Optimal design of the Side Sensitive Modified Group Runs Double Sampling (SSMGRDS) X̅ scheme with estimated process parameters

The previous studies on the side sensitive modified group runs double sampling (SSMGRDS) (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) scheme under Case-U in monitoring the silicon epitaxial process.