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Double sampling \(\overline{X} \) control chart for a first-order autoregressive moving average process model

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Abstract

In this paper, we consider the double sampling (DS) \(\overline{X} \)control chart for monitoring processes in which the observations can be represented as a first-order autoregressive moving average (ARMA(1, 1)) model. The properties of the DS \(\overline{X} \) control chart with the sampling intervals driven by the rational subgroup concept are studied and compared with the Shewhart chart and the variable sample size (VSS) chart, both properly modified to account for the serial correlation. Numerical results show that the correlation within subgroups has a significant impact on the properties of the charts. For processes with low to moderate correlation levels, the DS \(\overline{X} \)chart is substantially more efficient in detecting process mean shifts.

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References

  1. Montgomery DC (2001) Introduction to statistical quality control, 4th ed. Wiley, New York

    Google Scholar 

  2. Tagaras G (1998) A survey of recent developments in the design of adaptive control charts. J Qual Technol 30:212–231

    Google Scholar 

  3. Reynolds MR Jr, Amin RW, Arnold JC, Nachlas JA (1988) \(\overline{X} \) charts with variable sampling intervals. Technometrics 30:181–192

    Article  MathSciNet  Google Scholar 

  4. Reynolds MR Jr (1989) Optimal variable sampling interval control charts. Seq Anal 8:361–379

    Article  MATH  MathSciNet  Google Scholar 

  5. Reynolds MR Jr (1996) Shewhart and EWMA variable sampling interval control charts with sampling at fixed times. J Qual Technol 28:199–212

    Google Scholar 

  6. Reynolds MR Jr (1996) Variable-sampling-interval control charts with sampling at fixed times. IIE Trans 28:497–510

    Article  Google Scholar 

  7. Runger GC, Pignatiello Jr JJ (1991) Adaptive sampling for process control. J Qual Technol 23:135–155

    Google Scholar 

  8. Amin RW, Miller RW (1993) A robustness study of \(\overline{X} \) charts with variable sampling intervals. J Qual Technol 25:36–44

    Google Scholar 

  9. Runger GC, Montgomery DC (1993) Adaptive sampling enhancements for Shewhart control charts. IIE Trans 25:41–51

    Article  Google Scholar 

  10. Prabhu SS, Runger GC, Keats JB (1993) \(\overline{X} \) chart with adaptive sample sizes. Int J Prod Res 31:2895–2909

    Article  Google Scholar 

  11. Costa AFB (1994) \(\overline{X} \) charts with variable sample size. J Qual Technol 26:155–163

    Google Scholar 

  12. Stoumbos ZG, Reynolds MR Jr (1996) Control charts applying a general sequential test at each sampling point. Seq Anal 15:159–183

    Article  MATH  MathSciNet  Google Scholar 

  13. Stoumbos ZG, Reynolds MR Jr (1997) Control charts applying a sequential test at fixed sampling intervals. J Qual Technol 29:21–40

    Google Scholar 

  14. Costa AFB, Rahim MA (2004) Joint \(\overline{X} \) and R charts with two-stage samplings. Qual Reliab Eng Int 20:699–708

    Article  Google Scholar 

  15. Prabhu SS, Montgomery DC, Runger GC (1994) A combined adaptive sample size and sampling interval \(\overline{X} \) control chart scheme. J Qual Technol 26:164–176

    Google Scholar 

  16. Costa AFB (1997) \(\overline{X} \) charts with variable sample size and sampling intervals. J Qual Technol 29:197–204

    Google Scholar 

  17. Costa AFB (1998) VSSI \(\overline{X} \) charts with sampling at fixed times. Commun Stat Theory Methods 27:2853–2869

    Article  MATH  Google Scholar 

  18. Costa AFB (1999) Joint \(\overline{X} \) and R charts with variable sample sizes and sampling intervals. J Qual Technol 31:387–397

    Google Scholar 

  19. Costa AFB (1998) Joint \(\overline{X} \) and R charts with variable parameters. IIE Trans 30:505–514

    Google Scholar 

  20. Costa AFB (1999) \(\overline{X} \) charts with variable parameters. J Qual Technol 31:408–416

    Google Scholar 

  21. Costa AFB (1999) AATS for the \(\overline{X} \) chart with variable parameters. J Qual Technol 31:455–458

    Google Scholar 

  22. Carot V, Jabaloyes JM, Carot T (2002) Combined double sampling and variable sampling interval \(\overline{X} \) chart. Int J Prod Res 40:2175–2186

    Article  MATH  Google Scholar 

  23. De Magalhães MS, Epprecht EK, Costa AFB (2001) Economic design of a Vp \(\overline{X} \) chart. Int J Prod Econ 74:191–200

    Article  Google Scholar 

  24. De Magalhães MS, Epprecht EK, Costa AFB (2002) Constrained optimization model for the design of an adaptive \(\overline{X} \) chart. Int J Prod Res 40:3199–3218

    Article  MATH  Google Scholar 

  25. Croasdale P (1974) Control charts for a double-sampling scheme based on average production run lengths. Int J Prod Res 12:585–592

    Article  Google Scholar 

  26. Daudin JJ (1992) Double sampling \(\overline{X} \) charts. J Qual Technol 24:78–87

    Google Scholar 

  27. Irianto D, Shinozaki N (1998) An optimal double sampling \(\overline{X} \) control chart. Int J Ind Eng 5:226–234

    Google Scholar 

  28. He D, Grigoryan A (2006) Joint statistical design of double sampling \(\overline{X} \) and s charts. Eur J Oper Res 168:122–142

    Article  MATH  MathSciNet  Google Scholar 

  29. He D, Grigoryan A (2005) Multivariate multiple sampling charts. IIE Trans 37:509–521

    Article  Google Scholar 

  30. Amin RW, Lee SJ (1999) The effects of autocorrelation and outliers on two-sided tolerance limits. J Qual Technol 31:286–300

    Google Scholar 

  31. Vander Wiel SA (1996) Monitoring processes that wander using integrated moving average models. Technometrics 38:139–151

    Article  MATH  Google Scholar 

  32. Reynolds MR Jr, Lu C-W (1997) Control charts for monitoring processes with autocorrelated data. Nonlinear Anal Theory Methods Appl 30:4059–4067

    Article  MATH  MathSciNet  Google Scholar 

  33. Van Brackle III LN, Reynolds MR Jr (1997) EWMA and CUSUM control charts in the presence of correlation. Commun Stat Simul Comput 26:979–1008

    Article  Google Scholar 

  34. Lu C-W, Reynolds MR Jr (1999) Control Charts for monitoring the mean and variance of autocorrelated processes. J Qual Technol 31:259–274

    Google Scholar 

  35. Alwan LC, Radson D (1992) Time-series investigation of subsample mean charts. IIE Trans 24:66–80

    Article  Google Scholar 

  36. Runger CG, Willemain TR (1995) Model-based and model-free control of autocorrelated processes. J Qual Technol 27:283–292

    Google Scholar 

  37. Runger CG, Willemain TR (1996) Batch-means control charts for autocorrelated data. IIE Trans 28:483–487

    Article  Google Scholar 

  38. Alwan LC (1992) Effects of autocorrelation on control chart performance. Commun Stat Theory Methods 21:1025–1049

    Article  MATH  Google Scholar 

  39. Vasilopoulos AV, Stamboulis AP (1978) Modification of control chart limits in the presence of data correlation. J Qual Technol 10:20–30

    Google Scholar 

  40. Alwan LC, Roberts HV (1988) Time-series modeling for statistical process control. J Bus Econ Stat 6:87–95

    Article  Google Scholar 

  41. Montgomery DC, Mastrangelo CM (1991) Some statistical process control methods for autocorrelated data. J Qual Technol 23:179–193

    Google Scholar 

  42. Box GEP, Kramer T (1992) Statistical process monitoring and feedback adjustment: a discussion. Technometrics 34:251–267

    Article  MathSciNet  Google Scholar 

  43. Superville CR, Adams BM (1994) An evaluation of forecast-based quality control schemes. Commun Stat Simul Comput 23:645–661

    Article  MATH  MathSciNet  Google Scholar 

  44. Zhang NF (1997) Detection capability of residual control chart for stationary process data. J Appl Stat 24:475–492

    Article  Google Scholar 

  45. Wardell DG, Moscowitz H, Plante RD (1992) Control charts in the presence of data correlation. Manage Sci 38:1084–1105

    Article  MATH  Google Scholar 

  46. Yashchin E (1993) Performance of CUSUM control schemes for serially correlated observations. Technometrics 35:37–52

    Article  MATH  MathSciNet  Google Scholar 

  47. Faltin FW, Mastrangelo CM, Runger GC, Ryan TP (1997) Considerations in the monitoring of autocorrelated and independent data. J Qual Technol 29:131–133

    Google Scholar 

  48. Reynolds MR Jr, Arnold JC, Baik JW (1996) Variable sampling interval \(\overline{X} \) charts in the presence of correlation. J Qual Technol 28:12–30

    Google Scholar 

  49. Wardell DG, Moscowitz H, Plante RD (1994) Run-length distributions of special-cause control charts for correlated processes. Technometrics 36:3–17

    Article  MATH  MathSciNet  Google Scholar 

  50. Apley DW, Lee HC (2003) Design of exponentially weighted moving average control charts for autocorrelated processes with model uncertainty. Technometrics 45:187–198

    Article  MathSciNet  Google Scholar 

  51. Apley DW, Tsung F (2002) The autoregressive T 2 chart for monitoring univariate autocorrelated processes. J Qual Technol 34:80–96

    Google Scholar 

  52. Jiang W, Tsui K-L, Woodall WH (2000) A new SPC monitoring method: the ARMA chart. Technometrics 42:399–410

    Article  Google Scholar 

  53. Box GEP, Jenkins GM, Reinsel GC (1994) Time series analysis: forecasting and control, 3rd edn. Prentice Hall, Englewood Cliffs, New Jersey

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Production (Departmento de Producão), UNESP—Sao Paulo State University, Avenida Ariberto Pereira da Cunha, 333, 12516-410, Guaratinguetá, SP, Brazil

    Antonio F. B. Costa & Fernando A. E. Claro

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  1. Antonio F. B. Costa
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  2. Fernando A. E. Claro
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Correspondence to Antonio F. B. Costa.

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Costa, A.F.B., Claro, F.A.E. Double sampling \(\overline{X} \) control chart for a first-order autoregressive moving average process model. Int J Adv Manuf Technol 39, 521–542 (2008). https://doi.org/10.1007/s00170-007-1230-6

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  • DOI: https://doi.org/10.1007/s00170-007-1230-6

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