Augmented Form¶

Definition¶

The augmented form of the course's initial LP models replaces each $\le$ functional constraint by an equality with a nonnegative slack variable. For $\boldsymbol{A}\boldsymbol{x}\le \boldsymbol{b}$ $b$ , $\boldsymbol{x}\ge0$ $0$ , this gives

\boldsymbol{A}\boldsymbol{x}+\boldsymbol{s}=\boldsymbol{b},\qquad \boldsymbol{x}\ge0,\quad \boldsymbol{s}\ge0,

with the same objective in the original variables.

Interpretation¶

The augmented form rewrites each $\le$ constraint as an equation by giving the unused amount a name, the slack variable. Adding slacks changes the algebraic representation, not the allowed original decisions: each feasible choice of the original variables determines exactly one value of every slack.

Three inequality constraints of a production problem turned into equations by adding slack variables s1, s2 and s3, shown side by side.

The figure applies this to the three constraints of the Giapetto model: every row gets its own slack, and the slacks receive the same nonnegativity requirement as $x_s$ and $x_t$ . At the production plan $(x_s,x_t)=(20,60)$ $(20, 60)$ , for example, the slacks are $s_1=0$ $0$ , $s_2=0$ and $s_3=20$ $20$ : finishing and carpentry hours are used up, while the soldier-demand limit has 20 units to spare.

The requirement $s_i\ge0$ is what keeps the meaning of the inequality: $s_1=100-2x_s-x_t\ge0$ holds exactly when $2x_s+x_t\le100$ $100$ . Without it, the equation could be satisfied by any $x_s$ and $x_t$ and would restrict nothing. The objective gives every slack coefficient zero, since unused capacity earns nothing.

Equations are what the simplex method needs: it fixes some variables at zero and solves the equations for the others, which gives a basic solution. Constraints of type $\ge$ are turned into equations with a subtracted surplus variable instead, and rows without an obvious starting value then receive an artificial variable.

Key Results¶

Auxiliary variables count as variables of the equality system from which a basis is chosen.
For $m$ independent equalities in $n$ augmented variables, a basis contains $m$ variables.
Artificial variables used for other constraint types require a feasibility phase; they cannot simply remain free in the final model.

Examples¶

In the figure, $s_1$ measures unused finishing hours, $s_2$ unused carpentry hours, and $s_3$ unused soldier-demand allowance. The three equations contain five variables, so a basis sets two nonbasic variables to zero.

Two original variables and two slack variables give four augmented variables.
The constraints $2x_1+x_2+s_1=10$ and $x_1+3x_2+s_2=15$ have initial solution $(0,0,10,15)$ .

Prerequisites¶

Sources¶

OR Modelling · Week 2

Aliases: slack form

Last updated: 20 October 2026 03:02