The effect of measurement errors on the performance of the homogenously weighted moving average X¯ monitoring scheme
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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References
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)....
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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....
[...]
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....
[...]
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....
[...]
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)....
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...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…...
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...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)....
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...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…...
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Frequently Asked Questions (11)
Q2. How can the negative effect of multiple measurements be reduced?
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.
Q3. What is the effect of the measurement errors on the proposed HWMA scheme?
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.
Q4. What are the parameters that are required to implement the CUSUM scheme?
The implementation of the CUSUM ̅ scheme requires two important parameters known as the reference and control limits parameters denoted by and , respectively.
Q5. How many measurements should be used in the HWMA scheme?
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.
Q6. What are the parameters that are required for the HWMA scheme?
the EWMA ̅ scheme also requires two parameters known as the smoothing and control limits parameter denoted by and , respectively.
Q7. What is the used method to reduce measurement inaccuracy?
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).
Q8. What is the %Decrease in the performance of the HWMA scheme?
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.
Q9. What is the effect of measurement errors on the performance of the HWMA scheme?
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.
Q10. What is the effect of measurement errors on Shewhart monitoring schemes?
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.
Q11. What are the other alternative approaches of control charts?
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).