Minimum ratio test
Minimum Ratio Test
The minimum ratio test computes the largest feasible simplex step and chooses the leaving variable.
Formula
\[\theta^*=\min_{i:u_i>0}\frac{x_{B(i)}}{u_i},\qquad u=B^{-1}A_j.\]Reason
Every positive $u_i$ creates an upper bound on $\theta$; the smallest bound is the first variable to hit zero.
Zero Ratio
If the minimum ratio is zero, the pivot is degenerate.
No Positive u
If no $u_i>0$, no leaving variable exists. With negative reduced cost, the LP is unbounded below.
Checklist
Use only positive $u_i$ and choose the smallest ratio.
See Also
Exam checkpoint
For simplex questions in minimization form, negative reduced costs indicate possible improvement. Use $u=B^{-1}A_j$, apply the ratio test only to positive components of $u$, then update the basis.