Solution

This is the cost matrix.

47	58	79	24	53	36	94	70
11	97	97	62	45	38	6	78
93	22	11	28	73	62	67	63
95	8	26	55	89	61	60	69
25	81	45	3	11	11	65	88
85	22	62	48	31	99	40	95
75	19	39	2	67	80	18	4
46	50	42	25	24	86	61	68

Subtract row minima

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

23	34	55	0	29	12	70	46	(-24)
5	91	91	56	39	32	0	72	(-6)
82	11	0	17	62	51	56	52	(-11)
87	0	18	47	81	53	52	61	(-8)
22	78	42	0	8	8	62	85	(-3)
63	0	40	26	9	77	18	73	(-22)
73	17	37	0	65	78	16	2	(-2)
22	26	18	1	0	62	37	44	(-24)

Subtract column minima

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

18	34	55	0	29	4	70	44
0	91	91	56	39	24	0	70
77	11	0	17	62	43	56	50
82	0	18	47	81	45	52	59
17	78	42	0	8	0	62	83
58	0	40	26	9	69	18	71
68	17	37	0	65	70	16	0
17	26	18	1	0	54	37	42
(-5)					(-8)		(-2)

Cover all zeros with a minimum number of lines

A total of 7 lines are required to cover all zeros.

18	34	55	0	29	4	70	44	x
0	91	91	56	39	24	0	70	x
77	11	0	17	62	43	56	50	x
82	0	18	47	81	45	52	59
17	78	42	0	8	0	62	83	x
58	0	40	26	9	69	18	71
68	17	37	0	65	70	16	0	x
17	26	18	1	0	54	37	42	x
	x

Create additional zeros

The number of lines is smaller than 8. The smallest uncovered element is 9. We subtract this value from all uncovered elements and add it to all elements covered twice.

18	43	55	0	29	4	70	44
0	100	91	56	39	24	0	70
77	20	0	17	62	43	56	50
73	0	9	38	72	36	43	50
17	87	42	0	8	0	62	83
49	0	31	17	0	60	9	62
68	26	37	0	65	70	16	0
17	35	18	1	0	54	37	42

Cover all zeros with a minimum number of lines

A total of 7 lines are required to cover all zeros.

18	43	55	0	29	4	70	44	x
0	100	91	56	39	24	0	70	x
77	20	0	17	62	43	56	50	x
73	0	9	38	72	36	43	50
17	87	42	0	8	0	62	83	x
49	0	31	17	0	60	9	62
68	26	37	0	65	70	16	0	x
17	35	18	1	0	54	37	42
	x			x

Create additional zeros

The number of lines is smaller than 8. The smallest uncovered element is 1. We subtract this value from all uncovered elements and add it to all elements covered twice.

18	44	55	0	30	4	70	44
0	101	91	56	40	24	0	70
77	21	0	17	63	43	56	50
72	0	8	37	72	35	42	49
17	88	42	0	9	0	62	83
48	0	30	16	0	59	8	61
68	27	37	0	66	70	16	0
16	35	17	0	0	53	36	41

Cover all zeros with a minimum number of lines

A total of 7 lines are required to cover all zeros.

18	44	55	0	30	4	70	44
0	101	91	56	40	24	0	70	x
77	21	0	17	63	43	56	50	x
72	0	8	37	72	35	42	49
17	88	42	0	9	0	62	83	x
48	0	30	16	0	59	8	61
68	27	37	0	66	70	16	0	x
16	35	17	0	0	53	36	41
	x		x	x

Create additional zeros

The number of lines is smaller than 8. The smallest uncovered element is 4. We subtract this value from all uncovered elements and add it to all elements covered twice.

14	44	51	0	30	0	66	40
0	105	91	60	44	24	0	70
77	25	0	21	67	43	56	50
68	0	4	37	72	31	38	45
17	92	42	4	13	0	62	83
44	0	26	16	0	55	4	57
68	31	37	4	70	70	16	0
12	35	13	0	0	49	32	37

Cover all zeros with a minimum number of lines

A total of 7 lines are required to cover all zeros.

14	44	51	0	30	0	66	40
0	105	91	60	44	24	0	70	x
77	25	0	21	67	43	56	50	x
68	0	4	37	72	31	38	45
17	92	42	4	13	0	62	83
44	0	26	16	0	55	4	57
68	31	37	4	70	70	16	0	x
12	35	13	0	0	49	32	37
	x		x	x	x

Create additional zeros

The number of lines is smaller than 8. The smallest uncovered element is 4. We subtract this value from all uncovered elements and add it to all elements covered twice.

10	44	47	0	30	0	62	36
0	109	91	64	48	28	0	70
77	29	0	25	71	47	56	50
64	0	0	37	72	31	34	41
13	92	38	4	13	0	58	79
40	0	22	16	0	55	0	53
68	35	37	8	74	74	16	0
8	35	9	0	0	49	28	33

Cover all zeros with a minimum number of lines

A total of 8 lines are required to cover all zeros.

10	44	47	0	30	0	62	36	x
0	109	91	64	48	28	0	70	x
77	29	0	25	71	47	56	50	x
64	0	0	37	72	31	34	41	x
13	92	38	4	13	0	58	79	x
40	0	22	16	0	55	0	53	x
68	35	37	8	74	74	16	0	x
8	35	9	0	0	49	28	33	x

The optimal assignment

Because there are 8 lines required, an optimal assignment exists among the zeros.

10	44	47	0	30	0	62	36
0	109	91	64	48	28	0	70
77	29	0	25	71	47	56	50
64	0	0	37	72	31	34	41
13	92	38	4	13	0	58	79
40	0	22	16	0	55	0	53
68	35	37	8	74	74	16	0
8	35	9	0	0	49	28	33

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

47	58	79	24	53	36	94	70
11	97	97	62	45	38	6	78
93	22	11	28	73	62	67	63
95	8	26	55	89	61	60	69
25	81	45	3	11	11	65	88
85	22	62	48	31	99	40	95
75	19	39	2	67	80	18	4
46	50	42	25	24	86	61	68

The total minimum cost is 133.

47	58	79	24	53	36	94	70
11	97	97	62	45	38	6	78
93	22	11	28	73	62	67	63
95	8	26	55	89	61	60	69
25	81	45	3	11	11	65	88
85	22	62	48	31	99	40	95
75	19	39	2	67	80	18	4
46	50	42	25	24	86	61	68

18	34	55	0	29	4	70	44	x
0	91	91	56	39	24	0	70	x
77	11	0	17	62	43	56	50	x
82	0	18	47	81	45	52	59
17	78	42	0	8	0	62	83	x
58	0	40	26	9	69	18	71
68	17	37	0	65	70	16	0	x
17	26	18	1	0	54	37	42	x
	x

18	43	55	0	29	4	70	44
0	100	91	56	39	24	0	70
77	20	0	17	62	43	56	50
73	0	9	38	72	36	43	50
17	87	42	0	8	0	62	83
49	0	31	17	0	60	9	62
68	26	37	0	65	70	16	0
17	35	18	1	0	54	37	42

18	43	55	0	29	4	70	44	x
0	100	91	56	39	24	0	70	x
77	20	0	17	62	43	56	50	x
73	0	9	38	72	36	43	50
17	87	42	0	8	0	62	83	x
49	0	31	17	0	60	9	62
68	26	37	0	65	70	16	0	x
17	35	18	1	0	54	37	42
	x			x

18	44	55	0	30	4	70	44
0	101	91	56	40	24	0	70
77	21	0	17	63	43	56	50
72	0	8	37	72	35	42	49
17	88	42	0	9	0	62	83
48	0	30	16	0	59	8	61
68	27	37	0	66	70	16	0
16	35	17	0	0	53	36	41

18	44	55	0	30	4	70	44
0	101	91	56	40	24	0	70	x
77	21	0	17	63	43	56	50	x
72	0	8	37	72	35	42	49
17	88	42	0	9	0	62	83	x
48	0	30	16	0	59	8	61
68	27	37	0	66	70	16	0	x
16	35	17	0	0	53	36	41
	x		x	x

14	44	51	0	30	0	66	40
0	105	91	60	44	24	0	70
77	25	0	21	67	43	56	50
68	0	4	37	72	31	38	45
17	92	42	4	13	0	62	83
44	0	26	16	0	55	4	57
68	31	37	4	70	70	16	0
12	35	13	0	0	49	32	37

14	44	51	0	30	0	66	40
0	105	91	60	44	24	0	70	x
77	25	0	21	67	43	56	50	x
68	0	4	37	72	31	38	45
17	92	42	4	13	0	62	83
44	0	26	16	0	55	4	57
68	31	37	4	70	70	16	0	x
12	35	13	0	0	49	32	37
	x		x	x	x

10	44	47	0	30	0	62	36
0	109	91	64	48	28	0	70
77	29	0	25	71	47	56	50
64	0	0	37	72	31	34	41
13	92	38	4	13	0	58	79
40	0	22	16	0	55	0	53
68	35	37	8	74	74	16	0
8	35	9	0	0	49	28	33

10	44	47	0	30	0	62	36	x
0	109	91	64	48	28	0	70	x
77	29	0	25	71	47	56	50	x
64	0	0	37	72	31	34	41	x
13	92	38	4	13	0	58	79	x
40	0	22	16	0	55	0	53	x
68	35	37	8	74	74	16	0	x
8	35	9	0	0	49	28	33	x

47	58	79	24	53	36	94	70
11	97	97	62	45	38	6	78
93	22	11	28	73	62	67	63
95	8	26	55	89	61	60	69
25	81	45	3	11	11	65	88
85	22	62	48	31	99	40	95
75	19	39	2	67	80	18	4
46	50	42	25	24	86	61	68

18	34	55	0	29	4	70	44	x
0	91	91	56	39	24	0	70	x
77	11	0	17	62	43	56	50	x
82	0	18	47	81	45	52	59
17	78	42	0	8	0	62	83	x
58	0	40	26	9	69	18	71
68	17	37	0	65	70	16	0	x
17	26	18	1	0	54	37	42	x
	x

18	43	55	0	29	4	70	44
0	100	91	56	39	24	0	70
77	20	0	17	62	43	56	50
73	0	9	38	72	36	43	50
17	87	42	0	8	0	62	83
49	0	31	17	0	60	9	62
68	26	37	0	65	70	16	0
17	35	18	1	0	54	37	42

18	43	55	0	29	4	70	44	x
0	100	91	56	39	24	0	70	x
77	20	0	17	62	43	56	50	x
73	0	9	38	72	36	43	50
17	87	42	0	8	0	62	83	x
49	0	31	17	0	60	9	62
68	26	37	0	65	70	16	0	x
17	35	18	1	0	54	37	42
	x			x

18	44	55	0	30	4	70	44
0	101	91	56	40	24	0	70
77	21	0	17	63	43	56	50
72	0	8	37	72	35	42	49
17	88	42	0	9	0	62	83
48	0	30	16	0	59	8	61
68	27	37	0	66	70	16	0
16	35	17	0	0	53	36	41

18	44	55	0	30	4	70	44
0	101	91	56	40	24	0	70	x
77	21	0	17	63	43	56	50	x
72	0	8	37	72	35	42	49
17	88	42	0	9	0	62	83	x
48	0	30	16	0	59	8	61
68	27	37	0	66	70	16	0	x
16	35	17	0	0	53	36	41
	x		x	x

14	44	51	0	30	0	66	40
0	105	91	60	44	24	0	70
77	25	0	21	67	43	56	50
68	0	4	37	72	31	38	45
17	92	42	4	13	0	62	83
44	0	26	16	0	55	4	57
68	31	37	4	70	70	16	0
12	35	13	0	0	49	32	37

14	44	51	0	30	0	66	40
0	105	91	60	44	24	0	70	x
77	25	0	21	67	43	56	50	x
68	0	4	37	72	31	38	45
17	92	42	4	13	0	62	83
44	0	26	16	0	55	4	57
68	31	37	4	70	70	16	0	x
12	35	13	0	0	49	32	37
	x		x	x	x

10	44	47	0	30	0	62	36
0	109	91	64	48	28	0	70
77	29	0	25	71	47	56	50
64	0	0	37	72	31	34	41
13	92	38	4	13	0	58	79
40	0	22	16	0	55	0	53
68	35	37	8	74	74	16	0
8	35	9	0	0	49	28	33

10	44	47	0	30	0	62	36	x
0	109	91	64	48	28	0	70	x
77	29	0	25	71	47	56	50	x
64	0	0	37	72	31	34	41	x
13	92	38	4	13	0	58	79	x
40	0	22	16	0	55	0	53	x
68	35	37	8	74	74	16	0	x
8	35	9	0	0	49	28	33	x

Solve an assignment problem online

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

47	58	79	24	53	36	94	70
11	97	97	62	45	38	6	78
93	22	11	28	73	62	67	63
95	8	26	55	89	61	60	69
25	81	45	3	11	11	65	88
85	22	62	48	31	99	40	95
75	19	39	2	67	80	18	4
46	50	42	25	24	86	61	68

18	34	55	0	29	4	70	44	x
0	91	91	56	39	24	0	70	x
77	11	0	17	62	43	56	50	x
82	0	18	47	81	45	52	59
17	78	42	0	8	0	62	83	x
58	0	40	26	9	69	18	71
68	17	37	0	65	70	16	0	x
17	26	18	1	0	54	37	42	x
	x

18	43	55	0	29	4	70	44
0	100	91	56	39	24	0	70
77	20	0	17	62	43	56	50
73	0	9	38	72	36	43	50
17	87	42	0	8	0	62	83
49	0	31	17	0	60	9	62
68	26	37	0	65	70	16	0
17	35	18	1	0	54	37	42

18	43	55	0	29	4	70	44	x
0	100	91	56	39	24	0	70	x
77	20	0	17	62	43	56	50	x
73	0	9	38	72	36	43	50
17	87	42	0	8	0	62	83	x
49	0	31	17	0	60	9	62
68	26	37	0	65	70	16	0	x
17	35	18	1	0	54	37	42
	x			x

18	44	55	0	30	4	70	44
0	101	91	56	40	24	0	70
77	21	0	17	63	43	56	50
72	0	8	37	72	35	42	49
17	88	42	0	9	0	62	83
48	0	30	16	0	59	8	61
68	27	37	0	66	70	16	0
16	35	17	0	0	53	36	41

18	44	55	0	30	4	70	44
0	101	91	56	40	24	0	70	x
77	21	0	17	63	43	56	50	x
72	0	8	37	72	35	42	49
17	88	42	0	9	0	62	83	x
48	0	30	16	0	59	8	61
68	27	37	0	66	70	16	0	x
16	35	17	0	0	53	36	41
	x		x	x

14	44	51	0	30	0	66	40
0	105	91	60	44	24	0	70
77	25	0	21	67	43	56	50
68	0	4	37	72	31	38	45
17	92	42	4	13	0	62	83
44	0	26	16	0	55	4	57
68	31	37	4	70	70	16	0
12	35	13	0	0	49	32	37

14	44	51	0	30	0	66	40
0	105	91	60	44	24	0	70	x
77	25	0	21	67	43	56	50	x
68	0	4	37	72	31	38	45
17	92	42	4	13	0	62	83
44	0	26	16	0	55	4	57
68	31	37	4	70	70	16	0	x
12	35	13	0	0	49	32	37
	x		x	x	x

10	44	47	0	30	0	62	36
0	109	91	64	48	28	0	70
77	29	0	25	71	47	56	50
64	0	0	37	72	31	34	41
13	92	38	4	13	0	58	79
40	0	22	16	0	55	0	53
68	35	37	8	74	74	16	0
8	35	9	0	0	49	28	33

10	44	47	0	30	0	62	36	x
0	109	91	64	48	28	0	70	x
77	29	0	25	71	47	56	50	x
64	0	0	37	72	31	34	41	x
13	92	38	4	13	0	58	79	x
40	0	22	16	0	55	0	53	x
68	35	37	8	74	74	16	0	x
8	35	9	0	0	49	28	33	x