A Unified Framework for Statistical Quality Control: Classical, Memory-Type, Multivariate, and Bayesian Control Charts with Simulation-Based Evaluation

Statistical Quality Control (SQC) plays a central role in monitoring and improving the stability of production and service processes. This study presents a comprehensive and unified framework for statistical control charts, encompassing classical Shewhart charts, memory-type charts such as EWMA and CUSUM, multivariate control charts, and Bayesian control charts. The paper provides detailed mathematical formulations for each class of control charts, allowing clear theoretical comparison and practical implementation. To assess performance, a Monte Carlo simulation study is conducted using the Average Run Length (ARL) criterion under both in-control and out-of-control conditions. The simulation results highlight the relative strengths and limitations of each charting method, particularly in detecting small and moderate process shifts. In addition, realistic synthetic datasets are analyzed to demonstrate the practical applicability of the proposed framework and to illustrate how different control charts respond to process changes in real monitoring scenarios. The findings confirm that while classical Shewhart charts remain effective for detecting large shifts, memory-type and Bayesian charts provide superior sensitivity for small and gradual changes, especially in complex or uncertain environments. Overall, this study offers a systematic reference for researchers and practitioners seeking to select appropriate control charts based on process characteristics, data structure, and monitoring objectives.