Document Type : Article

**Authors**

Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan

**Abstract**

In this paper, we propose new Shewhart-EWMA (SEWMA) and Shewhart-CUSUM (SCUSUM) control charts using the varied L ranked set sampling (VLRSS) for monitoring the process mean, namely the SEWMA-VLRSS and SCUSUM-VLRSS charts. The run length characteristics of the proposed charts are computed using extensive Monte Carlo simulations. The proposed charts are compared with their existing counterparts in terms of the average and standard deviation of run lengths. It is found that, with perfect and imperfect rankings, the SEWMA-VLRSS and SCUSUM-VLRSS charts are more sensitive than their analogous charts based on simple random sampling, ranked set sampling (RSS) and median RSS schemes. A real dataset is also used to explain the implementation of the proposed control charts.

**Keywords**

- Average Run Length
- CUSUM
- Control chart
- EWMA
- Perfect and imperfect rankings
- Ranked set sampling
- Statistical process control

**Main Subjects**

References;

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25. Abujiya, M.R., Riaz, M., and Lee, M.H. "Improving the performance of combined Shewhart-cumulative sum control charts", Quality and Reliability Engineering International, 29(8), pp. 1193-1206 (2013).

26. Awais, M. and Haq, A. "An EWMA chart for monitoring process mean", Journal of Statistical Computation and Simulation, 88(5), pp. 1003-1025 (2018).

27. Awais, M. and Haq, A. "A new cumulative sum control chart for monitoring the process mean using varied L ranked set sampling", Journal of Industrial and Production Engineering, 35(2), pp. 74-90 (2018).

28. Haq, A. "An improved mean deviation exponentially weighted moving average control chart to monitor process dispersion under ranked set sampling", Journal of Statistical Computation and Simulation, 84(9), pp. 2011-2024 (2014).

29. Mehmood, R., Riaz, M., and Does, R.J.J.M. "Control charts for location based on different sampling schemes", Journal of Applied Statistics, 40(3), pp. 483- 494 (2013).

30. Mehmood, R., Riaz, M., and Does, R.J.J.M. "Quality quandaries: On the application of different ranked set sampling schemes", Quality Engineering, 26(3), pp. 370-378 (2014).

31. Haq, A., Brown, J., and Moltchanova, E. "New exponentially weighted moving average control charts for monitoring process dispersion", Quality and Reliability Engineering International, 30(8), pp. 1311-1332 (2014).

32. Haq, A., Brown, J., and Moltchanova, E. "A new maximum exponentially weighted moving average control chart for monitoring process mean and dispersion", Quality and Reliability Engineering International, 31(8), pp. 1587-1610 (2015).

33. Haq, A., Brown, J., and Moltchanova, E. "A new exponentially weighted moving average control chart for monitoring the process mean", Quality and Reliability Engineering International, 31(8), pp. 1623-1640 (2015).

34. Haq, A., Brown, J., and Moltchanova, E. "A new maximum exponentially weighted moving average control chart for monitoring process mean and dispersion", Quality and Reliability Engineering International, 31(8), pp. 1587-1610 (2015).

35. Haq, A., Brown, J., and Moltchanova, E. "New synthetic control charts for monitoring process mean and process dispersion", Quality and Reliability Engineering International, 31(8), pp. 1305-1325 (2015).

36. Haq, A., Brown, J., Moltchanova, E., and Al-Omari, A.I. "Effect of measurement error on exponentially weighted moving average control charts under ranked set sampling schemes", Journal of Statistical Computation and Simulation, 85(6), pp. 1224-1246 (2015).

37. Abbasi, S.A. and Riaz, M. "On dual use of auxiliary information for efficient monitoring", Quality and Reliability Engineering International, 32(2), pp. 705-714 (2016).

38. Abid, M., Nazir, H.Z., Riaz, M., and Lin, Z. "Use of ranked set sampling in nonparametric control charts", Journal of the Chinese Institute of Engineers, 39(5), pp. 627-636 (2016).

39. Abid, M., Nazir, H.Z., Riaz, M., and Lin, Z. "Investigating the impact of ranked set sampling in nonparametric CUSUM control charts", Quality and Reliability Engineering International, 33(1), pp. 203- 214 (2016).

40. Munir, W. and Haq, A. "New cumulative sum control charts for monitoring process variability", Journal of Statistical Computation and Simulation, 87(15), pp. 2882-2899 (2017).

41. Rhoads, T.R., Montgomery, D.C., and Mastrangelo, C.M. "A fast initial response scheme for the exponentially weighted moving average control chart", Quality Engineering, 9(2), pp. 317-327 (1996).

42. Steiner, S.H. "EWMA control charts with time-varying control limits and fast initial response", Journal of Quality Technology, 31(1), pp. 75-86 (1999).

43. David, H.A. and Nagaraja, H.N., Order Statistics, 3rd Edn., John Wiley & Sons, Inc., Hoboken, New Jersey (2003).

44. Montgomery, D.C., Introduction to Statistical Quality Control, 6th Edn., John Wiley & Sons, Inc., New York (2007).

2. Lucas, J.M. and Crosier, R.B. "Fast initial response for CUSUM quality-control schemes: Give your CUSUM a head start", Technometrics, 24(3), pp. 199-205 (1982).

3. Lucas, J.M. "Combined Shewhart-CUSUM quality control schemes", Journal of Quality Technology, 14(2), pp. 51-59 (1982).

4. Roberts, W.S. "Control chart tests based on geometric moving averages", Technometrics, 1(3), pp. 239-250 (1959).

5. Lucas, J.M. and Saccucci, M.S. "Exponentially weighted moving average control schemes: Properties and enhancements", Technometrics, 32(1), pp. 1-12 (1990).

6. Knoth, S. "Fast initial response features for EWMA control charts", Statistical Papers, 46(1), pp. 47-64 (2005).

7. Chiu, W.C. "Generally weighted moving average control charts with fast initial response features", Journal of Applied Statistics, 36(3), pp. 255-275 (2009).

8. Abbas, N., Riaz, M., and Does, R.J.M.M. "Mixed exponentially weighted moving average cumulative sum charts for process monitoring", Quality and Reliability Engineering International, 29(3), pp. 345-356 (2013).

9. Haq, A. "A new hybrid exponentially weighted moving average control chart for monitoring process mean", Quality and Reliability Engineering International, 29(7), pp. 1015-1025 (2013).

10. Haq, A., Brown, J., and Moltchanova, E. "Improved fast initial response features for exponentially weighted moving average and cumulative sum control charts", Quality and Reliability Engineering International, 30(5), pp. 697-710 (2014).

11. McIntyre, G.A. "A method for unbiased selective sampling, using ranked sets", Crop and Pasture Science, 3(4), pp. 385-390 (1952).

12. Dell, T.R. and Clutter, J.L. "Ranked set sampling theory with order statistics background", The International Biometric Society, 28(2), pp. 545-555 (1972).

13. Stokes, S.L. "Ranked set sampling with concomitant variables", Communications in Statistics Theory and Methods, 6(12), pp. 1207-1211 (1977).

14. Samawi, H.M., Ahmed, M.S., and Abu-Dayyeh, W. "Estimating the population mean using extreme ranked set sampling", Biometrical Journal, 38(5), pp. 577-586 (1996).

15. Muttlak, H.A. "Median ranked set sampling", Journal of Applied Statistical Science, 6(4), pp. 245-255 (1997).

16. Muttlak, H.A. "Investigating the use of quartile ranked set samples for estimating the population mean", Applied Mathematics and Computation, 146(2-3), pp. 437-443 (2003).

17. Al-Nasser, A.D. "L ranked set sampling: A generalization procedure for robust visual sampling", Communications in Statistics - Simulation and Computation, 36(1), pp. 33-43 (2007).

18. Haq, A., Brown, J., Moltchanova, E., and Al-Omari, A.I. "Varied L ranked set sampling scheme", Journal of Statistical Theory and Practice, 9(4), pp. 741-767 (2015).

19. Salazar, R.D. and Sinha, A.K. "Control chart x based on ranked set sampling", Comunicacion Tecnica, No. 1-97-09 (PE/CIMAT) (1997).

20. Muttlak, H.A. and Al-Sabah, W. "Statistical quality control based on ranked set sampling", Journal of Applied Statistics, 30(9), pp. 1055-1078 (2003).

21. Abujiya, M.R. and Muttlak, H. "Quality control chart for the mean using double ranked set sampling", Journal of Applied Statistics, 31(10), pp. 1185-1201 (2004).

22. Al-Omari, A.I. and Haq, A. "Improved quality control charts for monitoring the process mean, using doubleranked set sampling methods", Journal of Applied Statistics, 39(4), pp. 745-763 (2012).

23. Al-Sabah, W.S. "Cumulative sum statistical control charts using ranked set sampling data", Pakistan Journal of Statistics, 26(2), pp. 365-378 (2010).

24. Abujiya, M.R., Riaz, M., and Lee, M.H. "Enhancing the performance of combined Shewhart-EWMA charts", Quality and Reliability Engineering International, 29(8), pp. 1093-1106 (2013).

25. Abujiya, M.R., Riaz, M., and Lee, M.H. "Improving the performance of combined Shewhart-cumulative sum control charts", Quality and Reliability Engineering International, 29(8), pp. 1193-1206 (2013).

26. Awais, M. and Haq, A. "An EWMA chart for monitoring process mean", Journal of Statistical Computation and Simulation, 88(5), pp. 1003-1025 (2018).

27. Awais, M. and Haq, A. "A new cumulative sum control chart for monitoring the process mean using varied L ranked set sampling", Journal of Industrial and Production Engineering, 35(2), pp. 74-90 (2018).

28. Haq, A. "An improved mean deviation exponentially weighted moving average control chart to monitor process dispersion under ranked set sampling", Journal of Statistical Computation and Simulation, 84(9), pp. 2011-2024 (2014).

29. Mehmood, R., Riaz, M., and Does, R.J.J.M. "Control charts for location based on different sampling schemes", Journal of Applied Statistics, 40(3), pp. 483- 494 (2013).

30. Mehmood, R., Riaz, M., and Does, R.J.J.M. "Quality quandaries: On the application of different ranked set sampling schemes", Quality Engineering, 26(3), pp. 370-378 (2014).

31. Haq, A., Brown, J., and Moltchanova, E. "New exponentially weighted moving average control charts for monitoring process dispersion", Quality and Reliability Engineering International, 30(8), pp. 1311-1332 (2014).

32. Haq, A., Brown, J., and Moltchanova, E. "A new maximum exponentially weighted moving average control chart for monitoring process mean and dispersion", Quality and Reliability Engineering International, 31(8), pp. 1587-1610 (2015).

33. Haq, A., Brown, J., and Moltchanova, E. "A new exponentially weighted moving average control chart for monitoring the process mean", Quality and Reliability Engineering International, 31(8), pp. 1623-1640 (2015).

34. Haq, A., Brown, J., and Moltchanova, E. "A new maximum exponentially weighted moving average control chart for monitoring process mean and dispersion", Quality and Reliability Engineering International, 31(8), pp. 1587-1610 (2015).

35. Haq, A., Brown, J., and Moltchanova, E. "New synthetic control charts for monitoring process mean and process dispersion", Quality and Reliability Engineering International, 31(8), pp. 1305-1325 (2015).

36. Haq, A., Brown, J., Moltchanova, E., and Al-Omari, A.I. "Effect of measurement error on exponentially weighted moving average control charts under ranked set sampling schemes", Journal of Statistical Computation and Simulation, 85(6), pp. 1224-1246 (2015).

37. Abbasi, S.A. and Riaz, M. "On dual use of auxiliary information for efficient monitoring", Quality and Reliability Engineering International, 32(2), pp. 705-714 (2016).

38. Abid, M., Nazir, H.Z., Riaz, M., and Lin, Z. "Use of ranked set sampling in nonparametric control charts", Journal of the Chinese Institute of Engineers, 39(5), pp. 627-636 (2016).

39. Abid, M., Nazir, H.Z., Riaz, M., and Lin, Z. "Investigating the impact of ranked set sampling in nonparametric CUSUM control charts", Quality and Reliability Engineering International, 33(1), pp. 203- 214 (2016).

40. Munir, W. and Haq, A. "New cumulative sum control charts for monitoring process variability", Journal of Statistical Computation and Simulation, 87(15), pp. 2882-2899 (2017).

41. Rhoads, T.R., Montgomery, D.C., and Mastrangelo, C.M. "A fast initial response scheme for the exponentially weighted moving average control chart", Quality Engineering, 9(2), pp. 317-327 (1996).

42. Steiner, S.H. "EWMA control charts with time-varying control limits and fast initial response", Journal of Quality Technology, 31(1), pp. 75-86 (1999).

43. David, H.A. and Nagaraja, H.N., Order Statistics, 3rd Edn., John Wiley & Sons, Inc., Hoboken, New Jersey (2003).

44. Montgomery, D.C., Introduction to Statistical Quality Control, 6th Edn., John Wiley & Sons, Inc., New York (2007).

Transactions on Industrial Engineering (E)

November and December 2019Pages 3796-3818