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Journal ArticleDOI

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

01 Feb 2021-Transactions of the Institute of Measurement and Control (SAGE PublicationsSage UK: London, England)-Vol. 43, Iss: 3, pp 728-745
TL;DR: It is observed that as the smoothing parameter increases, measurement errors have a higher negative effect on the performance of the HWMA X ¯ scheme and it is shown that the negative effect of measurement errors is reduced by using multiple measurements and or by increasing the slope coefficient of the covariate error model.
Abstract: Monitoring schemes are typically designed under the assumption of perfect measurements. However, in real-life applications, data tend to be subjected to measurement errors, that is, a difference between the real quantities and the measured ones mostly exist even with highly sophisticated advanced measuring instruments. Thus, in this paper, the negative effect of measurement errors on the performance of the homogenously weighted moving average (HWMA) scheme is studied using the linear covariate error model for constant and linearly increasing variance. Monte Carlo simulations are used to evaluate the performance of the proposed HWMA scheme in terms of the run-length characteristics. It is observed that as the smoothing parameter increases, measurement errors have a higher negative effect on the performance of the HWMA X¯¯¯ scheme. More importantly, it is shown that the negative effect of measurement errors is reduced by using multiple measurements and/or by increasing the slope coefficient of the covariate error model. Moreover, the performance of the HWMA X¯¯¯ scheme is compared with the corresponding exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) X¯¯¯ schemes. An illustrative example is provided to help in implementing this monitoring scheme in a real-life situation.

Summary (3 min read)

Introduction

  • In statistical process monitoring (SPM), control charts are used to identify the causes of variation in the process.
  • Two sources of variation can be distinguished in SPM, namely the common (or chance) causes and the assignable (or special) causes of variation.
  • When the process runs in the presence of common causes only, the process is considered to be in-control (IC).
  • Raza et al. (2020) proposed a distribution-free HWMA scheme based on the sign and signed-rank statistics to monitor skewed and symmetric distributions observations.
  • 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) .

Design of the HWMA ̅ scheme

  • Abbas (2018) showed that Equation (1) can also be written as EQUATION From Equation ( 2), it can be seen that the HWMA ̅ statistic assigns weight to the current sample and a weight is equally distributed to the previous samples.
  • In case the process has been running for a long time (i.e. ), the term .

Covariate error model with a constant variance

  • Also, denotes the number of measurements taken in each sampled subgroup unit and is a random error due to the measurement error that is distributed independently of ; where is the variance of the measurement system.
  • Let represents the standardized ratio of the measurement system variability to the process variability.
  • As increases, the variance in the measurement error component decreases.
  • Hence, it is obvious that when the number of multiple components tends to infinity, the variance in the measurement component tends to zero.
  • The number of sets of measurements needs to be determined such that the maximum reduction in the variance of the measurement system is reached and, at the same time, minimizes the cost of using multiple measurements.

Sensitivity analysis

  • The effect of measurement errors and multiple measurements on the performance of the HWMA ̅ scheme is investigated in terms of the ARL and SDRL profiles for specific shifts and EARL profile for different ranges of shifts.
  • Next, the level of measurements errors ( ) indicates the level of severity of the measurement error, where = 0 implies perfect measurements (i.e. no measurement error), = 0.2 indicates lower level of measurement error, = 0.5 indicates moderate level of measurement errors and = 0.9 indicates higher level of measurement error.
  • This pattern holds for the EARLs which show that as increase, the performance of the HWMA ̅ scheme deteriorates.
  • Moreover, for each cluster of line graphs in Figures 1(a ) and (b), the smaller the value of , the lower are the ARL profiles as compared to those with higher values of .
  • As increases, the IC SDRL values increase towards the nominal value.

Table 1:

  • Thirdly, it is also shown in Table 3 that the ARLs and EARLs are lower when =4 than those when =1, indicating a reduction in the negative effect of measurement errors as increases.
  • Finally, although Table 3 is illustrated for =0.1 and =5 only, this pattern holds for other values of and , whenever 0 and 0.
  • A similar pattern is observed for the corresponding EARLs.
  • This shows that when increases there is a deterioration in the performance of the HWMA ̅ scheme.

Table 4:

  • Moreover, the %Decrease in the performance is larger for very small shifts when is small; however, it is smaller for moderate and large values of when is small and the converse is true for large values of .
  • The %Decrease in the performance of the HWMA ̅ scheme reaches its maximum point when 1 for moderate values of and =1.75 for large values of .
  • The minimum point is attained for very small shift values.
  • Note that for small values of , the %Decrease in the performance of the HWMA ̅ scheme reaches its maximum point in the interval 0 0.25.

Example 1: Yogurt cup filling process

  • In order to illustrate the implementation of the HWMA ̅ scheme with measurement errors, the data from Costa and Castagliola (2011) shown in Table 6 is used, assuming that =0 and =1 and that the data is subjected to a constant variance in the measurement system.
  • The data is based on a yogurt cup filling process where the quality characteristic is the weight of each yogurt cup.
  • The rest of the plotting statistics of the HWMA ̅ scheme with 2-measurements are empirically shown in To illustrate the negative effect of increasing measurement errors from =0 to =0.9 without the use of multiple measurements, consider the dataset from Montgomery (2013) on the inside diameters in millimeter (mm) of piston rings manufactured by a forging process.
  • It is observed that, for the same dataset, when =0 and 0.9, the HWMA ̅ scheme gives the first OOC signal at the sample number 12 and 13, respectively.
  • That is, HWMA ̅ scheme in Figure 9 shows that the control limits for =0.9 are wider than those of =0.

Conclusion

  • Most of the SPM schemes are based on the assumption of known process parameters under perfect measurements.
  • This paper contributes to the SPM literature with an extensive investigation of the performance (or sensitivity) of the HWMA ̅ scheme to monitor the process mean under the assumption of imperfect measurements using a constant and linearly increasing variance error model in the measurement system.
  • The HWMA scheme is superior to the EWMA scheme under small shifts only.
  • In terms of the overall performance measure, the HWMA scheme outperforms the EWMA scheme for small, small-to-moderate and small-to-large shifts in the process mean.
  • The latter performs better than the HWMA scheme under moderate, large and moderate-to-large shifts in the process mean.

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Submitted on 15 Jul 2021
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The eect of measurement errors on the performance of
the homogenously weighted moving average
monitoring scheme
Maonatlala Thanwane, Jean-Claude Malela-Majika, Philippe Castagliola,
Sandile Charles Shongwe
To cite this version:
Maonatlala Thanwane, Jean-Claude Malela-Majika, Philippe Castagliola, Sandile Charles Shongwe.
The eect of measurement errors on the performance of the homogenously weighted moving average
monitoring scheme. Transactions of the Institute of Measurement and Control, SAGE Publications,
2021, 43 (3), pp.728-745. �10.1177/0142331220973569�. �hal-03129234�

1
The effect of measurement errors on the performance of the homogenously weighted moving
average
monitoring scheme
Maonatlala Thanwane
1
, Jean-Claude Malela-Majika
1
, Philippe Castagliola
2
, Sandile Charles Shongwe
1
1
Department of Statistics, College of Science, Engineering and Technology, University of South Africa,
PO Box 392 UNISA 0003, Pretoria, South Africa
2
Département Qualité Logistique Industrielle et Organisation, Université de Nantes & LS2N UMR
CNRS 6004, Nantes, France.
Abstract
Monitoring schemes are typically designed under the assumption of perfect measurements. However, in
real-life applications, data tend to be subjected to measurement errors, i.e., a difference between the real
quantities and the measured ones mostly exist even with highly sophisticated advanced measuring
instruments. Thus, in this paper, the negative effect of measurement errors on the performance of the
homogenously weighted moving average (HWMA) scheme is studied using the linear covariate error
model for constant and linearly increasing variance. Monte Carlo simulations are used to evaluate the
performance of the proposed HWMA scheme in terms of the run-length characteristics. It is observed that
as the smoothing parameter increases, measurement errors have a higher negative effect on the
performance of the HWMA
scheme. More importantly, it is shown that the negative effect of
measurement errors is reduced by using multiple measurements and / or by increasing the slope
coefficient of the covariate error model. Moreover, the performance of the HWMA
scheme is compared
with the corresponding exponentially weighted moving average (EWMA) and Cumulative Sum
(CUSUM)
schemes. An illustrative example is provided to help in implementing this monitoring
scheme in a real-life situation.

2
Keywords: Homogenously Weighted Moving Average scheme; Linear covariate error model; Linearly
increasing variance; Measurement error; Multiple measurements.
Introduction
In statistical process monitoring (SPM), control charts are used to identify the causes of variation in the
process. Two sources of variation can be distinguished in SPM, namely the common (or chance) causes
and the assignable (or special) causes of variation. Common causes cannot be avoided, while assignable
causes of variation need to be reduced as much as possible. When the process runs in the presence of
common causes only, the process is considered to be in-control (IC). Otherwise, the process is said to be
out-of-control (OOC). When practitioners are interested in monitoring small-to-moderate shifts in the
process parameters, popular memory-type monitoring schemes such as the Cumulative Sum (CUSUM)
and exponentially weighted moving average (EWMA) schemes are mostly recommended; see for
example, Roberts (1959), Page (1961) and Montgomery (2013). Many authors devoted their valuable
time in improving the sensitivity of the CUSUM and EWMA schemes using various techniques. These
enhanced schemes include the double CUSUM (DCUSUM), hybrid EWMA (HEWMA), double EWMA
(DEWMA), Synthetic CUSUM, Synthetic EWMA, etc. For more details on the enhancement of memory-
type schemes, readers are referred to Waldmann (1996), Capizzi and Masarotto (2010), Abbas et al.
(2013), Haq et al. (2013), Ali and Haq (2017), Adeoti (2020), Malela-Majika (2020); just to cite a few.
For other alternative approaches of control charts, such as the use of divergence functions (e.g. parametric
and nonparametric Kullback-Leibler Divergence), see for instance Bakdi and Kouadri (2018), Bakdi et al.
(2019) and Bounoua et al. (2020).
More recently, Abbas (2018) developed a new memory-type scheme that allocates a specific weight to the
current sample and the remaining weight is distributed equally among the previous samples; this scheme

3
is known as homogeneously weighted moving average (HWMA) monitoring scheme. The HWMA
scheme is in its nature a memory-type scheme used to effectively monitor small-to-moderate shifts (see
for example the following articles on the HWMA-type monitoring schemes: Abbas (2018), Adegoke et al.
(2019a, b), Raza et al. (2020), Abid et al. (2020), Abbas et al. (2020) and Adeoti and Koleoso (2020)). To
provide an efficient and unbiased estimate of the process mean, Adegoke et al. (2019a) developed a
HWMA scheme to monitor the process mean that uses the auxiliary variable in the form of a bivariate
regression estimator. Next, Adegoke et al. (2019b) proposed a multivariate HWMA scheme for
monitoring the process mean vector when the underlying distribution parameters are known; more
recently though, Abbas et al. (2020) studied the same scheme when the underlying distribution
parameters are assumed unknown and they compared its performance against numerous well-known
multivariate schemes. Raza et al. (2020) proposed a distribution-free HWMA scheme based on the sign
and signed-rank statistics to monitor skewed and symmetric distributions observations. More recently,
Abid et al. (2020) proposed the double HWMA scheme for monitoring small shifts in the process mean
and they also investigated the effect of non-normality and parameter estimation on the performance of the
double HWMA scheme. Finally, Adeoti and Koleoso (2020) proposed a hybrid HWMA schemes for
monitoring the process mean and they also investigated the effect of non-normality. Note that the double
(hybrid) design of the HWMA scheme entails applying the same (different) smoothing parameter twice,
respectively. The key difference between the HWMA scheme proposed in this paper and the
abovementioned HWMA schemes is that it is not assumed that the observations have perfect or exact
measurements. That is, in this paper, the assumption given in the review paper by Maleki et al. (2017) is
followed: ‘… exact measurements in real-life applications are a rare phenomenon, even with highly
sophisticated advanced measuring instruments; hence, measurement errors tend to exist in any
manufacturing and service environment’. Therefore, this paper contributes to the SPM literature by

4
introducing an HWMA scheme that accounts for measurement errors in the process being monitored for a
univariate process mean.
A detailed early account of 60 articles on monitoring schemes with measurements errors are documented
in Maleki et al. (2017). To discuss a few, Linna and Woodall (2001) studied the effect of measurement
errors on Shewhart monitoring scheme and they reported that under measurement errors a monitoring
scheme is exposed to lose power in detecting parameters shifts. 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: Yeong et al. (2017), Cheng and Wang (2018), Salmasnia et al.
(2018), Tang et al. (2019), Riaz et al. (2019), Tran et al. (2019a, b, c, 2020), Nguyen et al. (2019), Zaidi et
al. (2019, 2020), Sabahno et al. (2019, 2020), Shongwe et al. (2020a, b, c), Asif et al. (2020), Noor-ul-
Amin et al. (2020).
A number of methods used as remedial approaches are outlined in the review article on measurement
errors by Maleki et al. (2017) for other remedial sampling strategies, see for instance the book by Aslam
and Ali (2019). The most used methodology to reduce measurement inaccuracy is by taking multiple
measurements of each item, which was first proposed by Linna and Woodall (2001). The multiple
measurements strategy reduces the effect of measurement errors on the performance of monitoring
schemes. That is, taking at least two measurements for each sampled unit effectively reduces the effect of
the measurement errors. The level of precision improves by taking and averaging several measurements.
Although it is preferable to maintain a larger number of multiple measurements for better results, one
needs to be mindful of additional implications such as costs and time to collect these observations. This is
so because, without measurement error, multiple measurements will become redundant in the monitoring
scheme methodology by only adding costs for measuring extra and useless observations.

Citations
More filters
Book ChapterDOI
01 Jan 1985
TL;DR: In this paper, the authors present a charting procedure for detecting abrupt changes in process level that remain with the process for about five sampling periods, which is known as cumulative sum control chart.
Abstract: The control charting procedures discussed in Chapters 6 and 10 are based on predetermined levels of significance at which warning or action is taken. Alternative conditions may arise where the evidence for talking action is based on the result of several samples, which individually would not be significant. In the case presented here, a differing procedure must be found for accumulating this evidence. A charting method to achieve this was proposed by Ewan and Kemp — ‘Sampling inspection of continuous processes with no autocorrelation between successive results’ (Biometrika 1960,47, 363). These charts are known as ‘cumulative sum’ charts and the values entered represent the accumulated sum of all data to date (instead of, as previously considered, the sample value just obtained). The cumulative sum control chart is particularly well-suited to detecting abrupt changes in process level that remain with the process for about five sampling periods. Usually, each point on the chart equals the previous point, plus the value of a statistic computed from the last subgroup, hence the name ‘cumulative sum’.

66 citations

Journal ArticleDOI
TL;DR: In this article , the performance of homogeneously weighted moving average (HWMA) type control charts has been investigated under zero and steady states at various shifts, and a detailed comparative analysis of the HWMA chart with time-varying limits is carried out using the Monte Carlo simulation method.
Abstract: In the recent literature of process monitoring, homogeneously weighted moving average (HWMA) type control charts have become quite popular. These charts are quite efficient for early detection of shifts, especially of smaller magnitudes, in process parameters such as location and dispersion. A recent study pointed out a few concerns related to HWMA charts that mainly relate to its steady-state performance. It needs to be highlighted that the initial studies on HWMA focused only on the zero-state performance of the chart relative to other well-known memory charts. This study reinvestigates the performance of the HWMA chart under zero and steady states at various shifts. Using the Monte Carlo simulation method, a detailed comparative analysis of the HWMA chart is carried out relative to the exponentially weighted moving average (EWMA) chart with time-varying limits. For several values of design parameters, the in-control and out-of-control performance of these charts is evaluated in terms of the average run length (ARL). It has been observed that the structure of the HWMA chart has the ability to safeguard the detection ability and the run-length properties under various delays in process shifts. More specifically, it has been found that HWMA chart is superior to the EWMA chart for several shift sizes under zero state and is capable of maintaining its dominance in case the process experiences a delay in shift. However, the steady-state performance depends on the suitable choice of design parameters. This study provides clear cut-offs where HWMA and EWMA are superior to one another in terms of efficient monitoring of the process parameters.

7 citations

Journal ArticleDOI
TL;DR: In this paper , the authors investigated the effect of measurement errors on the performance of four well-established combined charts for monitoring the mean of normally distributed processes (Shewhart-CUSUM, ShewhartCrosier’s CUSUM and GWMA) and found that measurement errors significantly reduce the power of the combined charts.

5 citations

References
More filters
Journal ArticleDOI
TL;DR: In this article, the development of process inspection schemes from the original methods of Shewhart to the new charts using cumulative sums, and surveys the present practice in the operation of schemes based on cumulative sums are presented.
Abstract: This paper, presented orally to the Gordon Research Conference on Statistics in Chemistry in July 1960, traces the development of process inspection schemes from the original methods of Shewhart to the new charts using cumulative sums, and surveys the present practice in the operation of schemes based on cumulative sums. In spitc of the completely different appearance of the visual records kept for Shewhart and cumulative sum charts, a continuous sequence of development from the one type of scheme to the other can be established. The criteria by which a particular process inspection scheme is chosen are also developed and the practical choice of schemes is described.

294 citations


"The effect of measurement errors on..." refers background in this paper

  • ...…interested in monitoring small-tomoderate shifts in the process parameters, popular memorytype monitoring schemes such as the cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) schemes are mostly recommended; see for example Montgomery (2013), Page (1961) and Roberts (1959)....

    [...]

Journal ArticleDOI
TL;DR: A mixed EWMA–CUSUM control chart for detecting a shift in the process mean is proposed and it is revealed that mixing the two charts makes the proposed scheme even more sensitive to the small shifts inThe process mean than the other schemes designed for detecting small shifts.
Abstract: The control chart is a very popular tool of statistical process control. It is used to determine the existence of special cause variation to remove it so that the process may be brought in statistical control. Shewhart-type control charts are sensitive for large disturbances in the process, whereas cumulative sum (CUSUM)–type and exponentially weighted moving average (EWMA)–type control charts are intended to spot small and moderate disturbances. In this article, we proposed a mixed EWMA–CUSUM control chart for detecting a shift in the process mean and evaluated its average run lengths. Comparisons of the proposed control chart were made with some representative control charts including the classical CUSUM, classical EWMA, fast initial response CUSUM, fast initial response EWMA, adaptive CUSUM with EWMA-based shift estimator, weighted CUSUM and runs rules–based CUSUM and EWMA. The comparisons revealed that mixing the two charts makes the proposed scheme even more sensitive to the small shifts in the process mean than the other schemes designed for detecting small shifts. Copyright © 2012 John Wiley & Sons, Ltd.

156 citations


"The effect of measurement errors on..." refers background in this paper

  • ...For more details on the enhancement of memory-type schemes, readers are referred to Abbas et al. (2013), Adeoti (2020), Ali and Haq (2017), Capizzi and Masarotto (2010), Haq et al. (2013), Malela-Majika (2020), Waldmann (1996), just to cite a few....

    [...]

Journal ArticleDOI
TL;DR: Significant measurement error often exists in quality control applications as mentioned in this paper, which is known to result in reduced power to detect a given change in the mean or variance of a quality chara...
Abstract: Significant measurement error often exists in quality control applications. Measurement error is known to result in reduced power to detect a given change in the mean or variance of a quality chara...

141 citations


"The effect of measurement errors on..." refers background or methods in this paper

  • ...The most used methodology to reduce measurement inaccuracy is by taking multiple measurements of each item, which was first proposed by Linna and Woodall (2001)....

    [...]

  • ...Based on the discussion in Linna and Woodall (2001) and Maravelakis et al....

    [...]

  • ...To discuss a few, Linna and Woodall (2001) studied the effect of measurement errors on Shewhart monitoring scheme and they reported that under measurement errors a monitoring scheme is exposed to lose power in detecting parameters shifts....

    [...]

  • ...(2020), Nawaz and Han (2020), Raza et al. (2020). Nawaz and Han (2020) studied the performance of the HWMA scheme using structured sampling techniques, that is, ranked set sampling (RSS), extreme RSS, median RSS and neoteric RSS. To provide an efficient and unbiased estimate of the process mean, Adegoke et al. (2019b) developed a HWMA scheme to monitor the process mean that uses the auxiliary variable in the form of a bivariate regression estimator....

    [...]

  • ...(2020), Nawaz and Han (2020), Raza et al. (2020). Nawaz and Han (2020) studied the performance of the HWMA scheme using structured sampling techniques, that is, ranked set sampling (RSS), extreme RSS, median RSS and neoteric RSS. To provide an efficient and unbiased estimate of the process mean, Adegoke et al. (2019b) developed a HWMA scheme to monitor the process mean that uses the auxiliary variable in the form of a bivariate regression estimator. Next, Adegoke et al. (2019a) proposed a multivariate HWMA scheme for monitoring the process mean vector when the underlying distribution parameters are known....

    [...]

Journal ArticleDOI
TL;DR: The study found that design parameters of the proposed chart can be adjusted to make it more robust to non-normality, and the superiority of the suggested chart is established over its competitors.

82 citations


"The effect of measurement errors on..." refers background or methods in this paper

  • ...For more details on the enhancement of memory-type schemes, readers are referred to Abbas et al. (2013), Adeoti (2020), Ali and Haq (2017), Capizzi and Masarotto (2010), Haq et al. (2013), Malela-Majika (2020), Waldmann (1996), just to cite a few....

    [...]

  • ...Note that the manner in which the smoothing parameter (l) is chosen depends on the size of the shifts that a practitioner prioritizes; see Abbas (2018)....

    [...]

  • ...For more details on the enhancement of memory-type schemes, readers are referred to Abbas et al. (2013), Adeoti (2020), Ali and Haq (2017), Capizzi and Masarotto (2010), Haq et al. (2013), Malela-Majika (2020), Waldmann (1996), just to cite a few. For other alternative approaches of control charts, such as the use of divergence functions (e.g. parametric and nonparametric Kullback-Leibler Divergence), see for instance Bakdi and Kouadri (2018), Bakdi et al. (2019) and Bounoua et al. (2020). More recently, Abbas (2018) developed a new memorytype scheme that allocates a specific weight to the current sample and the remaining weight is distributed equally among the previous samples; this scheme is known as the homogeneously weighted moving average (HWMA) monitoring scheme....

    [...]

  • ...Abbas (2018) showed that Equation (1) can also be written as Hi =l Xi + 1 l i 1 X i 1 + 1 l i 1 X i 2 + + 1 l i 1 X 2 + 1 l i 1 X 1 : ð2Þ From Equation (2), it can be seen that the HWMA X statistic assigns weight l to the current sample and a weight 1 lð Þ is equally distributed to the previous…...

    [...]

  • ...For more details on the enhancement of memory-type schemes, readers are referred to Abbas et al. (2013), Adeoti (2020), Ali and Haq (2017), Capizzi and Masarotto (2010), Haq et al....

    [...]

Journal ArticleDOI
TL;DR: A conceptual classification scheme based on content analysis method to analyze and categorize the researches which have explored the effect of measurement errors on different aspects of statistical process monitoring (SPM).

79 citations


"The effect of measurement errors on..." refers background or methods in this paper

  • ...A detailed early account of 60 articles on monitoring schemes with measurements errors is documented in Maleki et al. (2017)....

    [...]

  • ...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…...

    [...]

  • ...A number of methods used as remedial approaches are outlined in the review article on measurement errors by Maleki et al. (2017); for other remedial sampling strategies, see for instance the book by Aslam and Ali (2019)....

    [...]

  • ...That is, in this paper, the assumption given in the review paper by Maleki et al. (2017) is followed: ‘exact measurements in real-life applications are a rare phenomenon, even with highly sophisticated advanced measuring instruments; hence, measurement errors tend to exist in any manufacturing and…...

    [...]

Frequently Asked Questions (11)
Q1. What are the contributions mentioned in the paper "The effect of measurement errors on the performance of the homogenously weighted moving average x ̄ monitoring scheme" ?

Thus, in this paper, the negative effect of measurement errors on the performance of the homogenously weighted moving average ( HWMA ) scheme is studied using the linear covariate error model for constant and linearly increasing variance. 

Since multiple measurements increase cost and time in process monitoring, the negative effect of measurement errors can also be reduced by increasing the value of in the covariate error model. 

The negative effect of the measurement errors on the proposed HWMA ̅ scheme is reduced by using a multiple measurements strategy and / or by increasing the slope coefficient of the linear covariate error model. 

The implementation of the CUSUM ̅ scheme requires two important parameters known as the reference and control limits parameters denoted by and , respectively. 

Based on the sensitivity analysis, practitioners are not advised to use more than four measurements in the design of the HWMA ̅ scheme regardless of the level of measurement error. 

the EWMA ̅ scheme also requires two parameters known as the smoothing and control limits parameter denoted by and , respectively. 

The most used methodology to reduce measurement inaccuracy is by taking multiple measurements of each item, which was first proposed by Linna and Woodall (2001). 

For moderate values of = 0.5, when 0.2, 0.5 and 0.9, theexpected %Decrease in the performance of the HWMA ̅ scheme is 3.61%, 24.32% and 77.16%, respectively. 

In most of the cases, the elimination of the effect of measurement errors is almost impossible because in some situations, measurement costs need to be minimized and the use of large sample sizes must be avoided. 

To discuss a few, Linna and Woodall (2001) studied the effect of measurement errors on Shewhart monitoring scheme and they reported that under measurement errors a monitoring scheme is exposed to lose power in detecting parameters shifts. 

For other alternative approaches of control charts, such as the use of divergence functions (e.g. parametric and nonparametric Kullback-Leibler Divergence), see for instance Bakdi and Kouadri (2018), Bakdi et al. (2019) and Bounoua et al. (2020).