A MODEL-BASED HEURISTIC FOR THE IRREGULAR STRIP PACKING PROBLEM

Cherri, Luiz Henrique; Carravilla, Maria Antónia; Toledo, Franklina Maria Bragion

doi:10.1590/0101-7438.2016.036.03.0447

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Articles • Pesqui. Oper. 36 (3) • Sep-Dec 2016 • https://doi.org/10.1590/0101-7438.2016.036.03.0447 copy

A MODEL-BASED HEURISTIC FOR THE IRREGULAR STRIP PACKING PROBLEM

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

The irregular strip packing problem is a common variant of cutting and packing problems. Only a few exact methods have been proposed to solve this problem in the literature. However, several heuristics have been proposed to solve it. Despite the number of proposed heuristics, only a few methods that combine exact and heuristic approaches to solve the problem can be found in the literature. In this paper, a matheuristic is proposed to solve the irregular strip packing problem. The method has three phases in which exact mixed integer programming models from the literature are used to solve the sub-problems. The results show that the matheuristic is less dependent on the instance size and finds equal or better solutions in 87,5% of the cases in shorter computational times compared with the results of other models in the literature. Furthermore, the matheuristic is faster than other heuristics from the literature.

Keywords:
matheuristics; irregular strip packing; mixed integer programming models

Name	Piece types	Pieces quantity	Rotations	Board height	Reference
Albano	25	25	0,90,180,270	4900	(^{Albano & Sapuppo, 1980}1 ALBANO A & SAPUPPO G. 1980. Optimal allocation of two-dimensional irregular shapes using heuristic search methods. IEEE Transactions on Systems, Man, and Cybernetics, SMC-10(5): 242-248.)
Mao	9	20	0,90,180,270	2550	(^{Bounsaythip & Maouche, 1997}5 BOUNSAYTHIP C & MAOUCHE S 1997. Irregular shape nesting and placing with evolutionary approach. In Systems, Man, and Cybernetics, volume 4, pages 3425-3430.)
Marques	8	24	0,90,180,270	104	(^{Marques et al., 1991}22 MARQUES VMM, BISPO CFG & SENTIEIRO JJS. 1991. A system for the compaction of twodimensional irregular shapes based on simulated annealing. In: International Conference on Industrial Electronics, Control and Instrumentation, pages 1911-1916.)
Trousers	17	64	0,180	79	(^{Oliveira & Ferreira, 1993}23 OLIVEIRA JF & FERREIRA J 1993. Algorithms for nesting problems, applied simulated annealing. In Vidal RVV. (Ed., Lecture Notes in Economics andMaths Systems), 396: 255-274.)
Jakobs1	25	25	0,90,180,270	40	(^{Jakobs, 1996}18 JAKOBS S. 1996. On genetic algorithms for the packing of polygons. European Journal of Operational Research, 88(1): 165-181.)
Jakobs2	25	25	0,90,180,270	70	(^{Jakobs, 1996}18 JAKOBS S. 1996. On genetic algorithms for the packing of polygons. European Journal of Operational Research, 88(1): 165-181.)
Fu	12	12	0,90,180,270	38	(^{Fujita et al., 1981}13 FUJITA K, AKAGJI S & KIROKAWA N. 1981. Hybrid approach for optimal nesting using a genetic algorithm and a local minimization algorithm. Advances in Design Automation; American Society of Mechanical Engineers, 65: 477-484.)
Poly1a0	15	15	0	40	(^{Hopper, 2000}16 HOPPER E. 2000. Two-dimensional packing utilising evolutionary algorithms and other metaheuristic methods. PhD thesis, University of Wales, Cardiff.)
Shapes0	4	43	0	40	(^{Oliveira et al., 2000}24 OLIVEIRA JF, GOMES AM & FERREIRA JS. 2000. Topos - a new constructive algorithm for nesting problems. OR Spektrum, 22: 263-284.)
Shapes1	4	43	0,180	40	(^{Oliveira et al., 2000}24 OLIVEIRA JF, GOMES AM & FERREIRA JS. 2000. Topos - a new constructive algorithm for nesting problems. OR Spektrum, 22: 263-284.)
Shapes2	7	28	0,180	15	(^{Oliveira et al., 2000}24 OLIVEIRA JF, GOMES AM & FERREIRA JS. 2000. Topos - a new constructive algorithm for nesting problems. OR Spektrum, 22: 263-284.)
Blaze<i>	7	7´<i>	0	15	(^{Toledo et al., 2013}26 TOLEDO FMB, CARRAVILLA MA, RIBEIRO C, OLIVEIRA JF & GOMES AM. 2013. The dottedboard model: a new MIP model for nesting irregular shapes. International Journal of Production Economics, 145(2): 478-487.)
Shapes_T<i>	4	8	0	40	(^{Toledo et al., 2013}26 TOLEDO FMB, CARRAVILLA MA, RIBEIRO C, OLIVEIRA JF & GOMES AM. 2013. The dottedboard model: a new MIP model for nesting irregular shapes. International Journal of Production Economics, 145(2): 478-487.)
RCO<i>	7	<i>	0	15	(^{Toledo et al., 2013}26 TOLEDO FMB, CARRAVILLA MA, RIBEIRO C, OLIVEIRA JF & GOMES AM. 2013. The dottedboard model: a new MIP model for nesting irregular shapes. International Journal of Production Economics, 145(2): 478-487.)
Shapes_AV4	4	4	0	20	(^{Alvarez-Valdes et al., 2013}2 ALVAREZ-VALDES R, MARTINEZ A & TAMARIT J. 2013. A branch & bound algorithm for cutting and packing irregularly shaped pieces. International Journal of Production Economics, 145(2): 463- 477.)
Shapes_AV8	4	8	0	13	(^{Alvarez-Valdes et al., 2013}2 ALVAREZ-VALDES R, MARTINEZ A & TAMARIT J. 2013. A branch & bound algorithm for cutting and packing irregularly shaped pieces. International Journal of Production Economics, 145(2): 463- 477.)
Fu<i>	<i>	<i>	0	38	(^{Alvarez-Valdes et al., 2013}2 ALVAREZ-VALDES R, MARTINEZ A & TAMARIT J. 2013. A branch & bound algorithm for cutting and packing irregularly shaped pieces. International Journal of Production Economics, 145(2): 463- 477.)
threep<i>w<j>	3	3´<i>	0	<j>	(^{Alvarez-Valdes et al., 2013}2 ALVAREZ-VALDES R, MARTINEZ A & TAMARIT J. 2013. A branch & bound algorithm for cutting and packing irregularly shaped pieces. International Journal of Production Economics, 145(2): 463- 477.)

Instance	Constructive phase		Constructive + improvement			Add.	Time	3PM			Add.	Time
	Sol.	Time	Sol.	Time	Impr.	time	inc.(%)	Sol.	Time	Impr.	time	inc.(%)
Blaze1	18.0	0.1	12.0	1.9	33.3%	1.8	94.7%	7.4	23.3	38.3%	21.5	92.3%
Blaze2	16.0	1.0	14.0	8.7	12.5%	7.7	88.5%	14.0	68.9	0.0%	60.2	87.4%
Blaze3	24.0	1.6	21.0	38.8	12.5%	37.2	95.9%	20.5	340.2	2.6%	301.4	88.6%
Blaze4	32.0	2.2	29.0	35.0	9.4%	32.7	93.4%	27.9	517.6	3.9%	482.6	93.2%
Blaze5	40.0	2.7	34.0	153.5	15.0%	150.8	98.2%	34.0	395.2	0.1%	241.7	61.2%
Shapes_T2	30.0	1.2	16.0	7.0	46.7%	5.8	82.9%	14.0	8.1	12.5%	1.1	13.6%
Shapes_T4	30.0	4.1	30.0	10.4	0.0%	6.3	60.6%	26.0	201.5	13.3%	191.1	94.8%
Shapes_T5	36.0	7.5	31.0	45.6	13.9%	38.1	83.6%	31.0	106.0	0.0%	60.4	57.0%
Shapes_T7	60.0	9.9	42.0	116.1	30.0%	106.2	91.5%	42.0	176.3	0.0%	60.3	34.2%
Shapes_T9	71.0	10.2	48.0	170.6	32.4%	160.3	94.0%	48.0	292.3	0.0%	121.7	41.6%
RCO1	14.0	0.1	8.0	4.2	42.9%	4.1	97.6%	8.0	44.4	0.0%	40.3	90.8%
RCO2	16.0	0.5	16.0	2.1	0.0%	1.6	76.2%	15.0	254.5	6.3%	252.5	99.2%
RCO3	24.0	1.2	24.0	4.6	0.0%	3.4	73.9%	22.0	264.7	8.3%	260.1	98.3%
RCO4	32.0	2.1	30.0	33.0	6.3%	30.9	93.6%	29.0	83.1	3.3%	50.1	60.3%
RCO5	40.0	4.7	37.0	29.8	7.5%	25.0	83.9%	36.7	210.2	0.9%	180.5	85.9%
Shapes_AV4	24.0	0.6	24.0	2.1	0.0%	1.6	76.2%	24.0	2.2	0.0%	0.1	4.5%
Shapes_AV8	41.0	0.8	30.0	5.5	26.8%	4.7	85.5%	26.0	186.9	13.3%	181.4	97.1%
Fu5	20.0	0.1	20.0	0.3	0.0%	0.2	66.7%	17.9	1.0	10.6%	0.7	70.0%
Fu6	56.0	0.1	28.0	30.5	50.0%	30.4	99.7%	23.0	31.7	17.9%	1.2	3.8%
Fu7	70.0	0.1	28.0	3.9	60.0%	3.9	100.0%	24.0	5.0	14.3%	1.1	22.0%
Fu8	49.0	0.1	28.0	19.5	42.9%	19.4	99.5%	24.0	20.9	14.3%	1.4	6.7%
Fu9	56.0	0.1	30.0	26.7	46.4%	26.5	99.3%	25.0	52.3	16.7%	25.7	49.1%
Fu10	42.0	0.2	30.0	25.7	28.6%	25.5	99.2%	28.7	265.9	4.4%	240.3	90.4%
Fu12	45.0	0.4	42.0	2.1	6.7%	1.7	81.0%	33.5	186.7	20.4%	184.6	98.9%
threep1w7	6.5	0.6	6.5	1.4	0.0%	0.8	57.1%	6.0	2.5	7.7%	1.0	40.0%
threep2w7	13.5	0.3	11.5	2.4	14.8%	2.1	87.5%	9.3	12.4	18.9%	10.1	81.5%
threep3w7	20.0	0.3	17.0	1.1	15.0%	0.8	72.7%	13.5	183.1	20.4%	182.0	99.4%
threep2w9	12.0	0.1	10.0	0.6	16.7%	0.6	100.0%	8.0	36.2	20.0%	35.5	98.1%
threep3w9	18.0	0.3	13.0	1.9	27.8%	1.6	84.2%	11.0	191.2	15.4%	189.4	99.1%
Shapes0	68.0	33.9	60.0	178.6	11.8%	144.7	81.0%	60.0	239.1	0.0%	60.5	25.3%
Shapes1	62.0	424.4	58.0	1011.6	6.5%	587.2	58.0%	58.0	1132.7	0.0%	121.1	10.7%
Shapes2	31.0	7.2	28.0	130.1	9.7%	122.8	94.4%	27.6	310.7	1.5%	180.6	58.1%
Fu	40.0	5.6	40.0	9.6	0.0%	4.0	41.7%	32.0	252.3	20.0%	242.7	96.2%
Poly1a0	33.0	17.4	18.0	732.9	45.5%	715.5	97.6%	15.8	1048.8	12.2%	315.9	30.1%
Jakobs1	25.0	21.7	15.0	184.0	40.0%	162.4	88.3%	12.0	612.9	20.0%	428.8	70.0%
Jakobs2	36.0	34.1	30.0	645.2	16.7%	611.2	94.7%	26.0	1939.0	13.3%	1293.8	66.7%
Albano	12168.0	14.2	12168.0	342.1	0.0%	328.0	95.9%	10608.0	1614.1	12.8%	1272.0	78.8%
Mao	2315.0	21.0	2294.0	2321.8	0.9%	2300.9	99.1%	1927.2	2621.8	16.0%	300.0	11.4%
Marques	85.0	100.1	85.0	273.2	0.0%	173.1	63.4%	80.0	527.9	5.9%	254.7	48.2%
Trousers	439.0	229.9	296.0	1222.1	32.6%	992.2	81.2%	286.0	1403.7	3.4%	181.7	12.9%
Average					19.0%	171.8	85.3%			9.7%	200.8	61.7%

Instance	HS2¹		SCM²		DBM			3PM
	Solution	Time	Solution	Time	Solution	Time (find)	Time	Solution	Time
Blaze1	-	-	7.5	4.9	7.5	12.0	23.3	7.4	23.3
Blaze2	-	-	13.8	TL	14.0	15.1	15.2	14.0	68.9
Blaze3	-	-	20.7	TL	21.0	80.3	674.0	20.5	340.2
Blaze4	-	-	29.0	TL	27.0	1068.0	1239.2	27.9	517.6
Blaze5	-	-	37.7	TL	34.0	540.5	TL	34.0	395.2
Shapes_T2	-	-	14.0	6.6	16.0	0.5	1.7	14.0	8.1
Shapes_T4	-	-	28.0	TL	26.0	58.4	89.1	26.0	201.5
Shapes_T5	-	-	34.0	TL	30.0	340.6	365.0	31.0	106.0
Shapes_T7	-	-	52.0	TL	42.0	2901.0	TL	42.0	176.3
Shapes_T9	-	-	57.0	TL	49.0	3482.6	TL	48.0	292.3
RCO1	-	-	8.0	5.8	8.0	0.6	0.7	8.0	44.4
RCO2	-	-	15.0	TL	15.0	1.2	1.3	15.0	254.5
RCO3	-	-	23.0	TL	22.0	10.7	13.2	22.0	264.7
RCO4	-	-	32.3	TL	29.0	16.7	394.0	29.0	83.1
RCO5	-	-	38.0	TL	36.0	164.6	936.2	36.7	210.2
Albano	-	-	-	-	11088.0	592.4	592.4	10608.0	1614.1
Fu	-	-	-	-	35.0	53.1	53.1	32.0	252.3
Jakobs1	-	-	-	-	18.0	3285.3	TL	12.0	612.9
Jakobs2	-	-	-	-	30.0	596.2	TL	26.0	1939.0
Mao	-	-	-	-	2452.0	99.7	TL	1927.2	2621.8
Marques	-	-	-	-	88.0	1827.9	TL	85.0	527.9
Shapes0	-	-	78.0	TL	64.0	3590.5	TL	60.0	239.1
Shapes1	-	-	-	-	80.0	98.9	TL	58.0	1132.7
Shapes2	-	-	-	-	27.0	900.4	TL	27.6	310.7
Trousers	-	-	-	-	495.0	218.4	TL	286.0	1403.7
Poly1a0	16.6	TL	16.7	TL	17.0	3586.2	TL	15.8	1048.8
Shapes_AV4	24.0	0.0	-	-	24.0	1.7	1.7	24.0	2.2
Shapes_AV8	26.0	272.0	-	-	28.0	18.5	21.5	26.0	186.9
Fu5	17.9	0.1	17.9	76.3	20.5	2.8	3.4	17.9	1.0
Fu6	23.0	0.5	23.0	442.9	24.0	6.0	10.4	23.0	31.7
Fu7	24.0	1.0	24.0	TL	28.0	0.1	0.2	24.0	5.0
Fu8	24.0	1.3	24.0	TL	28.0	0.2	1.0	24.0	20.9
Fu9	25.0	70.0	25.0	TL	28.0	0.4	0.4	25.0	52.3
Fu10	28.7	3064.0	30.0	TL	30.0	0.8	0.9	28.7	265.9
Fu12	31.2	TL	34.4	TL	40.0	1.0	1.0	32.0	186.7
threep1w7	6.0	0.8	6.0	0.3	6.5	0.3	0.3	6.0	2.5
threep2w7	9.3	3.9	9.7	1.0	11.0	0.7	0.8	9.3	12.4
threep3w7	13.5	3394.0	14.0	852.7	14.5	1.3	1.3	13.5	183.1
threep2w9	8.0	8.5	8.0	3.1	8.5	1.4	1.6	8.0	36.2
threep3w9	11.0	TL	11.3	TL	13.0	0.2	0.2	11.0	191.2

Instance	3PM		(^{Leung et al., 2012}19 LEUNG SC, LIN Y & ZHANG D. 2012. Extended local search algorithm based on nonlinear programming for two-dimensional irregular strip packing problem. Computers & Operations Research, 39(3): 678-686.)		(^{Elkeran, 2013}10 ELKERAN A. 2013. A new approach for sheet nesting problem using guided cuckoo search and pairwise clustering. European Journal of Operational Research, 231(3): 757-769.)
	Solution	Time	Solution	Time	Solution	Time
Shapes0	60.0	239.1	59.7	10 × 1207.0	59.32	10 × 600.0
Shapes1	58.0	1132.7	53.7	10 × 1212.0	54.07	10 × 600.0
Shapes2	27.6	310.7	26.2	10 × 1205.0	26.21	10 × 600.0
Fu	32.0	252.3	31.7	10 × 600.0	31.46	10 × 600.0
Jakobs1	12.0	612.9	11.1	10 × 603.0	11.02	10 × 600.0
Jakobs2	26.0	1939.0	23.8	10 × 602.0	23.79	10 × 600.0
Albano	10608.0	1614.1	9969.5	10 × 1203.0	9959.24	10 × 600.0
Mao	1927.2	2621.8	1785.1	10 × 1204.0	1796.86	10 × 600.0
Marques	80.0	527.9	78.3	10 × 1204.0	77.37	10 × 600.0
Trousers	286.0	1403.7	246.7	10 × 1237.0	244.67	10 × 600.0

Instance	(^{Sato et al., 2012}25 SATO AK, MARTINS TC & TSUZUKI MSG 2012.An algorithm for the strip packing problem using collision free region and exact fitting placement. Computer-Aided Design, 44(8): 766-777.)		(^{Umetani et al., 2009}27 UMETANI S, YAGIURA M, IMAHORI S, IMAMICHI T, NONOBE K & IBARAKI T. 2009. Solving the irregular strip packing problem via guided local search for overlap minimization. International Transactions in Operational Research, 16(6): 661-683.)		(^{Imamichi et al., 2009}17 IMAMICHI T, YAGIURA M & NAGAMOCHI H. 2009. An iterated local search algorithm based on nonlinear programming for the irregular strip packing problem. Discrete Optimization, 6(4): 345- 361.)
	Solution	Time	Solution	Time	Solution	Time
Shapes0	61.1	7 × 21600.0	60.3	10 × 1200.0	60.2	10 × 1200.0
Shapes1	55.6	7 × 908175.0	55.0	10 × 1200.0	54.8	10 × 1200.0
Shapes2	26.7	7 × 5400.0	26.7	10 × 1200.0	26.4	10 × 1200.0
Fu	31.8	4 × 600.0	31.6	10 × 1200.0	32.6	10 × 600.0
Jakobs1	11.0	4 × 1800.0	11.0	10 × 1200.0	11.6	10 × 600.0
Jakobs2	22.8	7 × 5400.0	24.0	10 × 1200.0	24.0	10 × 600.0
Albano	10086.5	7 × 21600.0	9980.9	10 × 1200.0	9990.2	10 × 1200.0
Mao	1816.6	7 × 21600.0	1780.4	10 × 1200.0	1813.4	10 × 1200.0
Marques	78.9	7 × 5400.0	77.9	10 × 1200.0	79.7	10 × 1200.0
Trousers	248.3	7 × 86400.0	245.3	10 × 1200.0	245.6	10 × 1200.0

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