The previous studies on the side sensitive modified group runs double sampling (SSMGRDS) X scheme focused on the known process parameters assumption (Case-K). However, the process parameters in real-life scenarios are frequently undisclosed and require estimation using an appropriate in-control (IC) reference sample. Unfortunately, prior research works have revealed that a substantial quantity of reference samples is necessary for the scheme with unknown process parameters assumption (Case-U) to attain a comparable performance as the Case-K scheme. Given the challenges of obtaining a large number of IC samples, we resort to exploring optimal designs for the Case-U SSMGRDS X scheme, focusing on minimizing the average number of observations to signal (ANOS) in situations where the shift size is known. Moreover, we also investigate the expected ANOS (EANOS) since the shift size is commonly unknown in advance. The obtained optimal parameters for the SSMGRDS X scheme under Case-U ensure its performance is equivalent to the Case-K scheme, without requiring an extensive number of reference samples. Our study demonstrates the effectiveness of the SSMGRDS X scheme under Case-U in monitoring the silicon epitaxial process.

The analysis of an X$\bar{X} $ scheme often assumes that the process standard deviation is accurately assessed and remains constant. However, in practice, this is rarely true. Noting that the group runs (GR) scheme performs better than the synthetic scheme, in this research, we proposed the GR exponentially weighted moving average (GR EWMA) X$\bar{X} $ and t schemes and determined their true optimal parameters using the optimisation programmes. Our findings indicate that similar to the synthetic EWMA X$\bar{X} $ scheme, the proposed GR EWMA X$\bar{X} $ scheme is not resilient to errors in the estimation of the standard deviation of the process or when the standard deviation changes. Therefore, we also proposed the GR EWMA t scheme for surveilling the mean of a process. We demonstrate that this t scheme possesses the required robust characteristic. We showcase our developed schemes' superiority over existing schemes in a detailed performance comparison. An illustrative example related to the hard-baking process is utilised to demonstrate the applicability of the suggested schemes.