基于动态威布尔分布的假目标数量配比优化研究

doi:10.12305/j.issn.1001-506X.2026.07.21

摘要/Abstract

摘要：

针对复杂对抗环境下地空导弹系统假目标数量配比优化问题，提出一种基于生存概率、效费比和惩罚函数的多目标决策方法。首先，建立双参数可变的动态威布尔分布模型，刻画目标被发现识别概率随时间的变化规律，并结合火力分配构建目标生存概率计算模型。其次，综合考虑部署空间、时间及经济成本等约束条件，设置不同决策权重，构建融合作战价值、效费比和惩罚项的效用函数，进而建立真假目标数量配比优化模型。最后，结合典型应用场景仿真分析，验证模型与参数的有效性。结果表明，所提模型在持续对抗条件下能够更精确地评估系统生存概率并优化假目标配置数量，可为不同作战阶段提供差异化决策依据。

关键词: 双参数威布尔分布, 假目标, 效费比, 生存概率, 多目标优化

Abstract:

To address the optimization problem of decoy quantity ratio for surface-to-air missile system in complex adversarial environments, a multi-objective decision-making method based on survival probability, cost-effectiveness ratio, and penalty functions is proposed. Firstly, a dynamic Weibull distribution model with two variable parameters is established to characterize the time-varying probability of target detection and identificaiton, and a target survival probability calculation model is constructed in conjunction with firepower allocation. Secondly, by incorporating constraints such as deployment space, time, and economic costs, different decision-making weights are set to develop a utility function that integrates combat value, cost-effectiveness ratio, and penalty terms, thereby forming an optimization model for the allocation ratio between real and decoy targets. Finally, simulation analysis is conducted based on typical application scenarios to verify the effectiveness of the model and parameters. The results indicate that the proposed model can more accurately evaluate the survival probability of the system and optimize the number of decoy configurations under sustained adversarial conditions, providing differentiated decision-making basis for different combat stages.

Key words: two-parameter Weibull distribution, decoy, cost-effectiveness ratio, survival probability, multi-objective optimization

中图分类号:

李争浩, 李幸豪, 贾彦翔, 李健, 卞伟伟. 基于动态威布尔分布的假目标数量配比优化研究[J]. 系统工程与电子技术, 2026, 48(7): 2367-2377.

Zhenghao LI, Xinghao LI, Yanxiang JIA, Jian LI, Weiwei BIAN. Research on optimization of decoy quantity ratio based on dynamic Weibull distribution[J]. Systems Engineering and Electronics, 2026, 48(7): 2367-2377.

图/表 19

表1

图1

表2

表3

表4

表5

表6

图2

表7

图3

表8

图4

表9

图5

表10

表11

图6

表12

图7

参考文献 33

1	SHIMONI M, HAELTERMAN R, PERNEEL C. Hypersectral imaging for military and security applications: combining myriad processing and sensing techniques[J]. IEEE Geoscience and Remote Sensing Magazine, 2019, 7 (2): 101- 117. doi: 10.1109/MGRS.2019.2902525
2	张华. 2022年国外情报监视侦察装备发展研究[J]. 国防科技工业, 2023(1): 37−40.
	ZHANG H. Research on the development of foreign intelligence, surveillance and reconnaissance equipment in 2022[J]. Defence Science & Technology Industry, 2023(1): 37−40.
3	WANG F Y, LIU F R, ALI Z, et al. A polarization-insensitive and dynamic-tunable perfect absorber for a compatible infrared-visible multi-band stealth[J]. Optics Communications, 2025, 591, 132142. doi: 10.1016/j.optcom.2025.132142
4	GAN Y Y, LIU C T, LI H C, et al. An evaluation method of dynamic camouflage effect based on multifeature constraints[J]. IEEE Access, 2020, 8, 193845- 193855. doi: 10.1109/ACCESS.2020.3025801
5	HOSSAIN A. Advancement in UV-visible-IR camouflage textiles for concealment of defence surveillance against multidimensional combat backgrounds[J]. Journal of Materials Science: Materials in Engineering, 2024, 19, 45.
6	HE W T, WENG X L, LUO W, et al. Investigation of radar cross-section reduction for dihedral corner reflectors based on camouflage grass[J]. IEEE Antennas and Wireless Propagation Letters, 2021, 20 (12): 2447- 2451. doi: 10.1109/LAWP.2021.3114302
7	ZHANG X X, DING S, GE B F, et al. Resource allocation among multiple targets for a defender-attacker game with false targets consideration[J]. Reliability Engineering & System Safety, 2021, 211, 107617.
8	LI S Y, JIAO J N, WANG Z. Research on the detection algorithm of camouflage scattered landmines in vegetation environment based on polarization spectral fusion[J]. IEEE Geoscience and Remote Sensing Letters, 2024, 21, 6011305.
9	但兵兵, 朱万红, 桑杨阳, 等. 假目标配置数量模型及其效费比分析[J]. 指挥控制与仿真, 2012, 34 (4): 70- 73.
	DAN B B, ZHU W H, SANG Y Y, et al. Quantity of fault targets model and its effectiveness-cost ratio analysis[J]. Command Control & Simulation, 2012, 34 (4): 70- 73.
10	XU F. Research on target detection model of remote sensing image based on dual attention YOLO[C]//Proc. of the IEEE 7th International Conference on Information Systems and Computer Aided Education, 2024: 337−341.
11	TM D, VERMA R, RAJESH R, et al. Single shot radar target detection and localization using deep neural network[C]//Proc. of the IEEE International Conference on Electronics, Computing and Communication Technologies, 2022.
12	汤雪峰, 王可定, 陈徐均. 关于假目标伪装效能指标的计算与分析[J]. 军事运筹与系统工程, 2002 (2): 10- 13.
	TANG X F, WANG K D, CHEN X J. Calculation and analysis of camouflage effectiveness metrics for decoys[J]. Military Operations Research and Systems Engineering, 2002 (2): 10- 13.
13	宋志华, 张多林, 龚立占. 基于多目标规划的防空导弹假目标布置数量模型[J]. 弹箭与制导学报, 2006 (1): 493- 495.
	SONG Z H, ZHANG D L, GONG L Z. The number of fault air defense missile model based on multi-objective programming[J]. Journal of Projectiles, Rockets, Missiles and Guidance, 2006 (1): 493- 495.
14	YANG Y, SHEN T, FAN B X. Design and test of intelligent diffuse reflector false target jamming system[C]//Proc. of the 2nd International Conference on Networking Systems of AI, 2022: 241−248.
15	王亦之, 王斌, 赵禄达. 固定导弹阵地激光诱饵装备作战配置计算[J]. 航天电子对抗, 2021, 37 (1): 19- 24.
	WANG Y Z, WANG B, ZHAO L D. Operational configuration calculation of laserisco equipment for fixed missile positions[J]. Aerospace Electronic Warfare, 2021, 37 (1): 19- 24.
16	GONG Y H, YU R J, CHEN H Y, et al. Multifunctional metasurface for multispectral compatible camouflage of laser and infrared with thermal management[J]. Optics Letters, 2024, 49 (24): 6908- 6911. doi: 10.1364/OL.539341
17	COOLS K, MAATHUIS C, CUBBER G D, et al. Evaluation techniques for modern military camouflage[C]//Proc. of the International Conference on Military Technologies, 2025.
18	MASLOVSKIY A, VASILETS V, NECHITAYLO S, et al. The antiradar camouflage method for ground military objects[C]//Proc. of the IEEE 2nd Ukraine Conference on Electrical and Computer Engineering, 2019.
19	张最良, 邵国培, 刘湘伟. 军事运筹学[M]. 北京: 解放军出版社, 2000.
	ZHANG Z L, SHAO G P, LIU X W. Military operations research[M]. Beijing: PLA Press, 2000.
20	VALADARES D G D O, QUININO R C, CRUZ F R B, et al. Repairable system analysis using the discrete Weibull distribution [J]. IEEE Trans. on Reliability, 2023, 72(4): 1507−1514.
21	FENG X F, TANG J Y. Reliability assessment for dual constant-stress accelerated life test based on Weibull distribution with type-II censoring[J]. Quality and Reliability Engineering International, 2024, 41(3): 1025−1040.
22	刘新蕊, 王雪欣, 何非. 右删失数据下加权威布尔分布的估计研究[J]. 长春工业大学学报, 2024, 45 (5): 452- 456.
	LIU X R, WANG X X, HE F. Estimation for weighted Weibull distribution with right censored data[J]. Journal of Changchun University of Technology, 2024, 45 (5): 452- 456.
23	KAVITHA J C. , SUBITHA D. Lesion detection and classification in dermoscopic images using optimal threshold based on Newton Raphson iterative method[J]. Journal of Intelligent & Fuzzy Systems, 2024, 46 (1): 753- 767.
24	CARVALHO S F, REZENDE R B, SANTOS D P R A, et al. Modeling seed longevity and percentile prediction: a sigmoidal function approach in Soybean, Maize, and Tomato[J]. AgriEngineering, 2024, 7 (1): 5. doi: 10.3390/agriengineering7010005
25	CHEN B, PANG G M, XIANG Z T, et al. Modeling dual-layer interdependent command and control networks for integrated reconnaissance-strike and OODA-loop capabilities[J]. IEEE Trans. on Network Science and Engineering, 2024, 11 (6): 5744- 5759. doi: 10.1109/TNSE.2024.3443191
26	FARLIK J. Simulation of surface-to-air missile units: cluster design[C]//Proc. of the International Conference on Military Technologies, 2015.
27	GONG H, FU Y K, ZHANG Y, et al. Research on ground-to-air force deployment based on multi-objective evolutionary algorithm with game mechanism[C]//Proc. of the 2nd International Conference on Artificial Intelligence and Automation Control, 2024: 387−391.
28	NA H, LEE J I, Optimal arrangement of missile defense systems considering kill probability[J]. IEEE Trans. on Aerospace and Electronic Systems, 2020, 56(2): 972−983.
29	BOGDANOWICZ Z R, TOLANO A, PATEL K, et al. Optimization of weapon-target pairings based on kill probabilities[J]. IEEE Trans. on Cybernetics, 2013, 43 (6): 1835- 1844. doi: 10.1109/TSMCB.2012.2231673
30	GUO P S, LI W M, LI L Y, et al. Optimization of the tactical disposition scheme for ground air defense based on fuzzy comprehensive evaluation[C]//Proc. of the 33rd Chinese Control and Decision Conference, 2021: 4000−4004.
31	WU D L. GNSS signal jamming as observed from radio occultation[J]. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 2024, 17: 8642−8645.
32	YAN J K, LIU H W, JIU B, et al. Joint detection and tracking processing algorithm for target tracking in multiple radar system[J]. IEEE Sensors Journal, 2015, 15 (11): 6534- 6541. doi: 10.1109/JSEN.2015.2461435
33	PASICHNYK N A, SHVOROV S A, OPRYSHKO O A, et al. Prospective of camouflage ground objects to counter UAVs[C]//Proc. of the IEEE 7th International Conference on Methods and Systems of Navigation and Motion Control, 2023.

动态类型	适用场景	数学形式	示例
分段常数	侦察强度变化较大（如增派飞机、卫星）	$ \beta \left( {{t}} \right) = \left\{ \begin{gathered} {\beta _1},{\text{ }}t \leqslant {T_1} \\ {\beta _2},{\text{ }}t \gt {T_1} \\ \end{gathered} \right. $	红方在${T_1} = {\text{6 h}}$后增派2架飞机，临时调转3颗侦察卫星
线性/非线性渐变	侦察强度持续增强或衰减（如延长侦察时间、加大侦察范围）	$\beta \left( t \right) = {\beta _0} + {{a}}t$或$\beta \left( t \right) = {\beta _0}{{\mathrm{e}}^{bt}}$	红方每小时加大${{a}} = 0.2{\text{ k}}{{\text{m}}^2}$侦察范围
周期性波动	红方周期性侦察	$\alpha \left( t \right) = {\alpha _0} + {{\gamma }}\sin \left( {\omega t + \varphi } \right)$	卫星、侦察机过境时间$t \in \left[ {{\text{6}},{\text{18}}} \right]$，侦察强度$\gamma = {\text{0}}{\text{.5}}$，其他时间降低
协变量驱动	外部因素影响（如天气、兵力）	$\beta \left( t \right) = {\beta _0} + { \lambda }x\left( t \right)$，其中$ {{x}}\left( t \right) $为协变量，如云层覆盖率	云层覆盖率每增加${\text{10\% }}$，$ \beta \left( {{t}} \right) $增加${\text{15\% }}$，侦察强度降低
机器学习预测	复杂非线性动态模式	$\beta \left( {{t}} \right) = {f_{{\theta}}}\left( {t,x\left( t \right)} \right)$，${f_{\theta }}$为神经网络、长短时记忆等预测模型	基于实时情报输入预测未来$ \beta \left( {{t}} \right) $

符号	定义
$ v $	武器型号数量
${\theta _k}\left( {{\theta _k} \gt {\theta _{k + 1}}} \right)$	$ k $型号真目标军事价值权重系数
${P_{{\text{t}},k}}\left( t \right),{P_{{\text{f}},k}}\left( t \right)$	$ k $型号真（假）目标被发现识别概率
${N_{{\text{t}},k}},{N_{{\text{f}},k}}$	$ k $型号真（假）目标数量
${L_{{\text{t}},k}}\left( t \right),{L_{{\text{f}},k}}\left( t \right)$	$ k $型号真（假）目标被打击的数量
${L_{\text{t}}}\left( t \right)$	打击对象的总数量
${\gamma }$	红方攻击使用导弹的数量
${L_{{w}}}\left( t \right)$	被分配$ w $枚导弹的目标总数
${L_{k,w}}\left( t \right)$	$ k $型号目标被分配$ w $枚导弹的数量
${P_h}$	导弹命中摧毁概率
${P_{d,{{w}}}}$	分配$ w $枚导弹的摧毁概率
${P_{{\text{s}},k}}$	$ k $型号真目标的生存概率

系统组成	功能定位	外形特征	重要程度
指控车	指挥中枢，整合雷达、协调发射^[31]	方舱结构，多天线	很重要
雷达车	探测/制导核心，依赖电磁信号^[32]	天线阵面典型特征	重要
发射车	导弹存储与发射载体，依赖雷达信息	圆柱筒弹外形	重要
电源车	电力供应节点，现代系统多设计冗余	方舱结构，但热量大	一般
运输车	导弹及设备运输，保障快速补给	特征一般	不重要

参数	阶段i
参数	1	2	3
数据量	2 000	2 000	2 000
删失比例/%	5.7	0.4	0.0

参数	阶段i
参数	1	2	3
${\alpha _{{\text{true}}}}$	1	1.5	2
${\alpha _{{\text{est}}}}$	0.991 6	1.470 2	1.959 1
${\delta _{{r}}}/\% $	−0.8	−2.0	−2.0
${\beta _{{\text{true}}}}$	25	20	15
${\beta _{{\text{est}}}}$	25.125 0	19.678 1	14.755 9
${\delta _{\text{r}}}/\% $	0.5	−1.6	−1.6