Surplus variables
Surplus Variables
A surplus variable converts a $\ge$ inequality into an equality.
It measures how much the left-hand side exceeds the required minimum.
Basic Rule
For a constraint:
\[a^Tx\ge b\]introduce a new variable:
\[s\ge 0\]and write:
\[a^Tx-s=b\]The surplus variable is:
\[s=a^Tx-b\]Because $a^Tx\ge b$, we have:
\[s\ge 0\]Interpretation
Surplus is the amount above a minimum requirement.
For a demand constraint:
\[\text{production}\ge \text{minimum demand}\]surplus means:
\[\text{extra production beyond minimum demand}\]Example
Suppose:
\[3x_1+2x_2\ge 60\]Add surplus variable $s_1\ge 0$:
\[3x_1+2x_2-s_1=60\]Then:
\[s_1=3x_1+2x_2-60\]If $s_1=0$, the requirement is exactly met.
If $s_1>0$, the requirement is exceeded.
Another Example
For:
\[x_1+x_2\ge 3\]standard-form equality is:
\[x_1+x_2-s=3\]with:
\[s\ge 0\]Surplus vs Slack
| Constraint type | Variable type | Equality form |
|---|---|---|
| $a^Tx\le b$ | slack | $a^Tx+s=b$ |
| $a^Tx\ge b$ | surplus | $a^Tx-s=b$ |
Slack is unused capacity.
Surplus is excess above a requirement.
Active Constraints
If a $\ge$ constraint is active, then its surplus variable is zero.
For:
\[a^Tx-s=b\]if:
\[a^Tx=b\]then:
\[s=0\]If:
\[a^Tx>b\]then:
\[s>0\]Matrix Form
A surplus variable adds a column with coefficient $-1$ in its constraint row.
Example:
\[3x_1+2x_2-s_1=60\]has row:
\[\begin{bmatrix}3&2&-1\end{bmatrix}\]if the variables are:
\[(x_1,x_2,s_1)\]Common Mistake
For a $\ge$ constraint, subtract surplus.
Correct:
\[a^Tx-s=b, \qquad s\ge 0\]Wrong:
\[a^Tx+s=b, \qquad s\ge 0\]because that would imply $a^Tx\le b$.
Note About Initial Bases
Slack variables in $\le$ constraints often create identity columns.
Surplus variables create $-1$ columns, so they do not automatically give the same simple initial basis.
This matters later in simplex initialization.
Checklist
You understand surplus variables if you can:
- convert $a^Tx\ge b$ into $a^Tx-s=b$
- explain surplus as amount above a minimum
- distinguish slack from surplus
- identify active constraints by zero surplus
- write surplus columns in matrix form
See Also
Exam checkpoint
For standard-form questions, use the course convention $\min c^Tx$ subject to $Ax=b$, $x\ge0$. Convert max objectives, add slack to $\le$, subtract surplus from $\ge$, and split free variables.