Monitoring gamma processes using optimal one-sided variable sampling interval multi-region run sum control charts: an application to the packaging industry

In industrial environment, the gamma distribution frequently arises in manufacturing processes such as semiconductor wafer fabrication, packaging operations, chemical processing, and reliability analysis of components. To monitor such cases, this article introduces adaptively optimized control charts for monitoring gamma processes in quality control. Particularly, one-sided variable-sampling-interval (VSI) multi-region run sum (RS) X ¯ charts are theoretically developed for cases where the quality characteristic follows a gamma distribution. The Markov chain framework is employed to formulate the analytical expressions for the time-to-signal metrics of the proposed control charts. Evaluations are conducted under both zero-state and steady-state scenarios in this article. To achieve the optimum detection speed, four unique optimization frameworks and algorithms of the proposed one-sided VSI multi-region RS X ¯ charts under the gamma distribution are formulated. Detailed implementation steps for each framework are also provided. The optimal charting parameters of the proposed charts are determined using iteration of a structured numerical scheme in combination with the nonlinear least-squares method. These parameters improve the efficiency of the gamma process monitoring by enabling early detection and prevention of the out-of-control conditions. From comparative analysis, the proposed one-sided VSI multi-region RS X ¯ charts demonstrate superior detection capability compared to the Shewhart X ¯ , VSI X ¯ and conventional multi-region RS X ¯ charts under the gamma distribution for all shift sizes, as evidenced by shorter time to signal. An application to packaging process data further illustrates the reliability and practical relevance of the proposed charts in industrial monitoring contexts.