Solution

This is the cost matrix.

2	19	70	16	75	66	39	96	38
64	75	31	89	28	62	51	50	13
52	46	37	57	99	16	28	7	63
69	92	91	55	81	30	69	11	40
63	48	76	33	40	30	84	42	95
40	94	54	51	93	29	87	88	7
59	20	24	23	1	28	43	4	5
27	38	95	21	77	29	65	14	95
15	72	42	23	14	46	15	25	86

Subtract row minima

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

0	17	68	14	73	64	37	94	36	(-2)
51	62	18	76	15	49	38	37	0	(-13)
45	39	30	50	92	9	21	0	56	(-7)
58	81	80	44	70	19	58	0	29	(-11)
33	18	46	3	10	0	54	12	65	(-30)
33	87	47	44	86	22	80	81	0	(-7)
58	19	23	22	0	27	42	3	4	(-1)
13	24	81	7	63	15	51	0	81	(-14)
1	58	28	9	0	32	1	11	72	(-14)

Subtract column minima

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

0	0	50	11	73	64	36	94	36
51	45	0	73	15	49	37	37	0
45	22	12	47	92	9	20	0	56
58	64	62	41	70	19	57	0	29
33	1	28	0	10	0	53	12	65
33	70	29	41	86	22	79	81	0
58	2	5	19	0	27	41	3	4
13	7	63	4	63	15	50	0	81
1	41	10	6	0	32	0	11	72
	(-17)	(-18)	(-3)			(-1)

Cover all zeros with a minimum number of lines

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

0	0	50	11	73	64	36	94	36	x
51	45	0	73	15	49	37	37	0	x
45	22	12	47	92	9	20	0	56
58	64	62	41	70	19	57	0	29
33	1	28	0	10	0	53	12	65	x
33	70	29	41	86	22	79	81	0	x
58	2	5	19	0	27	41	3	4	x
13	7	63	4	63	15	50	0	81
1	41	10	6	0	32	0	11	72	x
							x

Create additional zeros

The number of lines is smaller than 9. The smallest uncovered element is 4. We subtract this value from all uncovered elements and add it to all elements covered twice.

0	0	50	11	73	64	36	98	36
51	45	0	73	15	49	37	41	0
41	18	8	43	88	5	16	0	52
54	60	58	37	66	15	53	0	25
33	1	28	0	10	0	53	16	65
33	70	29	41	86	22	79	85	0
58	2	5	19	0	27	41	7	4
9	3	59	0	59	11	46	0	77
1	41	10	6	0	32	0	15	72

Cover all zeros with a minimum number of lines

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

0	0	50	11	73	64	36	98	36	x
51	45	0	73	15	49	37	41	0	x
41	18	8	43	88	5	16	0	52
54	60	58	37	66	15	53	0	25
33	1	28	0	10	0	53	16	65	x
33	70	29	41	86	22	79	85	0	x
58	2	5	19	0	27	41	7	4	x
9	3	59	0	59	11	46	0	77	x
1	41	10	6	0	32	0	15	72	x
							x

Create additional zeros

The number of lines is smaller than 9. The smallest uncovered element is 5. We subtract this value from all uncovered elements and add it to all elements covered twice.

0	0	50	11	73	64	36	103	36
51	45	0	73	15	49	37	46	0
36	13	3	38	83	0	11	0	47
49	55	53	32	61	10	48	0	20
33	1	28	0	10	0	53	21	65
33	70	29	41	86	22	79	90	0
58	2	5	19	0	27	41	12	4
9	3	59	0	59	11	46	5	77
1	41	10	6	0	32	0	20	72

Cover all zeros with a minimum number of lines

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

0	0	50	11	73	64	36	103	36	x
51	45	0	73	15	49	37	46	0	x
36	13	3	38	83	0	11	0	47
49	55	53	32	61	10	48	0	20
33	1	28	0	10	0	53	21	65
33	70	29	41	86	22	79	90	0	x
58	2	5	19	0	27	41	12	4	x
9	3	59	0	59	11	46	5	77
1	41	10	6	0	32	0	20	72	x
			x		x		x

Create additional zeros

The number of lines is smaller than 9. The smallest uncovered element is 1. We subtract this value from all uncovered elements and add it to all elements covered twice.

0	0	50	12	73	65	36	104	36
51	45	0	74	15	50	37	47	0
35	12	2	38	82	0	10	0	46
48	54	52	32	60	10	47	0	19
32	0	27	0	9	0	52	21	64
33	70	29	42	86	23	79	91	0
58	2	5	20	0	28	41	13	4
8	2	58	0	58	11	45	5	76
1	41	10	7	0	33	0	21	72

Cover all zeros with a minimum number of lines

A total of 9 lines are required to cover all zeros.

0	0	50	12	73	65	36	104	36	x
51	45	0	74	15	50	37	47	0	x
35	12	2	38	82	0	10	0	46	x
48	54	52	32	60	10	47	0	19	x
32	0	27	0	9	0	52	21	64	x
33	70	29	42	86	23	79	91	0	x
58	2	5	20	0	28	41	13	4	x
8	2	58	0	58	11	45	5	76	x
1	41	10	7	0	33	0	21	72	x

The optimal assignment

Because there are 9 lines required, an optimal assignment exists among the zeros.

0	0	50	12	73	65	36	104	36
51	45	0	74	15	50	37	47	0
35	12	2	38	82	0	10	0	46
48	54	52	32	60	10	47	0	19
32	0	27	0	9	0	52	21	64
33	70	29	42	86	23	79	91	0
58	2	5	20	0	28	41	13	4
8	2	58	0	58	11	45	5	76
1	41	10	7	0	33	0	21	72

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

2	19	70	16	75	66	39	96	38
64	75	31	89	28	62	51	50	13
52	46	37	57	99	16	28	7	63
69	92	91	55	81	30	69	11	40
63	48	76	33	40	30	84	42	95
40	94	54	51	93	29	87	88	7
59	20	24	23	1	28	43	4	5
27	38	95	21	77	29	65	14	95
15	72	42	23	14	46	15	25	86

The total minimum cost is 152.

2	19	70	16	75	66	39	96	38
64	75	31	89	28	62	51	50	13
52	46	37	57	99	16	28	7	63
69	92	91	55	81	30	69	11	40
63	48	76	33	40	30	84	42	95
40	94	54	51	93	29	87	88	7
59	20	24	23	1	28	43	4	5
27	38	95	21	77	29	65	14	95
15	72	42	23	14	46	15	25	86

0	0	50	11	73	64	36	94	36	x
51	45	0	73	15	49	37	37	0	x
45	22	12	47	92	9	20	0	56
58	64	62	41	70	19	57	0	29
33	1	28	0	10	0	53	12	65	x
33	70	29	41	86	22	79	81	0	x
58	2	5	19	0	27	41	3	4	x
13	7	63	4	63	15	50	0	81
1	41	10	6	0	32	0	11	72	x
							x

0	0	50	11	73	64	36	98	36
51	45	0	73	15	49	37	41	0
41	18	8	43	88	5	16	0	52
54	60	58	37	66	15	53	0	25
33	1	28	0	10	0	53	16	65
33	70	29	41	86	22	79	85	0
58	2	5	19	0	27	41	7	4
9	3	59	0	59	11	46	0	77
1	41	10	6	0	32	0	15	72

0	0	50	11	73	64	36	98	36	x
51	45	0	73	15	49	37	41	0	x
41	18	8	43	88	5	16	0	52
54	60	58	37	66	15	53	0	25
33	1	28	0	10	0	53	16	65	x
33	70	29	41	86	22	79	85	0	x
58	2	5	19	0	27	41	7	4	x
9	3	59	0	59	11	46	0	77	x
1	41	10	6	0	32	0	15	72	x
							x

0	0	50	11	73	64	36	103	36
51	45	0	73	15	49	37	46	0
36	13	3	38	83	0	11	0	47
49	55	53	32	61	10	48	0	20
33	1	28	0	10	0	53	21	65
33	70	29	41	86	22	79	90	0
58	2	5	19	0	27	41	12	4
9	3	59	0	59	11	46	5	77
1	41	10	6	0	32	0	20	72

0	0	50	11	73	64	36	103	36	x
51	45	0	73	15	49	37	46	0	x
36	13	3	38	83	0	11	0	47
49	55	53	32	61	10	48	0	20
33	1	28	0	10	0	53	21	65
33	70	29	41	86	22	79	90	0	x
58	2	5	19	0	27	41	12	4	x
9	3	59	0	59	11	46	5	77
1	41	10	6	0	32	0	20	72	x
			x		x		x

0	0	50	12	73	65	36	104	36
51	45	0	74	15	50	37	47	0
35	12	2	38	82	0	10	0	46
48	54	52	32	60	10	47	0	19
32	0	27	0	9	0	52	21	64
33	70	29	42	86	23	79	91	0
58	2	5	20	0	28	41	13	4
8	2	58	0	58	11	45	5	76
1	41	10	7	0	33	0	21	72

0	0	50	12	73	65	36	104	36	x
51	45	0	74	15	50	37	47	0	x
35	12	2	38	82	0	10	0	46	x
48	54	52	32	60	10	47	0	19	x
32	0	27	0	9	0	52	21	64	x
33	70	29	42	86	23	79	91	0	x
58	2	5	20	0	28	41	13	4	x
8	2	58	0	58	11	45	5	76	x
1	41	10	7	0	33	0	21	72	x

0	0	50	12	73	65	36	104	36
51	45	0	74	15	50	37	47	0
35	12	2	38	82	0	10	0	46
48	54	52	32	60	10	47	0	19
32	0	27	0	9	0	52	21	64
33	70	29	42	86	23	79	91	0
58	2	5	20	0	28	41	13	4
8	2	58	0	58	11	45	5	76
1	41	10	7	0	33	0	21	72

2	19	70	16	75	66	39	96	38
64	75	31	89	28	62	51	50	13
52	46	37	57	99	16	28	7	63
69	92	91	55	81	30	69	11	40
63	48	76	33	40	30	84	42	95
40	94	54	51	93	29	87	88	7
59	20	24	23	1	28	43	4	5
27	38	95	21	77	29	65	14	95
15	72	42	23	14	46	15	25	86

2	19	70	16	75	66	39	96	38
64	75	31	89	28	62	51	50	13
52	46	37	57	99	16	28	7	63
69	92	91	55	81	30	69	11	40
63	48	76	33	40	30	84	42	95
40	94	54	51	93	29	87	88	7
59	20	24	23	1	28	43	4	5
27	38	95	21	77	29	65	14	95
15	72	42	23	14	46	15	25	86

0	0	50	11	73	64	36	94	36	x
51	45	0	73	15	49	37	37	0	x
45	22	12	47	92	9	20	0	56
58	64	62	41	70	19	57	0	29
33	1	28	0	10	0	53	12	65	x
33	70	29	41	86	22	79	81	0	x
58	2	5	19	0	27	41	3	4	x
13	7	63	4	63	15	50	0	81
1	41	10	6	0	32	0	11	72	x
							x

0	0	50	11	73	64	36	98	36
51	45	0	73	15	49	37	41	0
41	18	8	43	88	5	16	0	52
54	60	58	37	66	15	53	0	25
33	1	28	0	10	0	53	16	65
33	70	29	41	86	22	79	85	0
58	2	5	19	0	27	41	7	4
9	3	59	0	59	11	46	0	77
1	41	10	6	0	32	0	15	72

0	0	50	11	73	64	36	98	36	x
51	45	0	73	15	49	37	41	0	x
41	18	8	43	88	5	16	0	52
54	60	58	37	66	15	53	0	25
33	1	28	0	10	0	53	16	65	x
33	70	29	41	86	22	79	85	0	x
58	2	5	19	0	27	41	7	4	x
9	3	59	0	59	11	46	0	77	x
1	41	10	6	0	32	0	15	72	x
							x

0	0	50	11	73	64	36	103	36
51	45	0	73	15	49	37	46	0
36	13	3	38	83	0	11	0	47
49	55	53	32	61	10	48	0	20
33	1	28	0	10	0	53	21	65
33	70	29	41	86	22	79	90	0
58	2	5	19	0	27	41	12	4
9	3	59	0	59	11	46	5	77
1	41	10	6	0	32	0	20	72

0	0	50	11	73	64	36	103	36	x
51	45	0	73	15	49	37	46	0	x
36	13	3	38	83	0	11	0	47
49	55	53	32	61	10	48	0	20
33	1	28	0	10	0	53	21	65
33	70	29	41	86	22	79	90	0	x
58	2	5	19	0	27	41	12	4	x
9	3	59	0	59	11	46	5	77
1	41	10	6	0	32	0	20	72	x
			x		x		x

0	0	50	12	73	65	36	104	36
51	45	0	74	15	50	37	47	0
35	12	2	38	82	0	10	0	46
48	54	52	32	60	10	47	0	19
32	0	27	0	9	0	52	21	64
33	70	29	42	86	23	79	91	0
58	2	5	20	0	28	41	13	4
8	2	58	0	58	11	45	5	76
1	41	10	7	0	33	0	21	72

0	0	50	12	73	65	36	104	36	x
51	45	0	74	15	50	37	47	0	x
35	12	2	38	82	0	10	0	46	x
48	54	52	32	60	10	47	0	19	x
32	0	27	0	9	0	52	21	64	x
33	70	29	42	86	23	79	91	0	x
58	2	5	20	0	28	41	13	4	x
8	2	58	0	58	11	45	5	76	x
1	41	10	7	0	33	0	21	72	x

0	0	50	12	73	65	36	104	36
51	45	0	74	15	50	37	47	0
35	12	2	38	82	0	10	0	46
48	54	52	32	60	10	47	0	19
32	0	27	0	9	0	52	21	64
33	70	29	42	86	23	79	91	0
58	2	5	20	0	28	41	13	4
8	2	58	0	58	11	45	5	76
1	41	10	7	0	33	0	21	72

2	19	70	16	75	66	39	96	38
64	75	31	89	28	62	51	50	13
52	46	37	57	99	16	28	7	63
69	92	91	55	81	30	69	11	40
63	48	76	33	40	30	84	42	95
40	94	54	51	93	29	87	88	7
59	20	24	23	1	28	43	4	5
27	38	95	21	77	29	65	14	95
15	72	42	23	14	46	15	25	86

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

The optimal assignment

2	19	70	16	75	66	39	96	38
64	75	31	89	28	62	51	50	13
52	46	37	57	99	16	28	7	63
69	92	91	55	81	30	69	11	40
63	48	76	33	40	30	84	42	95
40	94	54	51	93	29	87	88	7
59	20	24	23	1	28	43	4	5
27	38	95	21	77	29	65	14	95
15	72	42	23	14	46	15	25	86

0	0	50	11	73	64	36	94	36	x
51	45	0	73	15	49	37	37	0	x
45	22	12	47	92	9	20	0	56
58	64	62	41	70	19	57	0	29
33	1	28	0	10	0	53	12	65	x
33	70	29	41	86	22	79	81	0	x
58	2	5	19	0	27	41	3	4	x
13	7	63	4	63	15	50	0	81
1	41	10	6	0	32	0	11	72	x
							x

0	0	50	11	73	64	36	98	36
51	45	0	73	15	49	37	41	0
41	18	8	43	88	5	16	0	52
54	60	58	37	66	15	53	0	25
33	1	28	0	10	0	53	16	65
33	70	29	41	86	22	79	85	0
58	2	5	19	0	27	41	7	4
9	3	59	0	59	11	46	0	77
1	41	10	6	0	32	0	15	72

0	0	50	11	73	64	36	98	36	x
51	45	0	73	15	49	37	41	0	x
41	18	8	43	88	5	16	0	52
54	60	58	37	66	15	53	0	25
33	1	28	0	10	0	53	16	65	x
33	70	29	41	86	22	79	85	0	x
58	2	5	19	0	27	41	7	4	x
9	3	59	0	59	11	46	0	77	x
1	41	10	6	0	32	0	15	72	x
							x

0	0	50	11	73	64	36	103	36
51	45	0	73	15	49	37	46	0
36	13	3	38	83	0	11	0	47
49	55	53	32	61	10	48	0	20
33	1	28	0	10	0	53	21	65
33	70	29	41	86	22	79	90	0
58	2	5	19	0	27	41	12	4
9	3	59	0	59	11	46	5	77
1	41	10	6	0	32	0	20	72

0	0	50	11	73	64	36	103	36	x
51	45	0	73	15	49	37	46	0	x
36	13	3	38	83	0	11	0	47
49	55	53	32	61	10	48	0	20
33	1	28	0	10	0	53	21	65
33	70	29	41	86	22	79	90	0	x
58	2	5	19	0	27	41	12	4	x
9	3	59	0	59	11	46	5	77
1	41	10	6	0	32	0	20	72	x
			x		x		x

0	0	50	12	73	65	36	104	36
51	45	0	74	15	50	37	47	0
35	12	2	38	82	0	10	0	46
48	54	52	32	60	10	47	0	19
32	0	27	0	9	0	52	21	64
33	70	29	42	86	23	79	91	0
58	2	5	20	0	28	41	13	4
8	2	58	0	58	11	45	5	76
1	41	10	7	0	33	0	21	72

0	0	50	12	73	65	36	104	36	x
51	45	0	74	15	50	37	47	0	x
35	12	2	38	82	0	10	0	46	x
48	54	52	32	60	10	47	0	19	x
32	0	27	0	9	0	52	21	64	x
33	70	29	42	86	23	79	91	0	x
58	2	5	20	0	28	41	13	4	x
8	2	58	0	58	11	45	5	76	x
1	41	10	7	0	33	0	21	72	x

0	0	50	12	73	65	36	104	36
51	45	0	74	15	50	37	47	0
35	12	2	38	82	0	10	0	46
48	54	52	32	60	10	47	0	19
32	0	27	0	9	0	52	21	64
33	70	29	42	86	23	79	91	0
58	2	5	20	0	28	41	13	4
8	2	58	0	58	11	45	5	76
1	41	10	7	0	33	0	21	72

2	19	70	16	75	66	39	96	38
64	75	31	89	28	62	51	50	13
52	46	37	57	99	16	28	7	63
69	92	91	55	81	30	69	11	40
63	48	76	33	40	30	84	42	95
40	94	54	51	93	29	87	88	7
59	20	24	23	1	28	43	4	5
27	38	95	21	77	29	65	14	95
15	72	42	23	14	46	15	25	86