流形对齐与双预测自适应动态多目标优化算法

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

摘要/Abstract

摘要：

针对动态多目标优化问题中现有预测型算法在复杂非线性环境和小样本条件下易出现预测失效、流形拓扑失真和环境响应与算子配置缺乏自适应性的问题，提出一种流形对齐与双预测自适应动态多目标优化算法（manifold alignment and dual prediction-based adaptive dynamic multi-objective evolutionary algorithm， MADP-DMOEA）。在环境变化时将决策变量流形解耦为中心与形状，利用二分图最优匹配实现相邻时刻流形的有效对齐，并依据几何位移判定环境变化强度；构建双预测模型，在剧烈变化时采用广义回归神经网络进行中心外推，在平缓变化时结合回声状态网络估计认知不确定性并自适应调节扰动尺度；设计基于动态收敛反馈的算子调度策略和决策空间拓扑保持的环境选择机制。基准测试结果表明：该算法在复杂时变环境下具有较好的收敛精度、分布多样性和动态跟踪能力，能够有效提升Pareto最优解集的追踪效果。

关键词: 动态多目标优化, 流形对齐, 自适应策略, 广义回归神经网络, 二分图匹配

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

中图分类号:

TP301

岳雪芝,熊超,蒋延申等 . 流形对齐与双预测自适应动态多目标优化算法[J]. 系统仿真学报, 2026, 38(10): 2887-2904.

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

图/表 11

图1

图2

图3

表1

MIGD 平均值和标准差

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

表1

图4

图5

图6

图7

图8

表2

表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