Monitoring the covariance matrix of bivariate processes with the DVMAX control charts

Abstract

Two versions of Phase II attribute+variable (DVMAX) control charts are investigated for monitoring the covariance matrix (Formula presented.) of bivariate processes. Monitoring always starts with an attribute chart employing the Max D control chart and, depending on the outcome, a variable control chart named VMAX chart is run at a second stage to check for process stability. In the first version, denoted as the (Formula presented.) chart, two independent samples are used at the two stages of the same inspection; with the second version, denoted as the (Formula presented.) chart, the same sample is used at both the first and second stage of the same inspection. This approach, based on the implementation of two types of charts, can be designed to be more advantageous than a single variable control chart in terms of detection speed of a shift in the covariance matrix. In general, we conclude that the (Formula presented.) control charts not only shows the best statistical performance but also presents a lower average sampling cost. A numerical example illustrates the implementation of the proposed control charts.