Opportunistic Maintenance Strategy for Complex Two-dimensional Warranty Equipment Considering Failure Correlation

doi:10.12382/bgxb.2022.0101

Abstract

Abstract:

To address the problem that isolated maintenance to individual components during the warranty period reduces equipment availability and increases equipment warranty costs, the optimal opportunistic maintenance strategy for multi-component systems is explored by combining the characteristics of complex equipment and the current situation of two-dimensional warranty, with the availability of equipment during the warranty period as the main quantitative analysis index. Based on the failure correlation analysis of the multi-component system, the opportunistic maintenance strategy is applied in combination with the preventive maintenance plan of each single component, and the two-dimensional warranty availability model of the multi-component system is established by introducing the reliability threshold at the time of opportunistic maintenance and optimizing the combined preventive maintenance work of each single component. In the case analysis, the genetic algorithm is used to find the optimal opportunistic maintenance strategy for a new armored assault vehicle power system. The results of the comparative analysis show that the opportunistic maintenance strategy increases the power system availability by 10% and reduces the warranty cost by 9.3%, which fully verifies the effectiveness of the strategy and improves the military and economic benefits of equipment maintenance and support. This strategy can provide a scientific basis and quantitative analysis method for the development of the opportunistic maintenance strategy for multi-component systems using two-dimensional warranty service.

Key words: equipment support, complex equipment, failure dependence, two-dimensional warranty, preventive maintenance, opportunistic maintenance

DONG Enzhi, CHENG Zhonghua, WANG Rongcai, ZHANG Xiaona, ZHANG Yuexing, WU Weiyi, YUE Shuai. Opportunistic Maintenance Strategy for Complex Two-dimensional Warranty Equipment Considering Failure Correlation[J]. Acta Armamentarii, 2023, 44(6): 1688-1703.

Figures/Tables 22

Fig.1 Failure rate function fitting under different utilization rates

Fig.2 Failure chain model

Table 1 Component number

部件	编号
A	1
B	2
C	3
D	4

Fig.3 Schematic diagram of component reliability

Fig.4 Schematic diagram of warranty period under different utilization rates

Fig.5 Algorithm flow chart

Fig.6 Failure chain model of the power system

Table 2 Parameter setting

部件	θ₀_i	θ₁_i	θ₂_i	θ₃_i	S_c_i/元	S_p_i/元	S_r_i/元	T_c_i/d	T_p_i/d	T_r_i/d	$R m i n i$
1	0.10	0.06	0.03	0.10	1620	510	140000	1.0	2.0	2.0	0.6
2	0.12	0.05	0.05	0.12	1040	210	113200	2.0	3.0	1.0	0.5
3	0.08	0.06	0.04	0.08	2510	150	148600	0.5	1.0	1.0	0.5
4	0.07	0.04	0.03	0.07	1450	300	52410	2.0	2.5	1.5	0.6

Table 2 Parameter setting

部件	θ₀_i	θ₁_i	θ₂_i	θ₃_i	S_c_i/元	S_p_i/元	S_r_i/元	T_c_i/d	T_p_i/d	T_r_i/d	$R m i n i$
1	0.10	0.06	0.03	0.10	1620	510	140000	1.0	2.0	2.0	0.6
2	0.12	0.05	0.05	0.12	1040	210	113200	2.0	3.0	1.0	0.5
3	0.08	0.06	0.04	0.08	2510	150	148600	0.5	1.0	1.0	0.5
4	0.07	0.04	0.03	0.07	1450	300	52410	2.0	2.5	1.5	0.6

Fig.7 Schematic diagram of changes in unit time warranty cost of each component

Table 3 Preventive maintenance interval of each component (Time)

部件	T_k_,_i/d	$C a i$ / (元·d^-1)	最优预防性维修次数
1	85,76,72,61,54,41,30	1082.7	6
2	43,32,20,15,12	652.4	4
3	32,30,27,25,16,12	1624.7	5
4	120,109,76,52	320.6	3

Table 3 Preventive maintenance interval of each component (Time)

部件	T_k_,_i/d	$C a i$ / (元·d^-1)	最优预防性维修次数
1	85,76,72,61,54,41,30	1082.7	6
2	43,32,20,15,12	652.4	4
3	32,30,27,25,16,12	1624.7	5
4	120,109,76,52	320.6	3

Table 4 Preventive maintenance interval of each component (Usage)

部件	行驶里程/km	$C a i$ / (元·d^-1)	最优预防性维修次数
1	2329,2082,1973,1671, 1479,1123,822	1082.7	6
2	1178,877,548,411,329	652.4	4
3	877,822,740,685,438,329	1624.7	5
4	3288,2986,2082,1425	320.6	3

Table 4 Preventive maintenance interval of each component (Usage)

部件	行驶里程/km	$C a i$ / (元·d^-1)	最优预防性维修次数
1	2329,2082,1973,1671, 1479,1123,822	1082.7	6
2	1178,877,548,411,329	652.4	4
3	877,822,740,685,438,329	1624.7	5
4	3288,2986,2082,1425	320.6	3

Table 5 Preventive maintenance time of each component

部件	预防性维修时刻/d
1	85,161,233,294,348,389,419,504,580,652,713,730
2	43,75,95,110,122,165,197,217,232,244,287,319,339,354,366,409,441,461,476,488,531,563,583,598,610, 653,685,705,720,730
3	32,62,89,114,130,142,174,204,231,256,272,284,316,346,373,398,414,426,458,488,515,540,556,568,600,630,657,682,698,710,730
4	120,229,305,357,477,586,662,714,730

Table 6 Mileage during preventive maintenance of each component

部件	预防性维修时里程/km
1	2329,4411,6384,8055,9534,10658,11479,13808,15890,17863,19534,20000
2	1178,2055,2603,3014,3342,4521,5397,5945,6356,6685,7863,8740,9288,9699,10027,11205,12082,12630,13041,13370,14548, 15425,15973,16384,16712,17890,18767,19315,19726,20000
3	877,1699,2438,3123,3562,3890,4767,5589,6329,7014,7452,7781,8658,9479,10219,10904,11342,11671,12548,13370,14110,14795,15233,15562,16438,17260,18000,18685,19123,19452,20000
4	3288,6274,8356,9781,13068,16055,18137,19562,20000

Table 7 Parameter setting of genetic algorithm

参数	数值
种群大小	50
精英数量	3
交叉概率	0.8
变异概率	0.01
进化代数	270

Fig.8 Programming steps of genetic algorithm

Fig.9 Schematic diagram of genetic algorithm iteration

Table 8 Opportunistic maintenance scheme of the power system

时刻/d	部件				时刻/d	部件
时刻/d	1	2	3	4	时刻/d	1	2	3	4
32	N	IM	PM	N	366	N	RM	IM	N
62	N	N	PM	N	389	PM	N	RM	N
75	IM	PM	N	N	409	IR	PM	IM	N
89	N	IM	PM	N	426	N	IM	RM	N
110	N	PM	IM	IM	458	N	IM	PM	IM
122	N	RM	PM	N	476	IM	PM	IM	N
142	IM	IM	RM	N	488	N	RM	N	N
174	N	N	PM	N	515	N	IM	PM	N
197	N	PM	PM	N	540	N	N	PM	N
217	IM	PM	IM	IM	556	IM	IM	PM	IM
232	N	PM	N	N	568	N	IM	RM	N
244	N	RM	PM	N	598	N	OM	IM	N
272	N	N	PM	N	610	N	RM	IM	N
284	IM	RM	IM	IM	652	PM	IM	IM	IM
316	N	IM	PM	N	682	N	IM	PM	N
339	IM	PM	IM	IR	698	IM	IM	PM	IR
354	N	PM	N	N	710	N	IM	RM	N

Fig.10 Comparison between the situations with and without opportunistic maintenance

Fig.11 Comparison between the situations with and without failure correlation

Fig.12 Warranty area under different warranty methods

Table 9 Comparison of warranty cost and availability under different warranty methods

使用率/ (10⁴km·a^-1)	保修成本/元		可用度
使用率/ (10⁴km·a^-1)	一维保修	二维保修	一维保修	二维保修
0.1	1956876	1956876	0.9125	0.9125
0.3	2365485	2365485	0.9023	0.9023
0.5	2423456	2423456	0.8945	0.8945
1.0	2435621	2435621	0.8923	0.8923
1.2	2569874	2436125	0.8812	0.8824
1.5	2657894	2468795	0.8713	0.8813
2.0	2785641	2316543	0.8645	0.8915
3.0	2845678	2245698	0.8566	0.8962
4.0	3125642	2147845	0.8478	0.9043
5.0	3541203	2056987	0.8346	0.9122
6.0	3642315	1987945	0.7523	0.9145
7.0	4123541	1985623	0.6487	0.9254
8.0	4136540	1972256	0.6178	0.9316

Fig.13 Comparison of availability and warranty cost under different warranty methods

References 38

[1]	张怀强, 李铁成, 杜军岗. 舰载导弹系统故障率及保修期模型优化[J]. 系统工程与电子技术, 2022, 44(1):347-356. doi: 10.12305/j.issn.1001-506X.2022.01.43
	ZHANG H Q, LI T C, DU J G. Optimization of failure rate and warranty period model of shipborne missile system[J]. Systems Engineering and Electronics, 2022, 44(1): 347-356. (in Chinese)
[2]	王双川, 贾希胜, 胡起伟, 等. 考虑随机共因失效的战时合成部队任务成功概率仿真评估[J]. 兵工学报, 2021, 42(1): 199-208. doi: 10.3969/j.issn.1000-1093.2021.01.023
	WANG S C, JIA X S, HU Q W, et al. Simulation evaluation of mission success probability of wartime synthetic force considering random common cause failure[J]. Acta Armamentarii, 2021, 42(1): 199-208. (in Chinese)
[3]	程中华, 王谦, 白永生, 等. 二维保修装备可用度优化模型[J]. 装甲兵工程学院学报, 2019, 33(3):34-39.
	CHENG Z H, WANG Q, BAI Y S, et al. Two dimensional warranty equipment availability optimization model[J]. Journal of Armored Forces Engineering College, 2019, 33(3): 34-39. (in Chinese)
[4]	李欣玥, 高勇民, 王维强, 等. 基于二维工龄更换的保修服务优化研究[J]. 航空科学技术, 2017, 28(10):66-72.
	LI X Y, GAO Y M, WANG W Q, et al. Research on warranty service optimization based on two-dimensional length of service replacement[J]. Aviation science and technology, 2017, 28(10): 66-72. (in Chinese)
[5]	李军亮, 陈跃良, 张勇, 等. 基于可用度的预防性维修间隔优化综述[J]. 系统工程理论与实践, 2021, 41(6):1611-1624. doi: 10.12011/SETP2019-1645
	LI J L, CHEN Y L, ZHANG Y, et al. Overview of preventive maintenance interval optimization based on availability[J]. System engineering theory and practice, 2021, 41(6): 1611-1624. (in Chinese)
[6]	WANG X L, XIE W. Two-dimensional warranty: a literature review[J]. Journal of Risk and Reliability, 2018, 232(1):1-15.
[7]	李龙辉, 刘子先, 张兆民, 等. 考虑预防性维护的二维基础质保策略与价格联合优化[J/OL]. 计算机集成制造系统(2021-09-11)[2022-04-07]. http://kns.cnki.net/kcms/detail/11.5946.TP.20210831.1822.009.html.
	LI L H, LIU Z X, ZHANG Z M, et al. Two dimensional basic quality assurance strategy and price joint optimization considering preventive maintenance[J/OL]. Computer integrated manufacturing system(2021-09-11)[2022-04-07]. http://kns.cnki.net/kcms/detail/11.5946.TP.20210831.1822.009.html. in Chinese)
[8]	苏春, 赵家彬. 基于产品使用率的柔性基本质保策略优化设计[J]. 上海交通大学学报, 2020, 54(11): 1182-1188.
	SU C, ZHAO J B. Optimal design of flexible basic quality assurance strategy based on product utilization[J]. Journal of Shanghai Jiaotong University, 2020, 54(11): 1182-1188. (in Chinese)
[9]	TALEIZADEH A A, MOKHTARZADEH M. Pricing and two-dimensional warranty policy of multi-products with online and offline channels using a value-at-risk approach[J]. Computers & Industrial Engineering, 2020, 148: 106674. doi: 10.1016/j.cie.2020.106674 URL
[10]	WANG X L, SU C. A two-dimensional preventive maintenance strategy for items sold with warranty[J]. International Journal of Production Research, 2016, 54(9): 5901-5915. doi: 10.1080/00207543.2016.1187314 URL
[11]	WANG R C, CHENG Z H, WANG Q. Research on availability model of two-dimensional warranty products based on incomplete maintenance[J]. IOP Conference Series: Materials Science and Engineering, 2021, 1043(3): 1-12.
[12]	WANG R C, CHENG Z H, RONG L Q, et al. Availability optimization of two-dimensional warranty products under imperfect preventive maintenance[J]. IEEE Access, 2021, 9: 8099-8109. doi: 10.1109/Access.6287639 URL
[13]	HUANG Y S, HUANG C D, HO J W. A customized two-dimensional extended warranty with preventive maintenance[J]. European Journal of Operations Research, 2017, 257(3):971-978. doi: 10.1016/j.ejor.2016.07.034 URL
[14]	TONG P, SONG X F, LIU Z X. A maintenance strategy for two-dimensional extended warranty based on dynamic usage rate[J]. International Journal of Production Research, 2017, 55(19):5743-5759. doi: 10.1080/00207543.2017.1330573 URL
[15]	SU C, WANG X L. A two-stage preventive maintenance optimization model incorporating two-dimensional extended warranty[J]. Reliability Engineering& System Safety, 2016, 155:169-178.
[16]	WANG D F, HE Z, HE S G, et al. Dynamic pricing of two-dimensional extended warranty considering the impacts of product price fluctuations and repair learning[J]. Reliability Engineering & System Safety, 2021, 210: 107516. doi: 10.1016/j.ress.2021.107516 URL
[17]	SHAHANAGHI K, NOOROSSANA R, JALALI-NAINI S G, et al. Failure modeling and optimizing preventive maintenance strategy during two-dimensional extended warranty contracts[J]. Engineering Failure Analysis, 2013, 28:90-102. doi: 10.1016/j.engfailanal.2012.09.006 URL
[18]	MITRA A. Warranty parameters for extended two-dimensional warranties incorporating consumer preferences[J]. European Journal of Operations Research, 2021, 291:525-535. doi: 10.1016/j.ejor.2019.12.035 URL
[19]	PENG W, ZHANG X L, HUANG H Z. A failure rate interaction model for two-component systems based on copula function with shock damage interaction[J]. Journal of Risk and Reliability, 2016, 230(3): 278-284.
[20]	SUN Y. Reliability prediction of complex repairable systems: an engineering approach[D]. Queensland: Queensland University of Technology, 2006.
[21]	SUN Y, MA L, MATHEW J, et al. An analytical model for interactive failures[J]. Reliability Engineering and System Safety, 2005, 91(5):495-504. doi: 10.1016/j.ress.2005.03.014 URL
[22]	ZHANG N, FOULADIRAD M, BARROS A, et al. Condition-based maintenance for a K-out-of-N deteriorating system under periodic inspection with failure dependence[J]. European Journal of Operational Research, 2020, 287(1):159-167. doi: 10.1016/j.ejor.2020.04.041 URL
[23]	韩思远. 考虑故障相关的双馈风电机组状态机会维修策略研究[D]. 兰州: 兰州交通大学, 2017.
	HAN S Y. Research on condition based opportunistic maintenance strategy of doubly fed wind turbine considering fault correlation[D]. Lanzhou: Lanzhou Jiaotong University, 2017. (in Chinese)
[24]	钱倩, 蒋祖华. 考虑预防维修时间及相关性的多部件系统维护策略[J]. 工业工程, 2020, 23(6):95-100.
	QIAN Q, JIANG Z H. Preventive maintenance strategy and maintenance time of multi-component system[J]. Industrial Engineering, 2020, 23(6): 95-100. (in Chinese)
[25]	王红, 杜维鑫, 刘志龙, 等. 联合故障与经济相关性的动车组多部件系统维护[J]. 上海交通大学学报, 2016, 50(5): 660-667.
	WANG H, DU W X, LIU Z L, et al. Multi component system maintenance of EMU based on joint fault and economic correlation[J]. Journal of Shanghai Jiaotong University, 2016, 50(5): 660-667. (in Chinese)
[26]	BERG M. Optimal replacement policies for two-unit machines with increasing running costs 1[J]. Stochastic Processes and their Applications, 1976, 4(1): 89-106. doi: 10.1016/0304-4149(76)90028-4 URL
[27]	van der DUYN SCHOUTEN F A, VANNESTE S G. Analysis and computation of (n,N)—strategies for maintenance of a two-components system[J]. European Journal of Operational Research, 1990, 48(2): 260-274. doi: 10.1016/0377-2217(90)90379-P URL
[28]	ZHENG X, FARD N. Hazard-rate tolerance method for an opportunistic replacement policy[J]. IEEE Transactions on Reliability, 1992, 41(1): 13-20. doi: 10.1109/24.126664 URL
[29]	CALDEIRA DUARTE J A C, CRAVEIRO J C T A, TRIGO T P. Optimisation of the preventive maintenance plan of a series components system[J]. International Journal of Pressure Vessels and Piping, 2006, 83(4): 244-248. doi: 10.1016/j.ijpvp.2006.02.016 URL
[30]	LAGGOUNEA R, CHATEAUNEUFB A, AISSANIA D. Opportunistic policy for optimal preventive maintenance of a multi-component system in continuous operating units[J]. Computers and Chemical Engineering, 2009, 33(9): 1499-1510. doi: 10.1016/j.compchemeng.2009.03.003 URL
[31]	苏春, 陈武. 基于滚动窗口方法的风力机动态机会维修优化[J]. 机械工程学报, 2014, 50(14):62-68.
	SU C, CHEN W. Dynamic opportunistic maintenance optimization of wind turbine based on rolling window method[J]. Journal of Mechanical Engineering, 2014, 50(14): 62-68. (in Chinese)
[32]	薛朝改, 谷雨, 曹武军, 等. 多部件重要度系统机会维修策略的优化[J]. 机械设计与制造, 2022(2):116-119.
	XUE C G, GU Y, CAO W J, et al. Optimization of opportunistic maintenance strategy for multi component importance system[J]. Mechanical Design and Manufacturing, 2022(2): 116-119. (in Chinese)
[33]	YUN W Y, KANG K M. Imperfect repair policies under two-dimensional warranty[J]. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 2007, 221(4): 239-247. doi: 10.1243/1748006XJRR55 URL
[34]	MALIK M A K. Reliable Preventive Maintenance Policy[J]. AIIE Transactions, 1979, 11(3): 221-228. doi: 10.1080/05695557908974463 URL
[35]	NAKAGAWA T. Sequential imperfect preventive maintenance policies[J]. IEEE Transactions on Reliability, 1988, 37(3): 295-298. doi: 10.1109/24.3758 URL
[36]	杨国军, 王红, 何勇, 等. 故障及经济相关下动车组系统动态成组维护策略[J]. 铁道科学与工程学报, 2021, 18(1):31-37.
	YANG G J, WANG H, HE Y, et al. Dynamic group maintenance strategy of EMU system under fault and economic correlation[J]. Journal of Railway Science and Engineering, 2021, 18(1): 31-37. (in Chinese)
[37]	董方岐, 刘子先, 郝剑虹, 等. 考虑预防性维修的二维质保期设计与优化研究[J]. 工业工程与管理, 2019, 24(4): 64-71.
	DONG F Q, LIU Z X, HAO J H, et al. Research on design and optimization of two-dimensional warranty period considering preventive maintenance[J]. Industrial Engineering and Management, 2019, 24(4): 64-71. (in Chinese)
[38]	张琳娜. 改进遗传算法在计算机数学建模中的应用研究[J]. 电子设计工程, 2021, 29(19):31-34.
	ZHANG L N. Application of improved genetic algorithm in computer mathematical modeling[J]. Electronic Design Engineering, 2021, 29(19): 31-34. (in Chinese)