Monitoring bivariate and trivariate mean vectors with a Shewhart chart

Abstract

In this article, we propose the use of the mean chart to control multivariate processes. The basic idea is to control the mean vector of bivariate (X, Y) and trivariate (X, Y, Z) processes by alternating the charting statistic of the Shewhart chart. If the mean of X observations was the charting statistic to obtain the current sample point, then the mean of Y observations will be the charting statistic to obtain the next sample point (for the trivariate case, the mean of Z observations will be the charting statistic to obtain the sample point subsequent to the next one). As a Shewhart chart, the signal is given anytime a sample point is plotted beyond the control limits, independent of the charting statistic in use. A fair comparison between the proposed chart and the Hotelling chart is based on an equal number of measurements per sample. The Shewhart chart with alternated charting statistic (ACS) always outperforms the Hotelling chart, except for specific types of disturbances in quality characteristics highly correlated (ρ = 0.7). The ACS chart is substantially easier to operate and faster than the Hotelling chart in signaling changes in the mean vector of bivariate and trivariate processes. Even with fewer measurements per sample, the trivariate ACS chart outperforms the Hotelling chart.