Advertisement

Statistical Process Control using Shewhart Control Charts with Supplementary Runs Rules

  • M. V. KoutrasEmail author
  • S. Bersimis
  • P. E. Maravelakis
Article

Abstract

The aim of this paper is to present the basic principles and recent advances in the area of statistical process control charting with the aid of runs rules. More specifically, we review the well known Shewhart type control charts supplemented with additional rules based on the theory of runs and scans. The motivation for this article stems from the fact that during the last decades, the performance improvement of the Shewhart charts by exploiting runs rules has attracted continuous research interest. Furthermore, we briefly discuss the Markov chain approach which is the most popular technique for studying the run length distribution of run based control charts.

Keywords

Statistical process control Control charts Shewhart Runs rules Scans rules Patterns Markov chain Average run length 

AMS 2000 Subject Classification

62N10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. A. Aerne, C. W. Champ, and S. E. Rigdon, “Evaluation of control charts under linear trend,” Communications in Statistics. Theory and Methods vol. 20 pp. 3341–3349, 1991.Google Scholar
  2. L. C. Alwan, C. W. Champ, and H. D. Maragah, “Study of the average run lengths for supplementary runs rules in the presence of autocorrelation,” Communications in Statistics. Theory and Methods vol. 23 pp. 373–391, 1994.zbMATHGoogle Scholar
  3. R. W. Amin and W. G. H. Letsinger, “Improved switching rules in control procedures using variable sampling intervals,” Communications in Statistics. Simulation and Computation vol. 20 pp. 205–230, 1991.zbMATHGoogle Scholar
  4. D. L. Antzoulakos and A. Rakintzis, “The Modified r|m Control Chart for Detecting Small Process Average Shifts,” (preprint) 2006.Google Scholar
  5. F. Aparisi, C. W. Champ, and J. C. Garcia Diaz, “A performance Hotelling’s T2 control chart with supplementary run rules,” Quality Engineering vol. 16 pp. 359–368, 2004.CrossRefGoogle Scholar
  6. N. Balakrishnan and M. V. Koutras, Runs and Scans with Applications, Wiley: New York, 2002.zbMATHGoogle Scholar
  7. A. F. Bissel, “An attempt to unify the theory of quality control procedures,” Bulletin in Applied Statistics vol. 5 pp. 113–128, 1978.Google Scholar
  8. P. D. Bourke, “Detecting a shift in fraction nonconforming using run-length control charts with 100% inspection,” Journal of Quality Technology vol. 23 pp. 225–238, 1991.Google Scholar
  9. C. W. Champ, “Steady-state run length analysis of a Shewhart quality control chart with supplementary runs rules,” Communications in Statistics. Theory and Methods vol. 21 pp. 765–777, 1992.zbMATHGoogle Scholar
  10. C. W. Champ and W. H. Woodall, “Exact results for Shewhart control charts with supplementary runs rules,” Technometrics vol. 29 pp. 393–399, 1987.zbMATHCrossRefGoogle Scholar
  11. C. W. Champ and W. H.Woodall, “A program to evaluate the run length distribution of a Shewhart control chart with supplementary runs rules,” Journal of Quality Technology vol. 22 pp. 68–73, 1990.Google Scholar
  12. C. W. Champ and W. H. Woodall, “Signal probabilities of runs rules supplementing a Shewhart control chart,” Communications in Statistics. Simulation and Computation vol. 26 pp. 1347–1360, 1997.zbMATHGoogle Scholar
  13. T. K. Das and V. Jain, “An economic design model for x-bar charts with random sampling policies,” IIE Transactions vol. 29 pp. 507–518, 1997.CrossRefGoogle Scholar
  14. T. K. Das, V. Jain, and A. Gosavi, “Economic design of dual-sampling-interval sampling policies for x-bar charts with and without run rules,” IIE Transactions vol. 30 pp. 515–523, 1997.Google Scholar
  15. R. B. Davis and T. C. Krehbiel, “Shewhart and zone control chart performance under linear trend,” Communications in Statistics. Simulation and Computation vol. 31 pp. 91–96, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  16. R. B. Davis and W. H. Woodall, “Performance of the control chart trend rule under linear shift,” Journal of Quality Technology vol. 20 pp. 260–262, 1990.Google Scholar
  17. R. B. Davis and W. H. Woodall, “Evaluating and improving the synthetic control chart,” Journal of Quality Technology vol. 34 pp. 200–207, 2002.Google Scholar
  18. R. B. Davis A. Homer, and W. H. Woodall, “Performance of the zone control chart,” Communications in Statistics. Theory and Methods vol. 19 pp. 1581–1587, 1990.Google Scholar
  19. R. B. Davis, J. Chun, and Y. Y. Guo, “Improving the performance of the zone control chart,” Communications in Statistics. Theory and Methods vol. 23 pp. 3557–3565, 1994.zbMATHGoogle Scholar
  20. C. Derman and S. M. Ross, Statistical Aspects of Quality Control, Academic: San Diego, CA, 1997.zbMATHGoogle Scholar
  21. J. J. Divoky and E. W. Taylor, “Detecting process drift with combinations of trend and zonal supplementary runs rules,” International Journal of Quality and Reliability Management vol. 12 pp. 60–71, 1995.CrossRefGoogle Scholar
  22. B. P. Dudding and W. J. Jannet, Quality Control Charts, B.S. 600 R. London: B.S.I., 1942.Google Scholar
  23. J. C. Fu and W. Y. W. Lou, Distribution Theory of Runs and Patterns and its Applications: A Finite Markov Chain Imbedding Approach, World Scientific: New Jersey, 2003.zbMATHGoogle Scholar
  24. J. C. Fu, G. Shmueli, and Y. M. Chang, “A unified Markov chain approach for computing the run length distribution in control charts with simple or compound rules,” Statistics and Probability Letters vol. 65 pp. 457–466, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  25. J. C. Fu, F. A. Spiring, and H. S. Xie, “On the average run lengths of quality control schemes using a Markov chain approach,” Statistics and Probability Letters vol. 56 pp. 369–380, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  26. J. Glaz, J. Naus, and S. Wallenstein, Scan Statistics, Springer: Berlin Heidelberg New York, 2001.zbMATHGoogle Scholar
  27. A. M. Hurwitz, and M. A. Mathur, “Very simple set of process control rules,” Quality Engineering vol. 1 pp. 21–29, 1992.Google Scholar
  28. A. H. Jaehn, “Zone control charts—SPC made easy,” Quality vol. 26 pp. 51–53, 1987.Google Scholar
  29. A. H. Jaehn, “Zone control charts find new applications,” ASQC Quality Congress Transactions pp. 890–895, 1989.Google Scholar
  30. M. B. C. Khoo, “Design of runs rules schemes,” Quality Engineering vol. 16 pp. 27–43, 2003.CrossRefGoogle Scholar
  31. M. B. C. Khoo and S. H. Quah, “Incorporating runs rules into Hotelling’s χ2 control charts,” Quality Engineering vol. 15 pp. 671–675, 2003.CrossRefGoogle Scholar
  32. M. Klein, “Modified Shewhart-EWMA control charts,” IIE Transactions vol. 29 pp. 1051–1056, 1997.CrossRefGoogle Scholar
  33. M. Klein, “Two alternatives to the Shewhart x-bar control chart,” Journal of Quality Technology vol. 32 pp. 427–431, 2000.Google Scholar
  34. M. V. Koutras, S. Bersimis, and D. L. Antzoulakos, “Improving the performance of the chi-square control chart via runs rules,” Methodology and Computing in Applied Probability vol. 8 pp. 409–426, 2006.zbMATHCrossRefGoogle Scholar
  35. V. Kuralmani, M. Xie, T. N. Goh, and F. F. Gan, “A conditional decision procedure for high yield processes,” IIE Transactions vol. 34 pp. 1021–1030, 2002.CrossRefGoogle Scholar
  36. A. C. Lowry, C. W. Champ, and W. H. Woodall, “The performance of control charts for monitoring process variation,” Communications in Statistics. Simulation and Computation vol. 24 pp. 409–437, 1995.zbMATHMathSciNetGoogle Scholar
  37. D. C. Montgomery, Introduction to Statistical Quality Control, Wiley: New York, 2001.Google Scholar
  38. P. G. Moore, “Some properties of runs in quality control procedures,” Biometrika vol. 45 pp. 89–95, 1958.Google Scholar
  39. F. Mosteller, “Note on application of runs to quality control charts,” Annals of Mathematical Statistics vol. 12 pp. 228–231, 1941.MathSciNetGoogle Scholar
  40. L. S. Nelson, “The Shewhart control chart—test for special causes,” Journal of Quality Technology vol. 16 pp. 237–239, 1984.Google Scholar
  41. L. S. Nelson, “Interpreting Shewharts x-bar chart,” Journal of Quality Technology vol. 17 pp. 114–116, 1985.Google Scholar
  42. P. R. Nelson and P. L. Stephenson, “Runs tests for group control charts,” Communications in Statistics. Theory and Methods vol. 25 pp. 2739–2765, 1996.zbMATHMathSciNetGoogle Scholar
  43. E. S. Page, “Continuous inspection schemes,” Biometrika vol. 41 pp. 100–115, 1954.zbMATHMathSciNetGoogle Scholar
  44. E. S. Page, “Control charts with warning lines,” Biometrics vol. 42 pp. 243–257, 1955.zbMATHMathSciNetGoogle Scholar
  45. A. C. Palm, “Tables of run length percentiles for determining the sensitivity of Shewhart control charts for averages with supplementary runs rules,” Journal of Quality Technology vol. 22 pp. 289–298, 1990.Google Scholar
  46. C. P. Quesenberry, “SPC Q charts for a binomial parameter p: Short or long runs,” Journal of Quality Technology vol. 23 pp. 239–246, 1991.Google Scholar
  47. S. W. Roberts, “Properties of control chart zone tests,” Bell System Technical Journal vol. 37 pp. 83–114, 1958.Google Scholar
  48. S. W. Roberts, “Control charts based on geometric moving averages,” Technometrics vol. 1 pp. 239–250, 1959.CrossRefGoogle Scholar
  49. W. A. Shewhart, Economic Control of Quality of Manufactured Products, Macmillan: New York, 1931.Google Scholar
  50. G. Shmueli and A. Cohen, “Run-length distribution for control charts with runs and scans rules,” Communication in Statistics. Theory and Methods vol. 32 pp. 475–495, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  51. E. Walker, J. W. Philpot, and J. Clement, “False signal rates for the Shewhart control chart with supplementary runs tests,” Journal of Quality Technology vol. 23 pp. 247–252, 1991.Google Scholar
  52. H. Weiler, “The use of runs to control the mean in quality control,” Annals of Mathematical Statistics vol. 48 pp. 816–825, 1953.Google Scholar
  53. Western Electric, Statistical Quality Control Handbook, Western Electric Corporation, Indianapolis, Ind., 1956.Google Scholar
  54. J. A. Westgard and T. Groth, “Power functions for statistical control rules,” Clinical Chemistry vol. 25 pp. 863–869, 1979.Google Scholar
  55. J. A. Westgard, P. L. Barr, M. R. Hunt, and T. Groth, “A multi-run Shewhart chart for quality control in clinical chemistry,” Clinical Chemistry vol. 27 pp. 493–501, 1981.Google Scholar
  56. D. J. Wheeler, “Detecting a shift in process average: tables of the power function for x-bar charts,” Journal of Quality Technology vol. 15 pp. 155–170, 1983.Google Scholar
  57. P. S. Wludyka and S. L. Jacobs, “Runs rules and p-charts for multi-stream binomial processes,” Communication in Statistics. Simulation and Computation vol. 31 pp. 97–142, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  58. J. Wolfowitz, “On the theory of runs with some applications to quality control,” Annals of Mathematical Statistics vol. 14 pp. 280–284, 1943.MathSciNetGoogle Scholar
  59. W. H. Woodall, “Controversies and contradictions in statistical process control,” Journal of Quality Technology vol. 32 pp. 341–350, 2000.Google Scholar
  60. Z. Wu and T. A. Spedding, “Synthetic control chart for detecting small shifts in the proces mean,” Journal of Quality Technology vol. 32 pp. 32–38, 2000 (a).Google Scholar
  61. Z. Wu and T. A. Spedding, “Implementing synthetic control charts,” Journal of Quality Technology vol. 32 pp. 74–78, 2000 (b).Google Scholar
  62. Z. Wu and S. H. Yeo, “Implementing synthetic control chart for attributes,” Journal of Quality Technology vol. 33 pp. 112–114, 2001.Google Scholar
  63. Z. Wu, S. H. Yeo, and T. A. Spedding, “A synthetic control chart for detecting fraction nonconforming increases,” Journal of Quality Technology vol. 33 pp. 104–111, 2001.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • M. V. Koutras
    • 1
    Email author
  • S. Bersimis
    • 1
  • P. E. Maravelakis
    • 1
  1. 1.Department of StatisticsUniversity of PiraeusPiraeusGreece

Personalised recommendations