- Published:
Bayesian reliability modeling and assessment solution for NC machine tools under small-sample data
Chinese Journal of Mechanical Engineering volume 28, pages 1229–1239 (2015)
Abstract
Although Markov chain Monte Carlo(MCMC) algorithms are accurate, many factors may cause instability when they are utilized in reliability analysis; such instability makes these algorithms unsuitable for widespread engineering applications. Thus, a reliability modeling and assessment solution aimed at small-sample data of numerical control(NC) machine tools is proposed on the basis of Bayes theories. An expert-judgment process of fusing multi-source prior information is developed to obtain the Weibull parameters’ prior distributions and reduce the subjective bias of usual expert-judgment methods. The grid approximation method is applied to two-parameter Weibull distribution to derive the formulas for the parameters’ posterior distributions and solve the calculation difficulty of high-dimensional integration. The method is then applied to the real data of a type of NC machine tool to implement a reliability assessment and obtain the mean time between failures(MTBF). The relative error of the proposed method is 5.8020×10-4 compared with the MTBF obtained by the MCMC algorithm. This result indicates that the proposed method is as accurate as MCMC. The newly developed solution for reliability modeling and assessment of NC machine tools under small-sample data is easy, practical, and highly suitable for widespread application in the engineering field; in addition, the solution does not reduce accuracy.
References
YANG Zhaojun, CHEN Chuanhai, CHEN Fei, et al. Progress in the research of reliability technology of machine tools[J]. Journal of Mechanical Engineering, 2013, 49(20): 130–139. (in Chinese)
KELLER A Z, KAMATH A R R, PERERA U D. Reliability analysis of CNC machine tools[J]. Reliability Engineering, 1982, 3(6): 449–473.
JIA Yazhou, WANG Molin, JIA Zhixin. Probability distribution of machining center failures[J]. Reliability Engineering and System Safety, 1995, 50(1): 121–125.
YANG Zhaojun, CHEN Chuanhai, CHEN Fei, et al. Reliability analysis of machining center based on the field data[J]. Eksploatacja i Niezawodnosc, 2013, 15(2): 147–155.
YANG Jianguo, WANG Zhiming, WANG Guoqiang, et al. Likelihood ratio test interval estimation of reliability indices for numerical control machine tools[J]. Journal of Mechanical Engineering, 2012, 48(2): 9–15. (in Chinese)
ZHANG Genbao, TANG Xianjin, LIAO Xiaobo, et al. Quantitative modeling and application of CNC machine failure distribution curve[J]. Journal of Chongqing University, 2013, 36(6): 119–123. (in Chinese)
CHEN Diansheng, WANG Tianmiao, WEI Hongxing. Sectional model involving two Weibull distributions for CNC lathe failure probability[J]. Journal of Beijing University of Aeronautics and Astronautics, 2005, 31(7): 766–769. (in Chinese)
ZHANG L F, XIE M, TANG L C. Bias correction for the least squares estimator of Weibull shape parameter with complete and censored data[J]. Reliability Engineering and System Safety, 2006, 91(8): 930–939.
MONTANARI G C, MAZZANTI G, CACCIARI M, et al. In search of convenient techniques for reducing bias in the estimation of Weibull parameters for uncensored tests[J]. IEEE Transactions on Dielectrics and Electrical Insulation, 1997, 4(3): 306–313.
FULTON J W, ABERNETHY R B. Likelihood adjustment: A simple method for better forecasting from small samples[C]//Proceedings of IEEE Annual Reliability and Maintainability Symposium, Los Angeles, CA, USA, January 24–27, 2000: 150–154.
HAMADA M S, WILSON A G, REESE C S, et al. Bayesian Reliability[M]. New York: Springer, 2008.
GUIKEMA S D, PATÉ-CORNELL M E. Bayesian analysis of launch vehicle success rates[J]. Journal of Spacecraft and Rockets, 2004, 41(1): 93–102.
GUIKEMA S D, PATÉ-CORNELL M E. Probability of infancy problems for space launch vehicles[J]. Reliability Engineering and System Safety, 2005, 87(3): 303–314.
GUIKEMA S D. A comparison of reliability estimation methods for binary systems[J]. Reliability Engineering and System Safety, 2005, 87(3): 365–376.
ANDERSON–COOK C M, GRAVES T, HAMADA M, et al. Bayesian stockpile reliability methodology for complex systems[J]. Military Operations Research, 2007, 12(2): 25–37.
PENG Weiwen, HUANG Hongzhong, LI Yanfeng, et al. Bayesian information fusion method for reliability assessment of milling head[J]. Journal of Mechanical Engineering, 2014, 50(6): 185–191. (in Chinese)
MING Zhimao, ZHANG Yunan, TAO Junyong, et al. Bayesian reliability assessment and prediction during product development[J]. Journal of Mechanical Engineering, 2010, 46(4):150–156. (in Chinese)
QUIGLEY L, BEDFORD T, WALLS L. Prior distribution elicitation, encyclopedia of statistics in quality and reliability[M]. New York: John Wiley & Sons Ltd., 2008.
WALLS L, QUIGLEY J. Building prior distributions to support Bayesian reliability growth modelling using expert judgement[J]. Reliability Engineering and System Safety, 2001, 74(2): 117–128.
MEYER M A, BOOKER J M. Eliciting and analyzing expert judgment: a practical guide[M]. London: Academic Press, 1991.
KAMINSKIY M P, KRIVTSOV V V. A simple procedure for Bayesian estimation of the Weibull distribution[J]. IEEE Transactions on Reliability, 2005, 54(4): 612–616.
GARTHWAITE P H, KADANE J B, O'HAGAN A. Statistical methods for eliciting probability distributions[J]. Journal of the American Statistical Association, 2005, 100(470): 680–701.
ALBERT I, DONNET S, GUIHENNEUC-JOUYAUX C, et al. Combining expert opinions in prior elicitation[J]. Bayesian Analysis, 2012, 7(3): 503–532.
KADANE J B, DICKEY J M, WINKLER R L, et al. Interactive elicitation of opinion for a normal linear model[J]. Journal of the American Statistical Association, 1980, 75(372): 845–854.
LOW-CHOY S, MURRAY J, JAMES A, et al. Indirect elicitation from ecological experts: from methods and software to habitat modelling and rock-wallabies[J]. The Oxford Handbook of Applied Bayesian Analysis, 2010: 511–544.
SOLAND R M. Bayesian analysis of the Weibull process with unknown scale and shape parameters[J]. IEEE Transactions on Reliability, 1969, 18(4): 181–184.
METROPOLIS N, ROSENBLUTH A W, ROSENBLUTH M N, et al. Equation of state calculations by fast computing machines[J]. The journal of chemical physics, 1953, 21(6): 1087–1092.
HASTINGS W K. Monte Carlo sampling methods using Markov chains and their applications[J]. Biometrika, 1970, 57(1): 97–109.
GELFAND A E, SMITH A F M. Sampling-based approaches to calculating marginal densities[J]. Journal of the American Statistical Association, 1990, 85(410): 398–409.
NEAL R M. Slice sampling[J]. The Annals of Statistics, 2003, 31(3): 705–767.
GUPTA A, MUKHERJEE B, UPADHYAY S K. Weibull extension model: A Bayes study using Markov chain Monte Carlo simulation[J]. Reliability Engineering and System Safety, 2008, 93(10): 1434–1443.
SOLIMAN A A, ABD-ELLAH A H, ABOU-ELHEGGAG N A, et al. Modified Weibull model: A Bayes study using MCMC approach based on progressive censoring data[J]. Reliability Engineering and System Safety, 2012, 100(2): 48–57.
LUNN D, JACKSON C, BEST N. et al. The BUGS book: a practical introduction to Bayesian analysis[M]. Boca Raton: CRC Press, 2012.
KRUSCHKE J K. Doing Bayesian data analysis: A tutorial with R and BUGS[M]. Burlington: Academic Press, 2011.
O'HAGAN A, BUCK C E, DANESHKHAH A, et al. Uncertain judgements: eliciting experts’ probabilities[M]. Chichester: John Wiley & Sons, Ltd., 2006.
BERGER J O. Statistical decision theory and Bayesian analysis[M]. Berlin: Springer, 1985.
RINNE H. The Weibull distribution: a handbook[M]. Boca Raton: CRC Press, 2009.
YOUNG G A, SMITH R L. Essentials of statistical inference[M]. New York: Cambridge University Press, 2005.
ZHANG L F, XIE M, TANG L C, A study of two estimation approaches for parameters of Weibull distribution based on WPP[J]. Reliability Engineering and System Safety, 2007, 92(3): 360–368.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by Research on Reliability Assessment and Test Methods of Heavy Machine Tools, China(State Key Science & Technology Project High-grade NC Machine Tools and Basic Manufacturing Equipment, Grant No. 2014ZX04014-011), Reliability Modeling of Machining Centers Considering the Cutting Loads, China(Science & Technology Development Plan for Jilin Province, Grant No. 3D513S292414), and Graduate Innovation Fund of Jilin University, China(Grant No. 2014053)
YANG Zhaojun, born in 1956, is currently a professor at College of Mechanical Science and Engineering, Jilin University, China. He received his PhD degree from Jilin University, China, in 1995. His research interests include reliability techniques and theories of CNC equipment.
KAN Yingnan, born in 1985, is currently a PhD candidate at College of Mechanical Science and Engineering, Jilin University, China. He received his master’s degree from Jilin University, China, in 2009. His research interests include reliability modeling methods of CNC equipment and Bayesian reliability theory.
CHEN Fei, born in 1970, is currently an associate professor at College of Mechanical Science and Engineering, Jilin University, China. She received her PhD degree from Jilin University, China, in 2009. Her research interests include reliability techniques and theories of CNC equipment.
XU Binbin, born in 1982, is currently a lecturer at Jilin University, China. She received her PhD degree from Jilin University, China, in 2011. Her research interests include reliability techniques and theories of CNC equipment.
CHEN Chuanhai, born in 1983, is currently a lecturer at Jilin University, China. He received his PhD degree from Jilin University, China, in 2013. His research interests include reliability techniques and theories of CNC equipment.
YANG Chuangui, born in 1987, is currently a master candidate at College of Mechanical Science and Engineering, Jilin University, China. He received his bachelor’s degree from Jilin University, China, in 2012.
Rights and permissions
About this article
Cite this article
Yang, Z., Kan, Y., Chen, F. et al. Bayesian reliability modeling and assessment solution for NC machine tools under small-sample data. Chin. J. Mech. Eng. 28, 1229–1239 (2015). https://doi.org/10.3901/CJME.2015.0707.088
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3901/CJME.2015.0707.088