Binary Variable¶

Definition¶

A binary variable is a decision variable restricted to $\{0,1\}$ . For a yes-or-no choice $j$ , the course convention is

y_j=\begin{cases}1,&\text{if decision }j\text{ is yes},\\0,&\text{if decision }j\text{ is no}.\end{cases}

Interpretation¶

A binary variable turns a yes-or-no decision into a number, 1 for yes and 0 for no, so that the decision can appear in linear objectives and constraints. Its practical meaning must be stated explicitly: which decision it describes and which value means yes.

A number line from 0 to 1 with the two allowed values of a binary variable as filled points, 0 for no and 1 for yes, and the whole interval shaded as its relaxation.

Only the two endpoints are allowed values. The light band between them is the relaxation $0\le y\le1$ $1$ , which also admits values such as $0.4$ that correspond to no decision at all.

The 0–1 coding gives linear expressions a logical meaning. The sum $\displaystyle \sum_jy_j$ counts the yes decisions, $\displaystyle \sum_jc_jy_j$ adds up the costs $c_j$ of the chosen alternatives only, $y_1+y_2\le1$ allows at most one of two alternatives, and $y_2\le y_1$ allows alternative 2 only if alternative 1 is chosen.

Key Results¶

The continuous bounds $0\le y_j\le1$ alone do not enforce binarity.
Summing binary variables counts the selected decisions.
Auxiliary binary variables can activate or relax constraints.

Examples¶

The red points are the allowed values 0 and 1; the shaded interval belongs to the continuous relaxation.

A station-location variable is 1 exactly when an ambulance station is built in that town.
$y_1+y_2\le1$ prohibits selecting both alternatives.

Prerequisites¶

Decision Variable

Sources¶

OR Modelling · Week 2

Aliases: 0-1 variable, binary decision variable, zero-one variable Symbols: sum

Last updated: 20 October 2026 03:02