Constraint¶

Definition¶

A constraint is a requirement that an allowed decision must satisfy. A linear functional constraint has the form

a_{i1}x_1+\cdots+a_{in}x_n\le b_i,

or the corresponding equality or $\ge$ inequality. The $a_{ij}$ are constraint coefficients and $b_i$ is the right-hand side.

Interpretation¶

Constraints translate capacity limits, demand requirements, and other restrictions into mathematics. Feasibility requires all constraints to hold together.

A linear program written out with its parts labelled: the objective, the functional constraints and the domain restrictions; decision variables are coloured blue and parameters red.

The objective function $f(\boldsymbol{x})$ is the quantity to maximize (for example, profit) or minimize (for example, cost). Functional constraints describe what an allowed decision must satisfy, such as resource limits or demand requirements. Domain restrictions specify signs or integrality.

In the diagram, blue identifies decision variables, whose values the model determines; red identifies model parameters, the given data $c_j$ , $a_{ij}$ and $b_i$ . In general, the model is $\max/\min\ f(\boldsymbol{x})$ subject to $\boldsymbol{x}\in F$ $F$ , where the constraints define the feasible set $F$ . A solution assigns values to all variables.

Key Results¶

A capacity limit commonly gives a $\le$ constraint; a minimum requirement commonly gives a $\ge$ constraint.
An equality represents an exact requirement or balance.
Multiplication by a negative number reverses an inequality.

Examples¶

$x_s+x_t\le80$ limits carpentry use in the Giapetto production model.
$800x_A+1000x_B\ge8000$ requires enough calories in the diet problem, where $x_A,x_B$ are the pounds of feeds A and B.

Prerequisites¶

Decision Variable

Sources¶

OR Modelling · Week 1

Symbols: element of

Last updated: 17 October 2026 21:53