Bayesian inference for ammunition demand based on Gompertz distribution

doi:10.23919/JSEE.2020.000035

Journal of Systems Engineering and Electronics ›› 2020, Vol. 31 ›› Issue (3): 567-577.doi: 10.23919/JSEE.2020.000035

• Systems Engineering • Previous Articles Next Articles

Bayesian inference for ammunition demand based on Gompertz distribution

Rudong ZHAO^1,²(), Xianming SHI^1,*(), Qian WANG³(), Xiaobo SU^1,⁴(), Xing SONG⁵()

¹ Department of Equipment Command and Management, Shijiazhuang Campus, Army Engineering University, Shijiazhuang 050003, China
² Unit 73127 of the PLA, Fuzhou 350503, China
³ The Ninth Comprehensive Training Base of Army, Zhangjiakou 075000, China
⁴ Army Infantry College of the PLA, Shijiazhuang 050003, China
⁵ Unit 68303 of the PLA, Golmud 816000, China

Received:2019-06-18 Online:2020-06-30 Published:2020-06-30
Contact: Xianming SHI E-mail:zrd13376475476@126.com;x.m.shi@126.com;18003131595@163.com;giantsu030700@sina.com;flying506@163.com
About author:ZHAO Rudong was born in 1995. He received his Bachelor's degree in management from Changchun Institute of Technology in 2017 and Master's degree in management from the Army Engineering University in 2019. His research interests are ammunition support and equipment management. E-mail: zrd13376475476@126.com|SHI Xianming was born in 1975. He received his B.E. degree from the Academy of Ordnance Engineering in 1996, M.E. degree from the National University of Science and Technology in 2002, and Ph.D. degree from National University of Science and Technology in 2006. He is currently an associate professor in Shijiazhuang Campus of Army Engineering University. His research interests are equipment support and systems engineering. E-mail: x.m.shi@126.com|WANG Qian born in 1989. He received his B.E. degree from the University of Defense Science and Technology in 2011 and M.S. degree in military science from the Army University of Engineering in 2019. He is a lecturer in the Ninth Comprehensive Training Base of Army. His research interest is equipment support. E-mail: 18003131595@163.com|SU Xiaobo was born in 1978. He received his B.E. degree from the Armored Corps College of Engineering in 2001, and M.S. degree in military science from the Academy of Ordnance Engineering in 2009. He is pursuing his Ph.D. degree at the Army Engineering University. He is a lecturer in the Army Infantry College of the PLA. His research interest is equipment support. E-mail: giantsu030700@sina.com|SONG Xing was born in 1991. He received his B.E. degree from the Academy of Ordnance Engineering in 2013, and M.S. degree in military science from the Army University of Engineering in 2019. His research interest is equipment management. E-mail: flying506@163.com
Supported by:
the Army Scientific Research(KYSZJWJK1744);the Army Scientific Research(012016012600B11403);This work was supported by the Army Scientific Research (KYSZJWJK1744; 012016012600B11403)

Abstract

Abstract:

Aiming at the problem that the consumption data of new ammunition is less and the demand is difficult to predict, combined with the law of ammunition consumption under different damage grades, a Bayesian inference method for ammunition demand based on Gompertz distribution is proposed. The Bayesian inference model based on Gompertz distribution is constructed, and the system contribution degree is introduced to determine the weight of the multi-source information. In the case where the prior distribution is known and the distribution of the field data is unknown, the consistency test is performed on the prior information, and the consistency test problem is transformed into the goodness of the fit test problem. Then the Bayesian inference is solved by the Markov chain-Monte Carlo (MCMC) method, and the ammunition demand under different damage grades is gained. The example verifies the accuracy of this method and solves the problem of ammunition demand prediction in the case of insufficient samples.

Key words: ammunition demand prediction, Bayesian inference, Gompertz distribution, system contribution, Markov chain-Monte Carlo (MCMC) method

Rudong ZHAO, Xianming SHI, Qian WANG, Xiaobo SU, Xing SONG. Bayesian inference for ammunition demand based on Gompertz distribution[J]. Journal of Systems Engineering and Electronics, 2020, 31(3): 567-577.

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References 25

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Target damage grade	Damage degree description
Zero damage $L_1 $	The target is intact or slightly damaged, and the overall combat effectiveness of the target is less than 5%.
Mild damage $L_2 $	The target is relatively lightly damaged. If it is not repaired in time, it will affect the performance of the combat technology, and the overall combat effectiveness of the target will be lost by 5% to 20%.
Moderate damage $L_3 $	The target is severely damaged, and special repairs and replacement parts are required. The overall combat effectiveness of the target is lost by 20% to 50%.
Severe damage $L_4 $	The target is severely damaged, and needs to be returned to the factory for overhaul, the repair cycle is long, and the overall combat effectiveness of the target is lost by 50% to 80%.
Destruction $L_5$	The target cannot be repaired or has no repair value, and the overall combat effectiveness of the target is lost by more than 80%.

Number of equipment	Ammunition quantity	Mild-damaged rate/%	Moderate-damaged rate/%	Severe-damaged rate/%	Destruction rate/%
6	240	15.0	12.0	20	53.0
6	252	20.0	25.0	13	52.0
6	263	15.0	15.0	17	53.0
6	126	15.0	10.0	75	0
6	138	10.0	15.0	65	10.0
6	152	6.0	5.0	57	32.0
6	84	5.0	59.0	4	33.0
6	93	8.0	56.0	4	32.0
6	102	0	66.7	0	33.4
6	36	58.0	6.0	4	32.0
6	48	59.0	5.0	4	33.0
6	58	66.7	0	0	33.4

Ammunition demand in mild damage			Ammunition demand in moderate damage			Ammunition demand in severe damage			Ammunition demand in destruction
Ammunition quantity	Frequency of use	Use probability	Ammunition quantity	Frequency of use	Use probability	Ammunition quantity	Frequency of use	Use probability	Ammunition quantity	Frequency of use	Use probability
2	5	0.026 315	12	9	0.053 571	21	3	0.030 928	38	9	0.084 906
3	16	0.084 211	13	22	0.130 952	22	5	0.051 546	39	16	0.150 943
4	28	0.147 368	14	30	0.178 571	23	9	0.092 784	40	23	0.216 981
5	42	0.221 053	15	42	0.250 000	24	13	0.134 021	41	18	0.169 811
6	51	0.268 421	16	21	0.125 000	25	19	0.195 876	42	13	0.122 642
7	20	0.105 263	17	19	0.113 095	26	21	0.216 495	43	10	0.094 340
8	16	0.084 211	18	12	0.071 429	27	15	0.154 639	44	8	0.075 472
9	7	0.036 842	19	8	0.047 619	28	7	0.072 165	45	6	0.056 604
10	5	0.026 316	20	5	0.029 762	29	5	0.051 546	46	3	0.028 302

Damage grade	OEL	Ammunition demand (maximum likelihood estimation)
Zero damage $L_1 $	Less than 5%	0
Mild damage $L_2 $	5% to 20%	5.615 789
Moderate damage $L_3 $	20% to 50%	15.303 570
Severe damage $L_4 $	50% to 80%	25.350 520
Destruction $L_5 $	More than 80%	41.132 080

Damage grade	OEL	Ammunition demand (maximum likelihood estimation)
Zero damage $L_1 $	Less than 5%	0
Mild damage $L_2 $	5% to 20%	6.752 9
Moderate damage $L_3 $	20% to 50%	15.401 8
Severe damage $L_4 $	50% to 80%	24.230 3
Destruction $L_5 $	More than 80%	41.536 0

Bayesian inference for ammunition demand based on Gompertz distribution

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Number	$\gamma _X^d$	Membership degree	$\eta _X^d$	Membership degree
1	$\gamma _X^1$	0.5	$\eta _X^1$	0.2
2	$\gamma _X^2$	0.6	$\eta _X^2$	0.3
3	$\gamma _X^3$	0.5	$\eta _X^3$	0.2
4	$\gamma _X^4$	0.4	$\eta _X^4$	0.5
5	$\gamma _X^5$	0.3	$\eta _X^5$	0.5
6	$\gamma _X^6$	0.5	$\eta _X^6$	0.2
7	$\gamma _X^7$	0.6	$\eta _X^7$	0.3
8	$\gamma _X^8$	0.7	$\eta _X^8$	0.1
9	$\gamma _X^9$	0.4	$\eta _X^9$	0.5
10	$\gamma _X^{10}$	0.7	$\eta _X^{10}$	0.1

Number	$\gamma _Y^d$	Membership degree	$\eta _Y^d$	Membership degree
1	$\gamma _Y^1$	0.5	$\eta _Y^1$	0.3
2	$\gamma _Y^2$	0.4	$\eta _Y^2$	0.5
3	$\gamma _Y^3$	0.7	$\eta _Y^3$	0.2
4	$\gamma _Y^4$	0.5	$\eta _Y^4$	0.4
5	$\gamma _Y^5$	0.7	$\eta _Y^5$	0.2
6	$\gamma _Y^6$	0.6	$\eta _Y^6$	0.3
7	$\gamma _Y^7$	0.8	$\eta _Y^7$	0.2
8	$\gamma _Y^8$	0.4	$\eta _Y^8$	0.5
9	$\gamma _Y^9$	0.6	$\eta _Y^9$	0.4
10	$\gamma _Y^{10}$	0.5	$\eta _Y^{10}$	0.2

Damage grade	Real ammunition demand	Ammunition demand from Bayesian inference	Relative error
Zero damage $L_1 $	0	0	0
Mild damage $L_2 $	6.752 9	6.494 6	-0.038 25
Moderate damage $L_3 $	15.401 8	15.596 8	0.012 661
Severe damage $L_4 $	24.230 3	24.852 8	0.025 691
Destruction $L_5 $	41.536 0	42.653 5	0.026 904

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