IMPROVED BIASED RANDOM KEY GENETIC ALGORITHM FOR THE TWO-DIMENSIONAL NON-GUILLOTINE CUTTING PROBLEM

Oliveira, Eliane Vendramini de; Romero, Rubén

doi:10.1590/0101-7438.2022.042.00260406

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ARTICLES • Pesqui. Oper. 42 • 2022 • https://doi.org/10.1590/0101-7438.2022.042.00260406 copy

IMPROVED BIASED RANDOM KEY GENETIC ALGORITHM FOR THE TWO-DIMENSIONAL NON-GUILLOTINE CUTTING PROBLEM

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

The two-dimensional cutting problem has a direct relationship with industry problems. There are several proposals to solve these problems. In particular, solution proposals using metaheuristics are the focus of this research. Thus, in this paper, we present a specialized biased random key genetic algorithm. Several tests were performed using known instances in the specific literature, and the results found by the metaheuristics proposed were, in many cases, equal or superior to the results already published in the literature. Another comparison of results presented in this paper is related to the results obtained by specialized metaheuristics and the results found by a mathematical model using commercial software. Once again, in this case, the genetic algorithm presented results equal to or very close to the optimum found by the mathematical model. In addition, the optimization proposal was extended to two-dimensional non-guillotine cutting without parts orientation.

Keywords:
Genetic algorithm; metaheuristics; two-dimensional cutting problem

	Fixed Orientation					Free Orientation
Test Case	Beasley ¹	GRASP ²	Tabu Search³	System I	System II	System III
1	247	247	247	247	234	259
2	230	230	230	230	230	250
3	164	164	164	164	156	193
4	358	358	358	358	358	370
5	268	268	268	268	268	268
6	289	289	289	289	289	298
7	834	834	834	834	828	856
8	430	430	430	430	430	430
9	924	924	924	924	924	924
10	1688	1688	1688	1688	1688	1786
11	1178	1178	1178	1178	1178	1272
12	1801	1865	1865	1865	1865	1932
13	1452	1452	1452	1452	1452	1452
14	1270	1270	1270	1270	1270	1431
15	2721	2726	2726	2726	2667	2793
16	1720	1860	1860	1860	1820	1840
17	27486	27589	27718	27718	27539	28090
18	21976	21976	22502	22502	22502	22852
19	23743	23743	24019	24019	24019	24811
20	31269	32893	32893	32893	32893	32893
21	26332	27923	27923	27923	27923	27983

R = (10x10)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	3	2	0	2	7
2	7	2	0	3	20
3	4	5	0	2	60
4	4	1	0	2	9
5	1	10	0	1	14
6	8	4	0	1	79
7	4	2	0	2	11
8	3	7	0	3	52
9	6	2	0	3	13
10	9	1	0	2	21

R = (10x10)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	1	10	0	3	28
2	5	3	0	2	40
3	9	3	0	3	63
4	6	1	0	3	13
5	3	8	0	3	31
6	4	1	0	1	10
7	6	3	0	2	44

R = (10x10)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	3	7	0	2	35
2	8	2	0	2	40
3	10	2	0	1	27
4	5	4	0	3	23
5	2	9	0	2	43

R = (15x10)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	10	3	0	2	34
2	9	3	0	1	48
3	12	2	0	3	72
4	11	3	0	1	91
5	12	3	0	3	37
6	11	1	0	3	15
7	2	10	0	1	36

R = (15x10)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	15	2	0	2	34
2	7	3	0	1	27
3	9	1	0	1	14
4	8	3	0	1	71
5	12	2	0	2	61

R = (15x10)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	2	9	0	1	50
2	10	2	0	3	29
3	2	4	0	1	11
4	2	8	0	2	30
5	3	8	0	1	46
6	4	1	0	1	7
7	6	4	0	3	32
8	10	3	0	1	74
9	11	2	0	1	48
10	4	5	0	1	32

R = (20x20)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	15	3	0	1	110
2	6	6	0	3	93
3	13	1	0	2	38
4	8	7	0	2	144
5	18	5	0	3	185
6	12	4	0	1	78
7	8	3	0	1	25

R = (20x20)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	1	9	0	3	26
2	16	3	0	1	113
3	18	3	0	1	127
4	20	2	0	1	104
5	3	1	0	2	4

R = (20x20)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	14	2	0	1	45
2	1	5	0	1	10
3	20	4	0	2	139
4	12	3	0	3	92
5	11	8	0	2	145
6	11	6	0	2	123
7	7	9	0	1	166
8	17	5	0	1	252
9	7	14	0	2	161
10	1	7	0	3	12

R = (30x30)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	3	23	0	3	118
2	29	5	0	1	244
3	21	2	0	3	81
4	17	11	0	3	371
5	14	7	0	2	254
6	8	5	0	2	78
7	21	8	0	1	293

R = (30x30)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	12	21	0	1	450
2	17	21	0	1	570
3	6	6	0	1	49
4	9	15	0	1	166
5	5	12	0	1	69
6	16	6	0	1	92
7	5	4	0	1	17

R = (30x30)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	16	6	0	1	132
2	24	1	0	3	29
3	6	28	0	3	351
4	8	23	0	3	275
5	5	1	0	3	10
6	6	26	0	2	259
7	2	30	0	2	110
8	9	11	0	3	240
9	4	30	0	1	265
10	16	13	0	1	507

R = (30x30)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	1	30	0	3	75
2	25	7	0	1	279
3	27	9	0	3	409
4	30	2	0	3	65
5	26	7	0	3	235

R = (30x30)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	22	21	0	1	828
2	12	21	0	1	450
3	11	18	0	1	335
4	17	21	0	1	570
5	8	21	0	1	268
6	9	17	0	1	236
7	5	18	0	1	128
8	6	6	0	1	49
9	9	15	0	1	166
10	1	19	0	1	22
11	5	12	0	1	69
12	22	21	0	1	530
13	5	6	0	1	33
14	16	6	0	1	92
15	5	4	0	1	17

R = (70x40)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	17	9	0	1	153
2	11	19	0	4	209
3	12	21	0	3	252
4	14	23	0	4	322
5	24	15	0	3	360
6	25	16	0	4	400
7	27	17	0	2	459
8	18	29	0	3	522
9	21	31	0	3	651
10	32	22	0	2	704
11	23	33	0	3	759
12	34	24	0	2	816
13	35	25	0	2	875
14	36	26	0	1	936
15	37	27	0	1	999
16	38	28	0	1	1064
17	39	29	0	1	1131
18	41	30	0	1	1230
19	43	31	0	1	1333

R = (40x70)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	31	43	0	4	500
2	30	41	0	2	480
3	29	39	0	4	460
4	28	38	0	4	440
5	27	37	0	3	420
6	26	36	0	4	410
7	25	35	0	3	400
8	24	34	0	4	380
9	33	23	0	4	360
10	22	32	0	3	340
11	31	21	0	3	320
12	29	18	0	3	300
13	17	27	0	2	280
14	15	24	0	2	240
15	16	25	0	4	260
16	15	24	0	1	240
17	23	14	0	4	220
18	21	12	0	3	180
19	19	11	0	4	160
20	9	17	0	1	140

R = (100x100)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	4	90	0	5	838
2	22	21	0	2	521
3	22	80	0	3	4735
4	1	88	0	5	181
5	6	40	0	5	706
6	100	9	0	5	2538
7	46	14	0	3	1349
8	10	96	0	1	1685
9	70	27	0	3	5336
10	57	18	0	1	1775
11	10	84	0	1	1131
12	100	1	0	5	129
13	2	41	0	5	179
14	36	63	0	2	6668
15	51	24	0	4	3551

R = (100x100)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	8	81	0	1	953
2	5	76	0	1	389
3	42	19	0	1	1668
4	6	80	0	1	676
5	41	48	0	1	3580
6	6	86	0	1	1416
7	58	20	0	1	3166
8	99	3	0	1	537
9	9	52	0	1	1176
10	100	14	0	1	3434
11	7	53	0	1	676
12	24	54	0	1	1408
13	23	77	0	1	2362
14	42	32	0	1	4031
15	17	30	0	1	1152
16	11	90	0	1	2255
17	26	65	0	1	3570
18	11	84	0	1	1913
19	100	11	0	1	1552
20	29	81	0	1	4559
21	10	64	0	1	713
22	25	48	0	1	1279
23	17	93	0	1	3989
24	77	31	0	1	4850
25	3	71	0	1	299
26	89	9	0	1	1577
27	1	6	0	1	12
28	12	99	0	1	2116
29	33	72	0	1	2932
30	21	26	0	1	1214

R = (100x100)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	3	98	0	1	756
2	34	36	0	1	2712
3	100	6	0	1	1633
4	49	26	0	1	2332
5	14	56	0	1	2187
6	100	3	0	1	470
7	10	90	0	1	1569
8	23	95	0	1	4947
9	10	97	0	1	2757
10	50	47	0	1	4274
11	41	45	0	1	4347
12	13	12	0	1	396
13	19	68	0	1	3866
14	50	46	0	1	5447
15	23	70	0	1	2906
16	28	82	0	1	6032
17	12	65	0	1	1799
18	9	86	0	1	929
19	21	96	0	1	5186
20	19	64	0	1	2120
21	21	75	0	1	1629
22	45	26	0	1	2059
23	19	77	0	1	2583
24	5	84	0	1	953
25	16	21	0	1	1000
26	23	69	0	1	2900
27	5	89	0	1	1102
28	22	63	0	1	2234
29	41	6	0	1	458
30	76	30	0	1	5458

R = (100x100)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	48	48	0	1	5145
2	6	85	0	2	874
3	100	14	0	1	2924
4	17	85	0	1	3182
5	69	20	0	1	2862
6	12	72	0	1	1224
7	5	48	0	3	531
8	1	97	0	3	249
9	66	36	0	2	6601
10	15	53	0	1	1005
11	29	80	0	3	6228
12	19	77	0	1	3362
13	97	7	0	1	907
14	7	57	0	2	473
15	63	37	0	2	6137
16	71	14	0	1	1556
17	3	76	0	3	313
18	34	54	0	1	4123
19	5	91	0	2	581
20	14	87	0	1	1999
21	62	28	0	3	5004
22	6	7	0	3	2040
23	20	71	0	1	3143
24	92	7	0	1	795
25	10	77	0	2	1460
26	99	4	0	3	841
27	14	44	0	2	1107
28	100	2	0	3	280
29	56	40	0	2	5898
30	86	14	0	1	2096
31	22	93	0	1	4411
32	13	99	0	3	3456
33	7	76	0	3	1406

R = (100x100)
Pieces	C _i	A _i	P _i	Q _i	V _i
1	8	81	0	3	953
2	5	76	0	4	389
3	42	19	0	4	1668
4	6	80	0	4	676
5	41	48	0	1	3580
6	6	86	0	5	1416
7	58	20	0	5	3166
8	99	3	0	5	537
9	9	52	0	5	1176
10	100	14	0	4	3434
11	7	53	0	5	676
12	24	54	0	1	1408
13	23	77	0	1	2362
14	42	32	0	5	4031
15	17	30	0	5	1152
16	11	90	0	4	2255
17	26	65	0	2	3570
18	11	84	0	3	1913
19	100	11	0	1	1552
20	29	81	0	1	4559
21	10	64	0	2	713
22	25	48	0	1	1279
23	17	93	0	4	3989
24	77	31	0	1	4850
25	3	71	0	5	299
26	89	9	0	4	1577
27	1	6	0	5	12
28	12	99	0	2	2116
29	21	26	0	5	1214

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[1] *Corresponding author