Simplex Pivot¶

Definition¶

A simplex pivot replaces the leaving basic variable with the entering variable. Divide the selected pivot row by the nonzero pivot element, then subtract appropriate multiples of that normalized row from every other row, including the objective row, so that the pivot column is a unit column.

Interpretation¶

A pivot is the algebra behind one step of the simplex method: one variable becomes basic, another becomes nonbasic, and the equations are rewritten so that the tableau is again in canonical form. The rewritten equations have exactly the same solutions as before; only their form changes.

Two simplex tableaus, before and after a pivot: the pivot element in row s3 and column x s is circled, and afterwards the column of x s is a unit column.

In the Giapetto example $x_s$ enters and $s_3$ leaves, so the pivot element is the $1$ in row $s_3$ , column $x_s$ . Afterwards the $x_s$ column is a unit column with its $1$ in the row now labelled $x_s$ . With $x_t=s_3=0$ the new tableau gives $x_s=40$ $40$ , $s_1=20$ $20$ , $s_2=40$ and $z=120$ $120$ : the solution has moved from the origin along an edge to the corner $(40,0)$ . The $-2$ under $x_t$ shows that $x_t$ enters next.

Each pivot exchanges one basic variable for another; geometrically it moves to an adjacent corner of the feasible region.

Each row operation divides an equation by a nonzero number or adds a multiple of one equation to another. Both steps can be undone, so no solution is gained or lost; the objective row is treated like any other equation. The pivot row must be the one chosen by the minimum-ratio test, because that keeps every right-hand side nonnegative. Pivoting on the row of $s_1$ instead, with ratio $50$ , would set $x_s=50$ and give $s_3=40-50=-10$ $- 10$ , an infeasible point.

The same elimination steps return in Linear Algebra as Gauss-Jordan elimination.

Key Results¶

The pivot element is the intersection of the entering column and leaving row.
Update all coefficients and the right-hand side during each row operation.
Update the basic-variable label as well as the numerical entries.

Examples¶

In the illustrated Giapetto pivot, the row operations are

\begin{aligned}R_{s_3}&\leftarrow R_{s_3}/1,\\R_z&\leftarrow R_z+3R_{s_3},\\R_{s_1}&\leftarrow R_{s_1}-2R_{s_3},\\R_{s_2}&\leftarrow R_{s_2}-R_{s_3}.\end{aligned}

The new point is $(x_s,x_t)=(40,0)$ with $z=120$ $120$ ; the next entering variable is $x_t$ .

Take the objective row $z-3x_1-2x_2=0$ with the constraints of the augmented-form example. Then $x_1$ enters and the minimum-ratio test picks the row of $s_1$ , so divide the pivot row $2x_1+x_2+s_1=10$ by 2.
Add three times that normalized row to $z-3x_1-2x_2=0$ to eliminate $x_1$ from the objective row.

Prerequisites¶

Minimum-Ratio Test
Simplex Tableau
Gauss-Jordan row operations

Sources¶

OR Modelling · Week 2

Aliases: pivot operation, basis exchange

Last updated: 20 October 2026 03:02