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

Abstract

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

Keywords: Average time to signal; compositional data; steady-state; variable sampling interval; zero-state.