We show that in the current practice of multivariate process variability monitoring, the reliability of Shewhart-type control charts cannot be measured except when the subgroup size n tends to infinity. However, the requirement of large n is meaningless not only in manufacturing industry where n is small but also in service industry where n is moderate. In this paper, we introduce a new definition of control limits in the two most appreciated control charts in the literature, i.e., the improved generalized variance chart (IGV-chart) and vector variance chart (VV-chart). With the new definition of control limits, the reliability of the control charts can be determined. Some important properties of new control limits will be derived and the computational technique of probability of false alarm will be delivered.
Fitting a time series model to the process data before applying a control chart to the residuals is essential to fulfill the basic assumptions of statistical process control (SPC). Autoregressive integrated moving average (ARIMA) model has been one of the well-established time series modeling approaches that is extensively used for this purpose and is widely recognized for its accuracy and efficiency. Nevertheless, the research community commented that its iterative stages are laborious and time-consuming. In addressing this gap, a novel time series modeling technique with its conceptual assumptions of attributes that was derived from the geometric Brownian motion (GBM) law was developed in this study. It was termed as the logarithmic return (LR) model. Then, the model was employed and tested on a real-world autocorrelated data, whereby the results were assessed and benchmarked with the ARIMA model. The findings for LR model reported a mean average percentage error that ranged between 1.5851% and 3.3793% (less than 10%), which were as accurate as the ARIMA model. The running time (in second of CPU time) taken by the LR model was at least 96.2% faster than the ARIMA model. Interestingly, the corresponding multivariate control chart constructed from the LR model also portrayed a similar general conclusion as that of its counterpart. The LR model was obviously parsimonious and easier to compute and took a shorter running time than the ARIMA model. Therefore, it possessed the potential as an alternative time series modeling methodology for the ARIMA model in the procedures of SPC.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.