefficients of entering variable a are found to be negative, it implies that the problem hasa)Infinite number of solutionsb)Infeasible solution c)Degeneracy d)Unbound solutionCorrect answer is option 'D'. 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 In the solution of a linear programming problem by Simplex method, if during iteration, all ratios of right-hand side bi to the coefficients of entering variable a are found to be negative, it implies that the problem hasa)Infinite number of solutionsb)Infeasible solution c)Degeneracy d)Unbound solutionCorrect answer is option 'D'. 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 In the solution of a linear programming problem by Simplex method, if during iteration, all ratios of right-hand side bi to the coefficients of entering variable a are found to be negative, it implies that the problem hasa)Infinite number of solutionsb)Infeasible solution c)Degeneracy d)Unbound solutionCorrect answer is option 'D'. Can you explain this answer?.
In the solution of a linear programming problem by Simplex method, if during iteration, all ratios of right-hand side bi to the coefficients of entering variable a are found to be negative, it implies that the problem hasa)Infinite number of solutionsb)Infeasible solution c)Degeneracy d)Unbound solutionCorrect answer is option 'D'. 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 In the solution of a linear programming problem by Simplex method, if during iteration, all ratios of right-hand side bi to the coefficients of entering variable a are found to be negative, it implies that the problem hasa)Infinite number of solutionsb)Infeasible solution c)Degeneracy d)Unbound solutionCorrect answer is option 'D'. 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 In the solution of a linear programming problem by Simplex method, if during iteration, all ratios of right-hand side bi to the coefficients of entering variable a are found to be negative, it implies that the problem hasa)Infinite number of solutionsb)Infeasible solution c)Degeneracy d)Unbound solutionCorrect answer is option 'D'. Can you explain this answer?.
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