Document Type : Article

**Authors**

^{1}
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China.; Department of Statistics, Government College University Faisalabad, Faisalabad 38000, Pakistan

^{2}
Department of Statistics, Faculty of Sciences, King Abdulaziz University, Jeddah 21551, Saudi Arabia

**Abstract**

Control chart is one of the important tools in statistical process control (SPC) that plays a significant role in monitoring and identifying disturbances of any production process. The Shewhart, cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) are commonly used control charts for detecting process shifts. CUSUM and EWMA chart are more sensitive in detecting smaller shifts whereas the typical Shewhart chart is known to be only sensitive to large process shifts. The present study incorporates ratio-type estimators based on auxiliary information in CUSUM structure as a substitute of simple mean estimator for monitoring process location. These estimators are more efficient than the simple mean estimator in the presence of high correlation between the study and the auxiliary variables. Average run length (ARL), standard deviation of run length (SDRL) and extra quadratic loss (EQL) are used to measure the performances of the proposed charts. The performance comparison of the proposed chart with the existing CUSUM, CUSUM-FIR and many other control charts are made by using out-of-control ARL. The comparison revealed the superiority of the suggested charts over the other existing charts. An illustrative example is also provided for the performance evaluation of the proposed charts.

**Keywords**

- Auxiliary variable
- Control Charts
- CUSUM Statistic
- Fast Initial Response (FIR)
- Ratio Estimator
- Average Run Length
- Extra Quadratic Loss

**Main Subjects**

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25. Raza, S.M.M. and Butt, M.M. "New Shewhart and EWMA type control charts using exponential type estimator with two auxiliary variables under two phase sampling", Pakistan Journal of Statistics and Operation Research, 14(2), pp. 367-386 (2018).

26. Hussain, S., Song, L., Ahmad, S., and Riaz, M. "New interquartile range EWMA control charts with applications in continuous stirred tank rector process", Arabian Journal for Science and Engineering, 44, pp. 2467-2485 (2019). https://doi.org/10.1007/s13369-018-3162-x.

27. Saghir, A., Ahmad, L., Aslam, M., and Jun, C.-H. "A EWMA control chart based on an auxiliary variable and repetitive sampling for monitoring process location", Communications in Statistics-Simulation and Computation, 48(7), pp. 2034-2045 (2019).DOI: 10.1080/03610918.2018.1433837.

28. Sanusi, R.A., Riaz, M., and Abbas, N. "Combined Shewhart CUSUM charts using auxiliary variable", Computers & Industrial Engineering, 105, pp. 329- 337 (2017).

29. Cochran, W.G., Sampling Techniques, John Wiley & Sons, 3rd Edn. (2007).

30. Qiu, P., Introduction to Statistical Process Control, CRC Press (2014).

31. TEAM, R.C. "R: A language and environment for statistical computing", R Foundation for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0 (2017), URL: http://www.R-project.org.

32. Abujiya, M.R., Lee, M.H., and Riaz, M. "Increasing the sensitivity of cumulative sum charts for location", Quality and Reliability Engineering International, 31(6), pp. 1035-1051 (2015).

33. Wu, Z., Yang, M., Jiang, W., and Khoo, M.B.C. "Optimization designs of the combined Shewhart- CUSUM control charts", Computational Statistics & Data Analysis, 53(2), pp. 496-506 (2008).

34. Ou, Y., Wu, Z., and Tsung, F. "A comparison study of effectiveness and robustness of control charts for monitoring process mean", International Journal of Production Economics, 135(1), pp. 479-490 (2012).

35. Nawaz, T., Raza, M.A., and Han, D. "A new approach to design efficient univariate control charts to monitor the process mean", Quality and Reliability Engineering International, 34(8), pp. 1732-1751 (2018). DOI: 10.1002/qre.2366.

36. 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).

37. Nazir, H.Z., Riaz, M., Does, R.J.M.M., and Abbas, Nasir. "Robust CUSUM control charting", Quality Engineering, 25(3), pp. 211-224 (2013).

38. Ahmad, S., Riaz, M., Abbasi, S.A., and Lin, Z. "On efficient median control charting", Journal of the Chinese Institute of Engineers, 37(3), pp. 358-375 (2014).

2. Page, E.S. "Continuous inspection schemes", Biometrika, 41(1/2), pp. 100-115 (1954).

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

4. Haq, A. and Shabbir, J. "Improved family of ratio estimators in simple and stratified random sampling", Communications in Statistics-Theory and Methods, 42(5), pp. 782-799 (2013).

5. Kadilar, C. and Cingi, H. "Ratio estimators in simple random sampling", Applied Mathematics and Computation, 151(3), pp. 893-902 (2004).

6. Kadilar, C. and Cingi, H. "New ratio estimators using correlation coefficient", Interstat, 4, pp. 1-11 (2006).

7. Gupta, S. and Shabbir, J. "On improvement in estimating the population mean in simple random sampling", Journal of Applied Statistics, 35(5), pp. 559- 566 (2008).

8. Al-Omari, A.I. "Ratio estimation of the population mean using auxiliary information in simple random sampling and median ranked set sampling", Statistics & Probability Letters, 82(11), pp. 1883-1890 (2012).

9. Grover, L.K. and Kaur, P. "A generalized class of ratio type exponential estimators of population mean under linear transformation of auxiliary variable", Communications in Statistics-Simulation and Computation, 43(7), pp. 1552-1574 (2014).

10. Singh, H.P., Pal, S.K., and Solanki, R.S. "A new class of estimators of finite population mean in sample surveys", Communications in Statistics-Theory and Methods, 46(6), pp. 2630-2637 (2017).

11. Irfan, M., Javed, M., and Lin, Z. "Efficient ratiotype estimators of finite population mean based on correlation coefficient", Scientia Iranica, 25(4), pp. 2361-2372 (2018).

12. Javed, M., Irfan, M., Pang, T., and Lin, Z. "On improved estimation of population mean using known coefficient of skewness of an auxiliary variable", Iranian Journal of Science and Technology, Transactions A: Science, pp. 1-11 (2018). DOI: 10.1007/s40995-018-0561-5.

13. Zhang, G. "Cause-selecting control charts-a new type of quality control charts", The QR Journal, 12(4), pp. 221-225 (1985).

14. Riaz, M. "Monitoring process mean level using auxiliary information", Statistica Neerlandica, 62(4), pp. 458-481 (2008).

15. Riaz, M. "Monitoring process variability using auxiliary information", Computational Statistics, 23(2), pp. 253-276 (2008).

16. Abbas, N., Riaz, M., and Does, R.J. "An EWMA-type control chart for monitoring the process mean using auxiliary information", Communications in Statistics- Theory and Methods, 43(16), pp. 3485-3498 (2014).

17. Ahmad, S., Abbasi, S.A., Riaz, M., and Abbas, N. "On efficient use of auxiliary information for control charting in SPC", Computers & Industrial Engineering, 67, pp. 173-184 (2014).

18. Riaz, M. "Control charting and survey sampling techniques in process monitoring", Journal of the Chinese Institute of Engineers, 38(3), pp. 342-354 (2015).

19. Hussain, S., Song, L., Ahmad, S., and Riaz, M. "On auxiliary information based improved EWMA median control charts", Scientia Iranica, 25(2), pp. 954-982 (2018).

20. Lee, H., Aslam, M., Shakeel, Q.-A., Lee, W., and Jun, C.-H. "A control chart using an auxiliary variable and repetitive sampling for monitoring process mean", Journal of Statistical Computation and Simulation, 85(16), pp. 3289-3296 (2015).

21. Riaz, M. "An improved control chart structure for process location parameter", Quality and Reliability Engineering International, 27(8), pp. 1033-1041 (2011).

22. Riaz, M., Mehmood, R., Ahmad, S., and Abbasi, S.A. "On the performance of auxiliary-based control charting under normality and nonnormality with estimation effects", Quality and Reliability Engineering International, 29(8), pp. 1165-1179 (2013).

23. Haq, A. "New EWMA control charts for monitoring process dispersion using auxiliary information", Quality and Reliability Engineering International, 33(8), pp. 2597-2614 (2017).

24. Arshad, W., Abbas, N., Riaz, M., and Hussain, Z. "Simultaneous use of runs rules and auxiliary information with exponentially weighted moving average control charts", Quality and Reliability Engineering International, 33(2), pp. 323-336 (2017).

25. Raza, S.M.M. and Butt, M.M. "New Shewhart and EWMA type control charts using exponential type estimator with two auxiliary variables under two phase sampling", Pakistan Journal of Statistics and Operation Research, 14(2), pp. 367-386 (2018).

26. Hussain, S., Song, L., Ahmad, S., and Riaz, M. "New interquartile range EWMA control charts with applications in continuous stirred tank rector process", Arabian Journal for Science and Engineering, 44, pp. 2467-2485 (2019). https://doi.org/10.1007/s13369-018-3162-x.

27. Saghir, A., Ahmad, L., Aslam, M., and Jun, C.-H. "A EWMA control chart based on an auxiliary variable and repetitive sampling for monitoring process location", Communications in Statistics-Simulation and Computation, 48(7), pp. 2034-2045 (2019).DOI: 10.1080/03610918.2018.1433837.

28. Sanusi, R.A., Riaz, M., and Abbas, N. "Combined Shewhart CUSUM charts using auxiliary variable", Computers & Industrial Engineering, 105, pp. 329- 337 (2017).

29. Cochran, W.G., Sampling Techniques, John Wiley & Sons, 3rd Edn. (2007).

30. Qiu, P., Introduction to Statistical Process Control, CRC Press (2014).

31. TEAM, R.C. "R: A language and environment for statistical computing", R Foundation for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0 (2017), URL: http://www.R-project.org.

32. Abujiya, M.R., Lee, M.H., and Riaz, M. "Increasing the sensitivity of cumulative sum charts for location", Quality and Reliability Engineering International, 31(6), pp. 1035-1051 (2015).

33. Wu, Z., Yang, M., Jiang, W., and Khoo, M.B.C. "Optimization designs of the combined Shewhart- CUSUM control charts", Computational Statistics & Data Analysis, 53(2), pp. 496-506 (2008).

34. Ou, Y., Wu, Z., and Tsung, F. "A comparison study of effectiveness and robustness of control charts for monitoring process mean", International Journal of Production Economics, 135(1), pp. 479-490 (2012).

35. Nawaz, T., Raza, M.A., and Han, D. "A new approach to design efficient univariate control charts to monitor the process mean", Quality and Reliability Engineering International, 34(8), pp. 1732-1751 (2018). DOI: 10.1002/qre.2366.

36. 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).

37. Nazir, H.Z., Riaz, M., Does, R.J.M.M., and Abbas, Nasir. "Robust CUSUM control charting", Quality Engineering, 25(3), pp. 211-224 (2013).

38. Ahmad, S., Riaz, M., Abbasi, S.A., and Lin, Z. "On efficient median control charting", Journal of the Chinese Institute of Engineers, 37(3), pp. 358-375 (2014).

Transactions on Industrial Engineering (E)

May and June 2020Pages 1593-1605