2. A team of 5 players, A, B, C, D and E, competes in a quiz. Each player must answer one of 5 rounds, \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }\) and T .
Each player must be assigned to exactly one round, and each round must be answered by exactly one player.
Player B cannot answer round Q, player D cannot answer round T, and player E cannot answer round R.
The number of points that each player is expected to earn in each round is shown in the table.
| \cline { 2 - 6 }
\multicolumn{1}{c|}{} | \(\mathbf { P }\) | \(\mathbf { Q }\) | \(\mathbf { R }\) | \(\mathbf { S }\) | \(\mathbf { T }\) |
| \(\mathbf { A }\) | 32 | 40 | 35 | 41 | 37 |
| \(\mathbf { B }\) | 38 | - | 40 | 27 | 33 |
| \(\mathbf { C }\) | 41 | 28 | 37 | 36 | 35 |
| \(\mathbf { D }\) | 35 | 33 | 38 | 36 | - |
| \(\mathbf { E }\) | 40 | 38 | - | 39 | 34 |
The team wants to maximise its total expected score.
The Hungarian algorithm is to be used to find the maximum total expected score that can be earned by the 5 players.
- Explain how the table should be modified.
- Reducing rows first, use the Hungarian algorithm to obtain an allocation which maximises the total expected score.
- Calculate the maximum total expected score.