Process Capability Indices for Shape Parameter of Weibull Distribution

Suboohi Safdar; Ejaz Ahmed

doi:10.4236/ojs.2014.43020

Home > Journals > Physics & Mathematics > OJS

OJS> Vol.4 No.3, April 2014

Share This Article:

Open Access

Process Capability Indices for Shape Parameter of Weibull Distribution

Abstract Full-Text HTML XML

Download as PDF (Size:483KB) PP. 207-219

DOI: 10.4236/ojs.2014.43020 4,187 Downloads 6,333 Views Citations

Author(s) Leave a comment

Suboohi Safdar, Ejaz Ahmed

Affiliation(s)

College of Computer Science and Information Systems, Institute of Business Management, Karachi, Pakistan.
Department of Statistics, University of Karachi, Karachi, Pakistan.

ABSTRACT

Process capability analysis is used to determine the process performance as capable or incapable within a specified tolerance. Basic indices C_p, C_pk, C_pm, C_pmk initially developed for normally distributed processes showed inappropriate for processes with non-normal distributions. A number of authors worked on non-normal distributions which were most notably those of Clements, Pearn and Chen, Montgomery and Johnson-Kotz-Pearn (JKP). Obtaining PCIs based on the parameters of non-normal distributions are completely disregarded and ignored. However parameters of some non-normal distributions have significance for knowing the status of process as capable or incapable. In this article we intend to work on the shape parameter of Weibull distribution to calculate PCIs. We work on two data sets for verification and validation purpose. Efficacy of the technique is assessed by bootstrapping the results of estimate and standard error of shape parameter.

KEYWORDS

Regression Coefficient; Non-Normal Distribution; Delta Method; Bootstrapping

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Safdar, S. and Ahmed, E. (2014) Process Capability Indices for Shape Parameter of Weibull Distribution. Open Journal of Statistics, 4, 207-219. doi: 10.4236/ojs.2014.43020.

References

[1]	Juran, J.M. (1974) Juran’s Quality Control Handbook. 3rd Edition. McGraw-Hill, New York.

[2]	Kane, V.E. (1986) Process Capability Indices. Journal of Quality Technology, 18, 41-52.

[3]	Chan, L.K., Cheng, S.W. and Spiring, F.A. (1988) A New Measure of Process Capability: Cpm. Journal of Quality Technology, 20, 162-175.

[4]	Pearn, W.L., Kotz, S. and Johnson, N.L. (1992) Distributional and Inferential Properties of Process Capability Indices. Journal of Quality Technology, 24, 216 -231.

[5]	Kotz, S. and Johnson, N.L. (1993) Process Capability Indices. Chapman & Hall, Suffolk.

[6]	Vannman, K. (1995) A Unified Approach to Capability Indices. Statistica Sinica, 5, 805-820.

[7]	Gunter, B.H. (1989) The Use and Abuse of Cpk. Quality Progress, 22, 108-109.

[8]	Boyles, R.A. (1994) Process Capability with Asymmetric Tolerance. Communications in Statistics: Simulations and Computation, 23, 615-643. http://dx.doi.org/10.1080/03610919408813190

[9]	Zwick, D. (1995) A Hybrid Method for Fitting Distributions to Data and It Use in Computing Process Capability Indices. Quality Engineering, 7, 601-613. http://dx.doi.org/10.1080/08982119508918806

[10]	Johnson, N.L. (1949) System of Frequency Curves Generated by Translation. Biometrika, 36, 149-176.

[11]	Box, G.E.P. and Cox, D.R. (1964) An Analysis of Transformation. Journal of Royal Statistical Society, 26, 211-243.

[12]	Clements, J.A. (1989) Process Capability Calculations for Non-Normal Distributions. Quality Progress, 22, 95-100.

[13]	English, J.R. and Taylor, G.D. (1993) Process Capability Analysis—A Robustness Study. International Journal of Production Research, 31, 1621-1635. http://dx.doi.org/10.1080/00207549308956813

[14]	Pearn, W.L. and Kotz, S. (1994) Application of Clements’ Method for Calculating Second and Third Generation Process Capability Indices for Non-Normal Pearsonian Populations. Quality Engineering, 7, 139-145. http://dx.doi.org/10.1080/08982119408918772

[15]	Schneider, H. and Pruett, J. (1995) Uses of Process Capability Indices in the Supplier Certification Process. Quality Engineering, 9, 225-235. http://dx.doi.org/10.1080/08982119508904621

[16]	Castagliola, P. (1996) Evaluation of Non-Normal Process Capability Indices Using Burr’s Distributions. Quality Engineering, 8, 587-593. http://dx.doi.org/10.1080/08982119608904669

[17]	Somerville, S.E. and Montgomery, D.C. (1996-1997) Process Capability Indices and Non-Normal Distributions. Quality Engineering, 19, 305-316. http://dx.doi.org/10.1080/08982119608919047

[18]	Kotz, S. and Lovelace, C.R. (1998) Process Capability Indices in Theory and Practice. Arnold, London.

[19]	Tong, L.I. and Chen, J.P. (1998) Lower Confidence Limits of Process Capability Indices for Non-Normal Process Distribution. International Journal of Quality and Reliability Management, 15, 907-919. http://dx.doi.org/10.1108/02656719810199006

[20]	Bai, D.S., Chang, Y.S. and Choi, S.I. (2002) Process Capability Indices for Skewed Populations. Quality and Reliability Engineering International, 18, 383-393. http://dx.doi.org/10.1002/qre.489

[21]	Liu, P.H. and Chen, F.L. (2006) Process Capability Analysis of Non-Normal Process Data Using the Burr XII Distribution. International Journal of Advanced Manufacturing Technology, 27, 975-984. http://dx.doi.org/10.1007/s00170-004-2263-8

[22]	Chang, L., Chen, S.C. and Hsieh, C.Y. (2007) Capability Indices for Non-Normal Populations. Journal of Statistics and Management Systems, 10, 183-193. http://dx.doi.org/10.1080/09720510.2007.10701247

[23]	Aleksander, K. and Thierry, P.L. (2013) Modeling of Stress Gradient Effect of Fatigue Life Using Weibull Based Distribution Function. Journal of Theoretical and Applied Mechanics, 51, 297-311.

[24]	Yavuz, A.A. (2013) Estimation of the Shape Parameter of the Weibull Distribution Using Linear Regression Methods: Non-Censored Samples. Quality and Reliability Engineering, 29, 1207-1219. http://dx.doi.org/10.1002/qre.1472

[25]	Ahmed, E. and Safdar, S. (2010) Process Capability Analysis for Non-Normal Data. Pakistan Business Review, 234243.

[26]	Kottegoda, N.T. and Renzo, R. (1997) Statistics, Probability and Reliability for Civil and Environmental Engineers. McGraw Hill International Editions, New York, 216-217.

[27]	Kendall, M.G. and Stuart, A. (1977) The Advanced Theory of Statistics. Macmillan, New York, Vol. 1, 168.

[28]	R Core Team (2013) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna. http://www.R-project.org/

[29]	Pearn, W.L. and Chen, K.S. (1997) Capability Indices for Non-Normal Distributions with an Application in Electrolytic Capacitor Manufacturing. Microelectronics Reliability, 37, 1853-1858. http://dx.doi.org/10.1016/S0026-2714(97)00023-1

[30]	Chang, P.L. and Lu, K.H. (1994) PCI Calculations for Any Shape of Distribution with Percentile. Quality World, Technical Section, 20, 110-114.

[31]	Pearn, W.L. and Chen, K.S. (1995) Estimating Process Capability Indices for Non-Normal Pearsonian Populations. Quality and Reliability Engineering International, 11, 386-388. http://dx.doi.org/10.1002/qre.1472

[32]	Lieblein, J. and Zelen, M. (1956) Statistical Investigation of the Fatigue Life of Deep-Groove Ball Bearings. Journal of Research of the National Bureau of Standards, 57, 273-315. http://dx.doi.org/10.6028/jres.057.033

[33]	Kao, J. (1959) A Graphical Estimation of Mixed Weibull Parameters in Life-Testing of Electron Tubes. Technometrics, 1, 389-407. http://dx.doi.org/10.1080/00401706.1959.10489870

[34]	Milkolaj, G. (1972) Environmental Applications of the Weibull Distribution Function. Oil Pollution Science, 176, 1019-1021.

[35]	Piotrovskii, V.K. (1987) The Use of Weibull Distribution to Describe the in Vivo Absorption Kinetics. Journal of Pharmacokinetics and Pharmacodynamics, 15, 681-686. http://dx.doi.org/10.1007/BF01068420

[36]	Padgett, W.J. and Spurrier, J.D. (1990) Shewhart-Type Charts of Percentiles of Strength Distributions. Journal of Quality Technology, 22, 283-288.

[37]	Corzo, O. and Bracho, N. (2008) Application of Weibull Distribution Model to Describe the Vacuum Pulse Osmotic Dehydration of Sardine Sheets. LWT-Food Science and Technology, 41, 1108-1115.

[38]	Basu, B., Tiwari, D., Kundu, D. and Prasad, R. (2009) Is Weibull Distribution the Most Appropriate Statistical Strength Distribution for Brittle Materials? Ceramics International, 35, 237-246. http://dx.doi.org/10.1016/j.ceramint.2007.10.003

[39]	Guo, B.C. and Wang, B.X. (2012) Control Charts for Monitoring the Weibull Shape Parameter Based on Type-II Censored Sample. Quality and Reliability Engineering International, 30, 13-24. http://dx.doi.org/10.1002/qre.1473