基于强化学习的海上要地群协同防空动态火力分配

doi:10.12382/bgxb.2022.1276

摘要/Abstract

摘要：

针对海上要地群协同防空作战动态火力分配问题,综合分析海上要地防空作战过程的特点,建立基于马尔可夫决策模型的动态火力分配问题,构建以海上要地毁伤期望、拦截成本为指标的优化模型。考虑到马尔可夫决策模型求解易陷入维数灾难的问题,提出利用近似动态规划方法来探究解的有效性,并给出基于强化学习的最小二乘时序差分算法来求解该问题。通过4种典型的攻防场景共80个案例仿真结果表明,相比传统的匹配算法、遗传算法和粒子群优化算法,新构建的模型和算法更加科学合理有效,可为海上要地群协同防空作战火力分配提供一定的理论依据。

关键词: 海上要地, 动态火力分配, 强化学习, 马尔可夫决策

Abstract:

For the dynamic firepower allocation in the cooperative air defense operation of strategic locations on the sea, the characteristics of air defense operations in strategic locations on the sea are comprehensively analyzed to establish the dynamic firepower allocation problem based on the Markov decision model, and an optimization model with the damage expectation and interception cost as the indexes is constructed. Considering the problem that the Markov decision model is easy to fall into the disaster of dimensionality, an approximate dynamic programming method is proposed to explore the validity of the solution, and a least squares temporal difference algorithm based on reinforcement learning is given to solve the problem. The simulated results of 80 cases in four typical offensive and defensive scenarios show that, compared with the traditional matching algorithm, genetic algorithm and particle swarm optimization algorithm, the proposed model and algorithmin this paper are more scientific, reasonable and effective, which can provide a certain basis for the firepower allocation in the cooperative air defense operations of strategic locations on the sea.

Key words: strategic location on the sea, dynamic firepower allocation, reinforcement learning, Markov decision

中图分类号:

E072

赵文飞, 陈健, 王, 滕克难. 基于强化学习的海上要地群协同防空动态火力分配[J]. 兵工学报, 2023, 44(11): 3516-3528.

ZHAO Wenfei, CHEN Jian, WANG Yan, TENG Kenan. Dynamic Firepower Allocation for Cooperative Air Defense of Strategic Locations on the Sea Based on Reinforcement Learning[J]. Acta Armamentarii, 2023, 44(11): 3516-3528.

图/表 14

图1 MDP概念图

Fig.1 Concept map of Markov decision process

表1 四类典型案例场景参数

Table 1 Scenario parameters of four typical cases

场景	要地导弹数量/发	要地价值系数
场景1	(30,20,15)	(15,20,10)
场景2	(10,10,10)	(15,20,10)
场景3	(30,20,15)	(15,10,20)
场景4	(10,10,10)	(15,10,20)

图2 海上要地群攻防部署示意图

Fig.2 Schematic diagram of offensive and defensive deployment of strategic locations on the sea

表2 特征函数

Table 2 Characteristic functions

φ(S)	Φ₀(S)	Φ₁(S)	Φ₂(S)	Φ₃(S)	Φ₄(S)	Φ₅(S)
1	1	A_t	$R t x$
2	1	A_t	$R t x$	$A t x$
3	1	$R t x$	$A t x$	$(R t x) 2$	$(A t x) 2$	$R t x A t x$

表2 特征函数

Table 2 Characteristic functions

φ(S)	Φ₀(S)	Φ₁(S)	Φ₂(S)	Φ₃(S)	Φ₄(S)	Φ₅(S)
1	1	A_t	$R t x$
2	1	A_t	$R t x$	$A t x$
3	1	$R t x$	$A t x$	$(R t x) 2$	$(A t x) 2$	$R t x A t x$

表3 仿真参数

Table 3 Simulation parameters

仿真情形	N	K	φ(S)	α
1	10	500	1	0.01
2	50	1000	2	0.3
3	100	1500	3	0.8

表4 Rt=(30,20,15), Ct=(15,20,10)仿真结果对比

Table 4 Comparison of simulated results in the case of Rt=(30,20,15), Ct=(15,20,10)

攻击向量	LSTD算法	GA	PSO算法	LSTD算法最优值v_L	GA最优值v_G	PSO算法最优值v_P	J_LG/%	J_LP/%
(1, 0, 0)	(2, 0, 0)	(3, 0, 0)	(3, 0, 0)	0.868	0.769	0.769	-12.87	-12.87
(0, 1, 0)	(1, 1, 0)	(2, 3, 0)	(2, 2, 0)	1.341	1.271	1.436	-5.51	6.62
(0, 0, 1)	(1, 0,1)	(2, 0, 3)	(2, 0, 2)	1.540	2.055	2.201	25.06	30.03
(2, 0, 0)	(4, 0, 0)	(6, 0, 0)	(6, 0, 0)	1.713	1.538	1.538	-11.38	-11.38
(0, 2, 0)	(0, 4, 0)	(4, 6, 0)	(4, 4, 0)	2.087	2.540	2.861	17.83	27.05
(0, 0, 2)	(2, 0, 4)	(3, 0, 6)	(3, 0, 4)	2.604	4.064	4.320	35.93	39.72
(1, 1, 0)	(2, 1, 0)	(5, 3, 0)	(5, 2, 0)	2.312	2.040	2.204	-13.33	-4.90
(1, 0, 1)	(3, 0, 2)	(5, 0,3)	(5, 0, 2)	2.512	2.823	2.970	11.02	15.42
(1, 1, 1)	(2, 1, 2)	(7, 3, 3)	(7, 2, 2)	3.936	4.094	4.406	3.86	10.67
(1, 2, 1)	(5, 4, 2)	(9, 6, 3)	(9, 4, 2)	5.215	5.363	5.831	2.76	10.56
(1, 1, 2)	(5, 2, 4)	(8, 3,6)	(8, 2, 4)	5.257	6.104	6.525	13.88	19.43
(2, 1, 1)	(4, 2, 2)	(10 3,3)	(9, 2, 2)	4.675	4.863	5.312	3.87	11.99
(2, 2, 1)	(5, 3, 1)	(12, 6,3)	(11, 4, 2)	6.158	6.132	6.737	-0.42	8.59
(1, 2, 2)	(6, 4, 4)	(10, 6, 6)	(10, 4, 4)	6.453	7.373	7.950	12.48	18.83
(1, 2, 1)	(4, 2, 2)	(9, 6, 3)	(9, 4, 2)	5.185	5.363	5.831	3.32	11.08
(3, 0, 0)	(6, 0, 0)	(9, 0, 0)	(9, 0, 0)	2.396	2.306	2.306	-3.90	-3.90
(3, 1, 1)	(8, 2, 2)	(13, 3, 3)	(11, 2, 2)	4.447	5.632	6.306	21.04	29.48
(1, 3, 1)	(5, 6, 2)	(11, 8, 3)	(11, 6, 2)	6.491	6.620	7.246	1.95	10.42
(1, 3, 3)	(7, 6, 6)	(14, 8, 8)	(13, 6, 6)	7.871	10.615	11.455	25.85	31.29
(2, 4, 1)	(4, 7, 2)	(17, 11, 3)	(15, 8, 2)	8.551	8.635	9.557	0.97	10.53

表5 Rt=(10,10,10), Ct=(15,20,10)仿真结果对比

Table 5 Comparison of simulated results in the case of Rt=(10,10,10), Ct=(15,20,10)

攻击向量	LSTD算法	GA	PSO算法	LSTD最优值v_L	GA最优值v_G	PSO最优值v_P	J_LG/%	J_LP/%
(1, 0, 0)	(2, 0, 0)	(3, 0, 0)	(3, 0, 0)	0.868	0.769	0.769	-12.87	-12.87
(0, 1, 0)	(1, 1, 0)	(2, 3, 0)	(2, 2, 0)	1.341	1.271	1.436	-5.51	6.62
(0, 0, 1)	(1, 0,1)	(2, 0, 3)	(2, 0, 2)	1.54	2.055	2.201	25.06	30.03
(2, 0, 0)	(4, 0, 0)	(6, 0, 0)	(6, 0, 0)	1.713	1.538	1.538	-11.38	-11.38
(0, 2, 0)	(1, 2, 0)	(4, 6, 0)	(4, 4, 0)	2.534	2.54	2.861	0.24	11.43
(0, 0, 2)	(2, 0, 4)	(3, 0, 6)	(3, 0, 4)	2.604	4.064	4.32	35.93	39.72
(1, 1, 0)	(2, 1, 0)	(5, 3, 0)	(5, 2, 0)	2.312	2.04	2.204	-13.33	-4.90
(1, 0, 1)	(2, 0, 2)	(5, 0,3)	(5, 0, 2)	2.631	2.823	2.97	6.80	11.41
(1, 1, 1)	(2, 1, 2)	(7, 3, 3)	(7, 2, 2)	3.936	4.094	4.406	3.86	10.67
(1, 2, 1)	(2, 4, 2)	(9, 6, 3)	(9, 4, 2)	5.243	5.363	5.831	2.24	10.08
(1, 1, 2)	(5, 2, 4)	(8, 3,6)	(8, 2, 4)	5.257	6.104	6.525	13.88	19.43
(2, 1, 1)	(2, 2, 1)	(10,3,3)	(9, 4, 2)	4.745	4.863	5.312	2.43	10.67
(2, 2, 1)	(5, 3, 1)	(10, 6,3)	(8, 4, 2)	6.158	6.327	6.821	2.67	9.72
(1, 2, 2)	(1, 3, 4)	(10, 6, 6)	(10, 4, 4)	6.542	7.373	7.95	11.27	17.71
(1, 2, 1)	(2, 2, 2)	(9, 6, 3)	(9, 4, 2)	5.213	5.363	5.831	2.80	10.60
(3, 0, 0)	(6, 0, 0)	(9, 0, 0)	(9, 0, 0)	2.396	2.306	2.306	-3.90	-3.90
(3, 1, 1)	(5, 1, 2)	(10, 3, 3)	(9, 2, 2)	4.812	6.059	6.403	20.58	24.85
(1, 3, 1)	(3, 3, 2)	(10, 8, 3)	(8, 6, 2)	6.621	6.678	7.681	0.85	13.80
(1, 3, 3)	(7, 6, 6)	(10, 8, 8)	(8, 6, 6)	7.871	11.14	11.851	29.34	33.58
(2, 4, 1)	(6, 4, 2)	(10, 10, 4)	(9, 8, 2)	8.887	9.98	10.134	10.95	12.31

表6 Rt=(30,20,15), Ct=(15,10,20)仿真结果对比

Table 6 Comparison of simulated results in the case of Rt=(30,20,15), Ct=(15,10,20)

攻击向量	LSTD算法	GA	PSO算法	LSTD最优值v_L	GA最优值v_G	PSO最优值v_P	J_LG/%	J_LP/%
(1, 0, 0)	(2, 0, 0)	(3, 0, 0)	(3, 0, 0)	0.891	0.769	0.769	-15.86	-15.86
(0, 1, 0)	(0, 1, 0)	(2, 2, 0)	(1, 1, 0)	0.791	1.118	1.436	29.25	44.92
(0, 0, 1)	(1, 0,2)	(2, 0, 4)	(1, 0, 2)	1.483	1.672	2.055	11.30	27.83
(2, 0, 0)	(4, 0, 0)	(3, 0, 0)	(3, 0, 0)	1.495	1.538	1.538	2.80	2.80
(0, 2, 0)	(1, 2, 0)	(3, 5, 0)	(2, 3, 0)	1.25	2.157	3.208	42.05	61.03
(0, 0, 2)	(2, 0, 4)	(4, 0, 9)	(2, 0, 4)	4.033	3.325	4.073	-21.29	0.98
(1, 1, 0)	(2, 1, 0)	(5, 2, 0)	(4, 1, 0)	1.8	1.887	2.377	4.61	24.27
(1, 0, 1)	(2, 0, 2)	(5, 0,4)	(3, 0, 2)	2.492	2.44	2.823	-2.13	11.73
(1, 1, 1)	(4,2, 2)	(7, 2, 4)	(4, 2, 2)	4.372	4.558	4.489	4.08	2.61
(1, 2, 1)	(3, 2, 2)	(8, 5, 4)	(5, 4, 2)	5.28	4.597	6.217	-14.86	15.07
(1, 1, 2)	(5, 2, 4)	(9, 2,9)	(5, 2, 4)	4.081	5.211	6.469	21.68	36.91
(2, 1, 1)	(4, 1, 1)	(10,2,4)	(6, 2, 2)	5.199	5.327	5.213	2.40	0.27
(2, 2, 1)	(5, 3, 2)	(11, 5,4)	(7, 4, 2)	5.264	5.366	7.128	1.90	26.15
(1, 2, 2)	(1, 3, 4)	(10, 5, 9)	(4, 4, 5)	6.751	7.25	7.88	6.88	14.33
(1, 2, 1)	(2, 2, 2)	(8,5, 4)	(8, 4, 2)	5.304	4.597	5.722	-15.38	7.31
(3, 0, 0)	(4, 0, 0)	(9, 0, 0)	(6, 0, 0)	2.47	2.506	2.601	1.44	5.04
(3, 1, 1)	(6, 2, 2)	(13, 2, 4)	(6, 2, 2)	5.824	5.996	6.154	2.87	5.36
(1, 3, 1)	(3, 6, 2)	(10, 7, 4)	(7, 5, 2)	6.055	5.652	7.031	-7.13	13.88
(1, 3, 3)	(8, 4, 6)	(14, 7, 13)	(8, 6,8)	9.134	9.650	9.698	5.35	5.82
(2, 4, 1)	(6, 4, 2)	(14, 9, 4)	(9, 8, 2)	7.437	7.490	10.191	0.71	27.02

表7 Rt=(10,10,10),Ct=(15,10,20)仿真结果对比

Table 7 Comparison of simulated results in the case of Rt=(10,10,10), Ct=(15,10,20)

攻击向量	LSTD算法	GA	PSO算法	LSTD最优值v_L	GA最优值v_G	PSO最优值v_P	J_LG/%	J_LP/%
(1, 0, 0)	(1, 0, 0)	(3, 0, 0)	(3, 0, 0)	0.94	0.769	0.769	-22.24	-22.24
(0, 1, 0)	(0, 2, 0)	(2, 2, 0)	(1, 1, 0)	0.846	1.118	1.436	24.33	41.09
(0, 0, 1)	(1, 0,2)	(2, 0, 4)	(1, 0, 2)	2.485	1.672	2.055	-48.62	-20.92
(2, 0, 0)	(3, 0, 0)	(3, 0, 0)	(3, 0, 0)	1.7	1.538	1.538	-10.53	-10.53
(0, 2, 0)	(0,2, 0)	(3, 5, 0)	(2, 3, 0)	1.472	2.157	3.208	31.76	54.11
(0, 0, 2)	(2, 0, 4)	(4, 0, 9)	(2, 0, 4)	4.025	3.325	4.073	-21.05	1.18
(1, 1, 0)	(1, 1, 0)	(5, 2, 0)	(4, 1, 0)	1.783	1.887	2.377	5.51	24.99
(1, 0, 1)	(1, 0, 2)	(5, 0,4)	(3, 0, 2)	2.438	2.44	2.823	0.08	13.64
(1, 1, 1)	(2,2, 2)	(7, 2, 4)	(4, 2, 2)	3.48	3.558	4.489	2.19	22.48
(1, 2, 1)	(3,4, 1)	(8, 5, 4)	(5, 4, 2)	4.1	4.597	6.217	10.81	34.05
(1, 1, 2)	(2, 2, 4)	(9, 2,9)	(5, 2, 4)	4.506	5.211	6.469	13.53	30.34
(2, 1, 1)	(6, 2, 2)	(10,2,4)	(6, 2, 2)	4.735	4.327	5.213	-9.43	9.17
(2, 2, 1)	(5, 3, 2)	(10, 5,4)	(7, 4, 2)	4.364	5.503	7.128	20.70	38.78
(1, 2, 2)	(3, 3,2)	(10, 5, 9)	(4, 4, 5)	5.972	6.25	7.88	4.45	24.21
(1, 2, 1)	(2, 2, 2)	(8,5, 4)	(8, 4, 2)	3.529	4.597	5.722	23.23	38.33
(3, 0, 0)	(3, 0, 0)	(9, 0, 0)	(6, 0, 0)	2.634	2.306	2.306	-14.22	-14.22
(3, 1, 1)	(6, 1, 2)	(10, 2, 4)	(6, 2, 2)	4.33	5.527	6.154	21.66	29.64
(1, 3, 1)	(2, 6, 2)	(10, 7, 4)	(7, 5, 2)	4.694	5.652	7.031	16.95	33.24
(1, 3, 3)	(5, 6, 8)	(10, 7, 10)	(8, 6,8)	9.767	9.850	9.898	0.84	1.32
(2, 4, 1)	(5, 6, 2)	(10, 9, 4)	(9, 8, 2)	7.481	8.222	10.191	9.01	26.59

图3 各类算法在80个测试案例中目标函数期望值

Fig.3 Expected values of objective functions of various algorithms in 80 test cases

表8 LSTD算法在各类仿真参数得到的最优目标函数值

Table 8 Optimal objective function values obtained by LSTD algorithm in various simulation parameters

仿真情形	R_t=(30,20,15), C_t=(15,20,10)	R_t=(10,10,10), C_t=(15,20,10)	R_t=(30,20,15), C_t=(15,10,20)	R_t=(10,10,10), C_t=(15,10,20)
1	5.14	5.23	5.45	5.44
2	4.88	5.01	5.31	5.29
3	4.93	4.94	5.29	5.24

图4 各算法迭代收敛曲线示意图1

Fig.4 Iterative convergence curves 1 of each algorithm

图5 各算法迭代收敛曲线示意图2

Fig.5 Iterative convergence curves 2 of each algorithm

图5 各算法时间消耗对比图

Fig.5 Comparison chart of time consumption of each algorithm

参考文献 29

[1]	马新星, 滕克难, 侯学隆. 岛礁防空火力单元配置距离计算模型[J]. 兵工自动化, 2017, 36(10):38-41.
	MA X X, TENG K N, HOU X L. Models to calculate the deployment distance of reef air defense fire units[J]. Ordnance Industry Automation, 2017, 36(10):38-41. (in Chinese)
[2]	韩锋, 陈岗. 岛礁防空的特点和对策[C]// 第四届中国指挥控制大会, 北京: 中国指挥与控制学会, 2016:332-336.
	HAN F, CHEN G. Characters and countermeasures of island air defense[C]// Proceedings of the 4th China command and Control Conference. Beijing: Chinsee Institute of Command and Control, 2016:332-336. (in Chinese)
[3]	LLOYD S P, WITSENHAUSEN H S. Weapons allocation is NP-complete[C]// Proceedings of the 1986 Summer Simulation Conference. Reno, NV, US: IEEE, 1986: 1054-1058.
[4]	沈培志, 杨历彪, 王培源. 多火力单元部署的地空导弹防空体系作战效能研究[J]. 火力与指挥控制, 2020, 45(1):1-6.
	SHEN P Z, YANG L B, WANG P Y. Research on operational effectiveness of surface-to-air missile air defense system deployed in multiple fire units[J]. Fire Control & Command Control, 2020, 45(1):1-6. (in Chinese)
[5]	KARASAKAI O. Air defense missile-target allocation models for a naval task group[J]. Computers & Operations Research, 2008, 35(6): 1759-1770. doi: 10.1016/j.cor.2006.09.011 URL
[6]	王小艺, 侯朝桢, 原菊梅. 防空火力分配建模及优化方法研究[J]. 控制与决策, 2006, 21(8):913-917.
	WANG X Y, HOU C Z, YUAN J M. Modeling and optimization method on antiaircraft firepower allocation[J]. Control and Decision, 2006, 21(8):913-917. (in Chinese)
[7]	王洁, 娄寿春, 王颖龙. 防空导弹混合部署火力单元间配置距离的量化[J]. 系统工程与电子技术, 2006, 28(2):263-265.
	WANG J, LOU S C, WANG Y L. Quantitative analysis of deployment distance between fire units based on the composite disposition of the air defense missiles[J]. Systems Engineering and Electronics, 2006, 28(2):263-265. (in Chinese)
[8]	JOHANSSON E, KARLSSON S. Deployment of air defense[D]. Gothenburg, Sweden: Chalmers University of Technology, 2019.
[9]	颜培远, 刘曙, 王君. 基于动态规划-遗传算法的防空部署优化模型[J]. 系统工程与电子技术, 2018, 40(10):2249-2255.
	YAN P Y, LIU S, WANG J. Optimization model of air defense disposition based on dynamic programming and genetic algorithm[J]. Systems Engineering and Electronics, 2018, 40(10): 2249-2255. (in Chinese)
[10]	MANNE A S. A target-assignment problem[J]. Operations Research, 1958, 6(3):346-351. doi: 10.1287/opre.6.3.346 URL
[11]	WACHOLDER E. A neural network-based optimization algorithm for the static weapon-target assignment problem[J]. Informs Journal on Computing, 1989, 1(4): 232-246. doi: 10.1287/ijoc.1.4.232 URL
[12]	KWON O, KANG D, LEE K, et al. Lagrangian relaxation approach to the targeting problem[J]. Naval Research Logistics, 1999, 46(6):640-653. doi: 10.1002/(ISSN)1520-6750 URL
[13]	AHUJA R K, KUMAR A, JHA K C, et al. Exact and heuristic algorithms for the weapon-target assignment problem[J]. Operations Research, 2007, 55 (6):1136-1146. doi: 10.1287/opre.1070.0440 URL
[14]	KLINE A, AHNER D, PACHTER M. A greedy hungarian algorithm for the weapon- target assignment problem[R]. OH, US: Air Force Institute of Technology, Air Force Institute of Technology Center for Operational Analysis, 2017.
[15]	LI J, XIN B, PARDALOS P M, et al. Solving bi-objective uncertain stochastic resource allocation problems by the CVaR-based risk measure and decomposition-based multi-objective evolutionary algorithms[J]. Annals of Operations Research, 2019, 296(1):1-28. doi: 10.1007/s10479-020-03848-6
[16]	LAI C M, WU T H. Simplified swarm optimization with initialization scheme for dynamic weapon-target assignment problem[J]. Applied Soft Computing, 2019, 82(5):1-15.
[17]	JOSEPH M L, MATTEW J R, BRIAN J L. Improving defensive air battle management by solving a stochastic dynamic assignment problem via approximate dynamic programming[J]. European Operational Research Societies, 2023, 305(3):1435-1449.
[18]	张安, 徐双飞, 毕文豪, 等. 空地多目标攻击武器-目标分配与制导序列优化[J/OL]. 兵工学报:1-12[2023-03-25]. http://kns.cnki.net/kcms/detail/11.2176.TJ.20220829.1712.007.html.
	ZHANG A, XU S F, BI W H, et al. Weapon-target assign- ment and guidance sequencc optimization in air-to-ground multi-target attack[J/OL]. Acta Armamentarii:1-12[2023-03-25]. http://kns.cnki.net/kcms/detail/11.2176.TJ.20220829.1712.007.html. (in Chinese)
[19]	JENKINS P R, ROBBINS M J, LUNDAY B J. Approximate dynamic programming for the military aeromedical evacuation dispatching, preemption-rerouting, and redeployment problem[J]. European Journal of Operational Research, 2021, 290(1):132-143. doi: 10.1016/j.ejor.2020.08.004 URL
[20]	AHNER D K, PARSON C R. Optimal multi-stage allocation of weapons to targets using adaptive dynamic programming[J]. Optimization Letters, 2015, 9(8):1689-1701. doi: 10.1007/s11590-014-0823-x URL
[21]	王, 赵文飞, 滕克难, 等. 不确定因素下海上要地防空动态火力分配模型[J]. 兵工学报, 2022, 43(11):2885-2896.
	WANG Y, ZHAO W F, TENG K N, et al. DWTA model of air defense in important place at sea under uncertain factors[J]. Acta Armamentarii, 2022, 43(11):2885-2896. (in Chinese)
[22]	赵文飞, 刘孝磊, 马翠玲, 等. 基于多目标模糊规划的海上要地动态火力分配[J/OL]. 系统工程与电子技术, 2023, 45(3):777-784.
	ZHAO W F, LIU X L, MA C L, et al. DWTA for strategic location on the sea based on multi-objective fuzzy programming[J]. Systems Engineering and Electronics, 2023, 45(3):777-784. (in Chinese)
[23]	朱建文, 赵长见, 李小平, 等. 基于强化学习的集群多目标分配与智能决策方法[J]. 兵工学报, 2021, 42(9):2040-2048.
	ZHU J W, ZHAO C J, LI X P, et al. Multi-target assignment and intelligent decision based on reinforcement learning[J]. Acta Armamentarii, 2021, 42(9):2040-2048. (in Chinese) doi: 10.3969/j.issn.1000-1093.2021.09.025
[24]	褚凯轩, 常天庆, 张雷. 基于改进人工蜂群算法的地面作战武器-目标分配[J/OL]. 兵工学报:1-13[2023-03-22]. http://kns.cnki.net/kcms/detail/11.2176.TJ.20230207.0851.001.html.
	CHU K X, CHANG T Q, ZHANG L. A WTA model based on shooting effectiveness.and improved artificial bee colony based on WTP library[J/OL]. Acta Armamentarii: 1-13[2023-03-22]. http://kns.cnki.net/kcms/detail/11.2176.TJ.20230207.0851.001.html. (in Chinese)
[25]	SUMMERS D S, ROBBINS M J, LUNDAY B J. An approximate dynamic programming approach for comparing firing policies in a networked air defense environment-science direct[J]. Computers & Operations Research, 2020, 117(5):1-15.
[26]	DAVIS M T, ROBBINS M J, LUNDAY B J. Approximate dynamic programming for missile defense interceptor fire control[J]. European Journal of Operational Research, 2016, 259(3):873-886. doi: 10.1016/j.ejor.2016.11.023 URL
[27]	谢俊洁. 空战仿真中的目标分配与火力分配方法[D]. 长沙: 国防科学技术大学, 2016.
	XIE J J. Target assignment and weapon-target assignment algorithms in air combat simulations[D]. Changsha: National University of Defense Technology, 2016. (in Chinese)
[28]	夏维, 刘新学, 范阳涛, 等. 基于改进型多目标粒子群优化算法的武器-目标分配[J]. 兵工学报, 2016, 37(11):2085-2093. doi: 10.3969/j.issn.1000-1093.2016.11.017
	XIA W, LIU X X, FAN Y T, et al. Weapon target assignment with an improved multi-objective particle swarm optimization algorithm[J]. Acta Armamentarii, 2016, 37(11): 2085-2093. (in Chinese) doi: 10.3969/j.issn.1000-1093.2016.11.017
[29]	SHERMAN J, MORRISON W J. Adjustment of an inverse matrix to changes in the elements of a given column or a given row in the original matrix[J]. Annals of Mathematical Statistics, 1949, 21:124-127. doi: 10.1214/aoms/1177729893 URL