Entering Variables

The entering variable is a nonbasic variable selected to become positive during a pivot.

Rule for Minimization

Choose a nonbasic variable $x_j$ with $\bar c_j<0$.

Why

Its simplex direction has objective change $c^Td=\bar c_j$, so negative reduced cost lowers the cost.

Candidate Rules

Possible rules: most negative reduced cost, largest expected decrease, or smallest index among eligible variables.

No Candidate

If no nonbasic reduced cost is negative, the current BFS is optimal.

Checklist

Find nonbasic variables and choose one with negative reduced cost.

See Also

• Reduced Costs
• Leaving Variables

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.