Revised Simplex Method

The revised simplex method performs simplex using the current basis matrix instead of a full tableau.

Current BFS

Compute $x_B=B^{-1}b$.

Multipliers

Compute $p^T=c_B^TB^{-1}$, usually by solving $B^Tp=c_B$.

Reduced Costs

For nonbasic $j$, compute $\bar c_j=c_j-p^TA_j$.

Direction

For entering $j$, solve $Bu=A_j$ to get $u=B^{-1}A_j$.

Ratio and Update

Apply the minimum ratio test and replace the leaving basis column.

Checklist

Use linear solves with $B$ instead of explicitly forming the full tableau.

See Also

• Reduced Costs
• Basis Matrices
• Full Tableau Simplex Method

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.