Simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear profiles with a new adaptive Shewhart-type control chart

There has been a growing interest in research regarding monitoring a process through a regression model (called a profile) rather than a simple quality characteristic. This paper proposes a new monitoring scheme to simultaneously monitor the multivariate multiple linear profiles' parameters. This scheme is based on the Shewhart control chart concept and only has one single (max-type) control chart for monitoring regression coefficients and error's variation, which uses a new statistic to improve the variability (error's variance-covariance matrix) shift detection in multivariate profiles. To increase the sensitivity and capability of the proposed scheme, especially in detecting small to moderate shift sizes, we add a variable parameters (VP) adaptive scheme to the developed control chart as well, considering that no adaptive monitoring schemes have so far been developed for monitoring the multivariate multiple linear profiles and neither are there any VP adaptive features for all profile monitoring schemes. Next, we develop a Markov chain model to compute the time to signal and run length performance measures. After that, we perform extensive numerical analyses to first compare the proposed control chart with the best available control charts and then evaluate its performance under different shift scenarios as well as different dimensions. The results show that the new monitoring scheme performs well compared to the best available monitoring schemes, and more importantly, it is more applicable in real practice. Finally, an illustrative example is presented to show the implementation of the proposed scheme in practice.