Optimal designs of the synthetic t chart with estimated process mean

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The synthetic t chart, which integrates a t chart and a conforming run length chart, is robust against changes in the process standard deviation. Traditionally, the synthetic t chart is studied by assuming that the in-control process mean is known. Practically, this is not always the case. The process mean is rarely known and it has to be estimated from a Phase-I dataset. Therefore, this paper presents the Markov chain approach for studying the run-length properties of the synthetic t chart with estimated process mean for both zero- and steady-state cases. The impact of the mean estimation on the synthetic t chart is evaluated and compared with its known-process-mean counterpart and the synthetic X¯ chart. For optimum implementation, this paper develops two optimal design strategies for the synthetic t chart with estimated process mean, by minimizing (i) the average run length (ARL) and (ii) the expected ARL, for deterministic and unknown shift sizes, respectively. By taking the number of Phase-I samples and sample sizes adopted in practice into consideration, tables listing the new optimal charting parameters of the proposed chart are provided in this paper. Comparative studies show that there are some potential benefits, especially the desirable robustness property, of the synthetic t chart with estimated process mean over the synthetic X¯, Shewhart X¯ and t charts with estimated process parameters or mean. The application of the synthetic t chart with estimated process mean is illustrated with real industrial data gathered from a silicon epitaxy process.