1. Reducing the effect of the autocorrelation on the performance of control charts using sampling strategies Antonio F. B. Costa Brazil Marcela A. G. Machado Brazil Session 3. Wednesday. July 13th. 15:45-17:15 pm

2. Shewhart model i=1, 2,... First-order autoregressive model - AR (1).

3. Autocorrelation Figure 1: Temperatures of a Chemical Bath. Xbar chart 210.00 215.00 220.00 225.00 230.00 235.00 240.00 05101520 Figure 2: Control charts for monitoring the Temperature of a Chemical Bath. Many false alarms (n=3. h=30 minutes)

4. 6 19 Figure 3: The effect of the autocorrelation on the ARL

5. References Chen Y.K., Chiou K.C. (2005) Optimal design of VSI Xbar control charts for monitoring correlated samples. Quality and Reliability Engineering International, 21: 757-768. Chen Y.K., Hsieh, K.L, Chang, C.C. (2007) Economic design of the VSSI Xbar control charts for correlated samples. International Journal of Production Economics; 107: 528-539. Costa, A. F. B., Claro, F. A. E. (2008). “Double sampling control Xbar chart for a first-order autoregressive and moving average process model”. The International Journal of Advanced Manufacturing Technology, 39: 521-542. Chau-Chen Torng, Pei-Hsi Lee, Huang-Sheng Liao, Nai-Yi Liao (2009) An economic design of double sampling X charts for correlated data using genetic algorithms Expert Systems with Applications, 36: 12621–12626 Lee, M. H. (2010) Economic Design of Cumulative Sum Control Charts for Monitoring a Process with Correlated Samples Communications in Statistics-Simulation and Computation 39:1909-1922

6. First strategy to reduce the effect of the autocorrelation 9:00 8:30 Case (B) Original samples mixed samples Case (A) 9:30 9:00

7. Figure 4: Case (A) and Case (B)  =0.2

8. Figure 5: Case (A) and Case (B)  =0.5

9. Figure 6: Case (A) and Case (B)  =0.7

10. Case (C) 8:008:309:00

11. Figure 7: Case (A) and Case (C)  =0.2

12. Figure 8: Case (A) and Case (C)  =0.5

13. Figure 9: Case (A) and Case (C)  =0.7

14. Time of production Skipping one item. Case (D) Sample items Time of production Skipping two items. Case (E) Sample items Second strategy to reduce the effect of the autocorrelation Costa A.F.B., Philippe C. (2011) Effect of measurement error and autocorrelation on the Xbar chart Journal of Applied Statistics, 38: 661-673

15. Figure 10: Cases A, D and E  =0.2

16. Figure 11: Cases A, D and E  =0.5

17. Figure 12: Cases A, D and E  =0.7

18. X1X1 X2X2 18 Monitoring the mean vector of bivariate processes The T 2 chart

19. Figure 14: The T 2 chart - Cases A and B -  =0.5

20. Wandering process mean Shewhart model Autocorrelated univariate processes – a second model

21. Figure 15: The Xbar chart with wandering mean  =0.4 8 25

22. Figure 16: The Xbar chart with wandering mean  =0.8 37

23. Percentage of reduction on the speed with which the Xbar chart signals Table 2: The VP scheme Table 1: The DS scheme

24. REYNOLDS JUNIOR, M. R.; ARNOLD, J. C.; BAIK, J. W. (1996) Variable sampling interval Xbar charts in the presence of correlation. Journal of Quality Technology, v. 28, n. 1, p. 12-29. LU, C. W.; REYNOLDS, M. R. (1999) EWMA control charts for monitoring the mean of autocorrelated processes. Journal of Quality Technology, v. 31, p. 166-188. LU, C. W.; REYNOLDS, M. R. (1999) Control chart for monitoring the mean and variance of autocorrelated processes. Journal of Quality Technology, v. 31, p. 259-274. LU, C. W.; REYNOLDS, M. R. (2001) Cusum charts for monitoring an autocorrelated process. Journal of Quality Technology, v.33, p. 316 ‑ 334. LIN, Y. C.; CHOU, C. Y. (2008) The variable sampling rate Xbar control charts for monitoring autocorrelated processes. Quality and Reliability Engineering International, v. 24, p. 855 ‑ 870. LIN, Y. C. (2009) The variable parameters control charts for monitoring autocorrelated processes. Communications in Statistics - Simulation and Computation, v. 38, p. 729 ‑ 749. COSTA, A. F. B.; MACHADO, M. A. G. (2011) Variable parameter and double sampling Xbar charts in the presence of correlation: the Markov chain approach. International Journal of Production Economics, v. 130, n. 2, p. 224-229.

25. Conclusions The autocorrelation reduces the performance of the control charts. The proposed sampling strategies counteract the undesired effect of the autocorrelation, when the dependence is between the observations. When the process mean wanders and  and  are large, the marginal gain in the speed with which the Xbar chart signals does not justify the use of the adaptive schemes. Future investigation: new strategies to improve the performance of the Xbar chart when the process mean wanders.

26. Symposium of Scientific Initiation 2010 - UNESP

27. Acknowledgements This work was supported by: 1) FAPESP – The State of São Paulo Research Foundation 2) CNPq – National Council for Scientific and Technological Development

28. Thanks a lot for your attention. I would appreciate your suggestions.