Solve an assignment problem online

Fill in the cost matrix of an assignment problem and click on 'Solve'. The optimal assignment will be determined and a step by step explanation of the hungarian algorithm will be given.

Fill in the cost matrix (random cost matrix):

Size: 3x3 4x4 5x5 6x6 7x7 8x8 9x9 10x10

Don't show the steps of the Hungarian algorithm
Maximize the total cost

This is the original cost matrix:

80	99	82	97	1	84	70	97
80	12	35	34	65	32	96	29
59	18	38	83	79	11	12	54
78	27	3	36	13	34	53	50
8	49	39	43	97	63	93	86
77	87	87	62	36	26	83	17
31	6	15	48	60	44	93	6
99	60	18	77	18	44	88	82

Subtract row minima

We subtract the row minimum from each row:

79	98	81	96	0	83	69	96	(-1)
68	0	23	22	53	20	84	17	(-12)
48	7	27	72	68	0	1	43	(-11)
75	24	0	33	10	31	50	47	(-3)
0	41	31	35	89	55	85	78	(-8)
60	70	70	45	19	9	66	0	(-17)
25	0	9	42	54	38	87	0	(-6)
81	42	0	59	0	26	70	64	(-18)

Subtract column minima

We subtract the column minimum from each column:

79	98	81	74	0	83	68	96
68	0	23	0	53	20	83	17
48	7	27	50	68	0	0	43
75	24	0	11	10	31	49	47
0	41	31	13	89	55	84	78
60	70	70	23	19	9	65	0
25	0	9	20	54	38	86	0
81	42	0	37	0	26	69	64
			(-22)			(-1)

Cover all zeros with a minimum number of lines

There are 7 lines required to cover all zeros:

79	98	81	74	0	83	68	96
68	0	23	0	53	20	83	17	x
48	7	27	50	68	0	0	43	x
75	24	0	11	10	31	49	47
0	41	31	13	89	55	84	78	x
60	70	70	23	19	9	65	0	x
25	0	9	20	54	38	86	0	x
81	42	0	37	0	26	69	64
		x		x

Create additional zeros

The number of lines is smaller than 8. The smallest uncovered number is 11. We subtract this number from all uncovered elements and add it to all elements that are covered twice:

68	87	81	63	0	72	57	85
68	0	34	0	64	20	83	17
48	7	38	50	79	0	0	43
64	13	0	0	10	20	38	36
0	41	42	13	100	55	84	78
60	70	81	23	30	9	65	0
25	0	20	20	65	38	86	0
70	31	0	26	0	15	58	53

Cover all zeros with a minimum number of lines

There are 7 lines required to cover all zeros:

68	87	81	63	0	72	57	85
68	0	34	0	64	20	83	17
48	7	38	50	79	0	0	43	x
64	13	0	0	10	20	38	36
0	41	42	13	100	55	84	78	x
60	70	81	23	30	9	65	0
25	0	20	20	65	38	86	0
70	31	0	26	0	15	58	53
	x	x	x	x			x

Create additional zeros

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

59	87	81	63	0	63	48	85
59	0	34	0	64	11	74	17
48	16	47	59	88	0	0	52
55	13	0	0	10	11	29	36
0	50	51	22	109	55	84	87
51	70	81	23	30	0	56	0
16	0	20	20	65	29	77	0
61	31	0	26	0	6	49	53

Cover all zeros with a minimum number of lines

There are 8 lines required to cover all zeros:

59	87	81	63	0	63	48	85	x
59	0	34	0	64	11	74	17	x
48	16	47	59	88	0	0	52	x
55	13	0	0	10	11	29	36	x
0	50	51	22	109	55	84	87	x
51	70	81	23	30	0	56	0	x
16	0	20	20	65	29	77	0	x
61	31	0	26	0	6	49	53	x

The optimal assignment

Because there are 8 lines required, the zeros cover an optimal assignment:

59	87	81	63	0	63	48	85
59	0	34	0	64	11	74	17
48	16	47	59	88	0	0	52
55	13	0	0	10	11	29	36
0	50	51	22	109	55	84	87
51	70	81	23	30	0	56	0
16	0	20	20	65	29	77	0
61	31	0	26	0	6	49	53

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

80	99	82	97	1	84	70	97
80	12	35	34	65	32	96	29
59	18	38	83	79	11	12	54
78	27	3	36	13	34	53	50
8	49	39	43	97	63	93	86
77	87	87	62	36	26	83	17
31	6	15	48	60	44	93	6
99	60	18	77	18	44	88	82

The optimal value equals 119.

80	99	82	97	1	84	70	97
80	12	35	34	65	32	96	29
59	18	38	83	79	11	12	54
78	27	3	36	13	34	53	50
8	49	39	43	97	63	93	86
77	87	87	62	36	26	83	17
31	6	15	48	60	44	93	6
99	60	18	77	18	44	88	82

79	98	81	74	0	83	68	96
68	0	23	0	53	20	83	17	x
48	7	27	50	68	0	0	43	x
75	24	0	11	10	31	49	47
0	41	31	13	89	55	84	78	x
60	70	70	23	19	9	65	0	x
25	0	9	20	54	38	86	0	x
81	42	0	37	0	26	69	64
		x		x

68	87	81	63	0	72	57	85
68	0	34	0	64	20	83	17
48	7	38	50	79	0	0	43
64	13	0	0	10	20	38	36
0	41	42	13	100	55	84	78
60	70	81	23	30	9	65	0
25	0	20	20	65	38	86	0
70	31	0	26	0	15	58	53

68	87	81	63	0	72	57	85
68	0	34	0	64	20	83	17
48	7	38	50	79	0	0	43	x
64	13	0	0	10	20	38	36
0	41	42	13	100	55	84	78	x
60	70	81	23	30	9	65	0
25	0	20	20	65	38	86	0
70	31	0	26	0	15	58	53
	x	x	x	x			x

59	87	81	63	0	63	48	85
59	0	34	0	64	11	74	17
48	16	47	59	88	0	0	52
55	13	0	0	10	11	29	36
0	50	51	22	109	55	84	87
51	70	81	23	30	0	56	0
16	0	20	20	65	29	77	0
61	31	0	26	0	6	49	53

59	87	81	63	0	63	48	85	x
59	0	34	0	64	11	74	17	x
48	16	47	59	88	0	0	52	x
55	13	0	0	10	11	29	36	x
0	50	51	22	109	55	84	87	x
51	70	81	23	30	0	56	0	x
16	0	20	20	65	29	77	0	x
61	31	0	26	0	6	49	53	x

59	87	81	63	0	63	48	85
59	0	34	0	64	11	74	17
48	16	47	59	88	0	0	52
55	13	0	0	10	11	29	36
0	50	51	22	109	55	84	87
51	70	81	23	30	0	56	0
16	0	20	20	65	29	77	0
61	31	0	26	0	6	49	53

80	99	82	97	1	84	70	97
80	12	35	34	65	32	96	29
59	18	38	83	79	11	12	54
78	27	3	36	13	34	53	50
8	49	39	43	97	63	93	86
77	87	87	62	36	26	83	17
31	6	15	48	60	44	93	6
99	60	18	77	18	44	88	82

80	99	82	97	1	84	70	97
80	12	35	34	65	32	96	29
59	18	38	83	79	11	12	54
78	27	3	36	13	34	53	50
8	49	39	43	97	63	93	86
77	87	87	62	36	26	83	17
31	6	15	48	60	44	93	6
99	60	18	77	18	44	88	82

79	98	81	74	0	83	68	96
68	0	23	0	53	20	83	17	x
48	7	27	50	68	0	0	43	x
75	24	0	11	10	31	49	47
0	41	31	13	89	55	84	78	x
60	70	70	23	19	9	65	0	x
25	0	9	20	54	38	86	0	x
81	42	0	37	0	26	69	64
		x		x

68	87	81	63	0	72	57	85
68	0	34	0	64	20	83	17
48	7	38	50	79	0	0	43
64	13	0	0	10	20	38	36
0	41	42	13	100	55	84	78
60	70	81	23	30	9	65	0
25	0	20	20	65	38	86	0
70	31	0	26	0	15	58	53

68	87	81	63	0	72	57	85
68	0	34	0	64	20	83	17
48	7	38	50	79	0	0	43	x
64	13	0	0	10	20	38	36
0	41	42	13	100	55	84	78	x
60	70	81	23	30	9	65	0
25	0	20	20	65	38	86	0
70	31	0	26	0	15	58	53
	x	x	x	x			x

59	87	81	63	0	63	48	85
59	0	34	0	64	11	74	17
48	16	47	59	88	0	0	52
55	13	0	0	10	11	29	36
0	50	51	22	109	55	84	87
51	70	81	23	30	0	56	0
16	0	20	20	65	29	77	0
61	31	0	26	0	6	49	53

59	87	81	63	0	63	48	85	x
59	0	34	0	64	11	74	17	x
48	16	47	59	88	0	0	52	x
55	13	0	0	10	11	29	36	x
0	50	51	22	109	55	84	87	x
51	70	81	23	30	0	56	0	x
16	0	20	20	65	29	77	0	x
61	31	0	26	0	6	49	53	x

59	87	81	63	0	63	48	85
59	0	34	0	64	11	74	17
48	16	47	59	88	0	0	52
55	13	0	0	10	11	29	36
0	50	51	22	109	55	84	87
51	70	81	23	30	0	56	0
16	0	20	20	65	29	77	0
61	31	0	26	0	6	49	53

80	99	82	97	1	84	70	97
80	12	35	34	65	32	96	29
59	18	38	83	79	11	12	54
78	27	3	36	13	34	53	50
8	49	39	43	97	63	93	86
77	87	87	62	36	26	83	17
31	6	15	48	60	44	93	6
99	60	18	77	18	44	88	82

80	99	82	97	1	84	70	97
80	12	35	34	65	32	96	29
59	18	38	83	79	11	12	54
78	27	3	36	13	34	53	50
8	49	39	43	97	63	93	86
77	87	87	62	36	26	83	17
31	6	15	48	60	44	93	6
99	60	18	77	18	44	88	82

79	98	81	74	0	83	68	96
68	0	23	0	53	20	83	17	x
48	7	27	50	68	0	0	43	x
75	24	0	11	10	31	49	47
0	41	31	13	89	55	84	78	x
60	70	70	23	19	9	65	0	x
25	0	9	20	54	38	86	0	x
81	42	0	37	0	26	69	64
		x		x

68	87	81	63	0	72	57	85
68	0	34	0	64	20	83	17
48	7	38	50	79	0	0	43
64	13	0	0	10	20	38	36
0	41	42	13	100	55	84	78
60	70	81	23	30	9	65	0
25	0	20	20	65	38	86	0
70	31	0	26	0	15	58	53

68	87	81	63	0	72	57	85
68	0	34	0	64	20	83	17
48	7	38	50	79	0	0	43	x
64	13	0	0	10	20	38	36
0	41	42	13	100	55	84	78	x
60	70	81	23	30	9	65	0
25	0	20	20	65	38	86	0
70	31	0	26	0	15	58	53
	x	x	x	x			x

59	87	81	63	0	63	48	85
59	0	34	0	64	11	74	17
48	16	47	59	88	0	0	52
55	13	0	0	10	11	29	36
0	50	51	22	109	55	84	87
51	70	81	23	30	0	56	0
16	0	20	20	65	29	77	0
61	31	0	26	0	6	49	53

59	87	81	63	0	63	48	85	x
59	0	34	0	64	11	74	17	x
48	16	47	59	88	0	0	52	x
55	13	0	0	10	11	29	36	x
0	50	51	22	109	55	84	87	x
51	70	81	23	30	0	56	0	x
16	0	20	20	65	29	77	0	x
61	31	0	26	0	6	49	53	x

59	87	81	63	0	63	48	85
59	0	34	0	64	11	74	17
48	16	47	59	88	0	0	52
55	13	0	0	10	11	29	36
0	50	51	22	109	55	84	87
51	70	81	23	30	0	56	0
16	0	20	20	65	29	77	0
61	31	0	26	0	6	49	53

80	99	82	97	1	84	70	97
80	12	35	34	65	32	96	29
59	18	38	83	79	11	12	54
78	27	3	36	13	34	53	50
8	49	39	43	97	63	93	86
77	87	87	62	36	26	83	17
31	6	15	48	60	44	93	6
99	60	18	77	18	44	88	82