Importance measures are effective tools to identify the weak components of multi-state systems from a reliability perspective. Nevertheless, the most reported works on importance measures were based on the premise that the deteriorating processes of systems and its components can be precisely known. In many engineering practices, due to limited data, imprecise information, and unknown failure/degradation mechanisms, it is inevitable that the estimated parameters may contain the epistemic uncertainty. By taking account of the epistemic uncertainty associated with the parameters of multi-state system degradation models, a new Birnbaum importance measure of multi-state systems under epistemic uncertainty is put forth. In the proposed method, the Dempster-Shafer evidence theory is used to quantify the epistemic uncertainty, while the Markov model is utilized to characterize the degradation process of multi-state systems. An illustrative example of a cutter feeding control system of machining tools is presented to demonstrate the impact of the epistemic uncertainty on the importance analysis and ranking of components.
[1] RAMIREZ-MARQUEZ J E, COIT D W. Composite importance measures for multi-state systems with multi-state components[J]. IEEE Transactions on Reliability, 2005, 54(3):517-529.
[2] ZIO E, PODOFILLINI L. Monte-Carlo simulation analysis of the effects on different system performance levels on the importance on multi-state components[J]. Reliability Engineering and System Safety, 2003, 82(1):63-73.
[3] 姚成玉, 吕军, 陈东宁, 等. 凸模型T-S故障树及重要度分析方法[J]. 机械工程学报, 2015, 51(24):184-192. YAO Chengyu, LÜ Jun, CHEN Dongning, et al. Convex model T-S fault tree and importance analysis methods[J]. Journal of Mechanical Engineering, 2015, 51(24):184-192.
[4] 韩烨. 基于FTA的数控冲床重要度分析[D]. 吉林:吉林大学, 2008. HAN Ye. FTA-based importance analysis of numerical control puncher[D]. Jilin:Jilin University, 2008.
[5] BIRNBAUM Z W. On the importance of different component in a multi-component system[M]. New York:Academic Press, 1969.
[6] BARLOW R E, PROSCHAN F. Importance of system components and failure tree events[J]. Stochastic Processes and Their Applications, 1975, 3(2):153-173.
[7] LAMBERT H E. Fault trees for decision making in systems analysis[D]. Livemore:University of California, 1975.
[8] 孟书, 申桂香, 陈炳焜, 等. 基于灰关联的加工中心可用性需求重要度研究[J]. 机械工程学报, 2016, 52(24):187-193. MENG Shu, SHEN Guixiang, CHEN Bingkun, et al. Analysis on requirement importance rating for availability of machining center based on grey relation method[J]. Journal of Mechanical Engineering, 2016, 52(24):187-193.
[9] NATVIG B. Multistate systems reliability theory with applications[M]. Norway:Wiley, 2011.
[10] 米金华. 认知不确定性下复杂系统的可靠性分析与评估[D]. 成都:电子科技大学, 2016. MI Jinhua. Reliability analysis and assessment of complex system under epistemic uncertainty[D]. Chengdu:University of Electronic Science and Technology of China, 2016.
[11] 刘宇. 多状态复杂系统可靠性建模及维修决策[D]. 成都:电子科技大学, 2010. LIU Yu. Multi-state complex system reliability modeling and maintenance decision[D]. Chengdu:University of Electronic Science and Technology of China, 2010.
[12] 钱文学, 尹晓伟, 谢里阳. 基于贝叶斯网络的多状态系统可靠性建模与评估[J]. 机械工程学报, 2009, 45(2):206-212. QIAN Wenxue, YIN Xiaowei, XIE Liyang. Multi-state system reliability modeling and assessment based on Bayesian networks[J]. Journal of Mechanical Engineering, 2009, 45(2):206-212.
[13] 段建国, 李爱平, 谢楠, 等. 可重构制造系统多状态可靠性建模与分析[J]. 机械工程学报, 2011, 47(17):104-111. DUAN Jianguo, LI Aiping, XIE Nan, et al. Multi-state reliability modeling and analysis of reconfigurable manufacturing systems[J]. Journal of Mechanical Engineering, 2011, 47(17):104-111.
[14] LIU Y, LIN P, LI Y F, et al. Bayesian reliability and performance assessment for multi-state systems[J]. IEEE Transactions on Reliability, 2015, 64(1):394-409.
[15] GRIFFITH W S. Multi-state reliability models[J]. Journal of Applied Probability, 1980, 17(3):735-744.
[16] LEVITIN G, LISNIANSKI A. Importance and sensitivity analysis of multi-state systems using universal generating function[J]. Reliability Engineering and System Safety, 1999, 65(3):271-282.
[17] SI S, DUI H, ZHANG S G, et al. The integrated importance measure of multi-state coherent systems for maintenance processes[J]. IEEE Transactions on Reliability, 2012, 61(2):266-273.
[18] SI S, DUI H, SUN S D. Component importance for multi-state system lifetimes with renewal functions[J]. IEEE Transactions on Reliability, 2014, 63(1):105-117.
[19] 姜潮, 张旺, 韩旭. 基于Copula函数的证据理论相关性分析模型及结构可靠性计算方法[J]. 机械工程学报, 2017, 53(16):199-209. JIANG Chao, ZHANG Wang, HAN Xu. A copula function based evidence theory model for correlation analysis and corresponding structural reliability method[J]. Journal of Mechanical Engineering, 2017, 53(16):199-209.
[20] MULA J, POLER R, GARCIA-SABATER J P. Material requirement planning with fuzzy constraints and fuzzy coefficients[J]. Fuzzy Sets and Systems, 2007, 158(7):783-793.
[21] SANKARARAMAN S, MAHADEVAN S. Likelihoodbased representation of epistemic uncertainty due to sparse point data and/or interval data[J]. Reliability Engineering and System Safety, 2011, 96(7):814-824.
[22] TONON F. Using random set theory to propagate epistemic uncertainty through a mechanical system[J]. Reliability Engineering and System Safety, 2004, 85(1):249-266.
[23] FERSON S, KREINOVICH V, GINZBURG L, et al. Constructing probability boxes and Dempster-Shafer structures[R]. Albuquerque, NM:Sandia National Laboratories, 2003.
[24] DESTERCKE S, SALLAK M. An extension of universal generating function in multi-state systems considering epistemic uncertainties[J]. IEEE Transactions on Reliability, 2013, 62(2):504-514.
[25] DENOEUX T. Reasoning with imprecise belief structures[J]. International Journal of Approximate Reasoning, 1999, 20(1):79-111.
[26] ICHIHASHI H, TANAKA H. Jeffrey-like rules of conditioning for the Dempster-Shafer theory of evidence[J]. International Journal of Approximate Reasoning, 1989, 3(2):143-156.
[27] XU H, SMETS P. Reasoning in evidential networks with conditional belief functions[J]. International Journal of Approximate Reasoning, 1996, 14(2):155-185.
[28] SMETS P. Belief functions:The disjunctive rule of combination and the generalized Bayesian theorem[J]. International Journal of Approximate Reasoning, 1993, 9(1):1-35.
[29] JIANG C, HAN X, LIU G R, et al. A nonlinear interval number programming method for uncertain optimization problems[J]. European Journal of Operational Research, 2008, 188(1):1-13.
[30] 李新兰, 姜潮, 韩旭. 基于区间的不确定多目标优化方法及应用[J]. 中国机械工程, 2011, 22(9):1100-1106. LI Xinlan, JIANG Chao, HAN Xu. An uncertainty multi-objective optimization based on interval analysis and its application[J]. China Mechanical Engineering, 2011, 22(9):1100-1106.
