ABSTRACT
Traditional process monitoring control charts (CCs) focused on sampling methods using fixed sampling intervals (FSIs). The variable sampling intervals (VSIs) scheme is receiving increasing attention, in which the sampling interval (SI) length varies according to the process monitoring statistics. A shorter SI is considered when the process quality indicates the possibility of an out-of-control (OOC) situation; otherwise, a longer SI is preferred. The VSI multivariate exponentially moving average for compositional data (VSI-MEWMACoDa) CC based on a coordinate representation using isometric log-ratio (ilr) transformation is proposed in this study. A methodology is proposed to obtain the optimal parameters by considering the zero-state (ZS) average time to signal (ZATS) and the steady-state (SS) average time to signal (SATS). The statistical performance of the proposed CC is evaluated based on a continuous-time Markov chain (CTMC) method for both cases, the ZS and the SS using a fixed value of in-control (IC) ATS0. Simulation results demonstrate that the VSI-MEWMACoDa CC has significantly decreased the OOC average time to signal (ATS) than the FSIMEWMACoDa CC. Moreover, it is found that the number of variables (d) has a negative impact on the ATS of the VSI-MEWMACoDa CC, and the subgroup size (n) has a mildly positive impact on the ATS of the VSI-MEWMACoDa CC. At the same time, the SATS of the VSI-MEWMACoDa CC is less than the ZATS of the VSI-MEWMACoDa CC for all the values of n and d. The proposed VSI-MEWMACoDa CC under steady-State performs effectively compared to its competitors, such as the FSI-MEWMACoDa CC, the VSI-T2CoDa CC and the FSI-T2CoDa CC. An example of an industrial problem from a plant in Europe is also given to study the statistical significance of the VSI-MEWMACoDa CC.
ABSTRACT
Recent researches on the control charts with unknown process parameters have noticed the large variability in the conditional in-control average run length (ARL) performance of control charts, especially when a small number of Phase I samples is used to estimate the process parameters. Some research works have been conducted on the conditional ARL performance of different types of control charts. In this paper, by simulating the empirical distribution of the conditional ARL and especially using the exceedance probability criterion (EPC), we study the conditional ARL performance of the synthetic [Formula: see text] chart. Our results show that a large amount of Phase I samples is needed to obtain a specified EPC of the synthetic chart. For the available number of Phase I samples, the control limits of the synthetic chart are adjusted using the EPC method to improve its conditional in-control performance. It is shown that, for small mean shift sizes, a tradeoff should be made between the conditional in-control and out-of-control performances. For moderate to large shifts, the conditional performance of the synthetic chart using the adjusted control limits is generally satisfied. By comparing the results with the ones using the bootstrap approach, it can also be concluded that the conditional performances of both approaches are comparable. While the method proposed in this paper requires much less computation work than the bootstrap approach.
