考虑指标协同效应重构的联合作战体系效能评估灰色主成分模型

doi:10.12305/j.issn.1001-506X.2025.05.18

Abstract

Abstract:

The number and dimensionality of system effectiveness evaluation indicators are numerous, and the indicators are interrelated and have synergistic effects, which increases the computational complexity of effectiveness evaluation. To address this issue, a grey principal components analysis (GPCA) reconstruction method of joint operation system effectiveness indicator considering synergism is proposed. Firstly, analyzing the operational mission, tasks, and processes of the joint air defense combat system of systems, the system effectiveness evaluation indicator under the joint operation mode is constructed, and the grey incidence model is used to analyze whether there is synergism between the indicators. Then, based on the discovery of synergism among indicators, three strategies are proposed to reconstruct the effectiveness evaluation indicator system of joint operation system. Combined with the GPCA method, a GPCA model with synergism is constructed to reduce the dimension of effectiveness evaluation indicator. Finally, this method is applied to the effectiveness evaluation case of a joint air defense operation system, and the indicators with synergism are selected. The effectiveness evaluation indicator system is reconstructed. The comparison and analysis of the calculation results and methods indicate that the proposed method can effectively find the synergism between the indicators, and the reconstructed evaluation index system maintains "isomorphism".

Key words: system-of-systems effectiveness evaluation, synergism, grey incidence, indicator reconstruction, grey principal component analysis (GPCA)

CLC Number:

N945.25

Ding CHEN, Zhigeng FANG, Baohua YANG, Feng YE, Na ZHANG, Jingru ZHANG. Grey principal component analysis model of effectiveness evaluation for joint operation system-of-systems considering indicator synergism reconstruction[J]. Systems Engineering and Electronics, 2025, 47(5): 1561-1574.

Figures/Tables 12

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评估指标	X₁	X₂	…	X_i	…	X_n
X₁	—	—	…	—	√	√
X₂	—	—	…	—	—	√
				√		
X_i	—	—	√	—	—	√
	√					
X_n	√	√	—	√	—	—

评估指标	仿真样本
评估指标	A₁	A₂	A₃	A₄	A₅	A₆	A₇	A₈	A₉	A₁₀	A₁₁	A₁₂
X₁(+)	550	400	650	400	450	400	500	500	400	400	450	400
X₂(+)	30	20	25	20	35	35	40	45	40	35	45	40
X₃(-)	15	10	40	20	25	30	44	35	50	35	40	45
X₄(-)	50	30	25	25	45	30	33	28	42	30	45	35
X₅(+)	750	900	800	950	1 000	850	900	1 000	850	1 000	750	800
X₆(+)	150	200	120	200	150	180	150	200	150	200	180	200
X₇(+)	20	35	25	30	25	35	40	30	35	40	40	30
X₈(+)	2	1	3	4	2	2	4	1	3	2	4	1
X₉(-)	1	2	2	1	1	2	2	2	1	2	2	1
X₁₀(-)	6	7	7	8	7	6	6	7	6	7	7	6
X₁₁(+)	0.3	0.35	0.4	0.34	0.42	0.40	0.38	0.4	0.35	0.38	0.42	0.4
X₁₂(+)	0.6	0.7	0.78	0.62	0.78	0.75	0.72	0.73	0.72	0.75	0.78	0.75
X₁₃(+)	0.82	0.85	0.8	0.75	0.85	0.87	0.90	0.88	0.85	0.87	0.78	0.80
样本效果	0.690 6	0.782 8	0.732 5	0.810 6	0.881 2	0.875	0.931 8	0. 916 2	0.865 6	0. 890 6	0.823 7	0.859 3

评估指标	X₁	X₂	X₃	X₄	X₅	X₆	X₇	X₈	X₉	X₁₀	X₁₁	X₁₂	X₁₃
X₁		√				√	√			√
X₂						√				√
X₃				√					√
X₄			√			√	√	√	√
X₅							√				√	√
X₆	√	√					√			√	√
X₇	√			√	√
X₈									√
X₉			√	√				√
X₁₀	√	√									√	√	√
X₁₁						√				√			√
X₁₂					√					√
X₁₃					√					√	√	√

协同效应指标	灰色关联度矩阵特征值
协同效应指标	μ₁	μ₂	μ₃	μ₄	μ₅
X₁X₂X₆	0.280 8	0.089 6	-0.17	-0.156 8	0.034 3
X₁X₂X₁₀	0.27	-0.278 7	-0.016 4	0.452 1	0.043
X₂X₁₀	0.279 2	-0.01	-0.251	-0.297 2	-0.066 8
X₃	0.253 2	0.446 1	0.222 8	-0.161	0.094 9
X₄	0.259 8	0.031 9	-0.438 8	-0.301 9	-0.095 4
X₅	0.278 4	-0.401 1	-0.147	0.051 6	0.040 8
X₇	0.264 4	-0.030 7	0.204 5	-0.401	0.105 7
X₈	0.224 1	0.666 7	-0.327 1	0.442 8	0.157
X₉	0.261 1	0.153 7	0.687 8	0.076 9	-0.072 1
X₁₀	0.276	-0.216 2	0.015 5	0.428 4	0.003 8
X₁₁	0.271 6	-0.009 8	0.034 1	0.025 8	-0.556 5
X₁₂	0.274 2	-0.032 3	0.008 2	0.043 1	-0.438 1
X₁₃	0.268 7	-0.177 5	0.020 9	-0.035 4	0.638 4
X₁₂X₁₃	0.274 8	-0.069 8	0.141 9	-0.113	0.149 7

协同效应指标	特征向量提取主成分
协同效应指标	F₁	F₂	F₃	F₄	F₅
X₁X₂X₆	0.296 8	0.060 8	-0.161 8	-0.139 9	-0.063 3
X₁X₂X₁₀	0.237 5	-0.05	0.064 3	0.540 5	-0.044 2
X₂X₁₀	0.294 9	-0.090 1	-0.260 3	-0.220 8	-0.149 2
X₃	0.224 3	0.375	0.218 3	-0.329 3	-0.023 9
X₄	0.280 3	-0.052 2	-0.450 8	-0.221 5	-0.166 5
X₅	0.315 7	-0.309 1	-0.104	0.239	-0.035 9
X₇	0.313 5	-0.149 2	0.188 2	-0.358 7	0.008
X₈	0.172 7	0.825 8	-0.265 1	0.196 5	0.041 6
X₉	0.134 3	0.169 2	0.702 1	-0.053	-0.183 3
X₁₀	0.224 3	-0.005 8	0.09	0.490 7	-0.086 4
X₁₁	0.060 7	-0.020 8	0.027 5	0.034 8	-0.615 9
X₁₂	0.104 9	-0.023 2	0.011 3	0.063 8	-0.504 4
X₁₃	0.481	-0.090 9	0.080 6	0.058 9	0.513 3
X₁₂X₁₃	0.312 2	-0.068 6	0.161 5	-0.0727	0.046 1

Grey principal component analysis model of effectiveness evaluation for joint operation system-of-systems considering indicator synergism reconstruction

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ε_ij	X₁	X₂	X₃	X₄	X₅	X₆	X₇	X₈	X₉	X₁₀	X₁₁	X₁₂	X₁₃
X₁	1	0.640 6	0.611 4	0.706 8	0.736 6	0.682 6	0.647 9	0.562 3	0.592 5	0.771 5	0.769 6	0.757 7	0.798 4
X₂		1	0.734 7	0.758 8	0.741 6	0.767 0	0.750 8	0.633 9	0.689 2	0.728 1	0.772 7	0.781 3	0.779 8
X₃			1	0.638 8	0.644 8	0.651 3	0.715 7	0.656 0	0.715 2	0.667 1	0.690 4	0.691 2	0.670 5
X₄				1	0.712 8	0.686 0	0.727 3	0.615 0	0.593 6	0.720 4	0.753 8	0.759 4	0.732 7
X₅					1	0.795 0	0.724 8	0.531 2	0.597 5	0.839 6	0.813 9	0.821 3	0.850 6
X₆						1	0.715 0	0.515 5	0.608 9	0.799 5	0.750 0	0.753 6	0.739 8
X₇							1	0.523 1	0.714 5	0.678 5	0.713 5	0.721 6	0.747 1
X₈								1	0.573 9	0.608 1	0.592 4	0.603 2	0.579 0
X₉									1	0.771 2	0.809 2	0.800 7	0.777 6
X₁₀										1	0.705 9	0.718 1	0.705 8
X₁₁											1	0.750 3	0.609 2
X₁₂												1	0.633 4
X₁₃													1

主成分		特征值及方差贡献率			主成分		特征值及方差贡献率
主成分		特征值	方差贡献率%	累积方差贡献率%	主成分		特征值	方差贡献率%	累积方差贡献率%
原指标体系	1	9.432 3	72.56	72.56	适度策略	1	10.261 7	73.30	73.30
	2	0.662 2	5.09	77.65		2	0.632 1	4.52	77.82
	3	0.510 2	3.92	81.57		3	0.501 1	3.58	81.40
	4	0.443 4	3.41	84.98		4	0.491 0	3.51	84.91
	5	0.423 0	3.25	88.23		5	0.444 0	3.17	88.08
保守策略	1	12.301 1	72.36	72.36	激进策略	1	8.045 0	73.14	73.14
	2	0.877 5	5.16	77.52		2	0.594 7	5.41	78.55
	3	0.605 5	3.56	81.08		3	0.484 9	4.41	82.96
	4	0.539 5	3.17	84.25		4	0.415 3	3.78	86.74
	5	0.513 9	3.02	87.27		—	—	—	—

主成分	本文方法			GPCA			PCA			KPCA
主成分	特征值	方差贡献率%	累积方差贡献率%	特征值	方差贡献率%	累积方差贡献率%	特征值	方差贡献率%	累积方差贡献率%	特征值	方差贡献率%	累积方差贡献率%
1	10.261 7	73.30	73.30	9.432 3	72.56	72.56	3.526 4	27.13	27.13	0.238 1	26.27	26.27
2	0.632 1	4.52	77.82	0.662 2	5.09	77.65	2.773 8	21.34	48.47	0.189 0	20.86	47.13
3	0.501 1	3.58	81.40	0.510 2	3.92	81.57	1.880 7	14.47	62.94	0.129 9	14.33	61.46
4	0.491 0	3.51	84.91	0.443 4	3.41	84.98	1.539 5	11.84	74.78	0.107 1	11.82	73.28
5	0.444 0	3.17	88.08	0.422 9	03.25	88.23	1.199 0	9.22	84.00	0.084 6	9.34	82.62
6	—	—	—	—	—	—	0.863 6	6.64	90.64	0.061 6	6.79	89.41

评价场景	本文方法	GPCA	PCA	KPCA
基准方法全部样本	88.11	83.22	65.04	66.44
基准方法前6个样本	99.30	97.20	91.60	83.90

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