Nonbasic Variable¶

Definition¶

A nonbasic variable is a variable outside the chosen basis and is set to zero when the associated basic solution is read. For $m$ independent equations in $n$ variables, there are $n-m$ nonbasic variables.

Interpretation¶

The nonbasic variables are the ones fixed at zero to read off a basic solution; with $m$ equations in $n$ variables there are $n-m$ of them. They are the directions in which the simplex method can move: it raises one nonbasic variable from zero, and the basic values change to keep the equations satisfied.

An optimal simplex tableau with the columns of the nonbasic variables s1 and s2, which are 0, highlighted.

At the optimal Giapetto plan $(20,60)$ the nonbasic variables are $s_1$ and $s_2$ , so the finishing and carpentry constraints are both used up. Their columns show what raising each one would do. In particular, the objective row reads $z+s_1+s_2=180$ $180$ , so raising either lowers $z$ , which is why this tableau is optimal.

Geometrically, a nonbasic slack at zero means that its constraint holds with equality, and a nonbasic decision variable at zero means that the point lies on an axis. The nonbasic variables therefore name the boundary lines that meet at the current corner. At each pivot, the entering variable is chosen among the nonbasic variables and becomes basic, and the leaving variable becomes nonbasic, so the method moves to a neighbouring corner.

Being nonbasic is a choice made before the equations are solved, not an observation that a value happens to be zero: a basic variable can be zero too.

Key Results¶

A zero value alone does not establish that a variable is nonbasic; basic variables can also be zero.
The objective row expresses the effect of changing nonbasic variables.
The entering variable becomes basic after a pivot.

Examples¶

The illustrated tableau has $n-m=5-3=2$ nonbasic variables, $s_1,s_2$ . Raising $s_1$ by 1 changes $x_t$ and $s_3$ by $+1$ , $x_s$ by $-1$ and $z$ by $-1$ , while the displayed basis remains feasible.

The original decision variables are nonbasic at the initial slack-basis solution.
The final two-variable example sets $s_1=s_2=0$ as its nonbasic variables.

Prerequisites¶

Basic Solution

Sources¶

OR Modelling · Week 2

Last updated: 20 October 2026 03:02