- Four workers, A, B, C and D, are each to be assigned to one of four tasks, P, Q, R and S.
Each worker must be assigned to one task, and each task must be done by exactly one worker.
Worker C cannot be assigned to task Q and worker D cannot be assigned to task S.
The time, in minutes, that each worker takes to complete each task is shown in the table below.
| \cline { 2 - 5 }
\multicolumn{1}{c|}{} | P | Q | R | S |
| A | 54 | 48 | 51 | 52 |
| B | 55 | 51 | 53 | 58 |
| C | 52 | - | 53 | 54 |
| D | 67 | 63 | 68 | - |
The Hungarian algorithm is to be used to find the minimum total time for the four workers to complete the tasks.
- Modify the table so that the Hungarian algorithm may be used.
- Reducing rows first, use the Hungarian algorithm to obtain an allocation that minimises the total time. You should explain how any initial row and column reductions are made and also how you determine if the table is optimal at each stage.