2 The following table shows the times taken, in minutes, by five people, Ron, Sam, Tim, Vic and Zac, to carry out the tasks \(1,2,3\) and 4 . Sam takes \(x\) minutes, where \(8 \leqslant x \leqslant 12\), to do task 2.
| Ron | Sam | Tim | Vic | Zac |
| Task 1 | 8 | 7 | 9 | 10 | 8 |
| Task 2 | 9 | \(x\) | 8 | 7 | 11 |
| Task 3 | 12 | 10 | 9 | 9 | 10 |
| Task 4 | 11 | 9 | 8 | 11 | 11 |
Each of the four tasks is to be given to a different one of the five people so that the total time for the four tasks is minimised.
- Modify the table of values by adding an extra row of non-zero values so that the Hungarian algorithm can be applied.
- Use the Hungarian algorithm, reducing columns first and then rows, to reduce the matrix to a form, in terms of \(x\), from which the optimum matching can be made.
- Hence find the possible way of allocating the four tasks so that the total time is minimised.
- Find the minimum total time.
- After special training, Sam is able to complete task 2 in 7 minutes and is assigned to task 2.
Determine the possible ways of allocating the other three tasks so that the total time is minimised.