Statistical design of an adaptive synthetic X over bar $\bar{X}$ control chart for autocorrelated processes

This paper aims to design a variable parameters synthetic X over bar $\bar{X}$ control chart for first-order AR(1) autocorrelated data following a Gaussian process. To improve the statistical performance in detecting small changes and maintaining a low false alarm rate, the variable parameters X over bar $\bar{X}$ control chart was combined with the Conforming Run Length (CRL) sub-chart to determine when to implement tight or relaxed control and intervene to identify an assignable cause. The statistical design of the proposed chart was performed under a discrete-time Markov chain approach and a non-linear programming mathematical model to obtain, using a genetic algorithm (GA), the values of the design parameters that minimise the average time to signal the out-of-control state (ATS1$\text{ATS}_{1}$). A sensitivity analysis was performed to examine the behaviour of the ATS1$\text{ATS}_{1}$ in the face on changes in the parameters. Also, a performance evaluation was conducted to compare the proposed chart with other adaptive synthetic charts, the synthetic X over bar $\bar{X}$ control chart and the chart developed by Costa and Machado. Our results indicate that this proposal is faster to detect small changes in the process mean considering the autocorrelation of the data.