Multivariate Bayesian Control Chart Based on Economic-Statistical Design with 2 and 3-Variable Sample Size

Some key factors in control chart use are associated with the speed of detecting process mean shifts, which are important in minimizing the number of false alarms and cost. Researchers proved that the Fixed Ratio Sampling (FRS) policy has some deficiency in detecting small and moderate shifts. Hence, they improved this issue by proposing the Variable Sample Size (VSS) strategy. They illustrated that this sampling model was more influential in detecting mean shifts. Subsequently, Variable Sample Size with Three-States (3-VSS) was investigated to improve the performance of VSS. It was shown that the performance of the 3-VSS model is more efficient than the VSS policy. In this paper, VSS and 3-VSS are considered in a multivariate Bayesian control chart based on the economic-statistical design. Due to the fact that the distribution of Bayesian statistic is hard to achieve, we apply the Monte Carlo method and employ the artificial bee colony (ABC) algorithm to obtain the optimal design parameters for VSS and 3-VSS models (sample sizes, sampling interval, warning limit(s), and control limit). Hotelling’s $$T^{2}$$ and FRS are the most popular multivariate control chart and sampling scheme, respectively. Thus, in order to demonstrate the statistical and economic desirability of VSS and 3-VSS multivariate Bayesian control charts, this case study is compared with VSS and 3-VSS Hotelling’s $$T^{2}$$ as well as FRS multivariate Bayesian control charts.