Dynamic Firepower Allocation for Cooperative Air Defense of Strategic Locations on the Sea Based on Reinforcement Learning

doi:10.12382/bgxb.2022.1276

Abstract

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

CLC Number:

E072

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.

Figures/Tables 14

Fig.1 Concept map of Markov decision process

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)

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

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	$R t x$	$A t x$	$(R t x) 2$	$(A t x) 2$	$R t x A t x$

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	$R t x$	$A t x$	$(R t x) 2$	$(A t x) 2$	$R t x A t x$

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

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

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

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

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

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

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

Fig.4 Iterative convergence curves 1 of each algorithm

Fig.5 Iterative convergence curves 2 of each algorithm

Fig.5 Comparison chart of time consumption of each algorithm

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