On auxiliary information-based control charts for autocorrelated processes with application in manufacturing industry

  • Shabbir AhmadEmail author
  • Muhammad Riaz
  • Shahid Hussain
  • Saddam Akber Abbasi


Multivariate autoregressive (MAR) models are an attractive choice for applications in the processes related to finance, medical, and industry. For the monitoring of such processes, control chart is the most important and widely used tool of statistical process control tool kit. Moreover, the presence of auxiliary information helps in better estimation of different process parameters. The literature on use of auxiliary variables in control charts assumes independence of observations. In practice, we may come across processes dealing with autocorrelated outcomes. In such situations, a control chart usually produces high false alarms and exhibits slow detection of shifts when the process is out-of-control. This study intends to suggest some auxiliary information-based Shewhart charts for autocorrelated univariate and bivariate AR(1) processes. The proposed structures take into account the autocorrelation structure and offer more effective designs of control charts for efficient process monitoring. The performance measures used in this study are based on run length measures such as average run length, extra quadratic loss, relative average run length and performance comparison index. A detailed performance analysis is carried out to sort out the best performing charts. In addition, we have considered an application from a manufacturing process to demonstrate the implementation of the proposed charting structures in real scenario.


Autoregressive AR(1) process Auxiliary variable Average run length ARL curves Control charts Location parameter Normal distribution 


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The author Muhammad Riaz is indebted to King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, Saudi Arabia, for providing excellent research facilities. The author Saddam Akber Abbasi would like to acknowledge the research support provided by Qatar University.

Funding information

Higher Education Commission (HEC) Pakistan provided funding for the project No. IPFP/HRD/HEC/2014/1630 under the Start-Up Research Grant Program (SRGP) (first author).


  1. 1.
    Jones LA, Champ CW, Rigdon SE (2004) The run length distribution of the CUSUM with estimated parameters. J Qual Tech 36(1):95–108Google Scholar
  2. 2.
    Riaz M (2008) Monitoring process mean level using auxiliary information. Stat Neerl 62(4):458–481MathSciNetGoogle Scholar
  3. 3.
    Riaz M (2008) Monitoring process variability using auxiliary information. Comput Stat 23(2):253–276MathSciNetzbMATHGoogle Scholar
  4. 4.
    Riaz M, Does RJMM (2009) A process variability control chart. Comput Stat 24(2):345–368MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ahmad S, Riaz M, Abbasi SA, Lin Z (2012) On efficient median control charting. J Chin Inst Eng:Accepted for PublicationGoogle Scholar
  6. 6.
    Ahmad S, Riaz M, Abbasi SA, Lin Z (2014) On median control charting under double sampling scheme. Eur J Ind Eng 8(4):478–512Google Scholar
  7. 7.
    Ahmad S, Riaz M, Abbasi SA, Lin Z (2013) On monitoring process variability under double sampling scheme. Int J Prod Econ 142(2):388–400. Google Scholar
  8. 8.
    Singh R, Mangat NS (1996) Elements of survey sampling. Kluwer Academic Publishers, The NetherlandszbMATHGoogle Scholar
  9. 9.
    Zhang GX (1985) Cause-selecting control charts a new type of quality control charts. The QR Journal 12(1):21–25Google Scholar
  10. 10.
    Hawkins DM (1993) Regression adjustment for variables in multivariate quality control. J Qual Tech 25(3):170–182Google Scholar
  11. 11.
    Wade MR, Woodall WH (1993) A review and analysis of cause-selecting control charts. J Qual Tech 25:161–169Google Scholar
  12. 12.
    Shu L, Tsung F, Tsui KL (2005) Effects of estimation errors on cause-selecting charts. IIE Trans 37(6):559–567Google Scholar
  13. 13.
    Riaz M, Abbasi SA, Ahmad S, Zaman B (2014) On efficient phase II process monitoring charts. Int J Adv Manuf Technol 70(9–12):2263–2274. Google Scholar
  14. 14.
    Montgomery DC (2009) Introduction to statistical quality control, 6th edn. John Willey and Sons, New YorkzbMATHGoogle Scholar
  15. 15.
    Abbasi SA, Miller A (2012) On proper choice of variability control chart for normal and non-normal processes. Qual Reliab Eng Int 28(3):279–296Google Scholar
  16. 16.
    Mason RL, Young JC (2002) Multivariate statistical process control with industrial applications. SIAM, Philadelphia, PAzbMATHGoogle Scholar
  17. 17.
    Alwan LC, Roberts HV (1988) Time-series process modeling for statistical control. J Bus Econ Stat 6(1):87–95Google Scholar
  18. 18.
    Montgomery DC, Mastrangelo CM (1991) Some statistical process control methods for autocorrelated data. J Qual Tech 23(3):179–193Google Scholar
  19. 19.
    Zhang NF (2000) Statistical control charts for monitoring the mean of a stationary process. J Stat Comput Simul 66(3):249–258MathSciNetzbMATHGoogle Scholar
  20. 20.
    Vanbrackle LN, Reynolds MR (1997) EWMA and CUSUM control charts in the presence of correlation. Communications in Statistics – Simulation and Computation 26(3):979–1008MathSciNetzbMATHGoogle Scholar
  21. 21.
    English JR, Lee SC, Martin TW, Tilmon C (2000) Detecting changes in autoregressive processes with X-bar and EWMA charts. IIE Trans 32(12):1103–1113Google Scholar
  22. 22.
    Karaoglan AD, Bayhan GM (2011) Performance comparison of residual control charts for trend stationary first order autoregressive processes. Gazi Univ J Sci 24(2):329–339Google Scholar
  23. 23.
    Areepong Y (2013) A comparison of performance of residual control charts for trend stationary AR (p) processes. Int J Pure Appl Math 85(3):583–592Google Scholar
  24. 24.
    Schmid W (1995) On the run length of a Shewhart chart for correlated data. Stat Pap 36(1):111–130MathSciNetzbMATHGoogle Scholar
  25. 25.
    Wieringa JE (1999) Statistical process control for serially correlated data. University of Groningen, PhD dissertationGoogle Scholar
  26. 26.
    Kramer H, Schmid W (1997) Control charts for time series. Nonlinear Analysis, Theory, Methods and Applications 30(7):4007–4016MathSciNetzbMATHGoogle Scholar
  27. 27.
    Chatfield C (1995) The analysis of time series an introduction, 5th edn. Champman and Hall/CRC, New YorkzbMATHGoogle Scholar
  28. 28.
    Alt FB (1985) Multivariate quality control. Encyclopedia of Statistical Science, vol 6, S. Kotz, N.L. Johnson (Eds.) edn. John Wiley, New YorkGoogle Scholar
  29. 29.
    Sukhatme PV, Sukhatme BV (1970) Sampling theory of surveys with application. Iowa Statistical University Press, New YorkzbMATHGoogle Scholar
  30. 30.
    Hartley HO, Ross A (1954) Unbiased ratio estimators. Nature 174:270–271Google Scholar
  31. 31.
    Riaz M (2011) An improved control chart structure for process location parameter. Qual Reliab Eng Int 27(8):1033–1041Google Scholar
  32. 32.
    Riaz M, Mehmood R, Ahmad S, Abbasi SA (2013) On the performance of auxiliary based control charting under normality and nonnormality with estimation effects. Qual Reliab Eng Int 29(8):1165–1179. Google Scholar
  33. 33.
    Riaz M (2015) Control charting and survey sampling techniques in process monitoring. J Chin Inst Eng 38(3):342–354Google Scholar
  34. 34.
    Wu Z, Jiao JX, Yang M, Liu Y, Wang ZJ (2009) An enhanced adaptive CUSUM control chart. IIE Trans 41(7):642–653Google Scholar
  35. 35.
    Riaz M, Does RJMM (2009) An alternative to the bivariate control chart for process dispersion. Qual Eng 21(1):63–71Google Scholar
  36. 36.
    Abbasi SA, Miller A (2013) MDEWMA chart: an efficient and robust alternative to monitor process dispersion. J Stat Comput Simul 83(2):247–268MathSciNetGoogle Scholar
  37. 37.
    Abbasi SA, Riaz M, Miller A (2012) Enhancing the performance of CUSUM scale chart. Comput Ind Eng 63(2):400–409Google Scholar
  38. 38.
    Ahmad S, Riaz M, Abbasi SA, Lin Z (2014) On efficient median control charting. J Chin Inst Eng 37(3):358–375Google Scholar
  39. 39.
    Mahmood T, Riaz M, Omar MH, Xie M (2018) Alternative methods for the simultaneous monitoring of simple linear profile parameters. Int J Adv Manuf Technol 97(5–8):2851–2871. Google Scholar
  40. 40.
    Haridy S, Maged A, Kaytbay S, Araby S (2017) Effect of sample size on the performance of Shewhart control charts. Int J Adv Manuf Technol 90(1):1177–1185. Google Scholar
  41. 41.
    Reynolds M Jr, Stoumbos Z (2004) Control charts and the efficient allocation of sampling resources. Technometrics 46(2):200–214MathSciNetGoogle Scholar
  42. 42.
    Ou Y, Wen D, Wu Z, Khoo M (2012) A comparison study on effectiveness and robustness of control charts for monitoring process mean and variance. Qual Reliab Eng Int 28(1):3–17Google Scholar
  43. 43.
    Ou YJ, Wu Z, Tsung F (2012) A comparison study of effectiveness and robustness of control charts for monitoring process mean. Int J Prod Econ 135(1):479–490Google Scholar
  44. 44.
    Ahmad S, Riaz M (2014) Process monitoring using quantiles control charts. J Test Eval 42(4):962–979Google Scholar
  45. 45.
    Zaman B, Abbas N, Riaz M (2016) Mixed CUSUM-EWMA chart for monitoring process dispersion. Int J Adv Manuf Technol 86(9):3025–3039. Google Scholar
  46. 46.
    Robert CP, Casella G (2005) Monte Carlo statistical methods (springer texts in statistics), Second edn. Springer-Verlag, New YorkGoogle Scholar
  47. 47.
    Robert CP, Casella G (2009) Introducing Monte Carlo methods with R (use R). Springer-Verlag, Berlin, HeidelbergzbMATHGoogle Scholar
  48. 48.
    Mundform DJ, Schaffer J, KM J (2011) Number of replications required in Monte Carlo simulation studies: a synthesis of four studies. J Mod App Stat Meth 10(1):4Google Scholar
  49. 49.
    Jones O, Maillardet R, Robinson A (2014) Introduction to scientific programming and simulation using R. Second Edition edn. Chapman \and Hall/CRC, London, New YarkGoogle Scholar
  50. 50.
    Khaliq Q, Riaz M, Ahmad S (2016) On designing a new Tukey-EWMA control chart for process monitoring. Int J Adv Manuf Technol 82:1):1–1)23. Google Scholar
  51. 51.
    Abbasi SA, Ahmad S, Riaz M (2017) On enhanced sensitivity of nonparametric EWMA control charts for process monitoring. Scientia Iranica 24(1):424–438Google Scholar
  52. 52.
    Riaz M, Mahmood T, Abbasi SA, Abbas N, Ahmad S (2017) Linear profile monitoring using EWMA structure under ranked set schemes. Int J Adv Manuf Technol 91:2751–2775. Google Scholar
  53. 53.
    Lucas JM, Saccucci MS (1990) Exponentially weighted moving average control schemes: properties and enhancements. Technometrics 32(1):1–12MathSciNetGoogle Scholar
  54. 54.
    Kim MJ (2005) Number of replications required in control chart Monte Carlo simulation studies. In: PhD dissertation. University of Northern Colorado, USAGoogle Scholar
  55. 55.
    Schaffer JR, Kim MJ (2007) Number of replications required in control chart Monte Carlo simulation studies. Commun Stat Simulation C 36(5):1075–1087MathSciNetzbMATHGoogle Scholar
  56. 56.
    Zaman B, Riaz M, Ahmad S, Abbasi SA (2015) On artificial neural networking based process monitoring under bootstrapping using runs rules schemes. Int J Adv Manuf Technol 76(1):311–327. Google Scholar
  57. 57.
    Abbass A (2018) Personal interview: concreat copression strength testing in cement factory. Quality Control Cell, Askri Cement Wah, PaksitanGoogle Scholar
  58. 58.
    Yeh IC (2006) Analysis of strength of concrete using design of experiments and neural networks. J Mater Civ Eng 18(4):597–604Google Scholar
  59. 59.
    Hussain S, Song L, Ahmad S, Riaz M (2018) A new auxiliary information based cumulative sum median control charts for monitoring location. Accepted, Frontiers of Information Technology and Electronic EngineeringGoogle Scholar
  60. 60.
    Duffuaa SO, Attia AM, Ghaithan AM (2018) Optimal design of cause selecting control charts for monitoring the processes of coating fire extinguishers: a case study. Qual Eng:1–27.
  61. 61.
    Selmi S, Ben Amara S, Ben Fredj N, Kobi A, Ben Salah I (2018) Process capability indices and mean and R control chart limit adjustments by taking into account measurement system errors. Int J Adv Manuf Technol 95(5):1919–1930. Google Scholar
  62. 62.
    Suman G, Prajapati D (2018) Control chart applications in healthcare: a literature review. Int J Metrol Qual Eng 9:5. Google Scholar
  63. 63.
    Riaz M (2008) A dispersion control chart. Communications in Statistics Simulation and Computation 37 (6):1239–1261Google Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  • Shabbir Ahmad
    • 1
    Email author
  • Muhammad Riaz
    • 2
  • Shahid Hussain
    • 3
  • Saddam Akber Abbasi
    • 4
  1. 1.Department of MathematicsCOMSATS University Islamabad, Wah CampusWah CanttPakistan
  2. 2.Department of Mathematics and StatisticsKing Fahad University of Petroleum and MineralsDhahranSaudi Arabia
  3. 3.Department of MathematicsCOMSATS University Islamabad, Attock CampusAttockPakistan
  4. 4.Department of Mathematics, Statistics and PhysicsQatar UniversityDohaQatar

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