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Traditional process monitoring control charts (CCs) focused on sampling methods using fixed sampling intervals ( $ \mathrm {FSI} $ FSI s). The variable sampling intervals ( $ \mathrm {VSI} $ VSI s) 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} $ MEWMA CoDa ) 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 $ ATS 0 . Simulation results demonstrate that the $ \mathrm {VSI} $ VSI - $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMA CoDa CC has significantly decreased the OOC average time to signal ( $ \mathrm {ATS} $ ATS ) than the $ \mathrm {FSI}\,\mathrm {MEWMA}\,\mathrm {CoDa} $ FSI MEWMA CoDa 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} $ MEWMA CoDa 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} $ MEWMA CoDa CC. At the same time, the $ \mathrm {SATS} $ SATS of the $ \mathrm {VSI} $ VSI - $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMA CoDa CC is less than the $ \mathrm {ZATS} $ ZATS of the $ \mathrm {VSI} $ VSI - $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMA CoDa CC for all the values of n and d. The proposed $ \mathrm {VSI} $ VSI - $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMA CoDa CC under steady-State performs effectively compared to its competitors, such as the $ \mathrm {FSI} $ FSI - $ \mathrm {MEWMA}\,\mathrm {CoDa} $ MEWMA CoDa CC, the $ \mathrm {VSI} $ VSI - $ T^2\,\mathrm {CoDa} $ T 2 CoDa CC and the $ \mathrm {FSI} $ FSI - $ T^2\,\mathrm {CoDa} $ T 2 CoDa 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} $ MEWMA CoDa CC. [ABSTRACT FROM AUTHOR]

The variable sampling interval (VSI) scheme is a well‐known technique for improving the detection ability of control charts (CCs). In the proposed study, measurement error (ME) has been applied to investigate the effectiveness of the Hotelling T2 charting scheme for compositional data (CoDa) using VSI. The current study considered the case of the monitoring phase, assuming that process parameters are known using the continuous times Markov chain model. The evaluation of the proposed scheme has been done using the average time to signal. The authors studied the impact of MEs on the performance of the Hotelling T2 CC for CoDa using VSI, and additionally, the authors also examined the impact of linearly covariate error model parameters on the performance of Hotelling T2 VSI CoDa CC. Six cases for the variance–covariance matrix were used to study the impact of different involved parameters. The effect of ME, sampling interval (SI), powering operator (b), error variance (σM)$\sigma _M)$, and subgroup size (m) has been studied on the performance of proposed CCs by increasing the value of one parameter while keeping the other parameters fixed. The ME and σ have a negative impact on the performance of the proposed CC, while SI, b and m have a positive impact on the performance of the proposed CC. Among all the six cases, the correlated cases (negatively and positively) with equal variance performed better than all the other four cases. In the end, an illustrative example of muesli production with p=3$p=3$ part CoDa (where p is the number of variables in CoDa vector) is provided for the practical implementation of the Hotelling T2 VSI CoDa CC in the presence of ME. [ABSTRACT FROM AUTHOR]

Control charts (CCs) are one of the main tools in Statistical Process Control that have been widely adopted in manufacturing sectors as an effective strategy for malfunction detection throughout the previous decades. Measurement errors (M.E's) are involved in the quality characteristic of interest, which can effect the CC's performance. The authors explored the impact of a linear model with additive covariate M.E on the multivariate cumulative sum (CUSUM) CC for a specific kind of data known as compositional data (CoDa). The average run length (ARL) is used to assess the performance of the proposed chart. The results indicate that M.E's significantly affects the multivariate CUSUM-CoDaCCs. The authors have used the Markov chain method to study the impact of different involved parameters using six different cases for the variance-covariance matrix (VCM) (i.e., uncorrelated with equal variances, uncorrelated with unequal variances, positively correlated with equal variances, positively correlated with unequal variances, negatively correlated with equal variances and negatively correlated with unequal variances). The authors concluded that the error VCM has a negative impact on the performance of the multivariate CUSUM-CoDa CC, as the ARL increases with an increase in the value of the error VCM. The subgroup size m and powering operator b positively impact the proposed CC, as the ARL decreases with an increase in m or b. The number of variables p also has a negative impact on the performance of the proposed CC, as the values of ARL increase with an increase in p. For the implementation of the proposal, two illustrated examples have been reported for multivariate CUSUM-CoDaCCs in the presence of M.E's. One deals with the manufacturing process of uncoated aspirin tablets, and the other is based on monitoring the machines involved in the muesli manufacturing process. [ABSTRACT FROM AUTHOR]

Conventionally, the standard process monitoring control charts ( CC s) focused on fixed sample size ( FSS ). An optimal statistical scheme is proposed in this study using a variable sample size (VSS) to enhance the performance of a multivariate exponentially weighted moving average ( MEWMA ) control chart ( CC ) for compositional data ( CoDa ) (i.e. VSSMEWMA - CoDa CC ) based on a coordinate representation using isometric log-ratio transformation ( ilrt). A methodology is proposed to obtain the optimal parameters by considering the zero-state ( ZS ) average run length ( ZARL ) and the steady-state ( SS ) average run length ( SARL ) conditions of the process. The statistical performance of the proposed CC is evaluated based on a continuous-time Markov chain ( CTMC ) method for both cases (i.e. the ZS and the SS ) using a fixed value of in-control (IC) average run length ARL 0 . For benchmarking reasons, the out-of-control ( OOC ) performance of the VSSMEWMA - CoDa CC is compared against the traditional MEWMA - CoDa CC with FSS in terms of ARL ; the proposed CC shows better performance than the FSSMEWMA - CoDa CC . The SARL and ZARL of the VSSMEWMA - CoDa CC are always less than that of the FSSMEWMA - CoDa CC at some certain level of shifts. The proposed VSSMEWMA - CoDa CC performs, on average, 15.25 % ( SS ) and 18.28 % (for ZS ) more effectively than FSSMEWMA - CoDa CC . Moreover, it is found that the number of variables (d) has a negative impact on the run length characteristics of the VSSMEWMA - CoDa CC . When the value of d increases, the OOC SARL and ZARL of the VSSMEWMA - CoDa CC also increase. The SARL of the VSSMEWMA - CoDa CC is less than the ZARL of the VSSMEWMA - CoDa CC for all the combinations of sample size (n) and d, i.e. under SS , the proposed CC performs on average 5.19 % (for d = 3) and 28.8 % (for d = 5) better than the ZS situation. An example of an industrial problem of grid production for a European plant is also given to study the statistical significance and implementation of the VSSMEWMA - CoDa CC over the existing FSSMEWMA - CoDa CC . [ABSTRACT FROM AUTHOR]