Review of Multivariate Dependent Degeneration Modeling Methods for Mechanical Products

doi:10.3901/JME.2023.20.470

Abstract

Abstract: Degradation modeling has been recognized as a critical approach for mechanical products in conducting reliability assessments and remaining useful life(RUL) predictions, as well as guiding the optimization of operation and maintenance actions. With the increasing complexity in structure and diversification in the function of modern mechanical products, more interest and attention have been paid to the modeling of multivariate degradation data as it can comprehensively reflect the health status of products. Especially, focusing on the stochastic dependence that is widespread in multivariate degradation processes, many practical and effective multivariate dependent degradation(MVDD) models have been proposed, based on which the accuracy of reliability assessment and RUL prediction of products is greatly enhanced. However, there is no systematic review focusing specifically on this topic. Therefore, the research status, recent advances, and applications of existing MVDD modeling methods are reviewed to facilitate their developments and applications. First, three commonly used MVDD modeling methods are introduced in detail, including their principles, advantages, and disadvantages. Then, extended studies of MVDD models involving the effects of multi-source variability, external covariates, and time-varying stochastic dependences are investigated. In addition, applications of MVDD models for accelerated test design, RUL prediction, and maintenance strategy optimization are discussed. Finally, several possible directions for further research are summarized.

Key words: multivariable degradation data, stochastic dependence, dependent degradation modeling, accelerated test design maintenance strategy optimization

CLC Number:

TB114

KONG Xuefeng, PAN Jun, QIAN Ping, WEI Yimin, CHEN Wenhua. Review of Multivariate Dependent Degeneration Modeling Methods for Mechanical Products[J]. Journal of Mechanical Engineering, 2023, 59(20): 470-488.

References

[1] YE Z，XIE M. Stochastic modeling and analysis of degradation for highly reliable products[J]. Applied Stochastic Models in Business and Industry，2015，31(1):16-32.
[2] HU C，YOUN B，KIM T，et al. A co-training-based approach for prediction of remaining useful life utilizing both failure and suspension data[J]. Mechanical Systems and Signal Processing，2015，62-63:75-90.
[3] ZHANG Z，SI X，HU C，et al. Degradation modeling–based remaining useful life estimation:A review on approaches for systems with heterogeneity[J]. Proceedings of the Institution of Mechanical Engineers，Part O:Journal of Risk and Reliability，2015，229(4):343-355.
[4] KANG R，GONG W，CHEN Y. Model-driven degradation modeling approaches:Investigation and review[J]. Chinese Journal of Aeronautics，2020，33(4):1137-1153.
[5] ZHANG H，CHEN M，SHANG J，et al. Stochastic process-based degradation modeling and RUL prediction:from Brownian motion to fractional Brownian motion[J]. Science China Information Sciences，2021，64(7):1-20.
[6] LI J，ZHANG X，ZHOU X，et al. Reliability assessment of wind turbine bearing based on the degradation- Hidden-Markov model[J]. Renewable Energy，2019，132:1076-1087.
[7] SHEN L，WANG Y，ZHAI Q，et al. Degradation modeling using stochastic processes with random initial degradation[J]. IEEE Transactions on Reliability，2019，68(4):1320-1329.
[8] LIN W，CHAI Y，LIU Q. Remaining useful life prediction of electronic products based on wiener degradation process[J]. IFAC-PapersOnLine，2019，52(24):24-28.
[9] LIU T，SUN Q，MENG J，et al. Degradation modeling of satellite thermal control coatings in a low earth orbit environment[J]. Solar Energy，2016，139:467-474.
[10] MU Z，RAN Y，ZHANG G，et al. Remaining useful life prediction method for machine tools based on meta-action theory[J]. Proceedings of the Institution of Mechanical Engineers，Part O:Journal of Risk and Reliability，2021，235(4):580-590.
[11] DAI Y，CHENG S，GAN Q，et al. Life prediction of Ni-Cd battery based on linear Wiener process[J]. Journal of Central South University，2021，28(9):2919-2930.
[12] HAO H，SU C，LI C. LED Lighting system reliability modeling and inference via random effects gamma process and copula function[J]. International Journal of Photoenergy，2015，2015:243648.
[13] YAZDAN MEHR M，BAHRAMI A，van DRIEL W D，et al. Degradation of optical materials in solid-state lighting systems[J]. International Materials Reviews，2020，65(2):102-128.
[14] PENG W，YE Z，CHEN N. Joint online RUL prediction for multivariate deteriorating systems[J]. IEEE Transactions on Industrial Informatics，2019，15(5):2870-2878.
[15] HUANG W，ASKIN R. Reliability analysis of electronic devices with multiple competing failure modes involving performance aging degradation[J]. Quality and Reliability Engineering International，2003，19(3):241-254.
[16] LI W，PHAM H. Reliability modeling of multi-state degraded systems with multi-competing failures and random shocks[J]. IEEE Transactions on Reliability，2005，54(2):297-303.
[17] DONG W，LIU S，BAE S，et al. Reliability modelling for multi-component systems subject to stochastic deterioration and generalized cumulative shock damages[J]. Reliability Engineering & System Safety，2021，205:107260.
[18] SUN F，FU F，LIAO H，et al. Analysis of multivariate dependent accelerated degradation data using a random-effect general Wiener process and D-vine Copula[J]. Reliability Engineering & System Safety，2020，204:107168.
[19] KEIZER M C A，FLAPPER S D P，TEUNTER R H. Condition-based maintenance policies for systems with multiple dependent components:A review[J]. European Journal of Operational Research，2017，261(2):405-420.
[20] 潘骏，王小云，陈文华，等. 基于多元性能参数的加速退化试验方案优化设计研究[J]. 机械工程学报，2012，48(2):30-35. PAN Jun，WAGN Xiaoyun，CHEN Wenhua，et al. Research on optimal design of accelerated degradation test plan based on multiple performance parameters[J]. Journal of Mechanical Engineering，2012，48(2):30-35.
[21] FANG G，PAN R，WANG Y. Inverse Gaussian processes with correlated random effects for multivariate degradation modeling[J]. European Journal of Operational Research，2022，300(3):1177-1193.
[22] PAN Z，SUN Q，FENG J. Reliability modeling of systems with two dependent degrading components based on gamma processes[J]. Communications in Statistics - Theory and Methods，2016，45(7):1923-1938.
[23] FANG G，PAN R，HONG Y. Copula-based reliability analysis of degrading systems with dependent failures[J]. Reliability Engineering & System Safety，2020，193:106618.
[24] BIAN L，GEBRAEEL N. Stochastic framework for partially degradation systems with continuous component degradation-rate-interactions[J]. Naval Research Logistics，2014，61(4):286-303.
[25] WANG P，COIT D W. Reliability prediction based on degradation modeling for systems with multiple degradation measures[C]//Annual Symposium Reliability and Maintainability，2004- RAMS. January 26-29，2004，Los Angeles，CA，USA，2004:302-307.
[26] 晁代宏，马静，陈淑英. 应用多元性能退化量评估光纤陀螺贮存的可靠性[J]. 光学精密工程，2011，19(1):35-40. CHAO Daihong，MA Jing，CHEN Shuying. Assessment of storage reliability for FOGs by multivariate degradation data[J]. Optics and Precision Engineering，2011，19(1):35-40.
[27] 孙绍辉，王华伟，陈福立. 多元退化信息的航空发动机可靠性预测[J]. 火力与指挥控制，2013，38(11):32-35. SUN Shaohui，WANG Huawei，CHEN Fuli. Reliability prediction of aircraft engine based on multivariate degradation information[J]. Fire Control & Command Control，2013，38(11):32-35.
[28] SHEN Y，ZHANG C，TAN Y，et al. Accelerated degradation testing for systems with multiple performance parameters[C]//2011 International Conference on Quality，Reliability，Risk，Maintenance，and Safety Engineering. June 17-19，2011，Xi'an，China，2011:292-296.
[29] BIRNBAUM Z W，SAUNDERS S C. A new family of life distributions[J]. Journal of Applied Probability，1969，6(2):319-327.
[30] PARK C，PADGETT W J. Accelerated Degradation models for failure based on geometric brownian motion and Gamma processes[J]. Lifetime Data Analysis，2005，11(4):511-527.
[31] KUNDU D，BALAKRISHNAN N，JAMALIZADEH A. Bivariate Birnbaum–Saunders distribution and associated inference[J]. Journal of Multivariate Analysis，2010，101(1):113-125.
[32] PAN Z，FENG J，SUN Q. Lifetime distribution and associated inference of systems with multiple degradation measurements based on gamma processes[J]. Eksploatacja i Niezawodność，2016，18(2):307-313.
[33] PAN Z，BALAKRISHNAN N. Reliability modeling of degradation of products with multiple performance characteristics based on gamma processes[J]. Reliability Engineering & System Safety，2011，96(8):949-957.
[34] NELSEN R B. An introduction to copulas[M]. 2nd ed. New York:Springer-Verlag，2006.
[35] YANG Q，HONG Y，ZHANG N，et al. A copula-based trend-renewal process model for analysis of repairable systems with multitype failures[J]. IEEE Transactions on Reliability，2017，66(3):590-602.
[36] WU S. Construction of asymmetric copulas and its application in two-dimensional reliability modelling[J]. European Journal of Operational Research，2014，238(2):476-485.
[37] XIE J，JIANG D，HAN Y，et al. Multi-performance degradation reliability assessment method for a cabin door locking mechanism[J]. Advances in Mechanical Engineering，2022，14(10):16878132221129742.
[38] LI H，LI R，LI H，et al. Reliability modeling of multiple performance based on degradation values distribution[J]. Advances in Mechanical Engineering，2016，8(10):1687814016673755.
[39] CHEN R，ZHANG C，WANG S，et al. Reliability estimation of mechanical seals based on bivariate dependence analysis and considering model uncertainty[J]. Chinese Journal of Aeronautics，2021，34(5):554-572.
[40] CAO G，LIU X，HU D，et al. Stochastic modeling of fatigue crack growth for bolt holes in turbine disc[J]. International Journal of Fatigue，2023，169:107504.
[41] LIU T，PAN Z，SUN Q，et al. Residual useful life estimation for products with two performance characteristics based on a bivariate Wiener process[J]. Proceedings of the Institution of Mechanical Engineers，Part O:Journal of Risk and Reliability，2017，231(1):69-80.
[42] HAO H，SU C. Bivariate nonlinear diffusion degradation process modeling via copula and MCMC[J]. Mathematical Problems in Engineering，2014，2014:510929.
[43] LI Y，BAI X，SHI S，et al. Dynamic fatigue reliability analysis of transmission gear considering failure dependence[J]. Computer Modeling in Engineering & Sciences，2022，130(2):1077-1092.
[44] XU D，XING M，WEI Q，et al. Failure behavior modeling and reliability estimation of product based on vine-copula and accelerated degradation data[J]. Mechanical Systems and Signal Processing，2018，113:50-64.
[45] XU D，WEI Q，ELSAYED E A，et al. Multivariate degradation modeling of smart electricity meter with multiple performance characteristics via vine copulas[J]. Quality and Reliability Engineering International，2017，33(4):803-821.
[46] BIAN L，GEBRAEEL N. Stochastic modeling and real-time prognostics for multi-component systems with degradation rate interactions[J]. IIE Transactions，2014，46(5):470-482.
[47] RASMEKOMEN N，PARLIKAD A K. Condition-based maintenance of multi-component systems with degradation state-rate interactions[J]. Reliability Engineering & System Safety，2016，148:1-10.
[48] ASSAF R，DO P，SCARF P，et al. Wear rate-state interaction modelling for a multi-component system:Models and an experimental platform[J]. IFAC- PapersOnLine，2016，49(28):232-237.
[49] 杨志远，赵建民，程中华. 退化相关多部件系统预防性维修决策模型[J]. 系统工程与电子技术，2018，40(4):823-832. YANG Zhiyuan，ZHAO Jianmin，CHENG Zhonghua. Preventive maintenance decision model of multi-component system with degradation interaction[J]. Systems Engineering and Electronics，2018，40(4):823-832.
[50] NIU H，ZENG J，SHI H，et al. Degradation modeling and remaining useful life prediction for a multi-component system with stochastic dependence[J]. Computers & Industrial Engineering，2023，175:108889.
[51] ZHAI Q，YE Z. A Multivariate Stochastic Degradation Model for Dependent Performance Characteristics[J]. Technometrics，2022，12:1-13.
[52] BUIJS F A，HALL J W，Van NOORTWIJK J M，et al. Time-dependent reliability analysis of flood defences using gamma processes[J]. Safety and reliability of engineering systems and structures，2005，1:2209-2216.
[53] MERCIER S，MEIER-HIRMER C，ROUSSIGNOL M. Bivariate Gamma wear processes for track geometry modelling，with application to intervention scheduling[J]. Structure and Infrastructure Engineering，2012，8(4):357-366.
[54] HUYNH K T，VU H C，NGUYEN T D，et al. A predictive maintenance model for k-out-of-n:F continuously deteriorating systems subject to stochastic and economic dependencies[J]. Reliability Engineering & System Safety，2022，226:108671.
[55] LIU B，PANDEY M D，WANG X，et al. A finite-horizon condition-based maintenance policy for a two-unit system with dependent degradation processes[J]. European Journal of Operational Research，2021，295(2):705-717.
[56] MERCIER S，PHAM H H. A preventive maintenance policy for a continuously monitored system with correlated wear indicators[J]. European Journal of Operational Research，2012，222(2):263-272.
[57] ZHOU W，XIANG W，HONG H. Sensitivity of system reliability of corroding pipelines to modeling of stochastic growth of corrosion defects[J]. Reliability Engineering & System Safety，2017，167:428-438.
[58] PANDEY M D，YUAN X，van NOORTWIJK J M. The influence of temporal uncertainty of deterioration on life-cycle management of structures[J]. Structure and Infrastructure Engineering，2009，5(2):145-156.
[59] GIORGIO M，GUIDA M，PULCINI G. An age- and state-dependent Markov model for degradation processes[J]. IIE Transactions，2011，43(9):621-632.
[60] BIAN L，GEBRAEEL N. Stochastic methodology for prognostics under continuously varying environmental profiles[J]. Statistical Analysis and Data Mining:The ASA Data Science Journal，2013，6(3):260-270.
[61] BIAN L，GEBRAEEL N，KHAROUFEH J P. Degradation modeling for real-time estimation of residual lifetimes in dynamic environments[J]. IIE Transactions，2015，47(5):471-486.
[62] KALBFLEISCH J D，PRENTICE R L. The statistical analysis of failure time data[M]. New York:John Wiley & Sons，2011.
[63] SI X，WANG W，HU C，et al. Estimating remaining useful life with three-source variability in degradation modeling[J]. IEEE Transactions on Reliability，2014，63(1):167-190.
[64] 庄东辰，茆诗松. 退化数据统计分析[M]. 北京:中国统计出版社，2013. ZHUANG Dongchen，MAO Shisong. Statistical analysis of degradation data[M]. Beijing:China Statistics Press，2013.
[65] SHAT H. Optimal design of stress levels in accelerated degradation testing for multivariate linear degradation models[J/OL]. arXiv:2102.09446，https://arxiv.org/abs/2106.09379
[66] CHEN X，SUN X. Reliability assessment for products with two performance characteristics based on marginal stochastic processes and copulas[J]. Communications in Statistics - Simulation and Computation，2022，51(7):3621-3644.
[67] LAWLESS J，CROWDER M. Covariates and random effects in a gamma process model with application to degradation and failure[J]. Lifetime Data Analysis，2004，10(3):213-227.
[68] WANG X，BALAKRISHNAN N，GUO B，et al. Residual life estimation based on bivariate non-stationary gamma degradation process[J]. Journal of Statistical Computation and Simulation，2015，85(2):405-421.
[69] RODRIGUEZ-PICON L A. Reliability assessment for systems with two performance characteristics based on gamma processes with marginal heterogeneous random effects[J]. Eksploatacja i Niezawodność，2017，19(1):8-18.
[70] SUN Q，YE Z，HONG Y. Statistical modeling of multivariate destructive degradation tests with blocking[J]. Technometrics，2020，62(4):536-548.
[71] LI Y，ZIO E，PAN E. An MEWMA-based segmental multivariate hidden Markov model for degradation assessment and prediction[J]. Proceedings of the Institution of Mechanical Engineers，Part O:Journal of Risk and Reliability，2021，235(5):831-844.
[72] YE X，HU Y，ZHENG B，et al. Reliability assessment of film capacitors oriented by dependent and nonlinear degradation considering three-source uncertainties[J]. Microelectronics Reliability，2021，126:114277.
[73] ZHENG B，CHEN C，ZHANG W，et al. Reliability estimation of complex systems based on a Wiener process with random effects and D-vine copulas[J]. Microelectronics Reliability，2022，138:114640.
[74] LU L，WANG B，HONG Y，et al. General path models for degradation data with multiple characteristics and covariates[J]. Technometrics，2021，63(3):354-369.
[75] ZHAI Q，YE Z. Robust degradation analysis with non-Gaussian measurement errors[J]. IEEE Transactions on Instrumentation and Measurement，2017，66(11):2803-2812.
[76] TSAI C，TSENG S，BALAKRISHNAN N. Mis-specification analyses of gamma and Wiener degradation processes[J]. Journal of Statistical Planning and Inference，2011，141(12):3725-3735.
[77] KONG X，YANG J，LI L. Remaining useful life prediction for degrading systems with random shocks considering measurement uncertainty[J]. Journal of Manufacturing Systems，2021，61:782-798.
[78] PRAJAPATI D，LING M H，SHING CHAN P，et al. Misspecification of copula for one-shot devices under constant stress accelerated life-tests[J]. Proceedings of the Institution of Mechanical Engineers，Part O:Journal of Risk and Reliability，2022:1748006X221108850.
[79] SHI Y，FENG Q，SHU Y，et al. Multi-dimensional Lévy processes with Lévy copulas for multiple dependent degradation processes in lifetime analysis[J]. Quality Engineering，2020，32(3):434-448.
[80] PALAYANGODA L K，NG H K T. Semiparametric and nonparametric evaluation of first-passage distribution of bivariate degradation processes[J]. Reliability Engineering & System Safety，2021，205:107230.
[81] DING Y，YANG Q，KING C，et al. A general accelerated destructive degradation testing model for reliability analysis[J]. IEEE Transactions on Reliability，2019，68(4):1272-1282.
[82] TSENG S，BALAKRISHNAN N，TSAI C. Optimal step-stress accelerated degradation test plan for Gamma degradation processes[J]. IEEE Transactions on Reliability，2009，58(4):611-618.
[83] LING M，TSUI K，BALAKRISHNAN N. Accelerated degradation analysis for the quality of a system based on the Gamma process[J]. IEEE Transactions on Reliability，2015，64(1):463-472.
[84] YE Z，CHEN L，TANG L，et al. Accelerated degradation test planning using the inverse Gaussian process[J]. IEEE Transactions on Reliability，2014，63(3):750-763.
[85] ESCOBAR L A，MEEKER W Q. A Review of accelerated test models[J]. Statistical Science，2006，21(4):552-577.
[86] LI X，XUE P. Multivariate storage degradation modeling based on copula function[J]. Advances in Mechanical Engineering，2014，6:503407.
[87] PENG W，LI Y，MI J，et al. Reliability of complex systems under dynamic conditions:A Bayesian multivariate degradation perspective[J]. Reliability Engineering & System Safety，2016，153:75-87.
[88] SUN F，LIU J，LI X，et al. Reliability analysis with multiple dependent features from a vibration-based accelerated degradation test[J]. Shock and Vibration，2016，2016:e2315916.
[89] YANG Z，LI S，CHEN C，et al. Reliability prediction of rotary encoder based on multivariate accelerated degradation modeling[J]. Measurement，2020，152:107395.
[90] HAJIHA M，LIU X，HONG Y. Degradation under dynamic operating conditions:Modeling，competing processes and applications[J]. Journal of Quality Technology，2021，53(4):347-368.
[91] SUN F，GUO H，WANG N，et al. Remaining useful life prediction for bivariate deteriorating systems under dynamic operational conditions[J]. Quality and Reliability Engineering International，2022，38(4):1729-1749.
[92] HAO S，YANG J，MA X，et al. Reliability modeling for mutually dependent competing failure processes due to degradation and random shocks[J]. Applied Mathematical Modelling，2017，51:232-249.
[93] WANG Y，PHAM H. Modeling the dependent competing risks with multiple degradation processes and random shock using time-varying copulas[J]. IEEE Transactions on Reliability，2012，61(1):13-22.
[94] PENG H，FENG Q，COIT D W. Reliability and maintenance modeling for systems subject to multiple dependent competing failure processes[J]. IIE Transactions，2010，43(1):12-22.
[95] FAN M，ZENG Z，ZIO E，et al. Modeling dependent competing failure processes with degradation-shock dependence[J]. Reliability Engineering & System Safety，2017，165:422-430.
[96] SONG S，COIT D W，FENG Q，et al. Reliability analysis for multi-component systems subject to multiple dependent competing failure processes[J]. IEEE Transactions on Reliability，2014，63(1):331-345.
[97] SONG S，COIT D W，FENG Q. Reliability analysis of multiple-component series systems subject to hard and soft failures with dependent shock effects[J]. IIE Transactions， 2016，48(8):720-735.
[98] PARK N，KIM J，KIM H，et al. Development of an algebraic model that predicts the maximum power output of solar modules including their degradation[J]. Renewable Energy，2017，113:141-147.
[99] HU M，WANG J，FU C，et al. Study on cycle-life prediction model of lithium-ion battery for electric vehicles[J]. International Journal of Electrochemical Science，2016，11(1):577-589.
[100] SUN Y，ZHANG Z，ZHANG Q，et al. Multiple failure mode reliability modeling and analysis in failure crack propagation based on time-varying copula[J]. Journal of Mechanical Science and Technology，2018，32(10):4637-4648.
[101] SUN F，LI H，CHENG Y，et al. Reliability analysis for a system experiencing dependent degradation processes and random shocks based on a nonlinear Wiener process model[J]. Reliability Engineering & System Safety，2021，215:107906.
[102] SUN F，WANG N，LI X，et al. A time-varying copula-based prognostics method for bivariate accelerated degradation testing[J]. Journal of Intelligent & Fuzzy Systems，2018，34(6):3707-3718.
[103] XU D，YUAN J，XING M. A Time-varying vine copula model for dependence analysis of failure system[C]//2018 International Conference on Sensing，Diagnostics，Prognostics，and Control (SDPC). August 15-17，2018，Xi'an，China，2018:437-442.
[104] YANG C，GU X，ZHAO F. Reliability analysis of degrading systems based on time-varying copula[J]. Microelectronics Reliability，2022，136:114628.
[105] PAN J，BAI G，CHEN W. Lifetime estimation of nitrile butadiene rubber O-rings under storage conditions using time-varying copula[J]. Proceedings of the Institution of Mechanical Engineers，Part O:Journal of Risk and Reliability，2018，232(6):635-646.
[106] SUN F，WANG N，LI X，et al. Remaining useful life prediction for a machine with multiple dependent features based on Bayesian dynamic linear model and copulas[J]. IEEE Access，2017，5:16277-16287.
[107] PATTON A J. Modelling asymmetric exchange rate dependence[J]. International Economic Review，2006，47(2):527-556.
[108] WANG C，XIA E，WANG K，et al. A mixed-copula-based integral method for reliability analysis of a novel multi-functional rescue end-effector[J]. Advances in Mechanical Engineering，2023，15(1):16878132221147463.
[109] LI Y，XU S，CHEN H，et al. A general degradation process of useful life analysis under unreliable signals for accelerated degradation testing[J]. IEEE Transactions on Industrial Informatics，2023，19(6):7742-7750.
[110] COLLINS D H，FREELS J K，HUZURBAZAR A V，et al. Accelerated test methods for reliability prediction[J]. Journal of Quality Technology，2013，45(3):244-259.
[111] LIMON S，YADAV O P，LIAO H. A literature review on planning and analysis of accelerated testing for reliability assessment[J]. Quality and Reliability Engineering International，2017，33(8):2361-2383.
[112] LI S，CHEN Z，LIU Q，et al. Modeling and analysis of performance degradation data for reliability assessment:A review[J]. IEEE Access，2020，8:74648-74678.
[113] TRUONG M，DO P，MENDIZABAL L，et al. A Bayesian-based optimization approach for accelerated degradation test plan of a LED component with self-heating impact[C]//2022 10th International Conference on Systems and Control (ICSC). November 23-25，2022，Marseille，France，2022:96-101.
[114] 葛蒸蒸，姜同敏，韩少华，等. 基于D优化的多应力加速退化试验设计[J]. 系统工程与电子技术，2012，34(4):846-853. GE Zhengzheng，JIANG Tongmin，HAN Shaohua，et al. Design of accelerated degradation testing with multiple stresses based on D optimality[J]. Systems Engineering and Electronics，2012，34(4):846-853.
[115] TSAI C，TSENG S，BALAKRISHNAN N. Optimal design for degradation tests based on gamma processes with random effects[J]. IEEE Transactions on Reliability，2012，61(2):604-613.
[116] TSAI C，TSENG S，BALAKRISHNAN N，et al. Optimal design for accelerated destructive degradation tests[J]. Quality Technology & Quantitative Management，2013，10(3):263-276.
[117] BARKER C T，NEWBY M J. Optimal non-periodic inspection for a multivariate degradation model[J]. Reliability Engineering & System Safety，2009，94(1):33-43.
[118] PAN Z，SUN Q. Optimal design for step-stress accelerated degradation test with multiple performance characteristics based on Gamma processes[J]. Communications in Statistics - Simulation and Computation，2014，43(2):298-314.
[119] WANG Y，CHEN X，TAN Y. Optimal design of step-stress accelerated degradation test with multiple stresses and multiple degradation measures[J]. Quality and Reliability Engineering International，2017，33(8):1655-1668.
[120] WANG Y，ZHANG C，ZHANG S，et al. Optimal design of constant stress accelerated degradation test plan with multiple stresses and multiple degradation measures[J]. Proceedings of the Institution of Mechanical Engineers，Part O:Journal of Risk and Reliability，2015，229(1):83-93.
[121] SHI Y，XIANG Y，LIAO Y，et al. Optimal burn-in policies for multiple dependent degradation processes[J]. IISE Transactions，2021，53(11):1281-1293.
[122] SI X，WANG W，HU C，et al. Remaining useful life estimation – A review on the statistical data driven approaches[J]. European Journal of Operational Research，2011，213(1):1-14.
[123] LEI Y，LI N，GUO L，et al. Machinery health prognostics:A systematic review from data acquisition to RUL prediction[J]. Mechanical Systems and Signal Processing，2018，104:799-834.
[124] FENG J，KVAM P，TANG Y. Remaining useful lifetime prediction based on the damage-marker bivariate degradation model:A case study on lithium-ion batteries used in electric vehicles[J]. Engineering Failure Analysis，2016，70:323-342.
[125] YAN B，WANG H，MA X. Correlation-driven multivariate degradation modeling and RUL prediction based on Wiener process model[J]. Quality and Reliability Engineering International，2022，10:1-27.
[126] ZHUANG X，YU T，SARAYGORD A S，et al. Remaining useful life prediction of a mechanism considering wear correlation of multiple joints[J]. Mechanical Systems and Signal Processing，2021，149:107328.
[127] CHEN S，WANG M，HUANG D，et al. Remaining useful life prediction for complex systems with multiple indicators based on particle filter and parameter correlation[J]. IEEE Access，2020，8:215145-215156.
[128] WANG X，BALAKRISHNAN N，GUO B. Residual life estimation based on nonlinear-multivariate Wiener processes[J]. Journal of Statistical Computation and Simulation，2015，85(9):1742-1764.
[129] WANG X，GUO B，CHENG Z. Residual life estimation based on bivariate Wiener degradation process with time-scale transformations[J]. Journal of Statistical Computation and Simulation，2014，84(3):545-563.
[130] LEE S，MOTAI Y，CHOI H. Tracking human motion with multichannel interacting multiple model[J]. IEEE Transactions on Industrial Informatics，2013，9(3):1751-1763.
[131] ZHANG Z，SI X，HU C，et al. Degradation data analysis and remaining useful life estimation:A review on Wiener-process-based methods[J]. European Journal of Operational Research，2018，271(3):775-796.
[132] DE JONGE B，SCARF P A. A review on maintenance optimization[J]. European Journal of Operational Research，2020，285(3):805-824.
[133] ALI A，ABDELHADI A. Condition-based monitoring and maintenance:State of the art review[J]. Applied Sciences，2022，12(2):688.
[134] RAO X，SHENG C，GUO Z，et al. A review of online condition monitoring and maintenance strategy for cylinder liner-piston rings of diesel engines[J]. Mechanical Systems and Signal Processing，2022，165:108385.
[135] DO P，ASSAF R，SCARF P，et al. Modelling and application of condition-based maintenance for a two-component system with stochastic and economic dependencies[J]. Reliability Engineering & System Safety，2019，182:86-97.
[136] XU J，LIANG Z，LI Y，et al. Generalized condition-based maintenance optimization for multi-component systems considering stochastic dependency and imperfect maintenance[J]. Reliability Engineering & System Safety，2021，211:107592.
[137] van HORENBEEK A，PINTELON L. A dynamic predictive maintenance policy for complex multi-component systems[J]. Reliability Engineering & System Safety，2013，120:39-50.
[138] SUN J，SUN Z，CHEN C，et al. Group maintenance strategy of CNC machine tools considering three kinds of maintenance dependence and its optimization[J]. The International Journal of Advanced Manufacturing Technology，2023，124(11):3749-3760.
[139] DONG W，LIU S，DU Y. Optimal periodic maintenance policies for a parallel redundant system with component dependencies[J]. Computers & Industrial Engineering，2019，138:106133.
[140] MAAROUFI G，CHELBI A，REZG N. A selective maintenance policy for multi-component systems with stochastic and economic dependence[C]//9th International Conference on Modeling，Optimization & Simulation. Jun 06-08，2012，Bordeaux，France，2012:hal-00728651.
[141] DAO C D，ZUO M J. Selective maintenance for multistate series systems with S-dependent components[J]. IEEE Transactions on Reliability，2016，65(2):525-539.
[142] SHAHRAKI A F，YADAV O P，VOGIATZIS C. Selective maintenance optimization for multi-state systems considering stochastically dependent components and stochastic imperfect maintenance actions[J]. Reliability Engineering & System Safety，2020，196:106738.
[143] Van HORENBEEK A，BURÉ J，CATTRYSSE D，et al. Joint maintenance and inventory optimization systems:A review[J]. International Journal of Production Economics，2013，143(2):499-508.
[144] FREDERICK D K，DECASTRO J A，LITT J S. User’s guide for the commercial modular aero-propulsion system simulation (C-MAPSS)[R]. NASA/TM2007- 215026，2007.
[145] KONG X，YANG J，LI L. Reliability analysis for multi-component systems considering stochastic dependency based on factor analysis[J]. Mechanical Systems and Signal Processing，2022，169:108754.