COMPARING A STATIC EQUILIBRIUM BASED METHOD WITH THE SUPPORT FACTOR FOR HORIZONTAL CARGO STABILITY IN THE CONTAINER LOADING PROBLEM

Oliveira, Liliane de Azevedo; de Lima, Vinícius Loti; Queiroz, Thiago Alves de; Miyazawa, Flávio Keidi

doi:10.1590/0101-7438.2021.041.00240379

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ARTICLES • Pesqui. Oper. 41 • 2021 • https://doi.org/10.1590/0101-7438.2021.041.00240379 copy

COMPARING A STATIC EQUILIBRIUM BASED METHOD WITH THE SUPPORT FACTOR FOR HORIZONTAL CARGO STABILITY IN THE CONTAINER LOADING PROBLEM

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

This paper presents an approach to deal with horizontal cargo stability in container loading problems. Cargo stability has been explored mainly with support factors that constrain the minimum area of each box’s faces to be supported by other boxes. On the other hand, we propose an approach based on the static equilibrium of rigid bodies to check the static stability of a given packing. The approach is used as a cutting plane routine in a branch-and-cut framework to the single container loading problem. This framework considers the resolution of an integer linear programming model to obtain feasible packings next checked with the proposed approach to avoid unstable packings. The computational experiments consider 180 benchmark instances on which stable solutions of the proposed approach have better container fill rates than the support factor approach. In terms of lateral support, the proposed approach provides the minimum value inferior to 70% on average, which is satisfactorily smaller and less restrictive than the full support. Results also indicate that more unstable solutions emerge from refined grids and fewer types of boxes available.

Keywords:
container loading problem; horizontal cargo stability; equilibrium of material bodies

			Unitary Discretization							CD							RRP
Group	Dim.	Type	JMY approach			Proposed approach				JMY approach			Proposed approach				JMY approach			Proposed approach
			gap	Time	Vol.	#Cuts	gap	Time	Vol.	gap	Time	Vol.	#Cuts	gap	Time	Vol.	gap	Time	Vol.	#Cuts	gap	Time	Vol.
			(%)	(s)	(%)		(%)	(s)	(%)	(%)	(s)	(%)		(%)	(s)	(%)	(%)	(s)	(%)		(%)	(s)	(%)
A ₅	10	Min.	0.00	0.10	76.00	0.00	0.00	0.20	77.20	0.00	0.10	76.00	0.00	0.00	0.07	77.20	0.00	0.05	76.00	0.00	0.00	0.06	77.20
		Avg.	0.00	6.15	88.75	50.30	1.11	391.22	90.04	0.00	2.74	88.75	58.90	0.00	15.14	90.74	0.00	2.35	88.75	58.90	0.00	17.99	90.74
		Max.	0.00	28.84	100.00	402.00	10.10	3600.08	100.00	0.00	10.58	100.00	402.00	0.00	92.60	100.00	0.00	12.82	100.00	402.00	0.00	124.72	100.00
	20	Min.	0.00	1.58	53.25	4.00	0.00	47.61	31.40	0.00	0.02	53.25	0.00	0.00	0.10	6.12	0.00	0.01	53.25	4.00	0.00	0.08	51.05
		Avg.	0.00	17.01	72.20	2125.40	14.30	2109.86	72.71	0.00	2.26	72.20	3130.80	20.60	2542.77	68.92	0.00	1.07	68.62	4641.40	12.97	2352.75	70.78
		Max.	0.00	76.84	87.00	10008.00	68.60	3601.37	90.10	0.00	15.40	87.00	17477.00	93.87	3603.93	90.10	0.00	1.07	68.62	28028.00	43.90	3632.73	88.25
	30	Min.	0.00	2.87	42.49	4.00	0.00	2568.87	9.48	0.00	0.00	42.49	4.00	0.00	0.13	29.67	0.00	0.00	42.49	4.00	0.00	0.09	34.20
		Avg.	0.00	7.64	58.58	15.00	34.60	3288.20	56.51	0.00	0.03	58.58	1211.30	12.91	1483.04	52.99	0.00	0.02	55.51	5054.90	8.45	1715.88	54.04
		Max.	0.00	16.12	76.18	28.00	90.50	3607.64	76.18	0.00	0.10	76.18	7769.00	41.90	3601.53	74.51	0.00	0.10	66.70	25885.00	35.00	3604.56	66.70
A ₁₀	10	Min.	0.00	2.35	96.80	0.00	0.00	0.33	98.40	0.00	0.62	96.80	0.00	0.00	0.12	98.40	0.00	0.84	96.80	0.00	0.00	0.13	98.40
		Avg.	0.00	214.63	99.22	3.40	0.00	27.52	99.53	0.00	225.53	99.21	2.20	0.00	29.56	99.53	0.00	221.73	99.20	2.20	0.00	31.21	99.53
		Max.	0.00	1268.58	100.00	12.00	0.00	193.64	100.00	0.00	1232.70	100.00	4.00	0.00	235.2	100.00	0.00	1208.86	100.00	4.00	0.00	252.34	100.00
	20	Min.	0.00	13.26	70.76	0.00	0.00	325.74	4.90	0.00	1.93	70.76	0.00	0.00	5.99	49.88	0.00	1.95	71.76	0.00	0.00	1.73	18.50
		Avg.	2.14	789.42	85.14	4.20	19.51	1966.89	77.19	0.00	190.96	85.26	4.10	3.60	1532.41	81.71	0.00	125.05	82.28	7.00	18.85	1723.18	75.96
		Max.	15.50	3600.07	94.30	12.00	97.00	3601.22	94.29	0.00	1088.87	94.30	17.00	21.60	3603.41	94.30	0.00	582.21	94.30	26.00	81.50	3602.53	94.30
	30	Min.	0.00	68.20	63.83	4.00	0.00	3409.98	5.40	0.00	0.45	63.83	4.00	0.00	0.33	58.32	0.00	1.08	61.89	4.00	0.00	0.03	45.47
		Avg.	0.00	683.61	76.02	4.00	41.77	3658.79	57.54	0.00	53.13	76.02	4.00	3.27	1774.98	79.04	0.00	17.84	73.83	4.00	6.28	1291.41	77.50
		Max.	0.00	3601.33	93.56	4.00	94.60	3954.21	93.56	0.00	382.04	93.56	4.00	17.50	3600.92	93.56	0.00	136.95	93.56	4.00	40.80	3601.68	93.56
A ₂₀	10	Min.	-	3600.00	-	0.00	0.00	1.84	99.60	0.00	16.86	98.40	0.00	0.00	1.70	99.60	0.00	16.97	98.40	0.00	0.00	1.70	99.60
		Avg.	-	3600.00	-	2.20	0.00	5.34	99.95	0.12	379.61	99.81	2.20	0.00	3.82	99.95	0.12	379.58	99.81	2.20	0.00	3.82	99.95
		Max.	-	3600.00	-	7.00	0.00	19.36	100.00	1.20	3601.01	100.00	7.00	0.00	11.30	100.00	1.20	3601.01	100.00	7.00	0.00	11.31	100.00
	20	Min.	-	3600.00	-	0.00	0.00	1588.51	19.25	0.00	37.24	100.00	0.00	0.00	11.33	90.64	0.00	183.03	81.93	0.00	0.00	79.78	90.64
		Avg.	-	3600.00	-	3.60	19.97	3123.07	79.14	0.47	1172.69	91.76	3.20	0.00	792.16	95.02	0.46	1021.18	91.53	3.20	1.36	730.02	95.01
		Max.	-	3600.00	-	4.00	80.70	3602.28	97.30	4.70	3601.03	97.75	4.00	0.00	3601.04	97.75	4.60	3601.12	97.75	4.00	0.00	3600.04	97.75
	30	Min.	-	3600.00	-	4.00	0.00	3257.34	7.70	0.00	51.63	77.96	4.00	0.00	37.24	86.24	0.00	27.09	77.96	4.00	0.00	21.18	86.42
		Avg.	-	3600.00	-	4.00	23.09	3643.17	75.58	0.28	806.01	88.51	4.00	1.27	1107.72	91.59	0.00	489.50	87.15	4.00	0.23	2366.33	91.27
		Max.	-	3600.00	-	4.00	92.30	4215.76	97.21	2.80	3601.03	92.89	4.00	2.70	2500.44	97.21	0.00	2663.68	92.89	4.00	1.20	14194.00	97.21

			Unitary Discretization							CD							RRP
Group	Dim.	Type	JMY approach			Proposed approach				JMY approach			Proposed approach				JMY approach			Proposed approach
			gap	Time	Vol.	#Cuts	gap	Time	Vol.	gap	Time	Vol.	#Cuts	gap	Time	Vol.	gap	Time	Vol.	#Cuts	gap	Time	Vol.
			(%)	(s)	(%)		(%)	(s)	(%)	(%)	(s)	(%)		(%)	(s)	(%)	(%)	(s)	(%)		(%)	(s)	(%)
B ₅	10	Min.	0.00	4.69	95.20	0.00	0.00	0.55	95.20	0.00	1.35	95.20	0.00	0.00	0.63	95.20	0.00	1.36	95.20	0.00	0.00	0.38	95.20
		Avg.	0.14	751.97	98.50	6653.10	0.00	468.01	98.59	0.14	853.12	98.51	804.90	0.00	447.45	98.59	0.14	754.74	98.51	833.20	0.00	435.36	98.59
		Max.	0.80	3600.34	100.00	50063.00	0.00	3600.01	100.00	0.80	3600.06	100.00	7062.00	0.00	3601.67	100.00	0.80	3600.01	100.00	7062.00	0.00	3600.00	100.00
	20	Min.	0.00	204.57	58.50	45.00	0.00	108.52	77.60	0.00	32.74	69.65	61.00	0.00	11.53	77.60	0.00	13.60	69.05	281.00	0.00	2.87	48.60
		Avg.	14.56	3260.49	83.36	13203.80	3.70	2553.87	95.03	11.36	3154.16	85.87	73058.20	0.67	2358.07	94.47	11.49	3081.18	85.23	72782.10	13.03	2301.38	90.53
		Max.	32.70	3600.13	99.20	46127.00	2.20	3600.62	100.00	30.50	3600.12	99.20	162107.00	6.70	3609.72	100.00	30.90	3600.14	99.20	160379.00	92.00	3613.29	100.00
	30	Min.	2.90	3600.18	24.69	115.00	2.00	1031.63	84.18	3.10	3600.01	24.69	59.00	0.00	1128.52	47.50	0.00	971.14	49.01	59.00	0.00	946.94	11.61
		Avg.	18.51	3601.29	78.97	7026.20	10.40	3600.54	88.95	16.50	3601.32	79.36	64928.60	7.27	3182.20	92.72	18.63	3217.22	79.19	76743.40	26.22	3335.01	71.42
		Max.	75.30	3603.98	95.64	55356.00	27.00	4215.76	97.21	75.30	3610.04	95.38	171815.00	15.81	3600.5	99.00	82.70	3606.36	95.64	168289.00	88.30	3601.62	99.00
B ₁₀	10	Min.	0.00	10.53	9 9.90	0.00	0.00	1.23	100.00	0.00	10.52	100.00	0.00	0.00	1.22	100.00	0.00	10.47	99.90	0.00	0.00	1.22	100.00
		Avg.	0.00	59.10	99.98	7.70	0.00	4.20	100.00	0.00	60.00	100.00	13.00	0.00	4.23	100.00	0.00	60.13	99.98	11.80	0.00	4.19	100.00
		Max.	0.00	273.47	100.00	22.00	0.00	7.07	100.00	0.00	276.49	100.00	56.00	0.00	7.06	100.00	0.00	278.84	100.00	56.00	0.00	7.06	100.00
	20	Min.	4.30	3600.12	60.00	42.00	0.00	3313.05	73.60	10.40	3600.13	60.00	32.00	0.00	1838.46	1.50	4.30	3600.92	60.00	45.00	0.00	2291.8	3.00
		Avg.	19.91	3602.11	81.73	12787.00	9.50	3571.42	90.07	24.48	3602.17	81.08	35940.30	18.28	3424.05	81.71	23.79	3603.25	81.01	33321.50	19.00	3469.40	81.95
		Max.	40.00	3605.10	95.63	54188.00	26.00	3600.64	100.00	40.30	3605.75	95.63	148589.00	98.50	3601.08	100.00	23.79	3603.25	81.01	159543.00	97.00	3600.63	100.00
	30	Min.	19.60	3660.61	68.47	13.00	9.30	3600.01	79.81	19.60	3622.96	68.47	46.00	9.30	3600.00	79.81	19.60	3606.69	68.47	7.00	9.00	3600.00	80.74
		Avg.	23.95	3724.07	76.00	270.10	15.17	3603.43	84.83	23.95	3646.85	76.00	42284.60	15.14	3600.01	84.83	25.58	3632.47	74.37	27184.60	15.02	3600.00	84.92
		Max.	31.50	3865.85	80.36	2295.00	20.19	3612.10	90.70	31.50	3671.19	80.36	136988.00	20.10	3600.03	90.70	31.50	3668.39	80.36	142789.00	19.20	3600.01	90.70
B ₂₀	10	Min.	0.00	32.06	100.00	0.00	0.00	0.72	100.00	0.00	32.13	100.00	0.00	0.00	0.72	100.00	0.00	32.15	100.00	0.00	0.00	0.72	100.00
		Avg.	0.00	87.24	100.00	3.50	0.00	4.79	100.00	0.00	87.18	100.00	4.60	0.00	4.79	100.00	0.00	87.35	100.00	4.60	0.00	4.81	100.00
		Max.	0.00	156.05	100.00	9.00	0.00	8.07	100.00	0.00	156.26	100.00	16.00	0.00	8.06	100.00	0.00	157.20	100.00	16.00	0.00	8.08	100.00
	20	Min.	0.50	3628.63	82.99	24.00	0.30	3600.01	92.03	0.50	3612.72	82.99	6.00	0.30	3600.00	92.03	0.50	3612.30	82.99	14.00	0.30	3600.00	92.03
		Avg.	6.89	3630.96	93.08	296.40	3.78	3600.13	96.18	6.89	3629.82	93.08	3986.70	3.78	3600.02	96.18	6.89	3631.28	93.08	4301.00	3.98	3600.22	96.18
		Max.	17.00	3635.48	99.45	1825.00	7.90	3600.37	99.70	17.00	3672.34	99.45	21945.00	7.90	3600.44	99.70	17.00	3678.31	99.45	24174.00	7.90	3600.72	99.70
	30	Min.	-	-	-	5.00	11.60	3600.18	81.46	-	3600.00	-	7.00	11.60	3600.01	81.46	-	3600.00	-	6.00	11.60	3600.01	81.46
		Avg.	20.71	1689.04	59.29	13.70	14.63	3608.24	85.31	-	3600.00	-	3751.10	14.72	3600.02	85.27	-	3600.00	-	14117.00	14.62	3600.02	85.32
		Max.	33.50	5176.78	85.40	23.00	18.50	3667.17	88.32	-	3600.00	-	24952.00	18.53	3600.05	88.32	-	3600.00	-	83909.00	18.5	3600.06	88.32

		Horizontal Stability
Group	Dim.	On the direction of z-axis			On the direction of x-axis			On the direction of y-axis
Group	Dim.	Min	Avg.	Max.	Min.	Avg.	Max.	Min.	Avg.	Max.
A ₅	10	75.00	98.82	100.00	70.00	98.89	100.00	81.67	99.17	100.00
	20	56.10	95.47	100.00	53.38	94.94	100.00	53.80	96.13	100.00
	30	58.81	94.50	100.00	37.30	93.05	100.00	48.43	94.52	100.00
A ₁₀	10	69.50	91.78	100.00	65.79	90.70	100.00	66.83	90.61	100.00
	20	59.76	78.30	93.98	41.78	70.28	90.36	38.69	67.05	91.82
	30	51.58	76.70	96.06	68.38	79.63	83.38	67.15	86.56	100.00
A ₂₀	10	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00
	20	54.28	97.53	100.00	54.41	97.17	100.00	69.84	97.40	100.00
	30	36.78	93.14	100.00	21.01	91.58	100.00	25.83	89.50	100.00
B ₅	10	94.67	99.41	100.00	83.29	98.38	100.00	94.17	99.69	100.00
	20	62.01	87.86	100.00	40.05	80.92	100.00	68.48	90.13	100.00
	30	58.77	81.67	96.46	43.94	72.56	90.47	52.68	74.79	87.82
B ₁₀	10	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00
	20	34.50	97.17	100.00	43.96	97.04	100.00	46.25	97.41	100.00
	30	27.85	91.54	100.00	29.14	91.16	100.00	33.23	90.62	100.00
B ₂₀	10	96.67	99.74	100.00	98.89	99.91	100.00	96.67	99.78	100.00
	20	64.98	91.81	100.00	63.38	89.56	97.78	69.53	91.72	100.00
	30	74.56	91.22	98.67	65.01	90.86	100.00	62.93	85.62	98.97

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[1] *Corresponding author