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Shan, TingTing; Huang, WeiDong

doi:https://doi.org/10.1155/2021/8405059

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 8405059 | https://doi.org/10.1155/2021/8405059

Treating Measurement Errors in the Run Rule Schemes Integrated with Shewhart Chart

TingTing Shan¹and WeiDong Huang¹

Academic Editor: Kauko Leiviskä

Received15 Apr 2021

Revised08 Jun 2021

Accepted11 Jul 2021

Published04 Aug 2021

Abstract

In modern quality control applications, there often exist significant measurement errors because observations are measured quickly in time order. As a result, the errors influence the power of a control chart to detect a given change in the process parameter(s) of a quality characteristic. In this paper, by using a covariate error model, the properties of the Shewhart chart integrated with run rules are investigated when errors exist in the measurement of quality characteristic. Two metrics, the average run length and 95% quantile of the run length, are adopted to evaluate the chart’s performance for different mean shifts and sample sizes. Numerous simulations are conducted, and the results indicate that the errors in the measurement significantly affect the performance of the run rule chart, especially when the errors are large. To reduce this negative effect on the run rule chart, measuring more times of each item in each subgroup and increasing the coefficient in the covariate error model are shown to be good choices for practitioners.

1. Introduction

As it is known in statistical process control (SPC), the Shewhart chart is effective in detecting large shifts in the process but is insensitive to small or moderate shifts. Even though the Shewhart-type charts still received much attention for the simple representation and implementation, as an intermediate solution, Western Electric [1] suggested the run rule scheme integrated with Shewhart-type charts to improve the chart’s performance. Then, run rules combined with control charts were studied by many researchers, such as Divoky and Taylor [2]; Champ and Woodall [3]; Klein [4]; Fu et al. [5]; and so on. More recent works on control charts integrated with run rule schemes can be found in Hu and Castagliola [6]; Chew et al. [7]; Shongwe and Graham [8]; Shongwe and Graham [8]; Shongwe et al. [9]; Khaw and Chew [10]; Chew et al. [11]; Shongwe [12]; Malela-Majika and Rapoo [13]; Malela-Majika et al. [14]; Hu et al. [15]; and Ali et al. [16]. All these existing research studies focused on how to improve the chart’s performance by incorporating run rules. They all assumed that observations are measured with no errors in the quality characteristic.

However, in the actual measuring equipment, an exact measurement of a quality characteristic is a rare phenomenon, causing measurement errors in practice. The properties of a control chart for monitoring observations with measurement errors differ from the chart for monitoring observations with precise measurements. Most existing works have only studied the measurement errors’ effect on a specific monitoring scheme. For example, by using a linear covariate model, Linna and Woodall [17] investigated the measurement errors’ effect on the Shewhart mean () and variance () charts. By extending the covariate model to the multivariate case, Linna et al. [18] studied the performance of multivariate control charts when measurement errors exist. Using the same linear model introduced in Linna and Woodall [17], the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts’ performance for monitoring observations with measurement errors was studied in Maravelakis et al. [19] and Maravelakis [20]. More recently, considering the autocorrelation and measurement errors, Costa and Castagliola [21] studied the Shewhart chart’s performance and suggested a new skipping sampling technique to reduce the negative effect of autocorrelation. Hu et al. [22, 23] investigated the overall performance of the synthetic and VSS chart when measurement errors exist. For a full account of publications on measurement errors prior to 2017, readers are referred to Maleki et al. [24]. For the most recent works on control charts with measurement errors, we suggest the research in Cheng and Wang [25]; Shongwe et al. [26]; Nguyen et al. [27]; Nguyen et al. [28]; Thanwane et al. [29]; Thanwane et al. [30]; and so on. Even the above research works are on the control charts with measurement errors. To the best of our knowledge, no research on the run rule schemes integrated with Shewhart chart with measurement errors can be found.

In this paper, we focus on the properties of 2-of-3 run rules integrated with Shewhart-type charts in the presence of measurement errors. The run rule schemes are set by using warning limits, and it gives a signal when the selected run rule pattern occurs. This research can be extended to other run rule schemes, which are not presented here. It is noted that this paper is mainly to investigate the run rule chart’s performance when measurement errors exist, not to increase the chart’s performance.

This paper is organized as follows. In Section 2, the linearly covariate error model used to represent the true value and the measurement value of observations is introduced. Then, the 2-of-3 run rule chart under this measurement error model is presented in Section 3. The detailed effect of measurement errors on the run rule chart is presented in Section 4. Finally, some conclusions and future research directions are given in Section 6.

2. The Linearly Covariate Measurement Error Model

In a process, consecutive items of the quality characteristic are collected at each sampling point . These items are assumed to be independent normal random variables with mean and standard deviation , i.e., . is the mean shift parameter in the process. Due to the measurement errors, the true value of quality characteristic is observed from measurements , with each being equal to the following equation:where and are two constants from the analysis of measurement device and is an independent random error term.

In practice, if there are errors in the measurements of observations, it is suggested to measure each item several times and average these measurements for each item. By doing this, a smaller variance in the measurement error component than for a single measurement can be obtained. This can be simplified to the following equation:where is the number of measurements per item. It can be derived that the expection and the variance of ,

Moreover, considering that the measurement error variance sometimes linearly depends on the process mean , is normally distributed with mean 0 and variance , with two known constants and . and are derived to be

3. The Shewhart Chart Integrated with Run Rules under Linearly Covariate Measurement Errors

Considering the constant measurement error variance in equations (3) and (4), by setting in the in-control state, two warning limits, (the lower warning limit) and (the upper warning limit), are given as follows:

Second, considering the linearly increasing measurement error variance in equations (5) and (6), the warning limits of the chart are given as

In the implementation of the Shewhart chart integrated with 2-of-3 run rules, if two out of the three successive points fall above or fall below , an out-of-control signal is triggered. By modeling the 2-of-3 run rule chart with a Markov chain, Figure 1 shows all the transient states corresponding to the 2-of-3 run rules.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

When a new sample is collected and the corresponding sample mean (point) falls into one of the above seven transient states or into the 8th state, i.e., the absorbing state, where two out of the three successive points fall outside [, ], then the run length properties of the Shewhart chart integrated with 2-of-3 run rules can be obtained by using the Markov chain matrix :with the probabilities , , and . refers to the transient probabilities between 7 states in the figure and the vector , with . For more details of the run rule charts, readers can refer to the related research works introduced in Section 1.

For the case in equations (3) and (4), the transient probabilities are equal towhere is the measurement error ratio and is the cumulative distribution function (c.d.f.) of the standard normal distribution.

For the case in equations (5) and (6), the transient probabilities are equal to

From equations (10)–(15), it is shown that the parameter A has no influence on the chart, while in practice, as it has been explained in Linna and Woodall [17], this parameter should be taken into account for the process location or the process capability.

Using the transition matrix in the Markov chain, the probability mass function (p.m.f.) and the c.d.f. of the run length distribution of the chart with measurement errors are given aswith being the initial probabilities of transient states.

Furthermore, the of the chart can be derived asand the quantile of the Shewhart chart with 2-of-3 run rules under linearly covariate measurement error model can be obtained as (see Gan [31])

With equations (18) and (19), any quantile of the run length distribution can be computed. By setting , the 95% quantile of the run length distribution can be obtained.

4. Influence of Linearly Covariate Measurement Errors on the Run Rule Shewhart Chart

Some characteristics of run length are usually used to show the control chart’s performance. Two metrics, and , are selected to evaluate the performance of the run rule chart. As it has been stated in Khoo et al. [32], by computing a higher percentile of the run length distribution, say in this paper, a practitioner can state with 95% confidence that when a process shift occurs (see the case , in Table 1), a signal will be triggered by the 50th sample. The information from the percentiles of the run length acts as practical guidance to study the behavior of control charts. The subscripts “0” and “1” of represent the corresponding in-control and out-of-control performance of the chart, respectively. Without loos of generality, it is assumed that . Consequently, the parameter of the run rule Shewhart chart with linearly covariate measurement errors satisfies the following constraint:

In this case, assuming in equations (10)–(15), we can compute the only value of W using the “vert solvevert” function in MATLAB, irrespective of , , , and .

4.1. A Constant Measurement Error Variance

It can be seen from equations (17)–(19) that and are functions of , , , , , and . Table 1 presents and of the run rule chart with linearly covariate error model for , , and when and . The case corresponds to the “no measurement error” case. For example, when , , and , the chart parameter . With this value, we have and at . As it can be seen from Table 1, for a specified , and increase when the measurement error ratio increases. This fact demonstrates the negative effect of the measurement errors on the run rule chart. For instance, for the example above but with , we have and . We can see that with the measurement error ratio increasing, larger and are needed to detect a shift.

The conclusions drawn above can also be seen from the four graphics in Figure 2 where and are plotted for , and when and . This figure confirms the conclusion that larger and are observed with increasing measurement errors.

(a)

(b)

(c)

(d)

Table 2 presents and of the chart for , , and when and . It can be seen that for specified and , the negative effect of measurement errors on the run rule chart decreases as increases. Moreover, through the comparison between Tables 1 and 2, it can be seen that when , the difference between and corresponding to (no measurement error) and is negligible. For instance, when , , , , and (Table 2), and , while in Table 1, for the same values of and , and when .

In Figure 3, and of the run rule chart in the presence of measurement errors are plotted for , , and when and . This figure also confirms the conclusion that with the value of increasing, both and decrease.

(a)

(b)

(c)

(d)

Table 3 presents and for , , and when and . It can be seen from Table 3 that as increases, the negative effect of the measurement errors on the run rule chart decreases. For instance, for the case where , , , and , and decrease down to and , respectively, when increases up to . So, it is concluded that if there is a relative high variability of the measurement error compared with the process variability, a large is needed to reduce the negative effect of measurement errors on the run rule chart.

In Figure 4, and of the run rule chart in the presence of measurement errors are plotted for , , and when and . This figure confirms the conclusion that by measuring the sample item several times, the negative effect of measurement errors on the run rule chart can be reduced to some extent.

(a)

(b)

(c)

(d)

Since measuring the item in each sample several times can reduce the measurement errors’ negative effect on control charts, the run length performance of run rule chart with linearly covariate error is presented in Figure 5, for the case of measurements per item, using , and when . It can be noted that for fixed , with measurements per item, and only have a slightly increasing trend with increasing. Moreover, through the comparisons between the four graphics in Figure 2, the negative effect of measurement errors on the run rule chart is reduced by taking multiple measurements of , especially for the cases when .

(a)

(b)

(c)

(d)

Finally, Figure 6 presents the run length performance of the run rule chart for , , and when . From Figure 3, it is known that increasing reduces the negative effect of measurement errors on the run rule chart. Meanwhile, through the comparison of Figures 3 and 6, it is noted that taking measurements and increasing enable the negative effect of measurement errors on the run rule chart to be diminished faster compared to the cases in Figure 3.

(a)

(b)

(c)

(d)

4.2. Linearly Increasing Measurement Error Variance

Considering the linearly increasing variance case, it can be seen that parameter C in equations (5) and (6) behaves like in equation (4). If there is a shift in the process mean, then it is amplified by parameter , which also changes the variance of the measurement error. So, the influence of on the run rule chart is investigated in Figure 7. In this figure, for specified values of , , , , , and are set for illustration. Through this figure, it is seen that increasing can significantly increase the values of and . This fact indicates the negative effect of linearly increasing variance on the run rule chart. For instance, if and , when increases up to , and increase up to and , respectively.

(a)

(b)

(c)

(d)

5. An Illustrative Example

Similar to the research work in Costa and Castagliola [21], a yogurt cup filling process is used here to illustrate the implementation of the chart integrated with run rules when measurement error exists. The critical quality characteristic in the filling process is the weight of each yogurt cup. From the historical database of yogurt cup weights (Phase I), the in-control mean and the standard deviation are estimated accurately. Moreover, an independent R&R (repeatability and reproducibility) study has showed an estimated measurement standard deviation , which leads to . By fixing the parameter settings and in the linearly covariate error model, the quality practitioner decides to use Shewhart chart integrated with 2-of-3 run rules to monitor the process. Taking yogurt cups and measuring each item times, in a sample, the parameter W of the run rule chart can be calculated as W = 1.9293 using equation (20). Then, the warning limits in equations (7) and (8) are given as follows:

In addition, 20 samples corresponding to a sequence of production (Phase II) are recorded in Table 4. We have 10 values for each sample, i.e., the weight of yogurt cups weighed times. The mean of these samples is presented in the right side of Table 4. In Figure 8, these mean values with the warning limits and are also plotted. As it can be seen in the figure, the Shewhart chart with 2-of-3 run rules actually triggers an out-of-control signal from samples 11 and 12 (shown in the frame), suggesting a downward shift in the process mean. One possible reason is that the pipe used for filling the cups is clogged.

6. Conclusion

The properties of Shewhart chart integrated with run rules are investigated when errors exist in the measurement of observations in a quality characteristic. By using a linearly covariate error model, of the chart is derived using a Markov chain approach. The simulations results show that of the run rule Shewhart chart is seriously affected by the measurement errors. With the measurement error increasing, the run rule Shewhart chart’s performance deteriorates. To reduce the negative effect of measurement errors, it is beneficial to measure each item in each subgroup more times and increase the coefficient in the linearly covariate error model whenever possible. Both are good alternatives to implement the run rule Shewhart chart. It should be careful that measuring each item more times practically causes more cost.

All the results are based on the covariate error model, and it would be interesting to extend our research to different control charts with some other measurement error models.

Data Availability

All relevant data are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the National Social Science Foundation of China (no. 20CRK004), Social Science Foundation of Jiangsu (no. 17GLC001), Humanity and Social Science Foundation of Ministry of Education of China (no. 19YJC630025), Key Research Base of Philosophy and Social Sciences in Jiangsu-Information Industry Integration Innovation and Emergency Management Research Center, and Excellent Innovation Teams of Philosophy and Social Science in Jiangsu Province (2017ZSTD022).

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Copyright © 2021 TingTing Shan and WeiDong Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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