Skip to main content
SpringerLink
Log in

Effect of phase I estimation error on the monitoring of simple linear profiles in phase II

  • ORIGINAL ARTICLE
  • Published:
The International Journal of Advanced Manufacturing Technology Aims and scope Submit manuscript

Abstract

The cases where a functional relationship between a response variable and one or more explanatory variables is used to monitor the quality of a product or process appear to be common in practice. This relationship usually is referred to as profile with parameters that are rarely known and have to be estimated prior to starting online monitoring. Phase II initiates after a statistically in-control condition is established, and parameters of the underlying distribution are estimated. It is obvious that phase II analysis of control charting is affected by errors in the estimated parameters. In this paper, the effect of estimation error on phase II analysis of simple linear profiles is studied. The in-control and out-of-control performance of the exponentially weighted moving average (EWMA)-3 method are evaluated using average run length (ARL) criterion. Overall, the in-control ARL in average decreases about 96.7 % when we have m = 100 samples in phase I to estimate the parameters. This value reduces to 2.4 % for m = 900 for a fixed sample size of n = 4. The behavior of the EWMA-3 chart in the out-of-control situation totally depends on the direction of the occurring shifts and whether the ill-estimated parameter is overestimated or underestimated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Del Castillo E (1996) Evaluation of run length distribution for \( \overline{X} \) charts with unknown process variance. J Qual Technol 28:116–122

    Google Scholar 

  2. Gupta S, Montgomery DC, Woodall WH (2006) Performance evaluation of two methods for online monitoring of linear calibration profiles. Int J Prod Res 38(44):349–364

  3. Jensen WA, Birch JB, Woodall WH (2008) Monitoring correlation within linear profiles using mixed models. J Qual Technol 40(2)167--183

  4. Jensen WA, Jones LA, Champ CW, Woodall WH (2006) Effects of parameter estimation on control chart properties: a literature review. J Qual Technol 38(4):349–364

  5. Jones LA (2002) The statistical design of EWMA control charts with estimated parameters. J Qual Technol 34(3):277–288

  6. Jones LA, Champ CW, Rigdon SE (2001) The performance of exponentially weighted moving average charts with estimated parameters. Technometrics 43(2):156–167

  7. Kang L, Albin SL (2000) On-line monitoring when the process yields a linear profile. J Qual Technol 32:418–426

    Google Scholar 

  8. Kim K, Mahmoud MA, Woodall WH (2003) On the monitoring of linear profiles. J Qual Technol 35:317–328

    Google Scholar 

  9. Mahmoud Mahmoud A, Parker Peter A, Woodall WH, Hawkins DM (2007) A change point method for linear profile data. Qual Reliab Eng Int 23:247–268. doi:10.1002/qre.788

    Article  Google Scholar 

  10. Mahmoud MA, Woodall WH (2004) Phase I analysis of linear profiles with calibration applications. Technometrics 46(4):380--391

  11. Montgomery DC (2013) Introduction to statistical quality control, 7th edition, Wiley, ISBN Number: 0471656313, 9780471656319, 978–0471656319

  12. NIST/SEMATECH e-handbook of statistical methods. Available online at: http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc37.htm

  13. Noorossana R, Fatemi A, Zerehsaz Y (2015) Phase II monitoring of linear profiles with random explanatory variable. Int J Adv Manuf Technol 76(5):779--787

  14. Noorossana, Zerehsaz Y (2015) Effect of measurement error on phase II monitoring of simple linear profiles. Int J Adv Manuf Technol 79(9):2031--2040

  15. Quesenberry CP (1993) The effect of sample size on estimated limits for \( \overline{X} \) and x control charts. J Qual Technol 25(4):237–247

  16. Shu L, Tsung F, Tsui K (2005) Effects of estimation errors on cause-selecting charts. IIE Trans 37(6):559–567

  17. Soleimani P, Noorossana R, Amiri A (2009) Simple linear profiles monitoring in the presence of within profile autocorrelation. Comput Ind Eng 57(3):1015–1021

  18. Woodall WH, Spitzner DJ, Montgomery DC, Gupta S (2004) Using control charts to monitor process and product quality profiles. J Qual Technol 36(3):309–320

    Google Scholar 

  19. Woodall WH (2007) Current research on profile monitoring. Revista Producão 17(3):420–425

    Google Scholar 

  20. Wu G, Zhao Y, Wang Z (2002) The median absolute deviations and their application to Shewhart \( \overline{X} \) control charts. Commun Stat-Simul Comput 31(3):425–442

  21. Yeh AB, Zerehsaz Y (2013) Phase I control of simple linear profiles with individual observations. Qual Reliab Eng Int 29(6):829–840

  22. Zou C, Tsung F, Wang Z (2007) Monitoring general linear profiles using multivariate exponentially weighted moving average schemes. Technometrics 49(4):395–408. doi:10.1198/004017007000000164

    Article  MathSciNet  Google Scholar 

  23. Zou C, Zhang Y, Wang Z (2006) A control chart based on a change-point model for monitoring linear profiles. IIE Trans 38(12):1093–1103, doi:10.1080/07408170600728913

  24. Zou C, Zhou C, Wang Z, Tsung F (2007) A self-starting control chart for linear profiles. J Qual Technol 39(4):364--375

Download references

Author information

Authors and Affiliations

  1. Industrial Engineering Department, Islamic Azad University-South Tehran Branch, Tehran, Iran

    R. Noorossana

  2. Industrial Engineering Department, Islamic Azad University-Science and Research Branch, Tehran, Iran

    Mozhdeh Aminmadani & Abbas Saghaei

Authors
  1. R. Noorossana
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mozhdeh Aminmadani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abbas Saghaei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Noorossana.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Noorossana, R., Aminmadani, M. & Saghaei, A. Effect of phase I estimation error on the monitoring of simple linear profiles in phase II. Int J Adv Manuf Technol 84, 873–884 (2016). https://doi.org/10.1007/s00170-015-7078-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00170-015-7078-2

Keywords

Search

Navigation