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.

54	71	84	60	1	45	1
30	15	3	47	90	3	84
9	66	11	48	95	39	27
48	58	9	72	21	87	57
48	57	41	15	74	53	66
29	18	86	92	65	30	69
48	72	9	88	22	59	93

Subtract row minima

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

53	70	83	59	0	44	0	(-1)
27	12	0	44	87	0	81	(-3)
0	57	2	39	86	30	18	(-9)
39	49	0	63	12	78	48	(-9)
33	42	26	0	59	38	51	(-15)
11	0	68	74	47	12	51	(-18)
39	63	0	79	13	50	84	(-9)

Subtract column minima

Because each column already contains a zero, subtracting the column minima has no effect.

Cover all zeros with a minimum number of lines

A total of 6 lines are required to cover all zeros.

53	70	83	59	0	44	0	x
27	12	0	44	87	0	81	x
0	57	2	39	86	30	18	x
39	49	0	63	12	78	48
33	42	26	0	59	38	51	x
11	0	68	74	47	12	51	x
39	63	0	79	13	50	84
		x

Create additional zeros

The number of lines is smaller than 7. The smallest uncovered element is 12. We subtract this value from all uncovered elements and add it to all elements covered twice.

53	70	95	59	0	44	0
27	12	12	44	87	0	81
0	57	14	39	86	30	18
27	37	0	51	0	66	36
33	42	38	0	59	38	51
11	0	80	74	47	12	51
27	51	0	67	1	38	72

Cover all zeros with a minimum number of lines

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

53	70	95	59	0	44	0	x
27	12	12	44	87	0	81	x
0	57	14	39	86	30	18	x
27	37	0	51	0	66	36	x
33	42	38	0	59	38	51	x
11	0	80	74	47	12	51	x
27	51	0	67	1	38	72	x

The optimal assignment

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

53	70	95	59	0	44	0
27	12	12	44	87	0	81
0	57	14	39	86	30	18
27	37	0	51	0	66	36
33	42	38	0	59	38	51
11	0	80	74	47	12	51
27	51	0	67	1	38	72

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

54	71	84	60	1	45	1
30	15	3	47	90	3	84
9	66	11	48	95	39	27
48	58	9	72	21	87	57
48	57	41	15	74	53	66
29	18	86	92	65	30	69
48	72	9	88	22	59	93

The total minimum cost is 76.

54	71	84	60	1	45	1
30	15	3	47	90	3	84
9	66	11	48	95	39	27
48	58	9	72	21	87	57
48	57	41	15	74	53	66
29	18	86	92	65	30	69
48	72	9	88	22	59	93

53	70	83	59	0	44	0	x
27	12	0	44	87	0	81	x
0	57	2	39	86	30	18	x
39	49	0	63	12	78	48
33	42	26	0	59	38	51	x
11	0	68	74	47	12	51	x
39	63	0	79	13	50	84
		x

53	70	95	59	0	44	0
27	12	12	44	87	0	81
0	57	14	39	86	30	18
27	37	0	51	0	66	36
33	42	38	0	59	38	51
11	0	80	74	47	12	51
27	51	0	67	1	38	72

53	70	95	59	0	44	0	x
27	12	12	44	87	0	81	x
0	57	14	39	86	30	18	x
27	37	0	51	0	66	36	x
33	42	38	0	59	38	51	x
11	0	80	74	47	12	51	x
27	51	0	67	1	38	72	x

53	70	95	59	0	44	0
27	12	12	44	87	0	81
0	57	14	39	86	30	18
27	37	0	51	0	66	36
33	42	38	0	59	38	51
11	0	80	74	47	12	51
27	51	0	67	1	38	72

54	71	84	60	1	45	1
30	15	3	47	90	3	84
9	66	11	48	95	39	27
48	58	9	72	21	87	57
48	57	41	15	74	53	66
29	18	86	92	65	30	69
48	72	9	88	22	59	93

54	71	84	60	1	45	1
30	15	3	47	90	3	84
9	66	11	48	95	39	27
48	58	9	72	21	87	57
48	57	41	15	74	53	66
29	18	86	92	65	30	69
48	72	9	88	22	59	93

53	70	83	59	0	44	0	x
27	12	0	44	87	0	81	x
0	57	2	39	86	30	18	x
39	49	0	63	12	78	48
33	42	26	0	59	38	51	x
11	0	68	74	47	12	51	x
39	63	0	79	13	50	84
		x

53	70	95	59	0	44	0
27	12	12	44	87	0	81
0	57	14	39	86	30	18
27	37	0	51	0	66	36
33	42	38	0	59	38	51
11	0	80	74	47	12	51
27	51	0	67	1	38	72

53	70	95	59	0	44	0	x
27	12	12	44	87	0	81	x
0	57	14	39	86	30	18	x
27	37	0	51	0	66	36	x
33	42	38	0	59	38	51	x
11	0	80	74	47	12	51	x
27	51	0	67	1	38	72	x

53	70	95	59	0	44	0
27	12	12	44	87	0	81
0	57	14	39	86	30	18
27	37	0	51	0	66	36
33	42	38	0	59	38	51
11	0	80	74	47	12	51
27	51	0	67	1	38	72

54	71	84	60	1	45	1
30	15	3	47	90	3	84
9	66	11	48	95	39	27
48	58	9	72	21	87	57
48	57	41	15	74	53	66
29	18	86	92	65	30	69
48	72	9	88	22	59	93

54	71	84	60	1	45	1
30	15	3	47	90	3	84
9	66	11	48	95	39	27
48	58	9	72	21	87	57
48	57	41	15	74	53	66
29	18	86	92	65	30	69
48	72	9	88	22	59	93

53	70	83	59	0	44	0	x
27	12	0	44	87	0	81	x
0	57	2	39	86	30	18	x
39	49	0	63	12	78	48
33	42	26	0	59	38	51	x
11	0	68	74	47	12	51	x
39	63	0	79	13	50	84
		x

53	70	95	59	0	44	0
27	12	12	44	87	0	81
0	57	14	39	86	30	18
27	37	0	51	0	66	36
33	42	38	0	59	38	51
11	0	80	74	47	12	51
27	51	0	67	1	38	72

53	70	95	59	0	44	0	x
27	12	12	44	87	0	81	x
0	57	14	39	86	30	18	x
27	37	0	51	0	66	36	x
33	42	38	0	59	38	51	x
11	0	80	74	47	12	51	x
27	51	0	67	1	38	72	x

53	70	95	59	0	44	0
27	12	12	44	87	0	81
0	57	14	39	86	30	18
27	37	0	51	0	66	36
33	42	38	0	59	38	51
11	0	80	74	47	12	51
27	51	0	67	1	38	72

54	71	84	60	1	45	1
30	15	3	47	90	3	84
9	66	11	48	95	39	27
48	58	9	72	21	87	57
48	57	41	15	74	53	66
29	18	86	92	65	30	69
48	72	9	88	22	59	93