\( \newcommand{\lra}{\leftrightarrow} \newcommand{\ll}{\mathcal L} \newcommand{\ff}{\mathcal F} \definecolor{cornellred}{RGB}{179,27,27} \newcommand{\cred}[1]{\textcolor{cornellred}{#1}} \newcommand{\tcred}[1]{\textcolor{cornellred}{\textrm{#1}}} \)

Example 2

Consider the surplus matrix
$\hspace{-6pt}\cred\ff$
$\cred 1$ $\cred 2$
$\cred\ll$ $\cred 1$ $10$ $9$
$\cred 2$ $9$ $3$

The optimal match is $\tiny 1\lra 2, 2\lra 1$ with a surplus of 18. The dual constraints are \begin{align*} \scriptscriptstyle w_{\tiny 1}+\pi_{\tiny 1}&\scriptscriptstyle\geq 10&\cred{\scriptscriptstyle w_{\tiny 1}+\pi_{\tiny 2}}&\cred{\scriptscriptstyle \geq 9}\\ \cred{\scriptscriptstyle w_{\tiny 2}+\pi_{\tiny 1}}&\cred{\scriptscriptstyle \geq 9}&\scriptscriptstyle w_{\tiny 2}+\pi_{\tiny 2}&\scriptscriptstyle \geq 3 \end{align*} and the binding constraints are in red.