Solve the following minimal assignment problem
Jobs Jobs Jobs
Machines I II III
$M_{1}$ $1$ $4$ $5$
$M_{2}$ $4$ $2$ $7$
$M_{3}$ $7$ $8$ $3$

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Solution

$\begin{matrix} I & I I & I I I M_{1} & 1 & 4 & 5 M_{2} & 4 & 2 & 7 M_{3} & 7 & 8 & 3 \end{matrix}$
Apply row reduction
$\begin{matrix} I & I I & I I I M_{1} & 0 & 3 & 4 M_{2} & 2 & 0 & 5 M_{3} & 4 & 5 & 0 \end{matrix}$
Apply column reduction (refer image)
Degeneracy check $=$ No. of crossed lines $= 3$
$=$ order of matrix
So, $M_{1} = I$
$M_{2} = I I$
$M_{3} = I I I$
Minimal time $= 1 + 2 + 3 = 6$

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