The Minimax control chart for multivariate quality control

Abstract

A new multicharacteristic control chart designed to detect shifts in the mean of a multivariate process is proposed. It is assumed that the correlation matrix is known and that the distribution of the data is multivariate normal. The new chart is based on the minimum standardized sample mean (Z_[1]) and the maximum standardized sample mean (Z_[p]) of p correlated quality variables or characteristics. For this reason the chart has been named the Minimax control chart. A method for calculating probabilities for the joint distribution of Z_[1] and Z_[p] is developed. This method is used to determine the position of the four control limits of the chart; the upper and lower control limits of Z_[1], and the upper and lower control limits of Z_[p]. The control limits of the chart are determined such that the chart has a fixed probability of Type I error.

The chart’s performance is compared to that of the Chi-squared control chart in terms of the average run length for several combinations of the parameters of the chart. Among these parameters are the sample size, the number of variables, the probability of Type I error, the correlation matrix, and the direction and magnitude of the shift in the mean. The proposed chart outperforms the Chi-squared chart in all the cases studied where the covariance matrix has non-negative elements.

The new chart provides an easy way for diagnosing the system when a signal occurs. That is to say, the chart provides a means to identify the source of the problem when a shift in the mean occurs. The criteria established for diagnosing the system is based on the positions of Z_[1] and Z_[p] in the Minimax chart. Thus, to diagnose the signals no further analysis is needed. The diagnosing criteria are shown to be particularly effective when the shifts in the mean are either axial or diagonal.