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

系统工程与电子技术 ›› 2025, Vol. 47 ›› Issue (5): 1561-1574.doi: 10.12305/j.issn.1001-506X.2025.05.18

• 系统工程 • 上一篇

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

陈顶¹, 方志耕², 杨保华³, 叶丰⁴^,*, 张娜⁵, 张靖如²

1. 上海机电工程研究所, 上海 201109
2. 南京航空航天大学经济与管理学院, 江苏南京 211106
3. 江苏师范大学商学院, 江苏徐州 221116
4. 军事科学院系统工程研究院, 北京 100101
5. 南京信息工程大学商学院, 江苏南京 210044

收稿日期:2022-11-24 出版日期:2025-06-11 发布日期:2025-06-18
通讯作者: 叶丰
作者简介:陈顶(1990—), 男, 高级工程师, 博士, 主要研究方向为体系工程、效能评估
方志耕(1962—), 男, 教授, 博士, 主要研究方向为质量与可靠性、灰色系统博弈理论、复杂装备研制管理
杨保华(1979—), 男, 副教授, 博士, 主要研究方向为灰色系统理论
叶丰(1981—), 男, 硕士, 副研究员，主要研究方向为体系建模、装备论证
张娜(1983—), 女, 副教授, 博士, 主要研究方向为质量与可靠性管理
张靖如(1997—), 女, 博士研究生, 主要研究方向为体系建模与评估

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

Ding CHEN¹, Zhigeng FANG², Baohua YANG³, Feng YE⁴^,*, Na ZHANG⁵, Jingru ZHANG²

1. Shanghai Electro-Mechanical Engineering Institute, Shanghai 201109, China
2. College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
3. School of Business, Jiangsu Normal University, Xuzhou 221116, China
4. Institute of System Engineering, Academy of Military Sciences, Beijing 100101, China
5. School of Business, Nanjing University of Information Science & Technology, Nanjing 210044, China

Received:2022-11-24 Online:2025-06-11 Published:2025-06-18
Contact: Feng YE

摘要/Abstract

摘要：

体系效能评估指标数量多、维数高, 指标之间关联, 且具有协同效应, 加大了效能评估的计算复杂性。针对这一问题, 建立考虑协同效应的联合作战体系效能指标灰色主成分分析(grey principal component analysis, GPCA)重构模型。首先, 分析联合防空作战体系的作战使命、任务、流程, 构建其效能评估指标体系, 并运用灰色关联模型分析指标间是否存在协同效应。其次, 基于指标间存在的协同效应, 给出3种重构效能评估指标体系的策略, 并结合GPCA方法, 构建具有协同效应的GPCA模型, 对评估指标体系进行降维。最后, 将所提方法应用于联合防空作战体系效能评估案例, 筛选出具有协同效应的指标, 重构效能评估指标体系。计算结果与方法对比分析表明, 所提方法能够有效发现指标间的协同效应, 重构后的评估指标体系保持了“同构性”。

关键词: 体系效能评估, 协同效应, 灰色关联, 指标重构, 灰色主成分

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)

中图分类号:

N945.25

陈顶, 方志耕, 杨保华, 叶丰, 张娜, 张靖如. 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[J]. 系统工程与电子技术, 2025, 47(5): 1561-1574.

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.

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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

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

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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