 # Pivot

This is the algebra associated with an iteration of Gauss-Jordan elimination, using the forward transformation. The tableaux for a pivot on element $LaTeX: a(p,q)$ $LaTeX: (\ne 0),$ which means nonbasic variable $LaTeX: x_q$ enters the basis in exchange for basic variable $LaTeX: x_p,$ are as follows:

Before pivot: $LaTeX: \begin{array}{ll|ll} Basic & & \multicolumn{2}{c}{Nonbasic} \\ Var. & Level & x_j & x_q \\ \hline \hline x_i & b_i & a(i,j) & a(i,q) \\ x_p & b_p & a(p,j) & a(p,q)^* \\ \hline \hline obj & -z & d_j & d_q \\ \hline \hline \end{array}$

After pivot: $LaTeX: \begin{array}{ll|lr} Basic & & \multicolumn{2}{c}{Nonbasic} \\ Var. & Level & x_j & x_q \\ \hline \hline x_i & b_i - b_p a(i,q)/a(p,q) & a(i,j) - a(i,q)a(p,j)/a(p,q) & -a(i,q)/a(p,q) \\ x_q & b_p/a(p,q) & a(p,j)/a(p,q) & 1/a(p,q) \\ \hline \hline obj & -a - b_p d_q/a(p,q) & d_j - a(p,j)d_q/a(p,q) & -d_q/a(p,q) \\ \hline \hline \end{array}$

A pivot is primal degenerate if the associated basic solution $LaTeX: (x)$ does not change (i.e., the nonbasic variable enters the basis, but its level remains at the same bound value, in which case no basic variable changes level). Similarly, the pivot is dual degenerate if the associated dual solution (i.e., pricing vector and reduced costs) does not change. For dealing with degenerate pivots, see Bland's rule and the TNP rule.