Surplus Variable¶

Definition¶

A surplus variable $S_i\ge0$ transforms a lower-bound constraint into an equality:

\sum_ja_{ij}x_j\ge b_i \Longleftrightarrow \sum_ja_{ij}x_j-S_i=b_i,\quad S_i\ge0.

It measures the excess of the left-hand side over the required minimum.

Interpretation¶

A surplus variable measures by how much a minimum requirement is exceeded: the amount $\displaystyle \sum_ja_{ij}x_j$ achieved minus the required $b_i$ . Subtracting it turns a $\ge$ constraint into an equation, just as an added slack does for a $\le$ constraint.

A feasible region above the line x A + 2x B = 6, with a point above the line that has surplus 4, a point on the line with surplus 0, and a point below it that is infeasible.

The figure shows $x_A+2x_B\ge6$ written as $x_A+2x_B-S=6$ $6$ . The point $(4,3)$ gives $10$ , so $S=4$ $4$ ; the point $(2,2)$ lies on the line with $S=0$ $0$ ; the point $(1,1)$ gives only $3$ and would need $S=-3$ $- 3$ , which the sign restriction forbids, so it is infeasible.

The minus sign follows from what is measured, and the nonnegativity constraint $S_i\ge0$ carries the original direction; writing $+S_i$ by mistake gives the opposite requirement $\displaystyle \sum_ja_{ij}x_j\le b_i$ $b_{i}$ . For a positive right-hand side $b_i>0$ $0$ , setting the decision variables to zero gives $S_i=-b_i<0$ $0$ , so the surplus alone does not provide a feasible starting basis for the simplex method; an artificial variable is commonly added. If $b_i=0$ $0$ , a zero surplus can instead occur in a degenerate feasible starting basis.

Key Results¶

The course uses capital $S_i$ for surplus and lower-case $s_i$ for slack.
A surplus variable alone generally does not produce an initial feasible slack-style basis.
An artificial variable may be added for the first simplex phase.

Examples¶

$2x_1+x_2\ge4$ becomes $2x_1+x_2-S_1=4$ $4$ .
At $(x_1,x_2)=(3,0)$ $(3, 0)$ , this surplus is 2.

Prerequisites¶

Sources¶

OR Modelling · Week 2

Aliases: excess variable Symbols: sum

Last updated: 27 October 2026 11:52