Minimum Ratio Test

Definition

The minimum ratio test is a method used in linear programming to determine which variable should leave the basis during the simplex algorithm process. This test helps identify the variable that will have the least impact on the objective function when another variable enters the solution set, ensuring optimality is maintained while adhering to constraints. The focus is on the ratios of the coefficients in the objective function relative to their respective constraints, facilitating efficient pivoting decisions.

5 Must Know Facts For Your Next Test

1. The minimum ratio test is applied during the pivoting step of the simplex method, where it helps decide which variable should leave the basis to maintain feasibility.
2. To perform the minimum ratio test, one calculates the ratios of the right-hand side values to their corresponding coefficients in the entering column, identifying the smallest positive ratio.
3. If all ratios are negative or zero, it indicates that no feasible solution exists or an unbounded solution may occur.
4. The variable associated with the minimum positive ratio will be removed from the current basis, allowing for an increase in the entering variable's value while maintaining feasibility.
5. The outcome of the minimum ratio test directly influences subsequent iterations in the simplex method, guiding towards finding an optimal solution efficiently.

Review Questions

• How does the minimum ratio test influence decision-making in linear programming when using the simplex method?
  • The minimum ratio test is critical in guiding which variable should exit the basis during each iteration of the simplex method. By evaluating ratios of coefficients and determining which one has the least impact on maintaining feasibility while allowing an entering variable to increase, it ensures optimal movement towards a solution. This test effectively balances constraint satisfaction with optimization goals, making it essential for efficient decision-making.
• Discuss how an unbounded solution is identified using the minimum ratio test and its implications for a linear programming problem.
  • An unbounded solution is identified during the minimum ratio test when all calculated ratios are negative or zero. This situation implies that increasing an entering variable can continue indefinitely without violating any constraints, indicating that there is no upper limit on the objective function. Recognizing this condition is vital as it reveals that adjustments to input parameters may be necessary to achieve a bounded feasible region.
• Evaluate how changes in constraint coefficients might affect the application of the minimum ratio test and subsequent outcomes in linear programming solutions.
  • Changes in constraint coefficients can significantly alter how the minimum ratio test is applied and its outcomes in linear programming solutions. If coefficients are adjusted, they can change the calculated ratios, potentially affecting which variable is deemed optimal to leave or enter the basis. This highlights how sensitive linear programming solutions are to modifications in constraints, necessitating a thorough re-evaluation through methods like the minimum ratio test to ensure continued optimality and feasibility.

Related terms

Simplex Method: An iterative algorithm used for solving linear programming problems by moving along the edges of the feasible region to reach the optimal vertex.

Feasible Region:

The set of all possible points that satisfy the constraints of a linear programming problem, representing potential solutions.

Basic Feasible Solution: A solution to a linear programming problem that satisfies all constraints and has a number of non-basic variables equal to zero, representing a vertex of the feasible region.