Simplex objective update for a priced slack variable ($z+{\sum }_j({\rho }_j-p{1}_{j=k})x_j=z_0$)

If a variable $x_k$ gains an objective coefficient $p$, the new objective is $z=z_{\mathrm{old}}+p{x}_k$. Substitute this in the old canonical objective equation and eliminate any remaining basic-column coefficients using the constraint rows. If $x_k$ is nonbasic, this simply subtracts $p$ from its objective-row coefficient. When a slack variable is sold, rename the old slack as the sales quantity; its equation already gives the correct equality and no new slack is added. Reoptimization can then start directly from the altered simplex tableau.