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

Abstract

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

CLC Number:

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.

Figures/Tables 19

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References 33

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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

Research on optimization of decoy quantity ratio based on dynamic Weibull distribution

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参数	动态威布尔	静态威布尔	指数分布
$\alpha $	$\alpha \left( t \right)$	${\alpha _1}$	1
$\beta $	$\beta \left( t \right)$	${\beta _1}$	${\beta _1}$

类型	参数	取值
定量	${{v}}$	5
	$\left\{ {{\theta _1},{\theta _2},{\theta _3},{\theta _4},{\theta _5}} \right\}$	$\left\{ {0.31,0.29,0.2,0.11,0.09} \right\}$
	$\left\{ {{N_{{\text{t}},1}},{N_{{\text{t}},2}},{N_{{\text{t}},3}},{N_{{\text{t}},4}},{N_{{\text{t}},5}}} \right\}$	$\left\{ {2,3,12,4,5} \right\}$
	$\left\{ {{N_{{\text{f}},1}},{N_{{\text{f}},2}},{N_{{\text{f}},3}},{N_{{\text{f}},4}},{N_{{\text{f}},5}}} \right\}$	$\left\{ {7,10,45,13,20} \right\}$
	${{P}_{h}}$	0.75
变量	${\gamma }$	17, 21, 25, 50, 100, 170

场景	权重系数
场景1	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.05,{\text{ }}0.45,{\text{ }}0.5} \right\}$
	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.1,{\text{ }}0.4,{\text{ }}0.5} \right\}$
	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.15,{\text{ }}0.35,{\text{ }}0.5} \right\}$
场景2	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.25,{\text{ }}0.3,{\text{ }}0.45} \right\}$
	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.35,{\text{ }}0.25,{\text{ }}0.4} \right\}$
	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.3,{\text{ }}0.35,{\text{ }}0.35} \right\}$
场景3	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.4,{\text{ }}0.28,{\text{ }}0.32} \right\}$
	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.4,{\text{ }}0.35,{\text{ }}0.25} \right\}$
	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.5,{\text{ }}0.3,{\text{ }}0.2} \right\}$
场景4	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.55,{\text{ }}0.3,{\text{ }}0.15} \right\}$
	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.55,{\text{ }}0.35,{\text{ }}0.1} \right\}$
	$\left\{ {{\omega _1},{\text{ }}{\omega _2},{\text{ }}{\omega _3}} \right\} = \left\{ {0.6,{\text{ }}0.3,{\text{ }}0.1} \right\}$

场景	真假目标数量配比	效费比	综合作战价值
初始	19:0	—	0.380
场景1	19:45	7.778	0.477
场景2	19:155	6.421	0.695
场景3	19:187	5.952	0.716
场景4	19:304	4.675	0.822

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