Process capability estimation for non-normal quality charactersitics using Clement, Burr and Box--Cox methods

Abstract

In today's competitive business environment, it is becoming more crucial than ever to assess precisely process losses due to non-compliance to customer specifications. To assess these losses, industry is widely using process capability indices for performance evaluation of their processes. Determination of the performance capability of a stable process using the standard process capability indices requires that the underlying process data should follow a normal distribution. However, if the data is non-normal, measuring process capability using conventional methods can lead to erroneous results. Different process capability indices such as Clements percentile method and data transformation method have been proposed to deal with the non-normal situation. Although these methods are practiced in industry, there is insufficient literature to assess the accuracy of these methods under mild and severe departures from normality. This article reviews the performances of the Clements non--normal percentile method, the Burr based percentile method and Box--Cox method for non-normal cases. A simulation study using Weibull, Gamma and lognormal distributions is conducted. Burr's method calculates process capability indices for each set of simulated data. These results are then compared with the capability indices obtained using Clements and Box--Cox methods. Finally, a case study based on real world data is presented. References I. W. Burr (1942). Cumulative frequency distribution. Ann Math Stat 13: 215--232. http://www.ams.org/mathscinet/pdf/6644.pdf I. W. Burr (1973). Parameters for a general system of distributions to match a grid of $\alpha _3$ and $\alpha _4$. Commun Stat 2:1--21. http://www.zentralblatt-math.org/zmath/search/?q=an:03413665&type=pdf&format=complete G. E. P. Box and D. R. Cox (1964). An analysis of transformation. J Roy Stat Soc B 26:211--252. http://www.jstor.org/view/00359246/di993152/99p02493/0 J. A. Clements (1989). Process capability calculations for non-normal distributions. Quality Progress 22:95--100. http://www.asq.org/qic/display-item/index.html?item=14059 N. L. Johnson (1949). System of frequency curves generated by methods of translation. Biometrika 36:149--176. http://www.jstor.org/view/00063444 /di 992300/99 p 0220l/0 S. Kotz and C. R. Lovelace (1998). Process capability indices in theory and practice. Arnold, London. http://www.amazon.com/Process-Capability-Indices-Theory-Practice/dp/0340691778 S. Kotz and N. L. Johnson (1993). Process capability indices. New York: Chapman and Hall. http://www.amazon.com/Process-Capability-Indices-Samuel-Kotz/dp/041254380X L. A. R. Rivera, N. F. Hubele and F. D. Lawrence (1995). $C_{pk}$ index estimation using data transformation. Comput Ind Engng 29: 55-58. http://www.ingentaconnect.com /content/els/03608352/1995/00000029/00000001/art00045 L. C. Tang, S. E. Than (1999). Computing process capability indices for non-normal data : a review and comparative study. Qual Reliab Engng Int 15: 339--353. doi:CCC 0748-8017/99/050339 D. Montgomery (1996). Introduction to Statistical Quality Control. 5th edition. Wiley, New York. http://bcs.wiley.com/he-bcs/Books?action=index&bcsId=2077&itemId=0471656313 J. O'Connell and Q. Shao (2004). Further investigation on a new approach in analyzing extreme events. CSIRO Mathematical and Information Sciences Report No. 04/41. http://www.cmis.csiro.au/techreports/docs/x0000ihw.pdf Pei--Hsi Liu and Fei--Long Chen (2006). Process capability analysis of non-normal process data using the Burr XII distribution. Int J Adv Manuf Technol 27: 975--984. doi:10.1007/s 00170-004-2263-8 S. Somerville and D. Montgomery (1996). Process capability indices and non-normal distributions. Quality Engineering 19(2):305--316. doi:10.1080/08982119608919047 F. K. Wang (2006). Quality evaluation of a manufactured product with multiple characteristics. Qual Relib Engng Int 22: 225--236. http://doi.wiley.com/10.1002/qre.712 H. H. Wu, J. S. Wang and T. L. Liu (1998). Discussions of the Clements-based process capability indices. In: Proceedings of the 1998 CIIE National Conference pp.561--566. doi:10.1007/s 00170-004-2263-8