성능변수와 대용변수를 이용한 3단계 X 관리도의경제적 설계
Economic Design of Three-Stage X Control Chart Based on both Performance and Surrogate Variables
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Purpose: Two-stage X chart is a useful tool for process control when a surrogate variable may be used together with a performance variable. This paper extends the two-stage X chart to a three stage version by decomposing the first stage into the preliminary stage and the main stage. Methods: The expected cost function is derived using Markov-chain approach. The optimal designs are found for numerical examples using a genetic algorithm combined with a pattern search algorithm and compared to those of the two-stage X chart. Sensitivity analysis is performed to see the parameter effects. Results: The proposed design outperforms the optimal design of the two-stage X chart in terms of the expected cost per unit time unless the correlation between the performance and surrogate variables is modest and the shift in process mean is smallish. Conclusion: Three-stage X chart may be a useful alternative to the two-stage X chart especially when the correlation between the performance and surrogate variables is relatively high and the shift in process mean is on the small side.
Abstract
Purpose: Two-stage X chart is a useful tool for process control when a surrogate variable may be used together with a performance variable. This paper extends the two-stage X chart to a three stage version by decomposing the first stage into the preliminary stage and the main stage. Methods: The expected cost function is derived using Markov-chain approach. The optimal designs are found for numerical examples using a genetic algorithm combined with a pattern search algorithm and compared to those of the two-stage X chart. Sensitivity analysis is performed to see the parameter effects. Results: The proposed design outperforms the optimal design of the two-stage X chart in terms of the expected cost per unit time unless the correlation between the performance and surrogate variables is modest and the shift in process mean is smallish. Conclusion: Three-stage X chart may be a useful alternative to the two-stage X chart especially when the correlation between the performance and surrogate variables is relatively high and the shift in process mean is on the small side.
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