Solution

This is the cost matrix.

48	89	94	65	40
47	56	82	60	5
24	32	9	5	75
18	88	90	53	36
13	60	57	73	3

Subtract row minima

For each row, the minimum element is subtracted from all elements in that row.

8	49	54	25	0	(-40)
42	51	77	55	0	(-5)
19	27	4	0	70	(-5)
0	70	72	35	18	(-18)
10	57	54	70	0	(-3)

Subtract column minima

For each column, the minimum element is subtracted from all elements in that column.

8	22	50	25	0
42	24	73	55	0
19	0	0	0	70
0	43	68	35	18
10	30	50	70	0
	(-27)	(-4)

Cover all zeros with a minimum number of lines

A total of 3 lines are required to cover all zeros.

8	22	50	25	0
42	24	73	55	0
19	0	0	0	70	x
0	43	68	35	18	x
10	30	50	70	0
				x

Create additional zeros

The number of lines is smaller than 5. The smallest uncovered element is 8. We subtract this value from all uncovered elements and add it to all elements covered twice.

0	14	42	17	0
34	16	65	47	0
19	0	0	0	78
0	43	68	35	26
2	22	42	62	0

Cover all zeros with a minimum number of lines

A total of 3 lines are required to cover all zeros.

0	14	42	17	0
34	16	65	47	0
19	0	0	0	78	x
0	43	68	35	26
2	22	42	62	0
x				x

Create additional zeros

The number of lines is smaller than 5. The smallest uncovered element is 14. We subtract this value from all uncovered elements and add it to all elements covered twice.

0	0	28	3	0
34	2	51	33	0
33	0	0	0	92
0	29	54	21	26
2	8	28	48	0

Cover all zeros with a minimum number of lines

A total of 4 lines are required to cover all zeros.

0	0	28	3	0	x
34	2	51	33	0
33	0	0	0	92	x
0	29	54	21	26	x
2	8	28	48	0
				x

Create additional zeros

The number of lines is smaller than 5. The smallest uncovered element is 2. We subtract this value from all uncovered elements and add it to all elements covered twice.

0	0	28	3	2
32	0	49	31	0
33	0	0	0	94
0	29	54	21	28
0	6	26	46	0

Cover all zeros with a minimum number of lines

A total of 4 lines are required to cover all zeros.

0	0	28	3	2
32	0	49	31	0
33	0	0	0	94	x
0	29	54	21	28
0	6	26	46	0
x	x			x

Create additional zeros

The number of lines is smaller than 5. The smallest uncovered element is 3. We subtract this value from all uncovered elements and add it to all elements covered twice.

0	0	25	0	2
32	0	46	28	0
36	3	0	0	97
0	29	51	18	28
0	6	23	43	0

Cover all zeros with a minimum number of lines

A total of 5 lines are required to cover all zeros.

0	0	25	0	2	x
32	0	46	28	0	x
36	3	0	0	97	x
0	29	51	18	28	x
0	6	23	43	0	x

The optimal assignment

Because there are 5 lines required, an optimal assignment exists among the zeros.

0	0	25	0	2
32	0	46	28	0
36	3	0	0	97
0	29	51	18	28
0	6	23	43	0

This corresponds to the following optimal assignment in the original cost matrix.

48	89	94	65	40
47	56	82	60	5
24	32	9	5	75
18	88	90	53	36
13	60	57	73	3

The total minimum cost is 151.

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Solution

Subtract row minima

Subtract column minima

Cover all zeros with a minimum number of lines

Create additional zeros

Cover all zeros with a minimum number of lines

Create additional zeros

Cover all zeros with a minimum number of lines

Create additional zeros

Cover all zeros with a minimum number of lines

Create additional zeros

Cover all zeros with a minimum number of lines

The optimal assignment