Exact Average Run Length Evaluation on One-Sided and Two-Sided
Extended EWMA Control Chart with Correlated Data
PHUNSA MONGKOLTAWAT, YUPAPORN AREEPONG*, SAOWANIT SUKPARUNGSEE
Department of Applied Statistics, Faculty of Applied Statistics,
King Mongkut’s University of Technology North Bangkok,
Bangkok 10800,
THAILAND
*Corresponding Author
Abstract: - The Extended Exponentially Weighted Moving Average (Extended EWMA) control chart is one of
the control charts that can rapidly detect a minor shift. Using the average run length (ARL), the control charts'
effectiveness can be evaluated. This research aims to derive the explicit formulations for the ARL on one-sided
and two-sided Extend EWMA control charts for the MA(1) model with exponential white noise, as they have
not been previously presented. The analytical solution accuracy was determined and compared to the numerical
integral equation (NIE) method. The results indicate that the ARL calculated using the explicit formula and the
NIE method are nearly identical, demonstrating the validity of the explicit formula. In addition, this study is
extended to compare the performance with the Exponentially Weighted Moving Average (EWMA) control
chart. The results show that the Extended EWMA control chart has superior efficacy to the EWMA control
chart. To demonstrate the efficacy of the proposed method, the analytical solution of ARL is finally applied to
real-world data on Thailand's monthly fuel price.
Key-Words: - Average Run Length, Moving Average process, explicit formula, The Extended Exponentially
Weighted Moving Average control chart, the EWMA control chart, the Extended EWMA, NIE
method, ARL.
Received: June 25, 2023. Revised: September 24, 2023. Accepted: October 15, 2023. Published: November 9, 2023.
1 Introduction
Control charts have been utilized in numerous
disciplines, including finance, economics, industry,
health care, and medicine. A study, introduced the
concept of the control chart in 1931, [1]. The
Shewhart control chart detects significant shifts in
the processes more effectively. Next, the
Exponentially Weighted Moving Average
(EWMA) control chart, [2], demonstrates that both
methods are effective at detecting shift size.
Numerous researchers have enhanced the EWMA
control chart, making it effective at detecting small
shifts rapidly for observations with both normal
distribution and exponential distribution. The
study, [3], proposed the Extended EWMA control
chart, which is an effective performance control
chart for detecting shift size in the monitored
process. The performance of these control charts is
also compared in terms of the average run length
(ARL), [4]. It consists of two components: The
ARL0 refers to the average run length when a
process is in the control state. ARL1 is the expected
number of observations taken from an out-of-
control process and should be as small as feasible.
Previous research has demonstrated that the
average run length (ARL) can be calculated using a
variety of methods. The study, [5], proposed
explicit formulas and numerical integral equations
of ARL for the SARX(P,r)L model based on the
CUSUM chart. The study, [6], proposed the ARL
of a multivariate EWMA by means of Monte Carlo
simulation. The study, [7], created the effectiveness
of the CUSUM control chart for trend stationary
seasonal autocorrelated data. The study, [8],
derived the explicit formula for the ARL on the
EWMA trend exponential AR(1) process. The
study, [9], derived the ARL for the Autoregressive
Moving Average (ARMA) process using both the
explicit formula and the NIE approach of the
EWMA control chart. The study, [10], analyzed the
ARL by applying the explicit formula on the
EWMA control chart. The focus of the analysis
was on a seasonal moving average model with
exponential white noise, specifically considering an
order q. Recently, [11], presented exact run length
computation on the EWMA control chart for
WSEAS TRANSACTIONS on MATHEMATICS
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Phunsa Mongkoltawat,
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stationary moving average process with exogenous
variables. The study, [12], derived designing the
performance of the EWMA control chart for the
seasonal moving average process with exogenous
variables.
However, the explicit formula for the ARL on
one-sided and two-sided Extended EWMA control
charts for the MA(1) has not previously been
studied. This study's objective is to derive the
explicit formula for the ARL on one-sided and two-
sided Extended EWMA for the MA(1). A
comparison was made between the explicit formula
and the NIE method for the ARL. In addition, the
capability of explicit formulas for deriving the
ARL on one-sided and two-sided Extended EWMA
was contrasted with the EWMA control chart for
both simulated and real-world data in the monthly
fuel price in Thailand.
2 Materials and Methods
The study, [2], proposed the EWMA control chart,
which can be detected variation in the process. The
EWMA control chart is depicted below;
1
1 , 1, 2,3,...
t t t
Z Z X t
(1)
where
t
X
is a process with mean,
is exponential
smoothing parameters with
01
and
0
Zu
is constant representing the initial value of the
EWMA control chart. The upper control limit
(UCL) and lower control limit (LCL) are
represented as follows:
0
/2
UCL LCL L
(2)
where
0
is the parameter the target mean of the
moving average process,
is the process standard
deviation parameter and L is the control limit
variable for the control limit of the moving average
process.
The study, [3], proposed the Extended EWMA.
The Extended EWMA statistic is.
1 2 1 1 2 1
1 , 1, 2,3,...
t t t t
E E X X t
(3)
where
1
and
2
are exponential smoothing
parameters with
21
01
and
0
Eu
is the
initial value of the Extended EWMA. The upper
control limit (UCL) and lower control limit (LCL)
are represented as follows
22
1 2 1 2 1 2
02
1 2 1 2
2 (1 )
/,
2( ) ( )
UCL LCL L
where
0
is the target mean parameter of the
moving average process,
is the parameter of the
process standard deviation and L is the variable
suitable for the control limit of the moving average
process. The moving average (MA(1)) process can
be characterized as
1t t t
X
(4)
where
t
is the parameter of the error term of time
and presumed to be exponential white noise,
is
the parameter of a constant and
1
is parameter of
moving average coefficient with
1
11
. A
describes the probability density function of
t
is
given by
1
()
x
f x e
.
3 The Explicit formulas of ARL on
Extended EWMA Control Chart
of MA(1) Process
3.1 The Exact Solution of ARL on one-sided
and two-side Extended EWMA for
MA(1) Process
Let H(u) represent the ARL for the moving average
(MA(1)) process. The Extended EWMA control
chart has identical statistics. From equation (3)
when t time equals one.
1 1 2 0 1 1 1 1 1 0 2 0
(1 )EXEX
(5)
where
1
and
2
are exponential smoothing
parameters with
21
01
and
0,Eu
0,Xv
is the initial value of the Extended
EWMA.
1 1 2 1 1 1 2 1 1
(1 ) ( )E u v
(6)
The upper limit and lower limit are LCL = a and
UCL = b, respectively. For the Extended EWMA
statistics
t
E
in an in-control;
1
a E b
H(u) denotes the ARL for the moving average
(MA(1)) process as follows:
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0
( ) ( ) , ,ARL H u E T E u
where
E
is the expectation value.
Consider the following Fredholm integral equation
of the second kind for the function
1 1 1
( ) 1 ( ) ( )H u H Z f d
:
1 2 1 1 1 2 1 1 1 1
( ) 1 ((1 ) ( ) ) ( )H u H u v f d
(7)
Therefore, the function H(u) is obtained as follows:
1 2 1 1 2
11
(1 ) ( )
1
( ) 1 ( ) ( )
b
a
y u v
H u H y f dy
(8)
If
( ), ( )
tt
Exp y Exp
then
1,0
y
y e y
1 2 1 1 2
11
(1 ) ( )
()
1
1
( ) 1
y y u v
b
a
H u e e dy
(9)
Assume the function G(u) to be
1 2 1 1 2
1
(1 ) ( )
()
y u v
G u e
1
1
1
1
1
( ) 1 ( ) ( )
()
( ) 1 ( ) , 0
y
b
a
y
b
a
H u H y e G u dy
Gu
H u H y e dy u b
Let
1
( ) , 0
y
b
a
d H y e dy u b
then
1
()
( ) 1 Gu
H u d
Thus
1
1
()
(1 )
y
b
a
Gy
d d e dy
, solving a constant d,
11
1
y
bb
aa
y
d e dy
Gude
Therefore
11
1 1 2 1 2 1 2
1 1 1
1
( ) ( ) ( )
12
()
1
1 ( )
ba
v b a
ee
d
e e e
(10)
and d and G(u) are substituted in the function H(u).
Therefore, substituting
is equal
0
when the
process is in control, the explicit formula of ARL0
for the two-sided Extended EWMA control chart
yields the following formula;
12
1 1 1
1 2 1 2
11
1 1 2
0 0 1 0 0 0 0
1 1 2
1 0 0 0 0
(1 )
12
0( ) ( )
12
()
()
( ) ( )
1
( ) ( )
uba
ba
nn
n
e e e
ARL
e e e
e
(11)
Consequently, substituting
is equal
1
in H(u),
the explicit formula of ARL1 for the two-sided
Extended EWMA control chart can be obtained in
the following
12
111
1 2 1 2
11
1 1 2
10 1 1 1 1 1
1 1 2
1 1 1 1 1
(1 )
12
1( ) ( )
12
()
()
( ) ( )
1
( ) ( )
uba
ba
nn
n
e e e
ARL
e e e
e
(12)
In addition, the explicit formula of ARL1 for the
one-sided Extended EWMA control chart can be
obtained as
12
11
12
1
1 1 2
10 1 1 1 1
1 1 2
1 1 1 1
(1 )
12
1()
12
()
()
( ) ( 1)
1
( ) ( 1)
ub
b
nn
n
ee
ARL
ee
e
(13)
Theorem 1.
The solution obtained by the ARL of the explicit
formulas demonstrates the existence of a unique
integral equation (NIE), as proven by Banach’s
fixed-point theorem. In this present study, let T
denote an operation within the set of all continuous
functions that are defined by.
1 2 1 1 2
11
(1 ) ( )
1
( ( )) 1 ( ) ( )
b
a
y u v
T H u H y f dy
(14)
According to Banach’s fixed-point theorem, if
an operator T meets the condition of being a
contraction, the fixed-point equation T(H(u)) =
H(u) has a unique solution, as stated. If equation
(14) exists and has a unique solution, the Banach
fixed-point theorem can be applied. The Banach
fixed-point theorem also termed the contraction
mapping theorem, appeared in explicit form in
Banach’s thesis in 1922, [13]. In general, it is
employed to demonstrate the existence of a
solution to an integral equation. Since then, due to
its simplicity and utility, it has become a widely
used instrument for solving existing problems in
numerous mathematical disciplines, [14]. The
specifics are listed below.
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Assume that
:T X X
is a contraction
mapping with contraction constant
0 1,s
such
that
1 2 1 2
()T L T L s L L
12
,.L L X
,
satisfies this condition. Then by, [15], there exists a
unique
(.)LX
such that
( ( )) ( )T L u L u
has a
unique fixed point in
.X
Proof Theorem 1: To demonstrate that T, as
defined by the equation T(H(u)) is a contraction
mapping for
12
, [ , ].H H G a b
that
1 2 1 2
()T H T H s H H
,
12
, [ , ].H H G a b
with
01s
under the norm
[ , ]
sup ( )
u a b
H H u
From
H(u) and T(H(u)).
12
()T H T H
1 2 1 1 2
11
1
(1 ) ( )
()
12
[ , ]
1
sup ( ( ) ( ))
u v y
b
u a b a
e H y H y e dy
1 2 1 1 2
11
1 2 1 1 2
1 1 1
1
(1 ) ( )
()
12
[ , ]
(1 ) ( )
()
12
[ , ]
12
1
sup ( ( ) ( ))
sup 1
uv y
b
u a b a
uv ba
u a b
e H y H y e dy
H H e e e
s H H
where
1 2 1 1 2
1 1 1
(1 ) ( )
()
[ , ]
sup 1
u v b a
u a b
s e e e
,
0 1,s
The uniqueness of the solution is
therefore ensured by Banach’s fixed-point theorem.
3.2 The NIE method of ARL of Extended
EWMA Control Chart of MA(1) Process
For the two-sided Extended EWMA control chart
for the moving average (MA(1)) process. Let HN(u)
be the estimated value of the ARL with the m linear
equation systems by using the composite midpoint
quadrature rule by, [16].
The ARL approximating NIE method on a two-
sided Extended EWMA is evaluated as follows;
1
m
b
jj
aj
L k f k dk w f x
The system of the m linear equation is shown as
1 1 1
1
m m m m m
L R L
or
1
11
1
m m m m m
L I R
1 1 1
, ,..., ,
T
m NIE NIE NIE m
L L x L x L x
1,1,...,1
m
I diag
and
1
1 1,1,...,1 .
T
m
Let
mm
R
be a matrix, the definition of the m to
mth element of the matrix R is given by
1 2 1 1 2
11
(1 ) ( )
1j
ij j
y u v
R w f
So, the solution of the numerical integral
equation can be explained as
1 2 1 1 2
1
11
(1 ) ( )
1
1
mj
Nj
j
y u v
H u w f
(15)
where yj is a set of the division point on the interval
[a,b] as
1, 1, 2,..., .
2
jj
y j w a j m
j
w
is a
weight of composite midpoint formula
.
jba
wm
4 Numerical Results
The relative mean index (RMI), [17], is used to test
the performance of a two-sided Extended EWMA
on varying bound control limits [a, b] and the
comparative performance of the ARL under various
λ conditions. The RMI can be computed as
1
( ) ( )
1,
()
nii
ii
RMI ARL c ARL s
n ARL s
(16)
where
()
i
ARL c
is the ARL of row i on the tested
control chart,
()
i
ARL s
is the lowest ARL of row i
from all the control charts such that a control chart
is more effective if the RMI value is lowest,
indicating that the control chart had the best
performance at change detection.
The ARL was approximated by the NIE method
using the composite midpoint rule on the Extended
EWMA with 1,000 nodes. The absolute percentage
difference to assess the veracity of ARL. When
ARL0 = 370,
= 0.5,
1
= 0.05, 0.10,
2
=
0.01, v = 1,
1
= 0.10, -0.10, 0.20, -0.20 and then
the initial parameter value was studied at
01
.
The out-of-control process
10
(1 )
is
computed by determining shift size
()
to be 0.01,
0.02, 0.03, 0.05, 0.100, 0.20, 0.30, 0.50, 1.00, 2.00
and 3.00. The upper control limit of one-sided and
two-sided Extended EWMA and EWMA control
charts are obtained in Table 1 (Appendix). The
results of ARL for the one-sided and two-sided
Extended EWMA using an explicit formula with
NIE are compared in Tables 2 (Appendix) and
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Table 3 (Appendix). Both ARL procedures yield
equivalent results.
Besides Table 4 (Appendix) and Table 5
(Appendix), the performance comparisons between
the Extended EWMA control chart and the EWMA
control chart when a = 0.0001 and ARL0 = 370 are
presented. The ARL1 value of the Extended
EWMA control chart is lower than the EWMA
control chart at all shift sizes. Moreover, it was
found that the Extended EWMA control chart had
the best performance because it gave the lowest
RMI. Therefore, it also can be concluded that the
Extended EWMA control chart performs better
than the EWMA and control chart.
5 Application to Real-world Data
The ARL was constructed using explicit formulas
on one-sided and two-sided Extended EWMA
control chart with ARL0 = 370 for
1
= 0.05, 0.10,
and
2
= 0.01, and its performance was compared
with the EWMA control chart using real-world data
on the monthly fuel price, Thailand between
January 2019 and May 2023. Based on the
autocorrelation function (ACF) and partial
autocorrelation function (PACF), this data
represents a stationary time series. The moving
average (MA(1)) process was obtained as
1
0.901
t t t
XX
and
(2.1262).
tExp
In Table 6 (Appendix), the upper control limits
for one-sided and two-sided Extended EWMA
control charts are obtained. In addition, the ARL of
the Extended EWMA control chart is evaluated and
compared with the EWMA control chart. The ARL
comparison of the one-sided and two-sided
Extended EWMA control chart for MA(1) using
NIE against the EWMA control chart is presented
in Table 7 (Appendix). The results found that one-
sided and two-sided Extended EWMA control
charts outperform the EWMA control chart with
small shift sizes detection as shown in Figure 1
(Appendix) and Figure 2 (Appendix). For the
various
1
values, the performance of control
charts performs better when
1
decreased.
6 Conclusions
In this specific study, when ARL0 = 370,
1
= 0.05,
0.10,
2
= 0.01. The ARL was used to evaluate
the efficacy of control charts. Using the numerical
integral equation (NIE) method, the explicit
formula is compared. Consequently, both methods
demonstrate that the ARL values are near, but the
explicit formula method can be calculated in less
time. The Extended EWMA control chart with
various
1
outperformed the EWMA control chart
for the moving average (MA(1)) procedure in terms
of performance. When considering the comparative
efficacy of the ARL under different smoothing
parameters, a smoothing parameter with a value of
0.05 is recommended. Eventually, the simulation
studies and the performance illustration with real-
world datasets using data on the monthly fuel price
yielded the same outcomes. Future research could
also evaluate the optimal parameters for MA(1)
processes when comparing the performance of the
Extended EWMA control chart with other control
charts. In addition, it is possible to develop
formulas for ARL values on the Extended EWMA
control chart to construct new control charts or
other interesting models.
Acknowledgement:
The authors would like to express our gratitude to
the National Research Council of Thailand (NRCT)
and King Mongkut’s University of Technology
North Bangkok for supporting the research fund
with contact no. N42A650318.
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APPENDIX
Table 1. Upper control limit of the Extended EWMA and the EWMA control charts for MA(1) when
= 0.5,
v = 1, a = 0.0001,
0
= 1 for ARL0 = 370
1
1
2
= 0.01
One-sided
Two-sided
Extended EWMA
EWMA
Extended EWMA
EWMA
b
h
b
h
0.0
5
0.1
-0.1
0.2
-0.2
6.930x10-8
5.680x10-8
7.660x10-8
5.140x10-8
0.937x10-7
0.767x10-7
1.036x10-7
0.694x10-7
1.000695x10-4
1.000569x10-4
1.000768x10-4
1.000515x10-4
1.000375x10-4
1.000307x10-4
1.000415x10-4
1.000278x10-4
0.1
0
0.1
-0.1
0.2
-0.2
2.980x10-3
2.430x10-3
3.300x10-3
2.200x10-3
4.459x10-3
3.637x10-3
4.940x10-3
3.285x10-3
3.080x10-3
2.531x10-3
3.400x10-3
2.300x10-3
4.120x10-3
3.380x10-3
4.560x10-3
3.068x10-3
Table 2. ARL comparison of one-sided Extended EWMA control chart for MA(1) using explicit formulas
against NIE method when
= 0.5, v = 1,
0
= 1 for ARL0 = 370
2
= 0.01
1
1
Shift
Size
Explicit
NIE
1
Shift
Size
Explicit
NIE
0.05
0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.04370
302.53400
248.32853
204.63007
140.53460
58.51039
13.12485
4.22723
1.38243
1.01131
1.00030
1.00004
370.04370
302.53400
248.32853
204.63007
140.53460
58.51039
13.12485
4.22723
1.38243
1.01131
1.00030
1.00004
0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.09883
302.87861
248.85166
205.25510
141.22196
59.04442
13.33050
4.30307
1.39545
1.01189
1.00032
1.00005
370.09883
302.87861
248.85166
205.25510
141.22196
59.04442
13.33050
4.30307
1.39545
1.01189
1.00032
1.00005
0.05
-0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.44574
302.26616
247.62957
203.66842
139.36295
57.53587
12.74016
4.08503
1.35815
1.01024
1.00026
1.00004
370.44574
302.26616
247.62957
203.66842
139.36295
57.53587
12.74016
4.08503
1.35815
1.01024
1.00026
1.00004
-0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.48339
301.99870
247.17154
203.09957
138.71968
57.02995
12.54729
4.01496
1.34645
1.00974
1.00024
1.00004
370.48339
301.99870
247.17154
203.09957
138.71968
57.02995
12.54729
4.01496
1.34645
1.00974
1.00024
1.00004
0.10
0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.77370
334.44627
302.26970
273.70977
225.67062
143.53268
64.92709
33.23285
11.61831
2.65469
1.22936
1.07846
370.77370
334.44627
302.26970
273.70977
225.67062
143.53268
64.92709
33.23285
11.61831
2.65469
1.22936
1.07846
0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.95318
334.95718
303.04052
274.68198
226.91094
144.97215
66.08522
34.03565
11.99830
2.74326
1.24573
1.08476
370.95318
334.95718
303.04052
274.68198
226.91094
144.97215
66.08522
34.03565
11.99830
2.74326
1.24573
1.08476
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.90
Phunsa Mongkoltawat,
Yupaporn Areepong, Saowanit Sukparungsee
E-ISSN: 2224-2880
825
Volume 22, 2023
2
= 0.01
1
1
Shift
Size
Explicit
NIE
1
Shift
Size
Explicit
NIE
0.10
-0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.23698
333.26829
300.59257
271.64938
223.10914
140.63698
62.64928
31.67539
10.89489
2.49066
1.19982
1.06723
370.23698
333.26829
300.59257
271.64938
223.10914
140.63698
62.64928
31.67539
10.89489
2.49066
1.19982
1.06723
-0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.87986
333.49996
300.49560
271.29140
222.38565
139.55407
61.69210
31.00027
10.57612
2.41850
1.18700
1.06239
370.87986
333.49996
300.49560
271.29140
222.38565
139.55407
61.69210
31.00027
10.57612
2.41850
1.18700
1.06239
Table 3. ARL comparison of two-sided Extended EWMA control chart for MA(1) using explicit formulas
against NIE method when
= 0.5, v = 1, a = 0.0001,
0
= 1 for ARL0 = 370
2
= 0.01
1
1
Shift
Size
Explicit
NIE
1
Shift
Size
Explicit
NIE
0.05
0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.36828
302.80600
248.55644
204.82159
140.67100
58.57160
13.13959
4.23157
1.38302
1.01133
1.00030
1.00004
370.36828
302.80600
248.55644
204.82159
140.67100
58.57160
13.13959
4.23157
1.38302
1.01133
1.00030
1.00004
0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.32315
303.06806
249.01202
205.39114
141.32054
59.09026
13.34211
4.30661
1.39595
1.01191
1.00032
1.00005
370.32315
303.06806
249.01202
205.39114
141.32054
59.09026
13.34211
4.30661
1.39595
1.01191
1.00032
1.00005
0.05
-0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.35672
302.19954
247.57981
203.63139
139.34279
57.53253
12.74124
4.08571
1.35831
1.01025
1.00026
1.00004
370.35672
302.19954
247.57981
203.63139
139.34279
57.53253
12.74124
4.08571
1.35831
1.01025
1.00026
1.00004
-0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.46257
301.98769
247.16732
203.09996
138.72503
57.03698
12.55049
4.01618
1.34666
1.00975
1.00024
1.00004
370.46257
301.98769
247.16732
203.09996
138.72503
57.03698
12.55049
4.01618
1.34666
1.00975
1.00024
1.00004
0.10
0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.40272
334.11506
301.97340
273.44417
225.45600
143.40273
64.87368
33.20800
11.61122
2.65387
1.22929
1.07844
370.40272
334.11506
301.97340
273.44417
225.45600
143.40273
64.87368
33.20800
11.61122
2.65387
1.22929
1.07844
0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
370.58201
334.62547
302.74346
274.41544
226.69512
144.84089
66.03085
34.01018
11.99096
2.74239
1.24565
1.08474
370.58201
334.62547
302.74346
274.41544
226.69512
144.84089
66.03085
34.01018
11.99096
2.74239
1.24565
1.08474
0.00
0.01
0.02
0.03
0.05
0.10
370.02283
333.07880
300.42461
271.50019
222.99076
140.56852
370.02283
333.07880
300.42461
271.50019
222.99076
140.56852
0.00
0.01
0.02
0.03
0.05
0.10
370.50878
333.16970
300.20105
271.02817
222.17418
139.42776
370.50878
333.16970
300.20105
271.02817
222.17418
139.42776
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.90
Phunsa Mongkoltawat,
Yupaporn Areepong, Saowanit Sukparungsee
E-ISSN: 2224-2880
826
Volume 22, 2023
2
= 0.01
1
1
Shift
Size
Explicit
NIE
1
Shift
Size
Explicit
NIE
0.10
-0.1
0.20
0.30
0.50
1.00
2.00
3.00
62.62368
31.66460
10.89241
2.49053
1.19984
1.06724
62.62368
31.66460
10.89241
2.49053
1.19984
1.06724
-0.2
0.20
0.30
0.50
1.00
2.00
3.00
61.64139
30.97715
10.56973
2.41779
1.18693
1.06237
61.64139
30.97715
10.56973
2.41779
1.18693
1.06237
Table 4. ARL comparison of the one-sided Extended EWMA for MA(1) against EWMA control charts when
= 0.5 , v = 1,
0
= 1 for ARL0 = 370
2
= 0.01
1
1
Shift
Size
Extended
EWMA
EWMA
1
Shift
Size
Extended
EWMA
EWMA
0.05
0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.04370
302.53400
248.32853
204.63007
140.53460
58.51039
13.12485
4.22723
1.38243
1.01131
1.00030
1.00004
0.0000
370.65340
303.93145
250.19939
206.75709
142.77649
60.19825
13.76760
4.46429
1.42335
1.01316
1.00036
1.00006
0.0179
0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.09883
302.87861
248.85166
205.25510
141.22196
59.04442
13.33050
4.30307
1.39545
1.01189
1.00032
1.00005
0.0000
370.81573
304.36470
250.79815
207.44788
143.51578
60.76510
13.98788
4.54673
1.43789
1.01384
1.00039
1.00006
0.0188
0.05
-0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.44574
302.26616
247.62957
203.66842
139.36295
57.53587
12.74016
4.08503
1.35815
1.01024
1.00026
1.00004
0.0000
370.58084
303.27284
249.17533
205.52183
141.40508
59.12026
13.34660
4.30739
1.39597
1.01191
1.00032
1.00005
0.0169
-0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.48339
301.99870
247.17154
203.09957
138.71968
57.02995
12.54729
4.01496
1.34645
1.00974
1.00024
1.00004
0.0000
370.57538
302.96924
248.68553
204.92400
140.73600
58.59340
13.14235
4.23189
1.38298
1.01132
1.00030
1.00004
0.0173
0.10
0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.77370
334.44627
302.26970
273.70977
225.67062
143.53268
64.92709
33.23285
11.61831
2.65469
1.22936
1.07846
0.0000
370.92612
335.97188
304.88166
277.17251
230.28954
149.11246
69.55004
36.48927
13.19227
3.03257
1.30114
1.10646
0.0512
0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.92318
334.95718
303.04052
274.68198
226.91094
144.97215
66.08522
34.03565
11.99830
2.74326
1.24573
1.08476
0.0000
370.98449
336.37964
305.56730
278.07707
231.49471
150.57828
70.78597
37.37372
13.63078
3.14221
1.32280
1.11508
0.0568
0.10
-0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
370.23698
333.26829
300.59257
271.64938
223.10914
140.63698
62.64928
31.67539
370.98535
335.31919
303.66321
275.50862
228.01420
146.30356
67.18474
34.80778
-0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
370.87986
333.49996
300.49560
271.29140
222.38565
139.55407
61.69210
31.00027
370.94168
334.92821
302.99791
274.62886
226.84395
144.89545
66.02437
33.99388
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.90
Phunsa Mongkoltawat,
Yupaporn Areepong, Saowanit Sukparungsee
E-ISSN: 2224-2880
827
Volume 22, 2023
2
= 0.01
1
1
Shift
Size
Extended
EWMA
EWMA
1
Shift
Size
Extended
EWMA
EWMA
0.50
1.00
2.00
3.00
RMI
10.89489
2.49066
1.19982
1.06723
0.0000
12.36960
2.83164
1.26237
1.09122
0.0508
0.50
1.00
2.00
3.00
RMI
10.57612
2.41850
1.18700
1.06239
0.0000
11.97882
2.73882
1.24493
1.08445
0.0532
Table 5. ARL comparison of the two-sided Extended EWMA for MA(1) against EWMA control charts when
= 0.5 , v = 1, a = 0.0001,
0
= 1 for ARL0 = 370
2
= 0.01
1
1
Shift
Size
Extended
EWMA
EWMA
1
Shift
Size
Extended
EWMA
EWMA
0.05
0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.36828
302.80600
248.55644
204.82159
140.67100
58.57160
13.13959
4.23157
1.38302
1.01133
1.00030
1.00004
0.0000
370.41143
303.49330
249.84379
206.46739
142.58211
60.12220
13.75313
4.46081
1.42301
1.01315
1.00036
1.00006
0.0170
0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.32315
303.06806
249.01202
205.39114
141.32054
59.09026
13.34211
4.30661
1.39595
1.01191
1.00032
1.00005
0.0000
370.61097
304.20274
250.66964
207.34560
143.45045
60.74287
13.98502
4.54640
1.43794
1.01385
1.00039
1.00006
0.0180
0.05
-0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.35672
302.19954
247.57981
203.63139
139.34279
57.53253
12.74124
4.08571
1.35831
1.01025
1.00026
1.00004
0.0000
370.80256
302.87129
248.85046
205.25800
141.22914
59.05243
13.33407
4.30445
1.39569
1.01190
1.00032
1.00005
0.0168
-0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.46257
301.98769
247.16732
203.09996
138.72503
57.03698
12.55049
4.01618
1.34666
1.00975
1.00024
1.00004
0.0000
370.46812
302.80588
248.55634
204.82151
140.67095
58.57158
13.13959
4.23157
1.38302
1.01133
1.00030
1.00004
0.0170
0.10
0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.40272
334.11506
301.97340
273.44417
225.45600
143.40273
64.87368
33.20800
11.61122
2.65387
1.22929
1.07844
0.0000
370.49889
335.41433
304.38512
276.72952
229.93514
148.90387
69.47016
36.45530
13.18472
3.03245
1.30129
1.10655
0.0558
0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.58201
334.62547
302.74346
274.41544
226.69512
144.84089
66.03085
34.01018
11.99096
2.74239
1.24565
1.08474
0.0000
370.97754
336.38138
305.57609
278.09154
231.51728
150.60922
70.81417
37.39474
13.64174
3.14515
1.32342
1.11533
0.0578
0.10
0.1
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
370.02283
333.07880
300.42461
271.50019
222.99076
140.56852
62.62368
31.66460
10.89241
370.33961
334.75044
303.16136
275.06501
227.66584
146.10801
67.11709
34.78228
13.36576
-0.2
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
370.50878
333.16970
300.20105
271.02817
222.17418
139.42776
61.64139
30.97715
10.56973
370.93295
334.93694
303.02046
274.66225
226.89214
144.95643
66.07514
34.02925
11.99548
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.90
Phunsa Mongkoltawat,
Yupaporn Areepong, Saowanit Sukparungsee
E-ISSN: 2224-2880
828
Volume 22, 2023
2
= 0.01
1
1
Shift
Size
Extended
EWMA
EWMA
1
Shift
Size
Extended
EWMA
EWMA
1.00
2.00
3.00
RMI
2.49053
1.19984
1.06724
0.0000
2.83221
1.26261
1.09133
0.0634
1.00
2.00
3.00
RMI
2.41779
1.18693
1.06237
0.0000
2.74263
1.24561
1.08471
0.0545
Table 6. Upper control limit of the Extended EWMA and the EWMA control charts for real-world data when
a = 0.0001,
0
= 2.1262,
1
= -0.901 for ARL0 = 370
1
2
= 0.01
One-sided
Two-sided
Extended EWMA
EWMA
Extended EWMA
EWMA
b
h
b
h
0.05
0.004441
0.004361
0.004545
0.001882
0.10
0.187616
0.188629
0.187750
0.167300
Table 7. ARL comparison of one-sided and two-sided Extended EWMA control chart for MA(1) using NIE
against EWMA control chart when a = 0.0001,
0
= 2.1262,
1
= -0.901 for ARL0 = 370
2
= 0.01
1
Shift Size
One-sided
Two-sided
Extended EWMA
EWMA
Extended EWMA
EWMA
0.05
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.06271
334.90280
303.79616
275.95855
229.01656
147.86130
68.54703
35.76452
12.83886
2.94598
1.28438
1.09987
0.0000
370.96000
335.64149
304.40145
276.45050
229.33199
147.92559
68.62827
35.85421
12.89523
2.96204
1.28771
1.10120
0.0023
370.05527
334.89911
303.79551
275.96037
229.02194
147.87061
68.63727
35.86091
12.89863
2.96290
1.28788
1.10127
0.0000
370.82145
335.59720
304.43282
276.54283
229.51081
148.19377
68.79191
35.94273
12.92745
2.96778
1.28859
1.10152
0.0018
0.10
0.00
0.01
0.02
0.03
0.05
0.10
0.20
0.30
0.50
1.00
2.00
3.00
RMI
370.00318
328.30959
292.81872
262.67743
214.90319
139.59408
71.39976
42.63037
19.86358
6.45577
2.51113
1.74422
0.0000
370.99628
328.37464
294.47922
263.88177
216.56967
141.35100
72.56040
43.35296
20.17653
6.52486
2.52208
1.74738
0.0088
370.23098
328.55482
293.71633
263.99881
216.64873
141.38453
72.56882
43.35576
20.17712
6.52504
2.52218
1.74745
0.0000
370.36147
329.32774
297.89702
265.43359
218.30872
142.94413
73.54454
43.94908
20.42483
6.57412
2.52733
1.74767
0.0082
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.90
Phunsa Mongkoltawat,
Yupaporn Areepong, Saowanit Sukparungsee
E-ISSN: 2224-2880
829
Volume 22, 2023
(a) (b)
Fig. 1: Real-world data of the ARL values on one-sided Extended EWMA and the EWMA control charts for
MA(1) with ARL0 = 370; (a)
1
= 0.05 and (b)
1
= 0.10
(a) (b)
Fig. 2: Real-world data of the ARL values on two-sided Extended EWMA and the EWMA control charts for
MA(1) with ARL0 = 370; (a)
1
= 0.05 and (b)
1
= 0.10
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Phunsa Mongkoltawat carried out the writing-
original draft preparation and simulation.
- Yupaporn Areepong has organized the
conceptualization, writing, review, editing, and
validation
- Saowanit Sukparungsee has implemented the
methodology and software.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The National Research Council of Thailand
(NRCT) and King Mongkut’s University of
Technology North Bangkok supported the research
fund with contact no. N42A650318
Conflicts of Interest
The authors declare no conflict of interest.
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.e
n_US
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.90
Phunsa Mongkoltawat,
Yupaporn Areepong, Saowanit Sukparungsee
E-ISSN: 2224-2880
830
Volume 22, 2023
