Shewhart-type monitoring schemes with supplementary w-of-w runs-rules to monitor the mean of autocorrelated samples

Self Cite

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.

Section: Processes With Measurement Errors and Autocorrelated Observamentioning

confidence: 91%

Section: Run‐length Properties Of the Rr W‐of‐w S‐and‐m Monitoring Schemementioning

confidence: 99%

“…Using the matrix Q of the transient states in Equation, Shongwe and Malela‐Majika showed that Equation becomes

boldR = ((), \begin{matrix} ς_{1} \\ ς_{2} \\ ς_{3} \\ ⋮ \\ ς_{w - 2} \\ ς_{w - 1} \\ ς_{w} \\ ς_{w + 1} \\ ς_{w + 2} \\ ⋮ \\ ς_{2 w - 3} \\ ς_{2 w - 2} \\ ς_{2 w - 1} \end{matrix}) bold= \frac{1}{D} ((), \begin{matrix} ((), 1 - p_{A}) ((), 1 - p_{C}^{w}) \\ ((), 1 - p_{A}^{2}) ((), 1 - p_{C}^{w}) \\ ((), 1 - p_{A}^{3}) ((), 1 - p_{C}^{w}) \\ ⋮ \\ ((), 1 - p_{A}^{w - 2}) ((), 1 - p_{C}^{w}) \\ ((), 1 - p_{A}^{w - 1}) ((), 1 - p_{C}^{w}) \\ ((), 1 - p_{A}^{w}) ((), 1 - p_{C}^{w}) \\ ((), 1 - p_{C}^{w - 1}) ((), 1 - p_{A}^{w}) \\ ((), 1 - p_{C}^{w - 2}) ((), 1 - p_{A}^{w}) \\ ⋮ \\ ((), 1 - p_{C}^{3}) ((), 1 - p_{A}^{w}) \\ ((), 1 - p_{C}^{2}) ((), 1 - p_{A}^{w}) \end{matrix})

…”

Section: Run‐length Properties Of the Rr W‐of‐w S‐and‐m Monitoring Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Shongwe

Malela‐Majika

Castagliola

2020

Self Cite

“…Shongwe et al presented four one‐sided Shewhart‐

\overset{X}{true}

charts with runs‐rules to monitoring the mean of autocorrelated normally distributed samples under an AR(1) model. Shongwe and Malela‐Majika proposed two Shewhart‐

\overset{X}{true}

charts with w‐of‐w runs‐rules based on the skipping sampling strategy for an AR(1) model. Li et al presented a nonparametric CUSUM chart with a wavelet decomposition for autocorrelated processes without distribution assumptions.…”

Section: Introductionmentioning

confidence: 99%

Exponentially weighted moving average chart with a likelihood ratio test for monitoring autocorrelated processes

Wang

Cheng

2019

In this article, an exponential weighted moving average chart based on a likelihood ratio test is developed to monitor the mean and variance shifts simultaneously for autocorrelated processes. A simple method is used to transform the positively autocorrelated data to the negatively autocorrelated data. The average run length of the proposed chart is derived from a simulation approach. The performance of our proposed chart is compared with some existing charts. The results show that the proposed chart provides better performance for detecting a wide range of shifts in the process mean and variance simultaneously. Additionally, the economic performance of different charts under the first‐order autoregressive model is provided. A real example of a stepper motor in the heating, ventilation, and air conditioning module is used to demonstrate the application of the proposed method.

Klein's supplementary runs rules on X¯$\overline{X}$ chart to control the mean of a Weibull random variable

Quinino

Fernandes

et al. 2021

Most of the papers found in the literature concerned monitoring a process mean of a quality characteristic normally distributed. However, in several practical situations (like in reliability studies), the engineering team also desires to monitor a process mean in which the quality characteristic follows other probabilistic distributions, such as the Weibull. In this paper, we monitor a process mean of a random variable following a Weibull distribution using the traditional X¯$\overline{X}$ control chart. To improve the performance, Klein's two of two supplementary run rule is included in the X¯$\overline{X}$ chart. Such adoption promotes better performance, producing faster signals of abnormal situations (in terms of average run length) than the traditional X¯$\overline{X}$ chart in a great variety of shifts. An empirical example illustrates the proposed scheme.