Question

A plant manager has four subordinates and four tasks to perform. The subordinates differ in efficiency and task differ in their intrinsic difficulty. Estimates of the time subordinate would take to perform tasks are given in the following table:

	I	II	III	IV
A	3	11	10	8
B	13	2	12	2
C	3	4	6	1
D	4	15	4	9

Complete the following activity to allocate tasks to subordinates to minimize total time.

Solution:

Step I: Subtract the smallest element of each row from every element of that row:

	I	II	III	IV
A	0	8	7	5
B	11	0	10	0
C	2	3	5	0
D	0	11	0	5

Step II: Since all column minimums are zero, no need to subtract anything from columns.

Step III: Draw the minimum number of lines to cover all zeros.

	I	II	III	IV
A	0	8	7	5
B	11	0	10	0
C	2	3	5	0
D	0	11	0	5

Since minimum number of lines = order of matrix, optimal solution has been reached

Optimal assignment is A →`square`  B →`square`

C →IV  D →`square`

Total minimum time = `square` hours.

Answer 1

Step I: Subtract the smallest element of each row from every element of that row:

	I	II	III	IV
A	0	8	7	5
B	11	0	10	0
C	2	3	5	0
D	0	11	0	5

Step II: Since all column minimums are zero, no need to subtract anything from columns.

Step III: Draw the minimum number of lines to cover all zeros.

	I	II	III	IV
A	0	8	7	5
B	11	0	10	0
C	2	3	5	0
D	0	11	0	5

Since minimum number of lines = order of matrix, optimal solution has been reached

	I	II	III	IV
A	0	8	7	5
B	11	0	10	0
C	2	3	5	0
D	0	11	0	5

Optimal assignment is A →I  B →II

C →IV  D →III

Total minimum time = 10 hours.