Robustness of Moving Average-Exponential Weighted Moving Average
Control Chart with the Light-Tailed Distribution
SUGANYA PHANTU1, YUPAPORN AREEPONG2, SAOWANIT SUKPARUNGSEE2,*
1Faulty of Science, Energy, and Environment,
King's Mongkut University of Technology North Bangkok,
Rayong 21120,
THAILAND
2Department of Applied Statistics, Faculty of Applied Science,
King Mongkut's University of Technology, North Bangkok,
Bangkok, 10800
THAILAND
*Corresponding Author
Abstract: - Control charts are the most significant statistical process control tools for ensuring industrial process
reliability and efficiency. This research uses moving average – exponentially weighted moving average (MA-
EWMA) control charts to investigate the dispersion characteristics. The control chart performance compares
the arithmetic mean of run length (AMRL) and standard deviation of run length (SDRL) profiles of MA-
EWMA and synthetic charts. Student's t-distributions are compared for dispersion processes. Finally, we
present a case to show how essential control charts are in practice.
Key-Words: - Process dispersion, Monte Carlo simulation, Variability chart, Average run length, Standard
deviation of run length, Monitoring, Sensitivity.
Received: April 5, 2024. Revised: August 5, 2024. Accepted: September 2, 2024. Published: October 22, 2024.
1 Introduction
Statistical Process Control (SPC) utilizes statistical
analysis to monitor and improve the quality of
processes across various industries, extending its
reach beyond traditional manufacturing
applications. Control charts, introduced by [1], serve
as the cornerstone of SPC methodology. These
charts effectively identify abnormal variations
within a process, ensuring consistent performance
and adherence to quality specifications. Initially
developed for the manufacturing sector, control
charts have seen widespread adoption in diverse
fields, including nuclear engineering, healthcare,
and education, [2], [3], [4], [5].
Prioritizing the analysis of dispersion
parameters before location parameters is crucial for
establishing a robust understanding of process
variability. Higher dispersion indicates a broader
process output. At the same time, low dispersion
suggests that the output is closely clustered around a
central trend. To characterize process features,
failing to assess and control dispersion before
estimating position parameters might drastically
impair their interpretability due to the potentially
deceptive effects of excessive variance. As a result,
a thorough examination of dispersion gives critical
insights into process variability and guides the
selection of the best position measurement for
robust characterization.
Although much control chart research focuses
on the fundamentals of normality, it is also helpful
for verification processes governed by standard
distributions, such as the Student's t-distribution and
the mixture-specific distribution, which are common
in industrial applications. Furthermore, control chart
features can be carefully chosen for processes with
non-normal distributions to achieve a given shift
magnitude. Conversely, control chart designs are
suited to processes with unknown distributions and
predefined target shift sizes. It is a daunting
challenge.
Shewhart chart is excellent at spotting
substantial process changes despite their reliance on
recent observations. It is, however, less responsive
to changes. Control charts with memory, such as
cumulative sum (CUSUM) and exponentially
weighted moving average (EWMA) control charts
[6], [7], can be used to address this problem. These
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graphics make use of both historical and current
data. Enhances ability to notice small process
changes. Although, [8] suggests moving average
(MA) charts as an alternative, their performance
may not always match that of CUSUM or EWMA
charts in all circumstances.
The never-ending quest for better process
parameter changes detection has driven the recent
development of advanced control charts.
Researchers used known approaches, such as
EWMA and MA charts, to present an innovative
control chart. [9] show a modified EWMA
(mEWMA) chart. It is specifically developed to
increase average verification while [10] used a
different strategy. [11], who pioneered the MA-
EWMA chart for processes with exponential
distributions, presented EWMA-MA charts, which
integrate strategic features from both control charts.
Their findings convincingly indicate the improved
effectiveness of MA-EWMA charts in identifying
parameter changes in a wide range of processes. It
includes symmetric and asymmetric distributions for
all change sizes.
The use of control charts is divided into two
situations: Phase I and Phase II where Phase I
retrospectively focuses on thoroughly understanding
the process and assessing its stability. This distance
ensures the process functions within the inherent
variability at the desired goal level. In addition,
Phase I includes estimating essential process
parameters and determining control limits.
Following this, Phase II, the prospective phase,
leverages the control chart to monitor processes in
real-time. Its primary objective is the detection of
incipient process shifts, facilitating the timely
implementation of corrective actions. Phase II
assesses control chart performance, particularly its
efficacy in identifying process changes. In this
paper, In Phase II, we focus on effective control
charts for process dispersion parameters to solve
problems with position parameters. For EWMA
charts, [12]. Leveraging the established framework
of EWMA charts, [13] introduced a ground-
breaking approach to process variability monitoring.
Their methodology centers on log-transformed
sample variance, explicitly targeting the detection of
nascent increases in variability that can critically
impact product quality. This innovative approach
outperforms traditional range or s2 chart by enabling
the swift identification of even minute standard
deviation increases within a normally distributed
process. [14], go into much detail about tracking
distributions via normalized transformation. Their
study looked at using EWMA control charts created
utilizing the sample variance transformed
logarithmically. They introduced a new control
chart known as the NEWMA chart. The strategy
includes a selective deletion of negative
observations. As a result, it can improve the
efficiency of detecting fragmentation changes,
particularly for little differences.
[15], extended the study of process variability
by employing one-sided and two-sided EWMA
charts. Their simulations confirmed the accuracy of
the preceding chart in detecting upward drift. The
chart below outperforms current approaches for
identifying shifts. [16], examine the choice of
control charts for variability. Eight configurations
were rigorously evaluated using standard deviation
estimators for normal and non-normal distributions.
They include calculation variables for control limits,
greatly aiding the operator in chart selection. [17],
compared the average performance (AMRL) of two
new memory charts (Float T-S^2 and U-S^2) to
CUSUM and EWMA charts. Their findings suggest
fragmentation changes are detected more accurately,
particularly for specific change sizes. [18],
evaluated the efficacy of moving average standard
deviation (MA-S) control charts in detecting process
variability changes. Their study compared the
performance of the MA-S chart to the standard S
control chart, which used a moving average of
sample standard deviations to measure process
variance.
This article offers a combination control chart
used to monitor process fragmentation. One
distinguishing characteristic is using EWMA and
MA statistics to estimate dispersion depending on
change magnitude. Identifying this constant will
mainly cause the control chart to differ. Monte
Carlo simulations are crucial. It gives essential
measures such as AMRL and SDRL. These
indicators enable us to evaluate the chart's
performance in various conditions.
Comprehensively, this guarantees that adequate
performance is evaluated under various scenarios.
As a result, this work presents an effective MA-
EWMA control chart for monitoring process
dispersion parameters. The design structure and
performance were investigated, particularly in
distinct primary conditions where the process
dispersion factors differed. The motivation and
inspiration for this investigation came from, [19].
Before moving on to the MA EWMA chart's basic
structure, we describe the robust estimators in the
next section. This paper is structured as follows:
Sec. 2 comprehensively develops the control chart
for standard deviation, detailing its construction;
Sec. 3 meticulously evaluates its performance,
scrutinizing effectiveness; Sec. 4 encompasses the
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simulation study; Sec. 5 includes comparative
analysis and Sec. 6 discusses the illustrating
example; and finally, Sec. 7 encapsulates the study's
findings, making a definitive statement about the
research.
2 Control Charts
This section outlines the theoretical basis for control
charts. It begins by outlining the assumptions
regarding the underlying data distribution,
specifically the Student's t distribution, followed by
a detailed explanation of control chart properties.
2.1 Distribution
This study investigates the performance
measurement of control charts utilizing the
arithmetic mean of run length and standard
deviation of the run length with the t-value
distribution shown below.
However,
t
has heavier tails, and the
parameter
controls the probability mass of the
tails. For
1,
the Student's t distribution
t
is
transformed into the standard Cauchy distribution,
which has very fat tails, whereas for
,
it is
transformed into the standard average distribution
N(0,1), which has thin tails. The probability density
function for the Student's t-distribution is:
2
1 / 2
2
1
21 ; .
2
t
f t t
(1)
The parameter
is the number of degrees of
freedom. The expectation of t distribution is 0, and
the variance of the distribution is
/ 2.
2.2 Control Chart
The author's research focuses on the performance of
the moving average control chart - exponential
moving average (MA-EWMAS) with standard
deviation. Then, compare the detection efficiency.
The Student's t-distribution is an example of a
symmetrical distribution. Control chart performance
is determined by the arithmetic mean of run length
when the manufacturing process is outside of the
arithmetic mean of run length control (AMRL1) and
the run length's standard deviation. The theories and
related research are discussed below.
2.2.1 S Chart
The standard deviation control chart (S-chart) is the
most basic chart for detecting variations in a
process's standard deviation. The standard deviation
control limit is computed using the probability limit,
or
3
in the approach, which is
3
SS
, where
S
and
S
are the process's mean and standard
deviation, [20].
Thus, the chart's upper and lower control limits
are given as (2). When
ˆ
is an unknown parameter,
it can be calculated by
4
/.Sc
2
4 1 4
ˆˆ
/1UCL LCL c B c
(2)
where
1
B
is the coefficient of control limit of the S
chart. The process is unstable if a sample point plot
is outside the control limit.
4
c
is the factor for
calculating the control limit of the control chart.
S
is the mean of the standard deviation of the process.
2.2.2 MA-S Chart
To detect process fluctuations, [18] investigated
standard moving average (MA) control charts. MA
charts may immediately identify departures from the
control for changes. Minor and Major Process
Variability MA charts have two possible statistical
values:
1 2 1
11
;
;.
i i i
ii i i
S S S ... S i
i
MA S S S ... S i
l
l
l
(3)
When l is the width of the MA-S chart. The
expectation of the MA-S statistic is denoted as
4
11
1
= ==
1ˆ
( )
ii
i j j
jj
E MA S E S c
i i ES
(4)
and the variance of MA statistic can be divided into
two cases as follows:
when
,il
2
11
1
=
1
( )
ii
i j j
jj
Var MA S S VV aar rS
ii
and when
,il
22
4
1
(1 )
= .
1
( )
i
ij
j i l
c
V Var MA S R
l
ar l
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Therefore, the variance of MA-S can be rewritten as
22
4
22
4
ˆ
(1 ) ,
= ( ) ˆ
(1 ) ,.
i
i
c
V
l
a
l
r MA i
Sl
ci
(5)
Therefore, the upper and lower control limits
are given as follows:
22
4
4
2
22
4
2
4
ˆ
(1 )
ˆ,
=
ˆ
(1
/
.
)
ˆ,
l
c
c
U
l
CL LC
i
Lc
Bi
Bi
l
c
(6)
Where
2
B
is the coefficient of control limit of the
MA chart.
2.2.3 EWMA-S chart
[6], introduced EWMA charts, which [13]
investigated further. EWMA charts are a great
alternative to Shewhart charts for detecting minor
changes in process parameters and
monitoring fluctuations. The procedure is based on
statistics, [20].
1
1 , 1, 2,...
i i i
EWMA S S EWMA i
(7)
Where
is the weighting parameter of the data
in the past having the value from 0 to 1, and
i
S
is
the average standard deviation at the time i. The
mean and variance of EWMA-S are:
4ˆ
i
E EWMA S c
and
22
4ˆ
1 / 2 .
i
Var EWMA S c
(8)
Therefore, the control limit of the EWMA-S chart is
as follows:
22
4 3 4
ˆˆ
/ 1 / 2 .UCL LCL c B c
(9)
Where
3
B
is the coefficient of control limit of the
EWMA-S chart.
2.2.4 MA-EWMAS Chart
The MA-EWMAS chart combines the MA-S and
EWMA-S charts, [11]. Let
i
Z
is statistical data for
the EWMA-S chart, which is input to the MA-S
chart. Thus, the statistics of the MA-EWMAS chart
are as follows:
1 2 1
11
;
;.
i
i i i
Si i i
Z Z Z ... Z i
i
l
MA EWMA Z Z ... Z li
l
(10)
When l is the width of the MA-EWMAS chart.
The mean and variance of statistics MA-EWMAS
are
4ˆ
()=
i
S
E MA EWMA c
and
22
4
22
4
()
ˆ
(1 ) ,
2
= ˆ
(1 .
2
),
i
S
c
VM
il
i
ar MA EW A
il
l
c
(11)
Therefore, the control limits of the MA-
EWMAS chart are as follows
22
4
4
22
4
4
4
4
ˆ
(1 )
ˆ,
2
=
ˆ
(1 )
ˆ,.
2
/
c
c
U
l
B
CL LCL c
c
Bi
i
il
l
(12)
Where
4
B
is the coefficient of control limit of the
MA-EWMAS chart.
2.3 The Performance of Control Chart
Commonly, the efficiency of control charts is
measured from the mean of run length or arithmetic
mean of run length (AMRL). The AMRL is the
estimated number of observations from a
process under control before the control chart
incorrectly flags out of control. It is divided into two
phases: Phase I is in control (represented by
AMRL0), and Phase II is out of control (represented
by AMRL1). The AMRL can be defined as follows:
1
.
T
i
i
AMRL RL T
(13)
In this case, the sample being examined before
the process surpasses the control limits for the first
time is indicated by RLi. T, set to 200,000, is the
number of experiment repetitions in the simulation
during round i.
The standard deviation of the run length
(SDRL) can be computed as follows:
22
i
SDRL E RL AMRL
(14)
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3 Analyses of Results
This study investigates the efficiency of control
charts in detecting changes in process standard
deviation. The analysis employed a student
distribution with 15, 30, and 50 degrees of freedom
and sample sizes of 5 and 10. Monte Carlo
simulations allow for the calculation of numerical
results. The in-control arithmetic mean of run length
(AMRL0) is set at 370. The AMRL1 measure is used
to determine the effectiveness of the control chart.
The figure displays the lowest AMRL1 readings,
regarded as the most effective at detecting changes.
The findings of the study are separated into two
parts:
3.1 Performance of MA-EWMA Chart
The run length evaluation method for a mixed
moving average – exponentially weighted moving
average control chart (MA-EWMA) for process
dispersion. The weighting factors
()
of the MA-
EWMA control charts were 0.50, and the width of
the moving average (l) was 2, 3, 5, 10, and 15. The
number of repetitions of the MA-EWMA chart is
5,000 iterations. The changing sizes of the process
were 1.10, 1.20, 1.30, 1.40, 1.50, 1.75, 2.00,
2.50, and 3.00. The estimated arithmetic mean of
run lengths (AMRLs) for each case can be explained
as follows:
Table 1 (Appendix) presents the arithmetic
mean of run length (AMRL) of the MA-EWMA
chart for t-distributed data with a parameter value of
15 and a subgroup size of 5. The AMRLs of the
MA-EWMA chart with smoothing parameters (l) of
2, 3, 5, 10, and 15 are compared. The results show
that when the process shift increases, the MA-
EWMA chart with a smoothing parameter (l) of 2 is
the most effective in detecting changes in the
standard deviation, as it has the lowest AMRL1
value.
Table 2 (Appendix) displays the AMRL values
of the MA-EWMA control chart based on data
distributed as the t-distribution, utilizing parameter
settings of 15 and a subgroup size of 10. This
research examines the AMRL performance of the
MA-EWMA control chart with smoothing
parameter (l) settings of 2, 3, 5, 10, and 15. The
results imply that the MA-EWMA control chart
with smoothing grows with the extent of the process
change. Because of the lowest AMRL1 value, the
parameter (l) set to 2 outperforms others in detecting
standard deviation differences.
Table 3 and Table 4 (Appendix) show the
arithmetic mean of run length (AMRL) of the MA-
EWMA charts when the data is assumed to follow a
t-distribution with parameters of 30 and subgroup
sizes of 5 and 10, respectively. Furthermore,
Table 5 and Table 6 (Appendix) display the
AMRL values of the MA-EWMA charts under the t
distribution assumption, with the parameter set to 50
and the subgroup sizes set to 5 and 10, respectively.
The results reveal that when process variability
grows, parameter (l) value 2 for MA-EWMA charts
performs the best in identifying changes in standard
deviation. This result mirrors the outcomes observed
when the t-distribution parameters were set at 15.
3.2 AMRL and SDRL Performance of the
Control Chart
To comparison, the MA-EWMA control chart is
compared with the S chart, moving average-
standard deviation (MA), exponentially weighted
moving average-standard deviation (EWMA),
measured by the out-of-control arithmetic mean of
run length (AMRL1), and standard deviation of run
length (SDRL). The parameter of in-control for the
MA-EWMA chart is given AMRL0 = 370,
0.5,
and l = 2. The parameter changes at
= 1.10, 1.20,
1.30, 1.40, 1.50, 1.75, 2.00, 2.50, and 3.00. The
results of comparing the efficiency of control charts
for measuring dispersion can be explained as
follows:
Table 7 (Appendix) specifies data assumed to
follow a t-distribution with parameters set at 15 and
subgroup sizes of 5. The numerical data analysis
reveals that when there is a change in the process,
the measurement of dispersion increases. The MA
chart is the most efficient control chart for detecting
these changes, exhibiting the lowest AMRL1 value.
Subsequently, Table 8 (Appendix) outlines data
presumed to conform to a student t-distribution with
parameters set at 15 and subgroup sizes of 10.
Analysis of the numerical data indicates that the
dispersion measurement falls within the range of
1.00 to 2.00 in the event of a process change. S
charts seem to be the best control charts for spotting
these changes. MA charts also outperform in
identifying changes, as evidenced by the lowest
AMRL1 value.
Table 9 (Appendix) displays the results from
numerical investigations expected to follow a t-
distribution with parameters set to 30 and a
subgroup size of 5, the dispersion metric increases
when the method changes. MA charts appear to be
the best control charts for detecting such changes. It
outperforms other charts, each with the lowest
AMRL1 value.
Table 10 (Appendix) demonstrates that with
data from the t-distribution and parameters set to 30
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with a subgroup size of 10, numerical analysis
suggests that S-charts are the most successful at
detecting changes within a range. Dispersion
measurement ranges from 1.00 to 2.50.
Additionally, when the dispersion measurement
reaches 3.00 or higher due to a process change, the
MA chart outperforms all presented charts,
exhibiting the lowest AMRL1 value in change
detection.
Table 11 and Table 12 (Appendix) compare
control charts' arithmetic mean of run length
outcomes under a t-distribution with parameters set
at 50 and subgroup sizes of 5 and 10, respectively.
The findings suggest that the most effective chart
consistently aligns with the scenario where the data
follows a student-t distribution with parameters set
at 30, regardless of the level of change.
3.3 A Real Application
This section illustrates the practical application of
an accelerometer dataset within the control chart
examined in this research. Accelerometers, versatile
devices with applications spanning manufacturing
vibration measurement, car accident detection,
pollution monitoring, scientific research, medicine,
and more, are the focus. For this study, we have
utilized a smartphone accelerometer dataset for
monitoring objectives, specifically implementing
control charts for accelerometer data. Following the
methodology outlined by [21], we have divided the
data into 10 subgroups, each comprising 40 entries.
The appropriate distribution for this data is a
student- t distribution with an average of 0.843 and
a standard deviation of 0.257. The performance
measurement results in detecting data changes
through graphical representations can be explained.
Figure 1 demonstrates that the S-chart statistic
falls between the upper and lower boundaries, which
leads to the conclusion that S chart cannot detect
changes in data. MA chart, like S chart, cannot
detect changes in data since the statistics do not
surpass the top and lower bounds, as seen in Figure
2.
Figure 3 indicates that the EWMA chart
cannot detect changes in the data. Finally, the
analysis detects variations in the standard deviation.
Figure 4 illustrates how the MA-EWMA chart
successfully identified nine process improvements
when comparing the performance of the charts
above. Finally, the MA-EWMA chart outperforms
other methods for detecting standard deviations.
4 Conclusion and Further Research
This research aims to construct the new mixed MA-
EWMA for monitoring the standard deviation and
examines the AMRL and SDRL of control charts for
a Student t-distribution method. The preliminary
study gives operators insight into control charts to
aid in process monitoring and prevent delays in
identifying process changes. The mixed MA-
EWMA chart is robust to the light-tail process as
student-t distribution to detect a process dispersion
based on standard deviation for large magnitudes of
shift sizes.
Fig. 1: Performance of S chart
Fig. 2: Performance of MA chart
Fig. 3: Performance of EWMA chart
0
1
2
3
13579111315171921232527293133353739
S chart
S LCL UCL
0
1
2
13579111315171921232527293133353739
MA chart (l= 5)
MA S LCL UCL
0
2
4
13579111315171921232527293133353739
EWMA chart
EWMA S LCL UCL
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Fig. 4: Performance of MA-EWMA chart
In addition, the results obtained by simulation
studies reveal that the mixed MA-EWMA based on
standard deviation outperformed as well as for MA
chart for moderate to large shift sizes when the
parameter of student-t distribution is increased. In
future research, the verification can be extended to
heavy-tail processes and compared with other
control charts. Other performance metrics include
the median run length (MRL) and the percentile of
the run length distribution. In most real-world
settings, comparing these charts helps determine
process parameters such as target mean and standard
deviation from Phase I datasets. Because so little
information is available, the process parameters
must be guessed.
Acknowledgement:
The Department of Applied Statistics is
acknowledged by the authors for providing
superhigh-performance computers and supporting
materials. Additionally, we are grateful to Thailand
Science Research and Innovation, the Ministry of
Higher Education, Science, and Research, and King
Mongkut's University of Technology North
Bangkok for their support of the research fund,
contract number KMUTNB-FF-67-B-18.
Declaration of Generative AI and AI-assisted
Technologies in the Writing Process
During the preparation of this work the authors used
Google Gemini in order to study the source and
importance of research. After using this tool/service,
the authors reviewed and edited the content as
needed and take full responsibility for the content of
the publication.
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Volume 19, 2024
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15) 2: e0228208, (2020), doi:
10.1371/journal.pone.0228208.
[12] I. M. Zwetsloot, M. Schoonhoven, and R. J.
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[13] S. V. Crowder, and M. D. Hamilton, "An
EWMA for monitoring a process standard
deviation," Journal of Quality Technology,
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10.1080/00224065.1992.11979369.
[14] L. Shu, and W. Jiang, "A new EWMA chart
for monitoring process dispersion," Journal
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(2008), doi:
10.1080/00224065.2008.11917737.
[15] L. Huwang, C. J. Huang and Y. H. Wang,
"New EWMA control chart for monitoring
process dispersion," Computational Statistics
and Data Analysis, 54(10), 2328-2342,
(2010), doi: 10.1016/j.csda.2010.03.011.
[16] S. A. Abbasi and A. Miller, "On proper
choice of variability control chart for normal
and non-normal processes," Quality and
Reliability Engineering International, 28(3),
279-296, (2012), doi: 10.1002/qre.1244.
[17] N. Abbas, M. Riaz, and R. J. Does,
"Memory-type control charts for monitoring
the process dispersion," Quality and
Reliability Engineering International, 30(5),
623-632, (2014), doi: 10.1002/qre.1514.
[18] O. A. Adeoti, and J. O. Olaomi, "A moving
average S control chart for monitoring
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10.1080/08982112.2015.1104542.
[19] S. Phantu, Y. Areepong, and S.
Sukparungsee, "Analysis process dispersion
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[20] D. C. Montgomery, "Statistical quality
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[21] M. Riaz, M. Abid, H.Z. Nazir, and S.A.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- P. S.: writing an original draft, software, data
analysis, data curation, proof, and validation. Y.
A.: investigation, methodology, validation,
reviewing
- S. S.: conceptualization, investigation, funding
acquisition, project administration, reviewing, and
editing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
Besides, I would like to thank King Mongkut's
University of Technology North Bangkok and
Thailand Science Research and Innovation, Ministry
of Higher Education, Science, and Research for
supporting the research fund with contract no.
KMUTNB-FF-67-B-18.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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APPENDIX
Table 1. AMRL1 of MA-EWMAS charts when the data are from t(15) with AMRL0 =370, n=5 and 𝜆=0.5
Shift size
l=2
l=3
l=5
l=10
l=15
B4 = 3.124
B4 = 3.113
B4 = 3.136
B4 = 3.125
B4 = 3.143
1.00
370.667
370.449
370.447
370.170
370.892
1.10
352.726
355.624
358.461
362.486
364.252
1.20
327.148
330.853
331.843
333.644
335.490
1.30
291.570
292.482
295.173
298.456
310.189
1.40
267.901
267.056
269.434
275.162
286.953
1.50
244.116
258.511
267.131
270.667
271.148
1.75
189.053
202.134
253.835
264.160
266.545
2.00
144.279
153.502
196.957
211.760
238.471
2.50
83.056
86.468
106.831
158.806
204.199
3.00
49.392
49.994
56.377
72.721
83.335
Note: Italics number represents the lowest AMRL1.
Table 2. AMRL1 of MA-EWMAS charts when the data are from t(15) with AMRL0 =370, n=10 and 𝜆=0.5
Shift size
l=2
l=3
l=5
l=10
l=15
B4 = 3.107
B4 = 3.115
B4 = 3.119
B4 = 3.121
B4 = 3.125
1.00
370.797
370.209
370.210
370.414
370.174
1.10
350.237
354.958
357.066
365.182
367.205
1.20
325.249
329.425
331.107
338.925
345.716
1.30
284.324
289.341
290.143
294.312
297.856
1.40
254.822
258.821
260.347
262.470
275.314
1.50
248.723
255.001
261.133
268.213
272.558
1.75
186.683
231.115
237.073
245.840
248.512
2.00
134.565
166.357
173.987
181.801
192.608
2.50
68.557
78.794
103.557
159.087
172.579
3.00
35.540
37.997
43.778
53.497
59.378
Note: Italics number represents the lowest AMRL1.
Table 3. AMRL1 of MA-EWMAS charts when the data are from t(30) with AMRL0 =370, n=5 and 𝜆=0.5
Shift size
l=2
l=3
l=5
l=10
l=15
B4 = 3.109
B4 = 3.112
B4 = 3.120
B4 = 3.124
B4 = 3.125
1.00
370.018
370.049
370.016
370.299
370.836
1.10
360.449
361.204
365.780
367.564
368.056
1.20
355.092
356.101
357.315
359.930
360.417
1.30
347.868
349.724
350.443
352.946
356.410
1.40
323.542
326.829
330.216
343.928
347.281
1.50
309.954
314.645
319.955
339.068
340.928
1.75
275.962
282.378
294.507
322.848
331.998
2.00
241.405
248.910
270.843
317.289
322.096
2.50
176.970
181.798
201.243
242.719
274.396
3.00
126.273
127.269
138.049
161.846
177.374
Note: Italics number represents the lowest AMRL1.
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Table 4. AMRL1 of MA-EWMAS charts when the data are from t(30) with AMRL0 =370, n=10 and 𝜆=0.5
Shift size
l=2
l=3
l=5
l=10
l=15
B4 = 3.105
B4 = 3.114
B4 = 3.118
B4 = 3.124
B4 = 3.126
1.00
370.015
370.641
370.534
370.521
370.296
1.10
351.967
353.187
356.822
360.478
361.854
1.20
344.630
346.859
349.558
352.064
358.416
1.30
325.418
330.714
333.561
336.205
342.758
1.40
312.546
316.852
320.141
324.189
337.286
1.50
309.064
315.986
317.717
319.902
327.354
1.75
271.972
293.404
308.508
312.539
318.041
2.00
232.434
255.135
293.359
304.849
308.863
2.50
160.521
174.281
202.015
252.182
287.580
3.00
104.469
106.640
119.966
136.785
144.904
Note: Italics number represents the lowest AMRL1.
Table 5. AMRL1 of MA-EWMAS charts when the data are from t(50) with AMRL0 =370, n=5 and 𝜆=0.5
Shift size
l=2
l=3
l=5
l=10
l=15
B4 = 3.109
B4 = 3.116
B4 = 3.119
B4 = 3.125
B4 = 3.126
1.00
370.524
370.568
370.268
370.173
370.096
1.10
349.786
352.748
357.809
361.270
364.593
1.20
347.152
349.617
352.641
355.276
358.947
1.30
343.268
345.501
348.286
350.421
352.119
1.40
339.107
341.226
343.028
347.564
349.871
1.50
336.352
338.328
341.881
340.819
345.469
1.75
315.385
318.340
327.536
336.919
338.771
2.00
293.496
296.135
307.660
321.987
327.538
2.50
245.854
247.676
260.307
288.648
306.944
3.00
198.407
199.040
209.393
232.726
246.549
Note: Italics number represents the lowest AMRL1.
Table 6. AMRL1 of MA-EWMAS charts when the data are from t(50) with AMRL0 =370, n=10 and 𝜆=0.5
Shift size
l=2
l=3
l=5
l =10
l =15
B4 = 3.108
B4 = 3.118
B4 = 3.121
B4 = 3.125
B4 = 3.128
1.00
370.637
370.020
370.342
370.084
370.026
1.10
352.411
355.967
357.187
359.721
360.187
1.20
349.724
351.092
355.131
357.413
358.943
1.30
343.528
345.201
348.664
350.217
352.323
1.40
339.642
340.829
342.511
345.976
348.665
1.50
335.655
337.017
339.812
341.139
342.956
1.75
312.999
320.704
330.587
334.313
338.704
2.00
287.991
297.356
316.148
329.476
332.221
2.50
233.555
243.308
262.604
294.001
315.031
3.00
179.770
186.303
198.041
214.766
222.886
Note: Italics number represents the lowest AMRL1.
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Table 7. Comparison AMRL1 of S, MA, EWMA, and MA-EWMAS charts from t(15) with l=2, 𝜆=0.5, and n=5
Shift size
S chart
MA chart
EWMA chart
MA-EWMAS chart
B1 = 3.156
B2 = 3.075
B3 = 3.154
B4 = 3.124
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
1.00
370.114
0.971
370.096
0.973
370.352
0.978
370.667
0.974
1.10
265.464
0.927
261.287
0.915
274.689
0.967
352.726
0.961
1.20
260.794
0.894
258.164
0.894
263.478
0.935
327.148
0.902
1.30
259.799
0.851
253.461
0.824
256.974
0.912
291.570
0.854
1.40
254.637
0.790
249.347
0.781
252.962
0.880
267.901
0.785
1.50
250.747
0.743
243.793
0.728
248.385
0.862
244.116
0.731
1.75
199.445
0.616
188.855
0.587
192.659
0.698
189.053
0.588
2.00
156.547
0.483
144.024
0.457
145.375
0.536
144.279
0.458
2.50
96.280
0.308
82.938
0.264
82.759
0.300
83.056
0.265
3.00
60.980
0.194
49.332
0.158
48.727
0.171
49.392
0.158
Note: Italics number represents the lowest AMRL1.
Table 8. Comparison AMRL1 of S, MA, EWMA, and MA-EWMAS charts from t(15) with l=2, 𝜆=0.5 and n=10
Shift size
S chart
MA chart
EWMA chart
MA-EWMAS chart
B1 = 3.158
B2 = 3.082
B3 = 3.167
B4 = 3.107
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
1.00
370.835
0.972
370.745
0.972
370.545
0.980
370.797
0.972
1.10
295.146
0.944
304.107
0.946
293.107
0.953
350.237
0.954
1.20
270.764
0.914
290.174
0.915
273.165
0.924
325.249
0.897
1.30
264.822
0.832
272.316
0.889
269.208
0.873
284.324
0.834
1.40
230.486
0.715
256.864
0.824
264.815
0.852
254.822
0.815
1.50
227.612
0.687
248.707
0.741
259.988
0.697
248.723
0.741
1.75
170.475
0.533
186.659
0.582
254.287
0.590
186.683
0.582
2.00
126.960
0.402
134.526
0.426
195.113
0.513
134.565
0.426
2.50
70.625
0.214
68.551
0.219
97.726
0.450
68.557
0.319
3.00
40.496
0.113
35.535
0.134
49.165
0.209
35.540
0.134
Note: Italics number represents the lowest AMRL1.
Table 9. Comparison AMRL1 of S, MA, EWMA, and MA-EWMAS charts from t(30) with l =2, 𝜆=0.5 and n=5
Shift size
S chart
MA chart
EWMA chart
MA-EWMAS chart
B1 = 3.164
B2 = 3.068
B3 = 3.159
B4 = 3.109
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
1.00
370.097
0.969
370.065
0.969
370.054
0.984
370.018
0.969
1.10
336.842
0.945
331.052
0.946
338.461
0.979
360.449
0.953
1.20
328.603
0.937
325.118
0.922
330.820
0.970
355.092
0.931
1.30
322.849
0.916
320.176
0.906
325.649
0.962
347.868
0.914
1.40
317.297
0.891
315.279
0.882
318.502
0.956
323.542
0.875
1.50
313.791
0.874
309.982
0.868
313.581
0.945
309.954
0.868
1.75
281.899
0.812
275.979
0.786
279.658
0.938
275.962
0.800
2.00
249.459
0.740
241.420
0.720
244.241
0.847
241.405
0.723
2.50
189.362
0.584
176.990
0.552
177.440
0.645
176.970
0.552
3.00
139.837
0.443
126.286
0.400
123.566
0.453
126.273
0.400
Note: Italics number represents the lowest AMRL1.
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Table 10. Comparison AMRL1 of S, MA, EWMA, and MA-EWMAS charts from t(30) with l=2, 𝜆=0.5 and
n=10
Shift size
S chart
MA chart
EWMA chart
MA-EWMAS chart
B1 = 3.168
B2 = 3.083
B3 = 3.176
B4 = 3.105
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
1.00
370.016
0.970
370.028
0.987
370.071
0.989
370.015
0.972
1.10
346.441
0.954
349.537
0.965
376.759
0.967
351.967
0.955
1.20
334.749
0.919
338.601
0.922
372.303
0.952
344.630
0.921
1.30
323.467
0.907
327.642
0.901
367.546
0.937
325.418
0.904
1.40
310.149
0.884
317.058
0.885
361.037
0.918
312.546
0.886
1.50
300.023
0.851
309.066
0.870
352.483
0.890
309.064
0.870
1.75
261.972
0.770
271.973
0.794
335.078
0.864
271.972
0.794
2.00
223.923
0.678
232.434
0.701
311.042
0.827
232.434
0.701
2.50
156.675
0.491
160.528
0.503
243.168
0.792
160.521
0.503
3.00
107.090
0.339
104.469
0.330
163.594
0.771
104.469
0.330
Note: Italics number represents the lowest AMRL1.
Table 11. Comparison AMRL1 of S, MA, EWMA, and MA-EWMAS charts from t(50) with l=2, 𝜆=0.5 and n=5
Shift size
S chart
MA chart
EWMA chart
MA-EWMAS chart
B1 = 3.172
B2 = 3.079
B3 = 3.165
B4 = 3.109
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
1.00
370.013
0.971
370.561
0.975
370.563
0.985
370.524
0.971
1.10
358.647
0.986
348.307
0.969
355.941
0.975
349.786
0.965
1.20
350.228
0.974
345.922
0.945
351.036
0.952
347.152
0.947
1.30
345.976
0.965
341.546
0.938
343.550
0.949
343.268
0.938
1.40
342.208
0.934
338.250
0.922
340.312
0.934
339.107
0.924
1.50
338.162
0.920
336.364
0.917
339.047
0.921
336.352
0.918
1.75
319.640
0.886
315.425
0.875
319.235
0.899
315.385
0.880
2.00
298.749
0.847
293.510
0.838
297.674
0.871
293.496
0.839
2.50
254.419
0.751
245.873
0.731
248.432
0.852
245.854
0.733
3.00
209.985
0.639
198.485
0.609
200.026
0.713
198.407
0.609
Note: Italics number represents the lowest AMRL1.
Table 12. Comparison AMRL1 of S, MA, EWMA, and MA-EWMAS charts from t(50) with l=2, 𝜆=0.5 and
n=10
Shift size
S chart
MA chart
EWMA chart
MA-EWMAS chart
B1 = 3.175
B2 = 3.087
B3 = 3.182
B4 = 3.108
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
AMRL
SDRL
1.00
370.833
0.972
370.229
0.971
370.524
0.992
370.637
0.972
1.10
345.411
0.961
347.023
0.965
349.602
0.984
352.411
0.968
1.20
341.869
0.955
343.556
0.957
344.856
0.962
349.724
0.960
1.30
337.528
0.930
340.528
0.934
340.929
0.950
343.528
0.954
1.40
333.142
0.917
338.641
0.920
339.528
0.934
339.642
0.936
1.50
331.990
0.909
335.231
0.914
336.430
0.911
335.655
0.915
1.75
308.280
0.868
312.643
0.874
313.138
0.893
312.999
0.875
2.00
282.932
0.816
287.609
0.825
288.414
0.871
287.991
0.826
2.50
230.997
0.696
233.311
0.701
234.775
0.862
233.555
0.701
3.00
180.136
0.561
179.585
0.557
180.647
0.850
179.770
0.557
Note: Italics number represents the lowest AMRL1.
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