3 A theatre company needs to employ three technicians for a performance. One will operate the lights, one the sound system and one the flying trapeze mechanism. Four technicians have applied for these tasks. The table shows how much it will cost the theatre company, in £, to employ each technician for each task. The theatre company wants to employ the three technicians for whom the total cost is least.
The Hungarian algorithm is to be used to find the minimum cost allocation, but before this can be done the table needs to be modified.

1. Make the necessary modification to the table. Working from your modified table, construct a reduced cost matrix by first reducing rows and then reducing columns. You should show the amount by which each row has been reduced in the row reductions and the amount by which each column has been reduced in the column reductions. Cross through the 0 's in your reduced cost matrix using the least possible number of horizontal or vertical lines. [You must ensure that the values in your table can still be read.]
2. Complete the application of the Hungarian algorithm to find a minimum cost allocation. Write a list showing which technician should be employed for each task. Calculate the total cost to the theatre company. Although Amir put in the lowest cost for operating the lighting, you should have found that he has not been allocated this task. Amir is particularly keen to be employed to operate the lights so is prepared to reduce his cost for this task.
3. Find a way to use two of Bex, Caz and Dee to operate the sound effects and the flying trapeze mechanism at the lowest cost. Hence find what Amir's new cost should be for the minimum total cost to the theatre company to be exactly \(\pounds 1\) less than your answer from part (ii).