Purpose: Two-stage □ 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 □ chart to a three stage version by decomposing the first stage into the p...
Purpose: Two-stage □ 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 □ 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 □ chart. Sensitivity analysis is performed to see the parameter effects. Results: The proposed design outperforms the optimal design of the two-stage 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 □ chart may be a useful alternative to the two-stage □ 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.