Reduced Costs

Reduced costs measure how the objective changes when a nonbasic variable increases from zero.

Formula

\(\bar c_j=c_j-c_B^TB^{-1}A_j.\) For all variables: \(\bar c^T=c^T-c_B^TB^{-1}A.\)

Basic Variables

Every basic variable has reduced cost zero because $B^{-1}A_{B(i)}=e_i$.

Minimization Meaning

$\bar c_j<0$ improves, $\bar c_j=0$ is neutral, and $\bar c_j>0$ worsens.

Multipliers

With $p^T=c_B^TB^{-1}$, reduced costs are $\bar c_j=c_j-p^TA_j$.

Checklist

Compute reduced costs and interpret their signs.

See Also

• Optimality Conditions
• Entering Variables
• Revised 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.