Synthetic double sampling (x)over-bar chart with estimated process parameters

The synthetic double sampling (SDS) (X) over bar chart comprises the double sampling (DS) (X) over bar and conforming run length (CRL) sub-charts. The SDS (X) over bar chart has been studied under the assumption of known process parameters in the literature. Nevertheless, in practice, process parameters are usually unknown and are estimated from an in-control Phase I dataset. This paper investigates the performance of the SDS (X) over bar chart with estimated process parameters, in terms of the average run length (ARL), average number of observations to signal (ANOS) and standard deviation of the run length (SDRL). The performance of the SDS (X) over bar chart with estimated process parameters by minimizing the out-of-control ARL and the out-of-control ANOS is compared with the corresponding chart's performance with known process parameters. In addition, the minimum number of Phase I samples required by the SDS (X) over bar chart with estimated process parameters so that it has approximately the same in-control ARL and ANOS performances as the chart with known process parameters is studied. The ARL, ANOS and SDRL properties of the SDS (X) over bar chart with estimated process parameters differ significantly from that of the chart with known process parameters. Therefore, suitable optimal charting parameters are introduced so that the SDS (X) over bar chart with estimated process parameters has an adequate performance as its known process parameters counterpart without having to use large number of Phase I samples and sample size