Mutually Exclusive Decisions¶

Definition¶

For binary variables $y_1,\ldots,y_k$ , mutually exclusive decisions allow at most one variable to equal 1:

\sum_{j=1}^ky_j\le1.

If one alternative must be chosen, use $\displaystyle \sum_{j=1}^ky_j=1$ instead.

Interpretation¶

Mutually exclusive decisions describe options of which at most one may be chosen, such as building at most one of two warehouses. Because each $y_j$ is $0$ or $1$ , the sum $\displaystyle \sum_jy_j$ counts the yes decisions, and the constraint is a statement about that count.

Two 2 by 2 tables of two binary decisions: at most one, where only both equal to 1 is forbidden, and exactly one, where both 0 and both 1 are forbidden.

For two decisions the tables list all four combinations. "At most one", $y_1+y_2\le1$ $1$ , forbids only $y_1=y_2=1$ $1$ ; "exactly one", $y_1+y_2=1$ $1$ , also forbids choosing neither. The inequality permits choosing none while the equality requires one choice, so the wording "at most one" or "exactly one" decides which version to use.

The same counting idea gives "at least $K$ of $N$ " in K-out-of-N constraints, and the exactly-one version selects a single listed value in a finite-value constraint.

Key Results¶

Binarity is essential to interpret the sum as the number of yes decisions.
Exactly one implies at most one, but the reverse is false.
This restriction concerns decisions, not an either-or condition on continuous inequalities.

Examples¶

With binaries $W_{LA}$ and $W_{SF}$ equal to 1 if a warehouse is built at location LA or SF, $W_{LA}+W_{SF}\le1$ permits at most one warehouse.
$y_1+y_2+y_3=1$ selects exactly one allowed quantity value.

Prerequisites¶

Binary Variable

Sources¶

OR Modelling · Week 3

Aliases: mutually exclusive alternatives Symbols: sum

Last updated: 20 October 2026 03:02