Investigating zero-state and steady-state performance of MEWMA-CoDa control chart using variable sampling interval

Traditional process monitoring control charts (CCs) focused on sampling methods using fixed sampling intervals ( $ \mathrm {FSI} $ FSIs). The variable sampling intervals ( $ \mathrm {VSI} $ VSIs) scheme is receiving increasing attention, in which the sampling interval ( $ \mathrm {SI} $ SI) length varies according to the process monitoring statistics. A shorter $ \mathrm {SI} $ SI is considered when the process quality indicates the possibility of an out-of-control (OOC) situation; otherwise, a longer $ \mathrm {SI} $ SI is preferred. The $ \mathrm {VSI} $ VSI multivariate exponentially moving average for compositional data ( $ \mathrm {VSI} $ VSI- $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMACoDa) CC based on a coordinate representation using isometric log-ratio ( $ \operatorname {ilr} $ ilr) transformation is proposed in this study. A methodology is proposed to obtain the optimal parameters by considering the zero-state ( $ \mathrm {ZS} $ ZS) average time to signal ( $ \mathrm {ZATS} $ ZATS) and the steady-state (SS) average time to signal ( $ \mathrm {SATS} $ SATS). The statistical performance of the proposed CC is evaluated based on a continuous-time Markov chain ( $ \mathrm {CTMC} $ CTMC) method for both cases, the $ \mathrm {ZS} $ ZS and the SS using a fixed value of in-control (IC) $ \mathrm {ATS}_0 $ ATS0. Simulation results demonstrate that the $ \mathrm {VSI} $ VSI- $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMACoDa CC has significantly decreased the OOC average time to signal ( $ \mathrm {ATS} $ ATS) than the $ \mathrm {FSI}\,\mathrm {MEWMA}\,\mathrm {CoDa} $ FSIMEWMACoDa CC. Moreover, it is found that the number of variables (d) has a negative impact on the $ \mathrm {ATS} $ ATS of the $ \mathrm {VSI} $ VSI- $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMACoDa CC, and the subgroup size (n) has a mildly positive impact on the $ \mathrm {ATS} $ ATS of the $ \mathrm {VSI} $ VSI- $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMACoDa CC. At the same time, the $ \mathrm {SATS} $ SATS of the $ \mathrm {VSI} $ VSI- $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMACoDa CC is less than the $ \mathrm {ZATS} $ ZATS of the $ \mathrm {VSI} $ VSI- $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMACoDa CC for all the values of n and d. The proposed $ \mathrm {VSI} $ VSI- $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMACoDa CC under steady-State performs effectively compared to its competitors, such as the $ \mathrm {FSI} $ FSI- $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMACoDa CC, the $ \mathrm {VSI} $ VSI- $ T^2\,\mathrm {CoDa} $ T2CoDa CC and the $ \mathrm {FSI} $ FSI- $ T^2\,\mathrm {CoDa} $ T2CoDa CC. An example of an industrial problem from a plant in Europe is also given to study the statistical significance of the $ \mathrm {VSI} $ VSI- $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMACoDa CC.