非正态过程能力指数构建与评价研究
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摘要
统计过程控制技术目前是生产过程中控制稳定产出的主要工具,在生产型企业中应用的非常广泛。传统的统计过程控制理论是基于正态分布这个基本假设的,而在实际生产过程中大量非正态过程传统SPC技术会失效。目前非正态过程能力指数的构建与评价研究是过程能力研究中的热点问题。
本文首先介绍了过程能力指数的基础理论和知识以及非正态过程能力研究的一些问题,并描述了非正态过程处理的实施步骤。接着详细探讨非正态过程能力研究的重要转换方法——Johnson转换和Box-Cox转换方法,并对这两种方法进行了比较分析。结合案例应用对Johnson转换的最佳转换参数进行了研究,还对过程能力指数的估计效果进行评价,即拟合有效性检验问题。这些方法可以用于指导生产实践。另外,本文还介绍了其他的转换方法和非正态过程能力的两个新指数,以及非转换方法——修正的加权方差方法估算非正态过程能力指数。以上这些方法都是为了更好的处理现实生产过程中的非正态现象,对于实践具有重要意义。
本文以非正态过程能力分析理论为主线,运用统计学原理、数学推导、比较分析和实证研究方法对非正态过程能力指数进行了详细的分析探讨,对不同的非正态过程能力评价方法进行理论研究和实证分析,构建了系统的非正态过程能力指数评价体系。
Statistical process control (SPC) is one of the main tools for controlling processes and widely used in the manufacturing industry. Traditional SPC theory is based on a fundamental assumption that the process data are normal. However, in many conditions the process data are non-normal which makes traditional SPC theory invalid. Therefore, the study on construction and appraisal of process capability index for non-normal processes is a hot issue to be solved.
This paper firstly introduces the basic theory and knowledge of process capability index and some issues in the study of non-normal process capability and describes how to handle non-normal processes. Then it discusses some important transformation methods for non-normal process capability, which are Johnson transformation and Box-Cox transformation methods, and does some competitions with these two methods. Study on the best transformation parameter has been done by analyzing some cases. All of these methods can be used directly in practice. This paper also introduces some other transformation methods, two new process capability indexes for non-normal processes and non-transformation methods——corrected weighted variance method. All these methods can contribute to the practice and do well to the non-normal processes.
The paper is based on the main line of process capability analysis methods for non-normal processes. Statistics principle, mathematics derivation, comparative analysis and case study are employed to carry out the thorough examination to the theory of process capability analysis and control methods for non-normal processes. All of these construct a systematic evaluation for capability index for non-normal processes.
本文首先介绍了过程能力指数的基础理论和知识以及非正态过程能力研究的一些问题,并描述了非正态过程处理的实施步骤。接着详细探讨非正态过程能力研究的重要转换方法——Johnson转换和Box-Cox转换方法,并对这两种方法进行了比较分析。结合案例应用对Johnson转换的最佳转换参数进行了研究,还对过程能力指数的估计效果进行评价,即拟合有效性检验问题。这些方法可以用于指导生产实践。另外,本文还介绍了其他的转换方法和非正态过程能力的两个新指数,以及非转换方法——修正的加权方差方法估算非正态过程能力指数。以上这些方法都是为了更好的处理现实生产过程中的非正态现象,对于实践具有重要意义。
本文以非正态过程能力分析理论为主线,运用统计学原理、数学推导、比较分析和实证研究方法对非正态过程能力指数进行了详细的分析探讨,对不同的非正态过程能力评价方法进行理论研究和实证分析,构建了系统的非正态过程能力指数评价体系。
Statistical process control (SPC) is one of the main tools for controlling processes and widely used in the manufacturing industry. Traditional SPC theory is based on a fundamental assumption that the process data are normal. However, in many conditions the process data are non-normal which makes traditional SPC theory invalid. Therefore, the study on construction and appraisal of process capability index for non-normal processes is a hot issue to be solved.
This paper firstly introduces the basic theory and knowledge of process capability index and some issues in the study of non-normal process capability and describes how to handle non-normal processes. Then it discusses some important transformation methods for non-normal process capability, which are Johnson transformation and Box-Cox transformation methods, and does some competitions with these two methods. Study on the best transformation parameter has been done by analyzing some cases. All of these methods can be used directly in practice. This paper also introduces some other transformation methods, two new process capability indexes for non-normal processes and non-transformation methods——corrected weighted variance method. All these methods can contribute to the practice and do well to the non-normal processes.
The paper is based on the main line of process capability analysis methods for non-normal processes. Statistics principle, mathematics derivation, comparative analysis and case study are employed to carry out the thorough examination to the theory of process capability analysis and control methods for non-normal processes. All of these construct a systematic evaluation for capability index for non-normal processes.
引文
[1] Juarn J.M., Gryna F.M. and Bingham R.S., Jr. Quality Control Handbook [M], McGraw-Hill, New York, 1974
[2] Kane V.E., Process Capability Indices [J], Journal of Quality Technology, 1986, 18(1): 41-52
[3] Hsiang T.C. and Taguchi G.A., Tutorial on quality control and assurance– The Taguchi methods, ASA Annual Meeting [C], Las Vegas, Nevada, 1985
[4] Pearn W.L., Kotz S and Johnson N.L., Distributional and inferential properties of process capability indices[J], Journal of Quality Technology, 1992, 4(4):216-231
[5] Gunter B.H., The use and abuse of Cpk[J], Quality Progress,1989, 22(3):108-109
[6] S.E. Somerville and D.C. Montgomery, Process capability indices and non-normal distribution [J], Quality Engineering, 1996,9: 305-316
[7] Box G.E.P. and Cox D.R., An analysis of transformations [J], Journal of the Royal Statistical Society, Series B, 1964, 26(2): 221-252
[8] Clements J.A., Process capability calculations for non-normal distributions [J], Quality Progress, 1989, 22(2): 95-100
[9] Kotz S. and Johnson N., Process Capability Indices - A Review [J], 1992-2000, Journal of Quality Technology, 2002, 34(1): 2-19
[10] Sundaraiyer V.H., Estimation of a process capability index for inverse Gaussian distributions [J], Communications in Statistics: Theory and Methods, 1996, 25(8): 2381-2398
[11] Pearn W.L., Kotz S., Application of Clements’method for calculating second and third generation process capability indices for non-normal Pearsonian populations [J], Quality Engineering, 1994,7(1): 139-145
[12] Clements J.A., Process capability calculations for non-normal distributions [J], Quality Progress, 1989, 22(2): 95-100
[13] Bittanti S, Lovera M, Moiraghi L, Application of non-normal process capability indices to semiconductor quality control [J], IEEE Transactions on Semiconductor, 1998, 11(2): 296-303
[14] Wright P.A., A process capability index sensitive to skewness [J], Journal of Statistical Computation and Simulation, 1995, 52(2): 195-203
[15] Chen H.F. and Kotz S., An asymptotic distribution of wright’s process capability index sensitive to skewness [J], Journal of Statistical Computation and Simulation, 1996, 55: 147-158
[16] Bai D.S. and Choi I.S., Process capability indices for skewed population [D], Master Thesis, Department of Industrial Engineering, Advanced Institute of Science and Technology, Taejon, South Korea, 1997
[17] Wu H.H., A weighted variance capability index for general non-normal processes [J], Quality and Reliability Engineering International, 1999, 15: 397-402
[18] Ding J.M., A Method of Estimating the Process Capability Index from the First Four Moments of Non-normal Data. Quality and Reliability Engineering International, 2004, 20: 787-805
[19] Johnson N.L. Systems of frequency curves generated by methods of translation [J], Biomertrika, 1949, 36: 149-176
[20] Farnum N.R., Using Johnson curves to describe non-normal process data [J], Quality Engineering, 1996, 9(2): 329-336
[21] Chou Y.M., Transforming non-normal data to normality in statistical process control [J], Journal of Quality Technology, 1998, 30(2): 133-141
[22] Polansky A.M., Chou Y.M. and Mason R.L., An algorithm for fitting transformations to non-normal data [J], Journal of Quality Technology, 1999, 31: 345-350
[23] K.S. Krishnamoorthi, Suraj Khatwani, A capability index for all occasions [A], Annual Quality Congress Proceedings [C], Milwaukee: American Society for Quality, 2000, 77-81
[24] Polansky A.M., A smooth nonparametric approach to process capability [J], Quality and Reliability Engineering International, 1998, 14: 43-48
[25] Polansky A.M., An algorithm for computing non-parametric capability estimate [J], Journal of Quality Technology, 2000, 32: 284-289
[26] Tang L.C., Computing process capability indices for non-normal data: a review and comparative study [J], Quality and Reliability Engineering International, 1999, 15: 339-353
[27]何桢,齐二石,张生虎,工序能力分析与评价中的几个问题[J],工业工程,2000,3(2):25-27
[28]郭正光,张国权,服义,关于非正态总体的工序能力指数Cp值计算的研究[J],华南农业大学学报(自然科学版),2004,25(1):110-111
[29]田志友,田澎,田浣尘,非正态过程能力指数研究中的几个问题[J],工业工程,2005,8(1):29-33
[30]周群艳,基于Johnson转换体系的非正态工序能力指数估计[J],系统工程,2004,22(5):98-102
[31]汤淑明,王飞跃,过程能力指数综述[J],应用概率统计,2004,20(2):207-216
[32]郑小林,郑希俊,余中华,基于约翰逊曲线拟合的非正态工序能力指数估算方法[J],机械科学与技术,2002,21(6):878-880
[33]张维铭,施雪忠,楼龙翔,非正态数据转换为正态数据的方法[J],浙江工程学院的学报,2000,17(3):204-207
[34]卓德保,最佳拟合非正态过程的质量控制[J],系统工程理论方法应用,2004,13(4):372-376
[35]卓德保,偏态过程的质量控制方法及其应用[J],世界标准化与质量管理,1999,2:9-12
[36]卓德保,刘晓芬,用约翰逊曲线拟合非正态过程数据的质量控制[J],系统工程理论与实践,1999,11:97-101
[37]卓德保,非正态分布条件下工序能力的度量[J],世界标准化与质量管理,1997,9(8):14-17
[38]杨剑锋,徐济超,王海宁,基于SWV方法的偏态过程能力分析[J],系统工程,2005,23(12):85-90
[39]张根保,何桢,刘英,质量管理与可靠性[M],北京:中国科学技术出版社,2005
[40]赵新华,工序能力指数在工序质量控制中的应用研究:[硕士学位论文],南京理工大学,1999
[41] Gunter B.H., The use and abuse of [J], Quality Progress, 1989, 22(3): 108-109 Cpk
[42] Kahneman D., P. Slovic, and A. Tversky(1982), Judgement under uncertainty: Heuristics and biases, Cambridge University Press
[43] Doubilet P., C.B. Begg, M.C. Weinstein, P. Braun and B.J. McNeil(1985),“Probabilistic sensitivity analysis using Monte Carlo simulation, a practical approach,”Medical Decision Making 5: 157-177
[44] Swain J.J., S. Venkatraman and J.R. Wilson(1998), Least squares estimation of distribution functions in Johnson’s translation system, Journal of Statistical Computation and Simulation 29: 271-279
[45] DeBrota C., S. D. Roberts, R.S. Dittus, and J.R. Wilson(1988), Visual interactive fitting of probability distributions, Simulation 52: 199-205
[46] Johnson N.L., System of frequency curves generated by methods of translation [J], Biometrika, 1949, 36: 149-176
[47]李选举,Box-Cox变换及其在MathCAD上的实现,数量经济技术经济研究[J],2000.4
[48] Johnson, N,L., Kotz, S. and Pearn, W.L. Flexible process capability indices[J]. Pakistan Journal of Statistics, 1994,10:23-31
[49] Hamaker H.C., Relative merits of using maximum errors versus 3σin describing the performance of laser-exposure retile writing systems[C], Optical/Laser Microlithography VIII (Proceedings of SPIE, vol.2440), Brumer TA(eds.). SPIE: Bellingham, WA, 1995, 550-559
[2] Kane V.E., Process Capability Indices [J], Journal of Quality Technology, 1986, 18(1): 41-52
[3] Hsiang T.C. and Taguchi G.A., Tutorial on quality control and assurance– The Taguchi methods, ASA Annual Meeting [C], Las Vegas, Nevada, 1985
[4] Pearn W.L., Kotz S and Johnson N.L., Distributional and inferential properties of process capability indices[J], Journal of Quality Technology, 1992, 4(4):216-231
[5] Gunter B.H., The use and abuse of Cpk[J], Quality Progress,1989, 22(3):108-109
[6] S.E. Somerville and D.C. Montgomery, Process capability indices and non-normal distribution [J], Quality Engineering, 1996,9: 305-316
[7] Box G.E.P. and Cox D.R., An analysis of transformations [J], Journal of the Royal Statistical Society, Series B, 1964, 26(2): 221-252
[8] Clements J.A., Process capability calculations for non-normal distributions [J], Quality Progress, 1989, 22(2): 95-100
[9] Kotz S. and Johnson N., Process Capability Indices - A Review [J], 1992-2000, Journal of Quality Technology, 2002, 34(1): 2-19
[10] Sundaraiyer V.H., Estimation of a process capability index for inverse Gaussian distributions [J], Communications in Statistics: Theory and Methods, 1996, 25(8): 2381-2398
[11] Pearn W.L., Kotz S., Application of Clements’method for calculating second and third generation process capability indices for non-normal Pearsonian populations [J], Quality Engineering, 1994,7(1): 139-145
[12] Clements J.A., Process capability calculations for non-normal distributions [J], Quality Progress, 1989, 22(2): 95-100
[13] Bittanti S, Lovera M, Moiraghi L, Application of non-normal process capability indices to semiconductor quality control [J], IEEE Transactions on Semiconductor, 1998, 11(2): 296-303
[14] Wright P.A., A process capability index sensitive to skewness [J], Journal of Statistical Computation and Simulation, 1995, 52(2): 195-203
[15] Chen H.F. and Kotz S., An asymptotic distribution of wright’s process capability index sensitive to skewness [J], Journal of Statistical Computation and Simulation, 1996, 55: 147-158
[16] Bai D.S. and Choi I.S., Process capability indices for skewed population [D], Master Thesis, Department of Industrial Engineering, Advanced Institute of Science and Technology, Taejon, South Korea, 1997
[17] Wu H.H., A weighted variance capability index for general non-normal processes [J], Quality and Reliability Engineering International, 1999, 15: 397-402
[18] Ding J.M., A Method of Estimating the Process Capability Index from the First Four Moments of Non-normal Data. Quality and Reliability Engineering International, 2004, 20: 787-805
[19] Johnson N.L. Systems of frequency curves generated by methods of translation [J], Biomertrika, 1949, 36: 149-176
[20] Farnum N.R., Using Johnson curves to describe non-normal process data [J], Quality Engineering, 1996, 9(2): 329-336
[21] Chou Y.M., Transforming non-normal data to normality in statistical process control [J], Journal of Quality Technology, 1998, 30(2): 133-141
[22] Polansky A.M., Chou Y.M. and Mason R.L., An algorithm for fitting transformations to non-normal data [J], Journal of Quality Technology, 1999, 31: 345-350
[23] K.S. Krishnamoorthi, Suraj Khatwani, A capability index for all occasions [A], Annual Quality Congress Proceedings [C], Milwaukee: American Society for Quality, 2000, 77-81
[24] Polansky A.M., A smooth nonparametric approach to process capability [J], Quality and Reliability Engineering International, 1998, 14: 43-48
[25] Polansky A.M., An algorithm for computing non-parametric capability estimate [J], Journal of Quality Technology, 2000, 32: 284-289
[26] Tang L.C., Computing process capability indices for non-normal data: a review and comparative study [J], Quality and Reliability Engineering International, 1999, 15: 339-353
[27]何桢,齐二石,张生虎,工序能力分析与评价中的几个问题[J],工业工程,2000,3(2):25-27
[28]郭正光,张国权,服义,关于非正态总体的工序能力指数Cp值计算的研究[J],华南农业大学学报(自然科学版),2004,25(1):110-111
[29]田志友,田澎,田浣尘,非正态过程能力指数研究中的几个问题[J],工业工程,2005,8(1):29-33
[30]周群艳,基于Johnson转换体系的非正态工序能力指数估计[J],系统工程,2004,22(5):98-102
[31]汤淑明,王飞跃,过程能力指数综述[J],应用概率统计,2004,20(2):207-216
[32]郑小林,郑希俊,余中华,基于约翰逊曲线拟合的非正态工序能力指数估算方法[J],机械科学与技术,2002,21(6):878-880
[33]张维铭,施雪忠,楼龙翔,非正态数据转换为正态数据的方法[J],浙江工程学院的学报,2000,17(3):204-207
[34]卓德保,最佳拟合非正态过程的质量控制[J],系统工程理论方法应用,2004,13(4):372-376
[35]卓德保,偏态过程的质量控制方法及其应用[J],世界标准化与质量管理,1999,2:9-12
[36]卓德保,刘晓芬,用约翰逊曲线拟合非正态过程数据的质量控制[J],系统工程理论与实践,1999,11:97-101
[37]卓德保,非正态分布条件下工序能力的度量[J],世界标准化与质量管理,1997,9(8):14-17
[38]杨剑锋,徐济超,王海宁,基于SWV方法的偏态过程能力分析[J],系统工程,2005,23(12):85-90
[39]张根保,何桢,刘英,质量管理与可靠性[M],北京:中国科学技术出版社,2005
[40]赵新华,工序能力指数在工序质量控制中的应用研究:[硕士学位论文],南京理工大学,1999
[41] Gunter B.H., The use and abuse of [J], Quality Progress, 1989, 22(3): 108-109 Cpk
[42] Kahneman D., P. Slovic, and A. Tversky(1982), Judgement under uncertainty: Heuristics and biases, Cambridge University Press
[43] Doubilet P., C.B. Begg, M.C. Weinstein, P. Braun and B.J. McNeil(1985),“Probabilistic sensitivity analysis using Monte Carlo simulation, a practical approach,”Medical Decision Making 5: 157-177
[44] Swain J.J., S. Venkatraman and J.R. Wilson(1998), Least squares estimation of distribution functions in Johnson’s translation system, Journal of Statistical Computation and Simulation 29: 271-279
[45] DeBrota C., S. D. Roberts, R.S. Dittus, and J.R. Wilson(1988), Visual interactive fitting of probability distributions, Simulation 52: 199-205
[46] Johnson N.L., System of frequency curves generated by methods of translation [J], Biometrika, 1949, 36: 149-176
[47]李选举,Box-Cox变换及其在MathCAD上的实现,数量经济技术经济研究[J],2000.4
[48] Johnson, N,L., Kotz, S. and Pearn, W.L. Flexible process capability indices[J]. Pakistan Journal of Statistics, 1994,10:23-31
[49] Hamaker H.C., Relative merits of using maximum errors versus 3σin describing the performance of laser-exposure retile writing systems[C], Optical/Laser Microlithography VIII (Proceedings of SPIE, vol.2440), Brumer TA(eds.). SPIE: Bellingham, WA, 1995, 550-559