Finite-Value Constraint¶

Definition¶

A linear expression can be restricted to one of the listed values $d_1,\ldots,d_N$ by introducing binary variables $y_i$ and writing

a_1x_1+\cdots+a_nx_n=\sum_{i=1}^Nd_iy_i,\qquad \sum_{i=1}^Ny_i=1,\qquad y_i\in\{0,1\}.

Interpretation¶

A finite-value constraint lets a linear expression take only a few listed values, such as $x_1+x_2\in\{3,5,8\}$ ${3, 5, 8}$ , which linear inequalities alone cannot express. Exactly one indicator selects the allowed value, so no separate alternative constraint is needed for every possible value.

$A number line for x1 + x2 with the three allowed values 3, 5 and 8 marked, one for each binary selector, and a light band of the values reachable if the selectors were fractions.$

The red points are the three allowed totals, each produced by one selector: $y_1=1$ gives $3$ , $y_2=1$ gives $5$ and $y_3=1$ gives $8$ , as in the first Example. The light band from $3$ to $8$ shows what the right-hand side $3y_1+5y_2+8y_3$ could reach if the selectors were fractions; $y_1=y_3=\tfrac{1}{2}$ $\frac{1}{2}$ , for instance, gives $5.5$ .

Each part of the formulation has a role. Because the $y_i$ are binary and sum to $1$ , they are mutually exclusive decisions of the exactly-one type: one $y_i$ equals $1$ , the others are $0$ , and $\displaystyle \sum_id_iy_i$ collapses to the single selected value $d_i$ . The values $d_i$ are fixed data, which keeps every term $d_iy_i$ linear.

A common slip is to write $\displaystyle \sum_iy_i\le1$ instead. Then all selectors may be $0$ , and the expression may equal $0$ , a value that is not on the list. The parity constraint can be read as a finite-value constraint with listed values $0$ and $2$ , written more compactly with one auxiliary variable.

Key Results¶

The sum-to-one equality ensures that a value is actually selected.
The $d_i$ are fixed data, so $d_iy_i$ is linear.
Relaxing the indicators to continuous fractions would generally allow values between the listed values.

Examples¶

The light band shows intermediate totals made possible by fractional selector values; the binary selectors permit only the listed totals.

For allowed totals 3, 5, or 8, write $x_1+x_2=3y_1+5y_2+8y_3$ with $y_1+y_2+y_3=1$ and $y_i\in\{0,1\}$ ${0, 1}$ .
An even number of three yes decisions can take only the values 0 and 2.

Prerequisites¶

Parity Constraint

Sources¶

OR Modelling · Week 3

Symbols: element of, sum

Last updated: 27 October 2026 11:52