On monitoring the process mean of autocorrelated observations with measurement errors using the w‐of‐w runs‐rules scheme

Abstract

Runs‐rules have been widely used since the 1950s in industrial and nonindustrial process monitoring applications to improve the performance of basic and other traditional monitoring schemes. However, none of the studies on runs‐rules have accounted for a process with a combined effect of measurement errors and autocorrelation. Hence, in this paper, the use of the w‐of‐w runs‐rules to improve the performance of the Shewhart trueX¯ scheme using an additive model with a constant variance and a first‐order autoregressive model is proposed. To reduce the combined negative effect of measurement errors and autocorrelation, we implement a sampling strategy based on rational subgroups in which (a) multiple measurements per item are taken (instead of a standard single measurement) and (b) non‐neighboring observations are gathered. Moreover, the latter sampling strategy is incorporated into the values of probability elements of a Markov chain matrix which is used to derive some closed‐form expressions for the zero‐ and steady‐state run‐length distribution. The main finding of this study is that, with respect to some overall performance measures, the proposed scheme outperforms the existing Shewhart trueX¯ scheme by a significant margin. A real‐life example is used to illustrate the practical implementation of the proposed scheme.

Effect of measurement errors on triple sampling X¯${{\bar{\bm X}}}$ chart

“…The effect of ME on the variable parameters $$\bar{X}$$ chart's performance was discussed by Sabahno et al 18 . Recent works on the effects of ME on control charts were explored by Sabahno et al, 19 Shongwe et al, 20 Zaidi et al, 21 Arif et al, 22 Nguyen and Tran, 23 Saha et al, 24 Thanwane et al, 25 Mim et al 26 , and Tran et al 27…”

“…14 The synthetic X chart when ME are present was presented by Hu et al 15 The joint monitoring of the mean and variance under ranked-set sampling when ME are present was studied by Ghashghaei et al 16 Yeong et al 17 introduced the one-sided coefficient of variation chart when ME are present. The effect of ME on the variable parameters X chart's performance was discussed by Sabahno et al 18 Recent works on the effects of ME on control charts were explored by Sabahno et al, 19 Shongwe et al, 20 Zaidi et al, 21 Arif et al, 22 Nguyen and Tran, 23 Saha et al, 24 Thanwane et al, 25 Mim et al 26 , and Tran et al 27 Mim et al 28 pointed out that the mathematical formulae provided in He et al 9 in the computation of the ARL of the TS X chart was incorrect and subsequently, presented the correct formulae for computing the ARL of the TS X chart with known and estimated process parameters. An attractive feature of the TS X chart in the context of Statistical Process Control is that if the process quality is deemed as good based on information from the first and second samples, a third sample is not needed, which in turn saves production costs.…”

A New $p$-Control Chart with Measurement Error Correction

“…A typical phenomenon in manufacturer factories is measurement error, which reflects a fact that observed data are different from underlying true ones that are unobservable. In the presence of continuous random variables, some existing works have examined impacts of measurement error to different kinds of control charts, such as EWMA charts (e.g., Maravelakis et al 2004;Asif et al 2020;Nguyen et al 2020;Tran et al 2021), Shewhart control charts (e.g., Nguyen et al 2020;, multivariate process variability (e.g., Huwang and Hung 2007), first-order autoregressive model (e.g., Shongwe et al 2020), and multivariate control charts (e.g., ). However, while those works discussed measurement error effects on control limits, to the best of knowledge, none of them provided suitable strategies to correct for measurement error effects and adjust control limits.…”

The effect of measurement errors on the performance of the homogenously weighted moving average X¯ monitoring scheme

“…Next, Maravelakis et al (2004) and Maravelakis (2012) investigated the effect of measurement errors on the EWMA and CUSUM schemes, respectively; with the effect of a two-component measurement error on the EWMA scheme investigated in Abbasi (2016). For some recent discussions on measurement errors published after the review paper of Maleki et al (2017), see for instance Asif et al (2020), Cheng and Wang (2018), Nguyen et al (2019), Noor-ul-Amin et al (2020), Riaz et al (2019), Sabahno et al (2019, 2020), Salmasnia et al (2018), Shongwe et al (2020a, b, c), Tang et al (2019), Tran et al (2019a, b, c, 2020), Yeong et al (2017), Zaidi et al (2019, 2020).…”