Discussion
Four workers are required to be assigned to 4 tasks. The number of hours required to complete the tasks by each of the workers is given in the following table (rows are tasks, columns are workers) :
| 20 | 30 | 40 | 50 |
| 40 | 50 | 60 | 70 |
| 70 | 80 | 90 | 80 |
| 30 | 50 | 80 | 40 |
Then, which of the following is true ?
Answer: A
The minimum time is 200 hours.
One optimal assignment takes the diagonal: 20 + 50 + 90 + 40 = 200. The Hungarian method confirms the minimum. Subtracting row minima (20, 40, 70, 30) and then column minima leaves
\[ \begin{pmatrix} 0 & 0 & 0 & 20 \\ 0 & 0 & 0 & 20 \\ 0 & 0 & 0 & 0 \\ 0 & 10 & 30 & 0 \end{pmatrix} \]
and a full assignment through zeros exists (for example the diagonal), so the minimum total is the sum of the subtracted amounts, \(20 + 40 + 70 + 30 + 0 + 10 + 20 + 10 = 200\). Several assignments reach 200. So 210 is not the minimum, and the statements about a basic feasible solution do not describe this result.
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