On designing efficient multivariate exponentially weighted moving average control chart for compositional data using variable sample size

Abstract

Conventionally, the standard process monitoring control charts (CCs) 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 ARL0. 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.

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