Attribute charts for monitoring the mean vector of bivariate processes Antonio F. B. Costa UNESP São Paulo State University Linda Lee Ho USP São Paulo University

Univariate Case Montgomery’s Book: X~N(  0 =50;  0 =2)  1 =52 (power=0.50) LSL=44 USL=56 The chart requires samples of size 9 The np chart requires samples of size 60 Go-No-Go Gauge

The second-class unit is not necessarely non conforming Warning Limits Hu, Z. and Jiao, J. (2008) A control chart for monitoring mean based on attibute inspection, IJPR. The np chart might be designed to signal an out of control condition even before any defective is produced. Discriminating Limits (0)

Walter Andrew Shewhart Shewhart model Wandering process mean REYNOLDS JUNIOR, M. R.; ARNOLD, J. C.; BAIK, J. W. (1996) Variable sampling interval X charts n the presence of correlation. Journal of Quality Technology, v. 28, n. 1, p

LU, C. W.; REYNOLDS, M. R. (1999) EWMA control charts for monitoring the mean of autocorrelated processes. Journal of Quality Technology, v. 31, p 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 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 X 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 X charts in the presence of correlation: the Markov chain approach. International Journal of Production Economics, v. 130, n. 2, p

The np chart proposed by Hu and Jiao to control a wandering mean Univariate Case (3)

When the process mean wanders, a simple way to study the chart’s performance is building a Markov chain that allows us to express the ARL as a function of the expected number the transient states are visited. (6) (7)

(8)

Assumptions: X 1 and X 2 follow a Normal distribution with in-control mean values, and covariance matrix  The assignable cause shifts the mean vector  to without changing 

First proposed attribute chart is the np a chart, its monitoring statistic is A: the number of sample units with a second or a third-class classification The np a chart with Upper Control Limit- UCL=m signals when A>m

The second proposed attribute chart is the np b chart, its monitoring statistic is B=B1+2*B2, being B1 (B2) the number of sample units with a second (third)-class classification. The np b chart with Upper Control Limit- UCL=m signals when B>m

Table 4: The ARL values (n=6 and  =0 )

Table 5: The ARL values (n=6 and  =0.3 )

Table 6: The ARL values (n=6 and  =0.5 )

Table 7: The ARL values (n=6 and  =0.8 )

Table 8 – Synthetic np a and np b control charts (L=5, n=6,  =0.3 )

Table 9– Synthetic np a and np b control charts (L=5, n=6,  =0.8 )

Table 10 -The np a and T 2 control charts

Table 11 -The synthetic np a and T 2 control charts (L=5)

Wandering mean vector

Thank you for your attention