Simplex Tableau¶

Definition¶

A simplex tableau records the variable coefficients, right-hand sides, and basic-variable labels of the current simplex equations. In the course convention, the first row writes the objective as

z+\sum_jq_jx_j=z_0.

In canonical form (also called proper form), basic-variable columns form unit columns and have zero objective-row coefficients.

Interpretation¶

A simplex tableau is the bookkeeping of the simplex method: it writes the current equations as a table of coefficients, one row per equation, with the objective as an extra equation on top. The table makes the row operations of a pivot, and reading off the current solution, mechanical.

An initial simplex tableau with the objective row highlighted and the columns of the slack variables forming an identity matrix.

This is the starting tableau of the Giapetto model. The orange row is the objective $z-3x_s-2x_t=0$ $0$ , with all variable terms on the left. The highlighted entries show that the columns of $s_1$ , $s_2$ and $s_3$ are unit columns, each with its $1$ in a different row, so these slacks are the basic variables named in the first column.

A canonical tableau is read in two directions. For a variable named in the basic-variable column, its unit column identifies its row, and that row's right-hand side is its value when the nonbasic variables are zero. A nonbasic column can also happen to be a unit column, so the column shape alone does not identify the chosen basis. Along the objective row, the right-hand side is the current value $z_0$ , and the entry $q_j$ under a nonbasic variable means that $z$ changes by $-q_j$ per unit increase of $x_j$ , which is what the optimality test reads.

The signs cause most slips. The row $z-3x_s-2x_t=0$ means $z=3x_s+2x_t$ $2 x_{t}$ , so the negative entries mark profitable directions, not losses. Writing the objective as $\displaystyle z=z_0-\sum_jq_jx_j$ shows the same information with opposite signs.

Key Results¶

Set nonbasic variables to zero and read basic values from the solution column.
The objective-row right-hand side is the current objective value.
A negative nonbasic coefficient indicates an improving direction for maximization.

Examples¶

The illustrated system is $2x_s+x_t+s_1=100$ $100$ , $x_s+x_t+s_2=80$ $80$ , $x_s+s_3=40$ $40$ , with objective $z-3x_s-2x_t=0$ $0$ . The orange objective row places coefficients on the left, and the blue basic columns form an identity matrix. Setting $x_s=x_t=0$ gives $s_1=100$ $100$ , $s_2=80$ $80$ , $s_3=40$ $40$ , $z=0$ $0$ .

$z-3x_1-2x_2=0$ gives initial objective coefficients $-3,-2$ .
The final main-example row is $z+(7/5)s_1+(1/5)s_2=17$ $17$ .

Prerequisites¶

Sources¶

OR Modelling · Week 2

Aliases: simplex table Keywords: tableau Symbols: sum

Last updated: 27 October 2026 11:52