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

Solution

This is the cost matrix.

91	92	55	44	3	53	6
61	67	77	20	7	3	42
56	57	74	78	93	76	65
14	84	24	14	33	43	5
97	71	67	20	24	87	71
91	4	65	10	72	11	89
31	80	14	99	1	58	30

Subtract row minima

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

88	89	52	41	0	50	3	(-3)
58	64	74	17	4	0	39	(-3)
0	1	18	22	37	20	9	(-56)
9	79	19	9	28	38	0	(-5)
77	51	47	0	4	67	51	(-20)
87	0	61	6	68	7	85	(-4)
30	79	13	98	0	57	29	(-1)

Subtract column minima

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

88	89	39	41	0	50	3
58	64	61	17	4	0	39
0	1	5	22	37	20	9
9	79	6	9	28	38	0
77	51	34	0	4	67	51
87	0	48	6	68	7	85
30	79	0	98	0	57	29
		(-13)

Cover all zeros with a minimum number of lines

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

88	89	39	41	0	50	3	x
58	64	61	17	4	0	39	x
0	1	5	22	37	20	9	x
9	79	6	9	28	38	0	x
77	51	34	0	4	67	51	x
87	0	48	6	68	7	85	x
30	79	0	98	0	57	29	x

The optimal assignment

Because there are 7 lines required, an optimal assignment exists among the zeros.

88	89	39	41	0	50	3
58	64	61	17	4	0	39
0	1	5	22	37	20	9
9	79	6	9	28	38	0
77	51	34	0	4	67	51
87	0	48	6	68	7	85
30	79	0	98	0	57	29

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

91	92	55	44	3	53	6
61	67	77	20	7	3	42
56	57	74	78	93	76	65
14	84	24	14	33	43	5
97	71	67	20	24	87	71
91	4	65	10	72	11	89
31	80	14	99	1	58	30

The total minimum cost is 105.

91	92	55	44	3	53	6
61	67	77	20	7	3	42
56	57	74	78	93	76	65
14	84	24	14	33	43	5
97	71	67	20	24	87	71
91	4	65	10	72	11	89
31	80	14	99	1	58	30

88	89	39	41	0	50	3	x
58	64	61	17	4	0	39	x
0	1	5	22	37	20	9	x
9	79	6	9	28	38	0	x
77	51	34	0	4	67	51	x
87	0	48	6	68	7	85	x
30	79	0	98	0	57	29	x

88	89	39	41	0	50	3
58	64	61	17	4	0	39
0	1	5	22	37	20	9
9	79	6	9	28	38	0
77	51	34	0	4	67	51
87	0	48	6	68	7	85
30	79	0	98	0	57	29

91	92	55	44	3	53	6
61	67	77	20	7	3	42
56	57	74	78	93	76	65
14	84	24	14	33	43	5
97	71	67	20	24	87	71
91	4	65	10	72	11	89
31	80	14	99	1	58	30

91	92	55	44	3	53	6
61	67	77	20	7	3	42
56	57	74	78	93	76	65
14	84	24	14	33	43	5
97	71	67	20	24	87	71
91	4	65	10	72	11	89
31	80	14	99	1	58	30

88	89	39	41	0	50	3	x
58	64	61	17	4	0	39	x
0	1	5	22	37	20	9	x
9	79	6	9	28	38	0	x
77	51	34	0	4	67	51	x
87	0	48	6	68	7	85	x
30	79	0	98	0	57	29	x

88	89	39	41	0	50	3
58	64	61	17	4	0	39
0	1	5	22	37	20	9
9	79	6	9	28	38	0
77	51	34	0	4	67	51
87	0	48	6	68	7	85
30	79	0	98	0	57	29

91	92	55	44	3	53	6
61	67	77	20	7	3	42
56	57	74	78	93	76	65
14	84	24	14	33	43	5
97	71	67	20	24	87	71
91	4	65	10	72	11	89
31	80	14	99	1	58	30

91	92	55	44	3	53	6
61	67	77	20	7	3	42
56	57	74	78	93	76	65
14	84	24	14	33	43	5
97	71	67	20	24	87	71
91	4	65	10	72	11	89
31	80	14	99	1	58	30

88	89	39	41	0	50	3	x
58	64	61	17	4	0	39	x
0	1	5	22	37	20	9	x
9	79	6	9	28	38	0	x
77	51	34	0	4	67	51	x
87	0	48	6	68	7	85	x
30	79	0	98	0	57	29	x

88	89	39	41	0	50	3
58	64	61	17	4	0	39
0	1	5	22	37	20	9
9	79	6	9	28	38	0
77	51	34	0	4	67	51
87	0	48	6	68	7	85
30	79	0	98	0	57	29

91	92	55	44	3	53	6
61	67	77	20	7	3	42
56	57	74	78	93	76	65
14	84	24	14	33	43	5
97	71	67	20	24	87	71
91	4	65	10	72	11	89
31	80	14	99	1	58	30