Real-time monitoring of linear profiles using a self-starting scheme with likelihood ratio

In modern industrial processes, product quality characteristics can sometimes be modeled as statistical regression relationships or functions, often referred to as profiles in the statistical process monitoring field. In practice, the corresponding process parameters are usually unknown and, most of the time, only a few number of samples can be used to estimate these regression coefficients, especially in the case of short run production processes. To adjust the monitoring of quality characteristic profiles under the impact of estimated process parameters in the case of short run production, a single self-starting chart based on the likelihood ratio test and recursive residuals is developed in this paper. This scheme can update the process parameters and monitor the process stability simultaneously, in real-time. Using Monte Carlo simulations, the run length performance of the proposed chart is obtained and compared with other existing charts under various shift case scenarios. Two real case studies from semiconductor and additive manufacturing processes are provided to illustrate and validate the proposed schemes.