Manifold Alignment and Dual Prediction-based Adaptive Dynamic Multi-objective Evolutionary Algorithm

doi:10.16182/j.issn1004731x.joss.26-0401

Abstract

Abstract:

To address prediction failure, manifold topology distortion, and insufficient adaptability in environmental response and operator configuration in existing prediction-based algorithms for dynamic multi-objective optimization under complex nonlinear environments and small-sample settings, this paper proposes a manifold alignment and dual prediction-based adaptive dynamic multi-objective evolutionary algorithm, termed MADP-DMOEA. When the environment changes, the algorithm decomposes the decision-variable manifold into center and shape components, effectively aligns manifolds at adjacent time instants via bipartite optimal matching, and identifies environmental change severity using geometric displacement. A dual prediction model is developed, in which a GRNN performs center extrapolation under severe changes, while an echo state network (ESN) estimates epistemic uncertainty to adaptively adjust the perturbation scale under mild changes. Operator scheduling based on dynamic convergence feedback and environmental selection that preserves decision-space topology are introduced. Experimental results demonstrate that the proposed algorithm achieves good convergence accuracy, diversity of solution distribution, and dynamic tracking performance in complex time-varying environments and effectively improves the tracking of the Pareto-optimal solution set.

Key words: dynamic multi-objective optimization, manifold alignment, adaptive strategy, generalized regression neural network, bipartite graph matching

CLC Number:

TP301

Yue Xuezhi, Xiong Chao, Jiang Yanshen, Guo Zhaolu, Peng Hu. Manifold Alignment and Dual Prediction-based Adaptive Dynamic Multi-objective Evolutionary Algorithm[J]. Journal of System Simulation, 2026, 38(10): 2887-2904.

Figures/Tables 11

Fig. 1

Fig. 2

Fig. 3

Table 1

Mean and standard deviation of MIGD

问题	( $τ t$ , $n t$ )	AE-DMOEA	DIP-DMOEA	KL-DMOEA	KTM-DMOEA	SVR-DMOEA	MADP-DMOEA
DF1	(5,5)	127.53×10⁻³ (7.73×10⁻³) -	350.04×10⁻³ (62.7×10⁻³) -	171.63×10⁻³ (11.5×10⁻³) -	179.31×10⁻³ (29.6×10⁻³) -	503.58×10⁻³ (109×10⁻³) -	64.403×10⁻³ (9.30×10⁻³)
	(5,10)	67.534×10⁻³ (6.52×10⁻³) -	108.45×10⁻³ (24.6×10⁻³) -	123.45×10⁻³ (9.80×10⁻³) -	106.49×10⁻³ (13.3×10⁻³) -	207.12×10⁻³ (71.2×10⁻³) -	43.648×10⁻³ (3.68×10⁻³)
	(10,5)	67.664×10⁻³ (5.29×10⁻³) -	98.053×10⁻³ (19.6×10⁻³) -	81.744×10⁻³ (4.69×10⁻³) -	63.693×10⁻³ (12.6×10⁻³) -	130.02×10⁻³ (39.2×10⁻³) -	20.520×10⁻³ (1.47×10⁻³)
	(10,10)	24.939×10⁻³ (1.71×10⁻³) -	30.945×10⁻³ (4.67×10⁻³) -	55.265×10⁻³ (4.57×10⁻³) -	43.344×10⁻³ (6.39×10⁻³) -	47.674×10⁻³ (11.9×10⁻³) -	15.305×10⁻³ (1.13×10⁻³)
DF2	(5,5)	103.53×10⁻³ (9.22×10⁻³) -	260.66×10⁻³ (24.9×10⁻³) -	122.09×10⁻³ (9.62×10⁻³) -	168.54×10⁻³ (16.9×10⁻³) -	312.95×10⁻³ (85.4×10⁻³) -	59.481×10⁻³ (4.78×10⁻³)
	(5,10)	103.20×10⁻³ (13.4×10⁻³) -	153.17×10⁻³ (18.1×10⁻³) -	94.084×10⁻³ (5.88×10⁻³) -	132.49×10⁻³ (16.6×10⁻³) -	217.30×10⁻³ (53.5×10⁻³) -	44.906×10⁻³ (4.26×10⁻³)
	(10,5)	123.74×10⁻³ (13.2×10⁻³) -	118.36×10⁻³ (17.4×10⁻³) -	61.215×10⁻³ (4.80×10⁻³) -	88.255×10⁻³ (6.38×10⁻³) -	147.99×10⁻³ (30.5×10⁻³) -	20.544×10⁻³ (1.87×10⁻³)
	(10,10)	122.00×10⁻³ (8.71×10⁻³) -	74.280×10⁻³ (6.87×10⁻³) -	48.699×10⁻³ (4.53×10⁻³) -	71.319×10⁻³ (7.67×10⁻³) -	98.023×10⁻³ (12.9×10⁻³) -	16.405×10⁻³ (1.51×10⁻³)
DF3	(5,5)	525.02×10⁻³ (43.6×10⁻³) -	343.26×10⁻³ (90.5×10⁻³) -	645.48×10⁻³ (74.3×10⁻³) -	469.16×10⁻³ (131×10⁻³) -	586.91×10⁻³ (99.8×10⁻³) -	172.85×10⁻³ (59.1×10⁻³)
	(5,10)	324.99×10⁻³ (47.6×10⁻³) -	297.12×10⁻³ (62.1×10⁻³) -	378.95×10⁻³ (29.8×10⁻³) -	407.47×10⁻³ (80.9×10⁻³) -	390.68×10⁻³ (34.6×10⁻³) -	157.04×10⁻³ (70.0×10⁻³)
	(10,5)	361.94×10⁻³ (28.0×10⁻³) -	180.04×10⁻³ (70.9×10⁻³) -	373.65×10⁻³ (21.1×10⁻³) -	177.04×10⁻³ (54.3×10⁻³) -	380.61×10⁻³ (45.2×10⁻³) -	57.229×10⁻³ (27.9×10⁻³)
	(10,10)	271.51×10⁻³ (46.3×10⁻³) -	142.94×10⁻³ (79.7×10⁻³) -	301.60×10⁻³ (26.5×10⁻³) -	234.99×10⁻³ (44.9×10⁻³) -	295.13×10⁻³ (34.3×10⁻³) -	44.586×10⁻³ (21.6×10⁻³)
DF4	(5,5)	304.60×10⁻³ (37.5×10⁻³) -	233.56×10⁻³ (34.0×10⁻³) -	289.96×10⁻³ (45.5×10⁻³) -	479.99×10⁻³ (39.3×10⁻³) -	428.25×10⁻³ (72.6×10⁻³) -	196.09×10⁻³ (22.3×10⁻³)
	(5,10)	249.55×10⁻³ (34.7×10⁻³) -	232.37×10⁻³ (24.2×10⁻³) -	284.09×10⁻³ (38.3×10⁻³) -	382.40×10⁻³ (50.0×10⁻³) -	430.28×10⁻³ (63.4×10⁻³) -	192.44×10⁻³ (18.4×10⁻³)
	(10,5)	166.04×10⁻³ (10.3×10⁻³) -	144.79×10⁻³ (6.04×10⁻³) -	164.06×10⁻³ (7.86×10⁻³) -	306.29×10⁻³ (10.2×10⁻³) -	198.79×10⁻³ (21.0×10⁻³) -	141.00×10⁻³ (2.46×10⁻³)
	(10,10)	160.79×10⁻³ (9.92×10⁻³) -	142.86×10⁻³ (6.79×10⁻³) =	172.41×10⁻³ (12.9×10⁻³) -	230.60×10⁻³ (10.2×10⁻³) -	200.18×10⁻³ (21.0×10⁻³) -	139.91×10⁻³ (3.47×10⁻³)
DF5	(5,5)	253.21×10⁻³ (36.6×10⁻³) -	192.29×10⁻³ (62.0×10⁻³) -	366.37×10⁻³ (30.6×10⁻³) -	68.456×10⁻³ (7.43×10⁻³) +	433.75×10⁻³ (88.6×10⁻³) -	76.205×10⁻³ (6.07×10⁻³)
	(5,10)	147.35×10⁻³ (19.3×10⁻³) -	71.692×10⁻³ (10.1×10⁻³) -	259.60×10⁻³ (23.6×10⁻³) -	47.529×10⁻³ (5.00×10⁻³) -	225.95×10⁻³ (55.6×10⁻³) -	44.710×10⁻³ (3.96×10⁻³)
	(10,5)	93.760×10⁻³ (9.83×10⁻³) -	41.117×10⁻³ (5.77×10⁻³) -	168.13×10⁻³ (10.1×10⁻³) -	27.849×10⁻³ (1.86×10⁻³) -	107.45×10⁻³ (28.4×10⁻³) -	22.592×10⁻³ (1.32×10⁻³)
	(10,10)	29.741×10⁻³ (3.17×10⁻³) -	21.237×10⁻³ (2.65×10⁻³) -	60.154×10⁻³ (5.09×10⁻³) -	22.804×10⁻³ (0.916×10⁻³) -	60.912×10⁻³ (17.1×10⁻³) -	14.932×10⁻³ (0.879×10⁻³)
DF6	(5,5)	5 266.2×10⁻³ (528×10⁻³) -	5 541.7×10⁻³ (779×10⁻³) -	7 241.5×10⁻³ (467×10⁻³) -	3 318.5×10⁻³ (836×10⁻³) +	9 373.3×10⁻³ (2 240×10⁻³) -	4 025.5×10⁻³ (723×10⁻³)
	(5,10)	4 424.5×10⁻³ (399×10⁻³) -	4 112.6×10⁻³ (1 010×10⁻³) =	5 659.0×10⁻³ (385×10⁻³) -	3 263.3×10⁻³ (845×10⁻³) +	6 466.5×10⁻³ (1 260×10⁻³) -	4 059.6×10⁻³ (762×10⁻³)
	(10,5)	1 612.6×10⁻³ (109×10⁻³) +	2 048.9×10⁻³ (581×10⁻³) =	2 448.9×10⁻³ (217×10⁻³) -	1 573.4×10⁻³ (740×10⁻³) +	3 271.3×10⁻³ (935×10⁻³) -	1 983.4×10⁻³ (533×10⁻³)
	(10,10)	1 496.3×10⁻³ (192×10⁻³) +	2 800.6×10⁻³ (760×10⁻³) =	1 995.9×10⁻³ (192×10⁻³) +	1 716.5×10⁻³ (675×10⁻³) +	2 633.3×10⁻³ (1 670×10⁻³) -	2 592.6×10⁻³ (470×10⁻³)
DF7	(5,5)	295.63×10⁻³ (72.7×10⁻³) -	585.85×10⁻³ (46.5×10⁻³) -	505.71×10⁻³ (38.8×10⁻³) -	388.94×10⁻³ (84.1×10⁻³) -	903.97×10⁻³ (434×10⁻³) -	90.228×10⁻³ (30.9×10⁻³)
	(5,10)	324.81×10⁻³ (53.0×10⁻³) -	341.78×10⁻³ (79.9×10⁻³) -	293.88×10⁻³ (49.2×10⁻³) -	308.30×10⁻³ (16.6×10⁻³) -	500.96×10⁻³ (86.2×10⁻³) -	121.45×10⁻³ (54.2×10⁻³)
	(10,5)	325.09×10⁻³ (63.5×10⁻³) -	429.50×10⁻³ (86.3×10⁻³) -	460.28×10⁻³ (47.8×10⁻³) -	305.09×10⁻³ (61.9×10⁻³) -	614.25×10⁻³ (61.2×10⁻³) -	70.331×10⁻³ (28.6×10⁻³)
	(10,10)	354.74×10⁻³ (32.2×10⁻³) -	163.36×10⁻³ (79.9×10⁻³) -	258.14×10⁻³ (46.2×10⁻³) -	280.14×10⁻³ (22.1×10⁻³) -	443.64×10⁻³ (39.8×10⁻³) -	86.641×10⁻³ (51.4×10⁻³)
DF8	(5,5)	90.196×10⁻³ (3.69×10⁻³) +	247.52×10⁻³ (55.2×10⁻³) -	95.146×10⁻³ (8.81×10⁻³) +	245.94×10⁻³ (34.7×10⁻³) -	101.35×10⁻³ (28.2×10⁻³) +	134.05×10⁻³ (15.4×10⁻³)
	(5,10)	83.857×10⁻³ (2.39×10⁻³) +	267.86×10⁻³ (81.2×10⁻³) -	86.523×10⁻³ (5.46×10⁻³) +	246.12×10⁻³ (49.4×10⁻³) -	95.397×10⁻³ (6.46×10⁻³) +	129.01×10⁻³ (12.0×10⁻³)
	(10,5)	83.411×10⁻³ (1.89×10⁻³) +	197.59×10⁻³ (42.5×10⁻³) -	84.297×10⁻³ (7.87×10⁻³) +	193.97×10⁻³ (23.4×10⁻³) -	92.922×10⁻³ (7.07×10⁻³) =	94.250×10⁻³ (9.90×10⁻³)
	(10,10)	76.290×10⁻³ (3.14×10⁻³) +	211.86×10⁻³ (52.5×10⁻³) -	71.465×10⁻³ (2.95×10⁻³) +	179.01×10⁻³ (24.9×10⁻³) -	88.229×10⁻³ (12.6×10⁻³) =	88.007×10⁻³ (8.98×10⁻³)
DF9	(5,5)	569.34×10⁻³ (31.5×10⁻³) -	1 844.6×10⁻³ (343×10⁻³) -	672.63×10⁻³ (36.7×10⁻³) -	463.79×10⁻³ (87.7×10⁻³) -	1 122.6×10⁻³ (405×10⁻³) -	240.79×10⁻³ (35.4×10⁻³)
	(5,10)	393.01×10⁻³ (34.3×10⁻³) -	1 040.6×10⁻³ (177×10⁻³) -	547.42×10⁻³ (41.8×10⁻³) -	318.46×10⁻³ (89.4×10⁻³) -	1 087.0×10⁻³ (315×10⁻³) -	149.06×10⁻³ (24.8×10⁻³)
	(10,5)	323.84×10⁻³ (25.2×10⁻³) -	1 116.6×10⁻³ (185×10⁻³) -	410.65×10⁻³ (30.5×10⁻³) -	237.52×10⁻³ (50.0×10⁻³) -	670.98×10⁻³ (218×10⁻³) -	124.48×10⁻³ (24.5×10⁻³)
	(10,10)	247.53×10⁻³ (26.3×10⁻³) -	606.94×10⁻³ (175×10⁻³) -	310.69×10⁻³ (26.7×10⁻³) -	129.86×10⁻³ (32.8×10⁻³) -	714.96×10⁻³ (103×10⁻³) -	58.476×10⁻³ (11.7×10⁻³)
DF10	(5,5)	89.831×10⁻³ (8.95×10⁻³) +	279.16×10⁻³ (45.2×10⁻³) +	114.24×10⁻³ (12.0×10⁻³) +	333.95×10⁻³ (35.4×10⁻³) -	145.24×10⁻³ (12.4×10⁻³) +	285.55×10⁻³ (22.1×10⁻³)
	(5,10)	101.91×10⁻³ (10.1×10⁻³) +	311.44×10⁻³ (48.6×10⁻³) =	118.81×10⁻³ (12.7×10⁻³) +	333.90×10⁻³ (54.1×10⁻³) =	173.38×10⁻³ (31.6×10⁻³) +	316.90×10⁻³ (18.9×10⁻³)
	(10,5)	77.491×10⁻³ (4.81×10⁻³) +	191.25×10⁻³ (24.4×10⁻³) -	80.807×10⁻³ (4.47×10⁻³) +	241.46×10⁻³ (28.4×10⁻³) -	110.62×10⁻³ (7.27×10⁻³) +	162.44×10⁻³ (14.0×10⁻³)
	(10,10)	87.178×10⁻³ (7.99×10⁻³) +	224.40×10⁻³ (54.7×10⁻³) -	89.907×10⁻³ (6.07×10⁻³) +	281.21×10⁻³ (55.5×10⁻³) -	126.33×10⁻³ (7.22×10⁻³) +	176.27×10⁻³ (14.5×10⁻³)
DF11	(5,5)	690.81×10⁻³ (3.62×10⁻³) -	707.05×10⁻³ (8.00×10⁻³) -	698.30×10⁻³ (3.44×10⁻³) -	652.58×10⁻³ (16.2×10⁻³) +	705.70×10⁻³ (8.63×10⁻³) -	682.21×10⁻³ (3.53×10⁻³)
	(5,10)	676.59×10⁻³ (3.29×10⁻³) -	688.89×10⁻³ (8.51×10⁻³) -	685.11×10⁻³ (5.15×10⁻³) -	619.55×10⁻³ (11.3×10⁻³) +	694.88×10⁻³ (13.1×10⁻³) -	668.25×10⁻³ (4.24×10⁻³)
	(10,5)	676.37×10⁻³ (1.96×10⁻³) -	668.39×10⁻³ (1.99×10⁻³) -	681.28×10⁻³ (2.51×10⁻³) -	582.12×10⁻³ (4.97×10⁻³) +	686.38×10⁻³ (2.63×10⁻³) -	661.71×10⁻³ (1.09×10⁻³)
	(10,10)	658.35×10⁻³ (2.20×10⁻³) -	652.11×10⁻³ (3.15×10⁻³) -	662.54×10⁻³ (2.62×10⁻³) -	574.61×10⁻³ (4.01×10⁻³) +	671.79×10⁻³ (4.28×10⁻³) -	646.92×10⁻³ (1.16×10⁻³)
DF12	(5,5)	556.79×10⁻³ (117×10⁻³) -	382.59×10⁻³ (45.2×10⁻³) -	383.41×10⁻³ (41.5×10⁻³) -	471.06×10⁻³ (183×10⁻³) -	636.29×10⁻³ (104×10⁻³) -	294.01×10⁻³ (23.0×10⁻³)
	(5,10)	515.69×10⁻³ (58.3×10⁻³) -	301.01×10⁻³ (26.7×10⁻³) -	356.63×10⁻³ (90.4×10⁻³) -	280.24×10⁻³ (69.4×10⁻³) -	589.03×10⁻³ (49.9×10⁻³) -	233.50×10⁻³ (11.6×10⁻³)
	(10,5)	419.44×10⁻³ (102×10⁻³) -	257.26×10⁻³ (30.5×10⁻³) -	315.98×10⁻³ (35.2×10⁻³) -	308.05×10⁻³ (155×10⁻³) -	616.16×10⁻³ (93.0×10⁻³) -	206.12×10⁻³ (15.0×10⁻³)
	(10,10)	230.46×10⁻³ (77.8×10⁻³) -	172.60×10⁻³ (12.2×10⁻³) -	154.60×10⁻³ (5.93×10⁻³) -	183.43×10⁻³ (34.4×10⁻³) -	478.36×10⁻³ (126×10⁻³) -	144.25×10⁻³ (8.06×10⁻³)
DF13	(5,5)	1 928.6×10⁻³ (137×10⁻³) -	467.58×10⁻³ (82.5×10⁻³) -	617.65×10⁻³ (49.4×10⁻³) -	260.88×10⁻³ (16.2×10⁻³) +	692.79×10⁻³ (154×10⁻³) -	327.05×10⁻³ (9.92×10⁻³)
	(5,10)	765.80×10⁻³ (167×10⁻³) -	349.76×10⁻³ (38.0×10⁻³) -	461.55×10⁻³ (29.4×10⁻³) -	240.10×10⁻³ (18.0×10⁻³) +	412.28×10⁻³ (118×10⁻³) -	297.58×10⁻³ (7.42×10⁻³)
	(10,5)	701.90×10⁻³ (93.9×10⁻³) -	289.16×10⁻³ (11.6×10⁻³) -	364.48×10⁻³ (25.8×10⁻³) -	164.89×10⁻³ (6.80×10⁻³) +	301.67×10⁻³ (36.0×10⁻³) -	250.08×10⁻³ (3.88×10⁻³)
	(10,10)	185.87×10⁻³ (15.0×10⁻³) +	259.37×10⁻³ (7.99×10⁻³) -	237.20×10⁻³ (12.6×10⁻³) =	164.16×10⁻³ (4.58×10⁻³) +	182.42×10⁻³ (14.1×10⁻³) +	242.56×10⁻³ (3.70×10⁻³)
DF14	(5,5)	1 552.6×10⁻³ (518×10⁻³) -	149.94×10⁻³ (33.6×10⁻³) -	711.51×10⁻³ (69.4×10⁻³) -	115.39×10⁻³ (9.80×10⁻³) -	827.00×10⁻³ (178×10⁻³) -	94.932×10⁻³ (2.86×10⁻³)
	(5,10)	1 327.5×10⁻³ (470×10⁻³) -	95.242×10⁻³ (6.46×10⁻³) -	549.77×10⁻³ (86.5×10⁻³) -	88.562×10⁻³ (8.35×10⁻³) -	650.84×10⁻³ (166×10⁻³) -	81.057×10⁻³ (2.07×10⁻³)
	(10,5)	942.87×10⁻³ (362×10⁻³) -	74.803×10⁻³ (3.70×10⁻³) -	532.05×10⁻³ (59.9×10⁻³) -	62.665×10⁻³ (2.46×10⁻³) +	696.39×10⁻³ (184×10⁻³) -	65.922×10⁻³ (1.38×10⁻³)
	(10,10)	579.63×10⁻³ (190×10⁻³) -	63.912×10⁻³ (1.54×10⁻³) -	328.78×10⁻³ (36.9×10⁻³) -	49.356×10⁻³ (2.15×10⁻³) +	615.16×10⁻³ (133×10⁻³) -	63.048×10⁻³ (1.39×10⁻³)
FDA1	(5,5)	173.65×10⁻³ (22.9×10⁻³) -	182.25×10⁻³ (55.1×10⁻³) -	279.78×10⁻³ (33.3×10⁻³) -	117.40×10⁻³ (22.2×10⁻³) -	351.72×10⁻³ (141×10⁻³) -	62.480×10⁻³ (5.71×10⁻³)
	(5,10)	73.190×10⁻³ (9.78×10⁻³) -	80.709×10⁻³ (21.3×10⁻³) -	167.96×10⁻³ (19.1×10⁻³) -	68.430×10⁻³ (18.1×10⁻³) -	111.94×10⁻³ (28.9×10⁻³) -	39.565×10⁻³ (3.30×10⁻³)
	(10,5)	73.122×10⁻³ (8.98×10⁻³) -	46.051×10⁻³ (10.1×10⁻³) -	109.53×10⁻³ (6.76×10⁻³) -	31.481×10⁻³ (6.86×10⁻³) -	76.388×10⁻³ (16.1×10⁻³) -	20.658×10⁻³ (1.09×10⁻³)
	(10,10)	22.130×10⁻³ (1.61×10⁻³) -	23.706×10⁻³ (4.07×10⁻³) -	37.311×10⁻³ (1.96×10⁻³) -	22.151×10⁻³ (5.97×10⁻³) -	37.336×10⁻³ (8.08×10⁻³) -	14.103×10⁻³ (0.629×10⁻³)
FDA2	(5,5)	43.662×10⁻³ (7.53×10⁻³) -	72.355×10⁻³ (12.5×10⁻³) -	80.498×10⁻³ (7.17×10⁻³) -	209.80×10⁻³ (98.0×10⁻³) -	151.05×10⁻³ (49.5×10⁻³) -	41.548×10⁻³ (17.8×10⁻³)
	(5,10)	26.216×10⁻³ (7.02×10⁻³) +	92.806×10⁻³ (160×10⁻³) -	47.569×10⁻³ (7.57×10⁻³) -	131.78×10⁻³ (97.1×10⁻³) -	108.70×10⁻³ (28.3×10⁻³) -	39.266×10⁻³ (25.3×10⁻³)
	(10,5)	14.196×10⁻³ (2.04×10⁻³) +	27.866×10⁻³ (10.4×10⁻³) -	26.667×10⁻³ (3.75×10⁻³) -	95.731×10⁻³ (9.77×10⁻³) -	39.193×10⁻³ (7.01×10⁻³) -	16.540×10⁻³ (5.23×10⁻³)
	(10,10)	11.009×10⁻³ (1.40×10⁻³) +	22.239×10⁻³ (6.05×10⁻³) -	15.048×10⁻³ (2.43×10⁻³) =	93.057×10⁻³ (136×10⁻³) -	31.679×10⁻³ (11.7×10⁻³) -	16.169×10⁻³ (12.6×10⁻³)
FDA3	(5,5)	177.53×10⁻³ (13.6×10⁻³) -	270.98×10⁻³ (60.6×10⁻³) -	323.46×10⁻³ (95.4×10⁻³) -	276.24×10⁻³ (68.8×10⁻³) -	330.24×10⁻³ (76.6×10⁻³) -	117.93×10⁻³ (4.06×10⁻³)
	(5,10)	85.778×10⁻³ (5.23×10⁻³) =	122.76×10⁻³ (12.5×10⁻³) -	158.33×10⁻³ (10.0×10⁻³) -	129.32×10⁻³ (9.55×10⁻³) -	188.76×10⁻³ (50.6×10⁻³) -	86.236×10⁻³ (4.50×10⁻³)
	(10,5)	88.417×10⁻³ (10.7×10⁻³) -	104.40×10⁻³ (7.86×10⁻³) -	122.78×10⁻³ (7.48×10⁻³) -	162.27×10⁻³ (7.59×10⁻³) -	137.56×10⁻³ (40.7×10⁻³) -	58.907×10⁻³ (2.46×10⁻³)
	(10,10)	51.496×10⁻³ (3.78×10⁻³) -	63.464×10⁻³ (3.32×10⁻³) -	58.455×10⁻³ (3.97×10⁻³) -	80.098×10⁻³ (4.88×10⁻³) -	85.275×10⁻³ (16.7×10⁻³) -	46.070×10⁻³ (1.88×10⁻³)
FDA4	(5,5)	687.52×10⁻³ (41.6×10⁻³) -	302.47×10⁻³ (46.6×10⁻³) -	283.77×10⁻³ (14.4×10⁻³) -	187.03×10⁻³ (11.6×10⁻³) -	628.10×10⁻³ (99.9×10⁻³) -	153.00×10⁻³ (11.2×10⁻³)
	(5,10)	365.77×10⁻³ (56.5×10⁻³) -	177.04×10⁻³ (16.1×10⁻³) -	217.84×10⁻³ (11.8×10⁻³) -	135.58×10⁻³ (10.7×10⁻³) =	507.00×10⁻³ (135×10⁻³) -	131.27×10⁻³ (4.12×10⁻³)
	(10,5)	405.91×10⁻³ (33.8×10⁻³) -	143.36×10⁻³ (11.7×10⁻³) -	171.30×10⁻³ (7.78×10⁻³) -	95.897×10⁻³ (3.54×10⁻³) +	337.59×10⁻³ (76.8×10⁻³) -	98.904×10⁻³ (1.50×10⁻³)
	(10,10)	134.30×10⁻³ (10.0×10⁻³) -	104.74×10⁻³ (4.04×10⁻³) -	148.37×10⁻³ (6.64×10⁻³) -	76.379×10⁻³ (4.37×10⁻³) +	160.15×10⁻³ (32.5×10⁻³) -	93.252×10⁻³ (1.03×10⁻³)
FDA5	(5,5)	684.91×10⁻³ (82.1×10⁻³) -	511.42×10⁻³ (82.5×10⁻³) -	383.54×10⁻³ (20.3×10⁻³) -	589.94×10⁻³ (67.5×10⁻³) -	765.51×10⁻³ (129×10⁻³) -	270.31×10⁻³ (9.83×10⁻³)
	(5,10)	410.47×10⁻³ (60.4×10⁻³) -	338.59×10⁻³ (28.4×10⁻³) -	292.98×10⁻³ (16.3×10⁻³) -	510.46×10⁻³ (123×10⁻³) -	538.59×10⁻³ (133×10⁻³) -	237.33×10⁻³ (8.37×10⁻³)
	(10,5)	450.84×10⁻³ (30.2×10⁻³) -	267.71×10⁻³ (43.4×10⁻³) -	225.71×10⁻³ (7.66×10⁻³) -	400.96×10⁻³ (77.1×10⁻³) -	420.86×10⁻³ (73.2×10⁻³) -	178.13×10⁻³ (2.90×10⁻³)
	(10,10)	207.88×10⁻³ (12.5×10⁻³) -	211.24×10⁻³ (15.8×10⁻³) -	200.23×10⁻³ (5.44×10⁻³) -	261.72×10⁻³ (15.6×10⁻³) -	221.15×10⁻³ (31.9×10⁻³) -	168.71×10⁻³ (3.13×10⁻³)
dMOP1	(5,5)	172.61×10⁻³ (52.3×10⁻³) -	143.68×10⁻³ (43.8×10⁻³) -	241.64×10⁻³ (57.4×10⁻³) -	192.91×10⁻³ (97.3×10⁻³) -	339.39×10⁻³ (158×10⁻³) -	25.886×10⁻³ (10.9×10⁻³)
	(5,10)	182.03×10⁻³ (74.5×10⁻³) -	157.84×10⁻³ (47.8×10⁻³) -	202.69×10⁻³ (56.3×10⁻³) -	173.16×10⁻³ (73.0×10⁻³) -	371.35×10⁻³ (177×10⁻³) -	34.323×10⁻³ (24.3×10⁻³)
	(10,5)	63.200×10⁻³ (25.6×10⁻³) -	37.108×10⁻³ (14.2×10⁻³) -	68.804×10⁻³ (29.3×10⁻³) -	71.951×10⁻³ (15.2×10⁻³) -	80.091×10⁻³ (36.3×10⁻³) -	10.496×10⁻³ (5.01×10⁻³)
	(10,10)	56.982×10⁻³ (21.7×10⁻³) -	43.303×10⁻³ (11.6×10⁻³) -	62.552×10⁻³ (31.9×10⁻³) -	68.012×10⁻³ (21.3×10⁻³) -	87.021×10⁻³ (31.9×10⁻³) -	9.681 4×10⁻³ (4.41×10⁻³)
dMOP2	(5,5)	127.70×10⁻³ (7.35×10⁻³) -	359.23×10⁻³ (47.9×10⁻³) -	172.49×10⁻³ (12.6×10⁻³) -	181.49×10⁻³ (34.5×10⁻³) -	500.04×10⁻³ (144×10⁻³) -	62.624×10⁻³ (6.65×10⁻³)
	(5,10)	69.029×10⁻³ (7.06×10⁻³) -	115.79×10⁻³ (27.1×10⁻³) -	127.00×10⁻³ (10.9×10⁻³) -	108.49×10⁻³ (13.6×10⁻³) -	228.40×10⁻³ (92.2×10⁻³) -	43.788×10⁻³ (5.65×10⁻³)
	(10,5)	66.998×10⁻³ (4.22×10⁻³) -	97.317×10⁻³ (12.4×10⁻³) -	83.783×10⁻³ (7.16×10⁻³) -	62.518×10⁻³ (8.44×10⁻³) -	138.10×10⁻³ (34.9×10⁻³) -	21.002×10⁻³ (1.43×10⁻³)
	(10,10)	25.609×10⁻³ (1.90×10⁻³) -	31.028×10⁻³ (5.80×10⁻³) -	56.046×10⁻³ (4.73×10⁻³) -	42.907×10⁻³ (5.55×10⁻³) -	48.354×10⁻³ (11.7×10⁻³) -	15.603×10⁻³ (1.10×10⁻³)
dMOP3	(5,5)	101.55×10⁻³ (5.43×10⁻³) -	256.87×10⁻³ (46.6×10⁻³) -	113.80×10⁻³ (6.88×10⁻³) -	129.49×10⁻³ (17.0×10⁻³) -	312.48×10⁻³ (69.1×10⁻³) -	44.054×10⁻³ (3.58×10⁻³)
	(5,10)	57.100×10⁻³ (4.95×10⁻³) -	86.755×10⁻³ (21.4×10⁻³) -	85.909×10⁻³ (6.29×10⁻³) -	65.180×10⁻³ (9.36×10⁻³) -	167.06×10⁻³ (68.5×10⁻³) -	31.956×10⁻³ (3.32×10⁻³)
	(10,5)	58.588×10⁻³ (6.83×10⁻³) -	76.181×10⁻³ (13.2×10⁻³) -	58.222×10⁻³ (4.15×10⁻³) -	41.385×10⁻³ (5.36×10⁻³) -	100.29×10⁻³ (23.8×10⁻³) -	17.010×10⁻³ (1.44×10⁻³)
	(10,10)	20.107×10⁻³ (1.87×10⁻³) -	26.143×10⁻³ (4.82×10⁻³) -	39.847×10⁻³ (2.73×10⁻³) -	24.144×10⁻³ (4.65×10⁻³) -	35.469×10⁻³ (7.39×10⁻³) -	12.748×10⁻³ (0.895×10⁻³)
+/-/=		14/73/1	1/82/5	9/77/2	17/69/2	7/79/2

Table 1

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Table 2

Table 3

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模型	MIGD(+/-/=)	MHV(+/-/=)
MADP-DMOEA vs MADP-Ⅰ	14/4/1	14/1/4
MADP-DMOEA vs MADP-Ⅱ	15/1/3	15/0/4
MADP-DMOEA vs MADP-Ⅲ	15/1/3	15/0/4
MADP-DMOEA vs MADP-Ⅳ	18/1/0	18/0/1
MADP-DMOEA vs MADP-Ⅴ	15/4/0	14/4/1