Abstract
The Hotelling’s χ2 control chart is one of the most widely used multivariate charting procedures for monitoring the vector of means of several quality characteristics. As a Shewhart-type control chart, it incorporates information pertaining to most recently inspected sample and subsequently it is relatively insensitive in quickly detecting small magnitude shifts in the process mean vector. A popular solution suggested to overcome this handicap was the use of runs and scans rules as criteria to declare a process out-of-control. During the last years, the examination of Hotelling’s χ2 control charts supplemented with various runs rules has attracted continuous research interest. In the present article we study the performance of the Hotelling’s χ2 control chart supplemented with a r-out-of-m runs rule. The new control chart demonstrates an improved performance over other competitive runs rules based control charts.
Similar content being viewed by others
References
Acosta-Mejia C, Pignatiello JJ (2009) ARL-design of S charts with k-of-k runs rules. Commun Stat-Simul C 38:1625–1639
Alt FB, Smith ND (1988) Multivariate process control. In: Krishnaiah PR, Rao CR (eds) Handbook of Statistics vol. 7. Elsevier, Amsterdam, pp 333–351
Antzoulakos DL (2001) Waiting times for patterns in a sequence of multistate trials. J Appl Probab 38:508–518
Antzoulakos DL, Rakitzis AC (2008) The modified r out of m control chart. Commun Stat-Simul C 37:396–408
Aparisi F, Champ CW, Diaz JCG (2004) A performance analysis of Hotelling’s χ2 control chart with supplementary runs rules. Qual Eng 16(3):13–22
Artiles-Leon N, David HT, Meeks HD (1996) Statistical optimal design of control charts with supplementary stopping rules. IIE Trans 28(3):225–236
Bersimis S, Psarakis S, Panaretos J (2007) Multivariate statistical process control charts: an overview. Qual Reliab Eng Int 23:517–543
Crosier RB (1988) Multivariate generalizations of cumulative sum quality-control schemes. Technometrics 30:291–303
Fu JC, Chang YM (2002) On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials. J Appl Probab 39:70–80
Hotelling H (1947) Multivariate quality control. In: Eisenhart C, Hastay M, Wallis WA (eds) Techniques of statistical Analysis. McGraw-Hill, New York, pp 111–184
Khoo MBC (2005) A nonrigorous approach of incorporating sensitizing rules into multivariate control charts. J Mod Appl Stat Methods 4(2):567–582
Khoo MBC, Quah SH (2003) Incorporating runs rules into Hotelling’s χ2 control charts. Qual Eng 15(4):671–675
Khoo MBC, Quah SH (2004) Alternatives to the multivariate control chart for process dispersion. Qual Eng 16(3):423–435
Khoo MBC, Quah SH, Low HC (2005) Powerful rules for the Hotelling’s χ2 control chart. Qual Eng 17:139–149
Kim Y-B, Hong J-S, Lie C-H (2009) Economic-statistical design of 2-of-2 and 2-of-3 runs rule scheme. Qual Reliab Eng Int 25:215–228
Koutras MV, Bersimis S, Antzoulakos DL (2006) Improving the performance of the chi-square control chart via runs rules. Methodol Comput Appl 8:409–426
Koutras MV, Bersimis S, Maravelakis PE (2007) Statistical process control using Shewhart control charts with supplementary runs rules. Methodol Comput Appl 9:207–224
Lim T-J, Cho M (2009) Design of control charts with m-of-m runs rules. Qual Reliab Eng Int doi:10.1002/qre.1023
Lowry CA, Woodall WH, Champ CW, Rigdon SE (1992) A multivariate exponentially weighted moving average control chart. Technometrics 34:46–53
Montgomery DC (2005) Introduction to statistical quality control. Wiley, New York
Palm AC (1990) Tables of run length percentiles for determining the sensitivity of Shewhart control charts for averages with supplementary runs rules. J Qual Technol 22:289–298
Pignatiello JJ Jr, Runger GC (1990) Comparisons of multivariate CUSUM charts. J Qual Technol 22:173–186
Zhang S, Wu Z (2005) Designs of control charts with supplementary runs rules. Comput Ind Eng 49:76–97
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rakitzis, A.C., Antzoulakos, D.L. Chi-square Control Charts with Runs Rules. Methodol Comput Appl Probab 13, 657–669 (2011). https://doi.org/10.1007/s11009-010-9178-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-010-9178-7