Basic Variable¶

Definition¶

A basic variable belongs to the selected basis of an equality-form LP. Its value is obtained from the constraint equations after the nonbasic variables have been set to zero. With $m$ independent equalities, the basis contains $m$ variables.

Interpretation¶

In a basic solution the variables fall into two groups: the nonbasic variables are fixed at zero, and the basic variables take whatever values the constraint equations then force. Being basic describes a variable's role in the current representation; it does not mean that the variable must be positive.

A simplex tableau in which the columns of the basic variables x t, s3 and x s are unit columns, with their values 60, 20 and 20 in the right-hand side column.

In this tableau each basic variable, $x_t$ , $s_3$ and $x_s$ , has a column with a single 1 in its own row and zeros elsewhere, including the $z$ -row. With the nonbasic $s_1=s_2=0$ $0$ , every row therefore reads "basic variable equals right-hand side": $x_t=60$ $60$ , $s_3=20$ $20$ , $x_s=20$ and $z=180$ $180$ . This is the optimal plan of the Giapetto model, with 20 units of soldier demand left unused.

There are as many basic variables as constraint equations, here three, regardless of how many decision variables the model has, and slack variables can be basic, as $s_3$ is here. At each simplex pivot the set of basic variables changes by one: the variable chosen by the entering variable rule becomes basic, and the one chosen by the minimum-ratio test becomes nonbasic.

A basic variable may still take the value zero: what makes it basic is that its value is solved from the equations rather than fixed at zero in advance.

Key Results¶

In a canonical tableau, a basic variable has coefficient 1 in its row and 0 in the other rows, including the objective row.
A zero-valued basic variable makes the basic feasible solution degenerate.
A pivot replaces one member of the basis.

Examples¶

In the illustrated canonical tableau, the basic variables are $x_t,s_3,x_s$ . Their unit columns let their values $60,20,20$ be read from the right-hand side when $s_1=s_2=0$ $0$ .

The initial simplex example uses the two slacks as basic variables.
At the optimal basis, $x_1=3$ and $x_2=4$ are basic.

Prerequisites¶

Basic Solution

Sources¶

OR Modelling · Week 2

Last updated: 20 October 2026 03:02