Statistical Design for Monitoring Process Mean on Modified EWMA

Control Chart based on Autocorrelated Data

SITTIKORN KHAMROD1, YUPAPORN AREEPONG1*, SAOWANIT SUKPARUNGSEE1,

RAPIN SUNTHORNWAT2

1Department of Applied Statistics, Faculty of Applied Statistics,

King Mongkut’s University of Technology North Bangkok,

Bangkok 10800,

THAILAND

2 Industrial Technology and Innovation Management Program, Faculty of Science and Technology,

Pathumwan Institute of Technology,

Bangkok 10330,

THAILAND

*Corresponding Author

Abstract: - This research endeavor is focused on establishing explicit formulas for the computation of the

average run length (ARL) within the context of a moving average process characterized by exogenous

variables, denoted as MAX(q,r), and subjected to exponential white noise. Additionally, we aim to conduct a

comparative analysis of their performance against the exponentially weighted moving average (EWMA) and

the modified exponentially weighted moving average (modified EWMA) methodologies. The evaluation of

their performance will be based on metrics such as the absolute percentage relative error (APRE) and the

relative mean index (RMI). Furthermore, we undertake a rigorous assessment of the accuracy of these explicit

formulas in relation to ARL by considering CPU time, utilizing the numerical integral equation (NIE) method

derived through the application of the Gauss-Legendre quadrature rule. This comparative evaluation is carried

out for both control chart methodologies. To ascertain the efficacy of our explicit formulas approach, we apply

it to two distinct datasets. The first dataset pertains to the closing price of natural gas, with the crude oil WTI

price serving as the exogenous variable. The second dataset encompasses the closing stock price of KTB Public

Company Limited, with daily foreign exchange rates for USD/JPY and EUR/USD as the exogenous variables.

The results of applying the ARL based on the explicit formulas to these two datasets demonstrate that, under

these conditions, the modified EWMA control chart outperforms the EWMA control chart.

Key-Words: - Average Run Length, Moving Average Process, Explanatory Variable, Explicit Formulas,

Modified Exponentially Weighted Moving Average.

Received: October 13, 2022. Revised: August 24, 2023. Accepted: September 23, 2023. Published: October 9, 2023.

1 Introduction

Currently, Statistical Process Control (SPC) stands

as a vital methodology employed to monitor and

control process variations, ensuring that various

industries such as manufacturing, healthcare,

medical sciences, finance analysis, and others

operate at their maximum potential to produce

conforming products. In particular, researchers

have conducted reviews on the advantages and

limitations of SPC in the context of quality

improvement, which has implications for financial

systems, healthcare, and the manufacturing

industry, [1], [2]. The foundational work in this

field began with, [3], who introduced the first

control chart, widely used for monitoring and

detecting significant process changes, especially

when observations follow a normal distribution.

The study, [4], subsequently proposed the

Cumulative Sum (CUSUM) control chart, while,

[5], presented the Exponentially Weighted Moving

Average (EWMA) control chart, which is

particularly adept at detecting subtle shifts in the

process means. The benefits of employing the

EWMA control chart have been extensively

documented. Building on this foundation, [6],

introduced the modified EWMA control chart, a

highly effective tool for detecting small, sudden

shifts in the process mean. Subsequently, [7],

further developed the Modified EWMA control

chart by introducing an additional constant factor,

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DOI: 10.37394/232018.2023.11.41

Sittikorn Khamrod, Yupaporn Areepong,

Saowanit Sukparungsee, Rapin Sunthornwat

E-ISSN: 2415-1521

450

Volume 11, 2023

denoted as 'k,' and incorporating an exponential

smoothing limiting factor. This modified approach

outperforms both existing EWMA control charts in

terms of Average Run Length (ARL).

The ARL is a measurement method utilized in

control charts to assess performance, and it can be

categorized into two components. Firstly, ARL0,

also known as in-control ARL, represents the

average number of data points before an out-of-

control condition is detected. Simultaneously,

ARL1, which pertains to out-of-control situations,

denotes the average number of data points that fall

outside control limits before the process is

recognized as out-of-control. The objective is to

keep ARL1 as small as possible. Various methods

can be employed to evaluate ARL, such as explicit

formulas, the Markov chain approach (MCA), or

numerical integral equations (NIE). Researchers

commonly employ these methods to determine

ARL in a variety of contexts. For instance, [8],

utilized the martingale approach to derive explicit

formulas for ARL and average delay time. They

compared these results to performance metrics

under the Exponentially Weighted Moving Average

(EWMA) and other measures in the exponential

distribution. The study, [9], employed Fredholm's

second-kind integral equations method to resolve

ARL in the context of the EWMA procedure for

AR(1) processes. The study, [10], considered the

numerical integral equation (NIE) method,

employing Gauss-Legendre quadrature rules, to

analyze the modified exponentially weighted

moving average (Modified EWMA) control chart

for MA(1) processes with exponential white noise.

The study, [11], derived approaches involving

Markov chains and integral equations to evaluate

ARL in the context of CUSUM and EWMA control

charts. The study, [12], employed the Fredholm

integral equation approach to establish an explicit

formula for calculating the ARL in CUSUM

control charts based on the SAR(P)L with a trend

process. Furthermore, researcher conducted a

comparative analysis with the NIE approach.

Building on this work, [13], derived an explicit

formula and extended the NIE method for ARL

calculations in CUSUM charts when dealing with

observations that follow seasonal autoregressive

models with exogenous variables, specifically

SARX(P,r)L with exponential white noise. In

another research endeavor, [14] introduced a novel

explicit formula for ARL in EWMA control charts,

focusing on stationary moving average processes

with exogenous variables represented as MAX(q,r).

Their approach made innovative use of the

Fredholm integral equation technique. The study,

[15], conducted an explicit formula was developed

for ARL in CUSUM control charts, considering a

seasonal autoregressive model with one exogenous

variable (SARX(1,1)L). They also compared their

results to those obtained through NIE, employing

various numerical integration techniques such as

the Gaussian rule, the Midpoint rule, and the

Trapezoidal rule. Furthermore, [16], introduced a

new solution for calculating ARL within the

context of the EWMA control chart, specifically

when the process adheres to the SMAX(Q,r)L

model. This explicit ARL solution for the

SMAX(Q,r)L process is analyzed using the

Fredholm integral equation method.

The observers typically encounter situations

governed by Stochastic processes, which involve

accidental time-space or time-series dynamics.

Moreover, these processes often originate from

econometric models, specifically the autoregressive

(AR) model and moving average (MA) model.

However, these situations tend to exhibit

unpredictability in their movement patterns, and the

error factors from discrepancies between actual

values and predictions. Consequently, they evolve

into seasonal moving average (SMA) models.

Subsequently, in cases where the time-series errors

follow a white noise pattern and there is auto-

correlated due to seasonal factors, it is referred to

as exponential white noise, [17], [18], [19].

Exogenous variables are defined as those not

directly influenced by other variables, and they find

frequent application in econometric models.

Furthermore, processes that incorporate exogenous

variables, especially multiple exogenous variables

simultaneously, often yield enhanced performance.

Consequently, the components of the AR, MA,

SAR, or SMA models incorporating exogenous

variables are designated as ARX, MAX, SARX,

and SMAX, respectively. The Numerical Integral

Equation (NIE) method is widely adopted for

explaining continuous distributions. Additionally,

researchers who explore these methods often make

comparisons with explicit formulas while assessing

the performance of a modified EWMA control

chart, [14], [15], [16], [20], [21], [22].

The preceding research underscores the

utilization of the Numerical Integral Equation

(NIE) method to conduct a comparative analysis of

explicit Average Run Length (ARL) formulas in

various control chart contexts, encompassing

CUSUM, EWMA, and modified EWMA control

charts, across different models such as AR(1),

MA(1), AR(p), and ARX(p,r). Notably, there has

been a gap in the exploration of the modified

EWMA control chart applied to the MAX(q,r)

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DOI: 10.37394/232018.2023.11.41

Sittikorn Khamrod, Yupaporn Areepong,

Saowanit Sukparungsee, Rapin Sunthornwat

E-ISSN: 2415-1521

451

Volume 11, 2023

process with exponential white noise.

Consequently, our primary objective revolves

around deriving explicit formulas and conducting

ARL comparisons using the NIE method. The

outcomes of this study reveal that explicit formulas

enable more rapid evaluation, particularly when

employing the Gauss-Legendre quadrature rule, for

a MAX(q,r) process with exponential white noise

when implemented within a modified EWMA

control chart. To provide empirical validation, real

data were employed in the observational process,

involving two distinct datasets. The first dataset

pertains to the closing price of natural gas, with the

crude oil WTI price serving as the exogenous

variable, covering the period from July 1st to

August 31, 2022. The second dataset involves the

closing stock price of KTB Public Company

Limited, with daily foreign exchange rates for

USD/JPY and EUR/USD as exogenous variables,

spanning from August 1st to September 15, 2022.

Furthermore, this study encompasses a comparative

analysis of the performance of the modified and

EWMA control charts based on metrics such as the

Absolute Percentage Relative Error (APRE) and

the Relative Mean Index (RMI).

2 Materials and Methods

The EWMA control chart used to monitor and

detect small changes in the process mean, [3], can

be derived by using the recursive equation.

1

(1 )

tt

t

E E Y





  

,

1, 2, 3,...t

(1)

where

Et

is the EWMA statistic,

01





is an

exponential smoothing parameter, and

Yt

is the

sequence of the MAX(q,r) process with exponential

white noise. The mean and variance of the EWMA

control chart are

 

0

EE

t





and

2

Var( ) ( ),

2

t

E







respectively. Therefore, the

general upper control limit (UCL) and lower

control limit (LCL) to detect the sequence are

respectively given by

012

UCL L



 

 

(2)

012

LCL L



 

 

(3)

where

0



is the target mean,



is the process

standard deviation, and

1

L

is an appropriate control

width limit.

The stopping time for the one-sided EWMA

control chart is given by

inf{ 0: }

t

ht E h



  

(4)

The study, [7], proposed a new structure for the

control statistics of the modified EWMA control

chart by using the following recursive equation:

11

(1 ) ( )

t t t

tt

Z Z Y k Y Y





    

(5)

where

01





is an exponential smoothing

parameter and

k

is a constant. The mean

and variance of the modified EWMA control

chart is

 

0

EZ

t





and

 

2

222

( ),

2

tkk

Var Z









respectively. Therefore, the general UCL and LCL

to detect the sequence are respectively given by

2

02 22

2kk

UCL L



 



 

(6)

2

02 22

2kk

LCL L



 



 

(7)

where

0



is the target mean,



is the process

standard deviation,

2

L

is an appropriate control

width limit,

t

Y

is the sequence of observations,

0

Zu

and

0

Yv

are the initial values, and

01





is an exponential smoothing parameter.

The stopping time for the one-sided modified

EWMA control chart is given by

inf{ 0: }t Z l

t

b



  

(8)

where

b



is the stopping time,

a

is the LCL, and

l

is the UCL.

3 The ARL of Modified EWMA

Control Chart

3.1 The Exact Solution of ARL the modified

EWMA Control Chart for MAX(q,r)

process with Exponential White Noise

A MAX (q,r) process with exponential white noise

can be derived as

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E-ISSN: 2415-1521

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t

Y

1 1 2 2 ...

t q t q

tt

       





     

1

r

i it

ix







(9)

where



is a constant,

t



is the white noise

process

~ ( )

tExp



,



is the MA coefficient with

an initial value of

0s





,

t

X

is an exogenous

variable, and



is the coefficient of

t

X

. Therefore,

modified EWMA statistics

 

t

Z

can be written as

 

11

11 1

1

...

tt

r

q t q i it

ti

Z Z k kY

kX

  

      













    

      

If

t

Y

signals the out-of-control state for

1

Z

when

0

Zu

, then

 

1 1 0

10 1

1

... r

q t q i it

i

Z u k kY

kX

  

      









    

      

If

1



is the in-control limit for

1

Z

, then

1.0lZ

Consider the following function:

1 1 1

( ) 1 ( ) ( ) ( )F u F Z f d







(10)

which is a Fredholm integral equation of the second

kind, [23]. Moreover,

()Fu

can be rewritten as

   

 



1

0

11

l

t

F u F u kY k y





     



 

11 1

... r

q t q i it

ti

kX

      













     

 

f y dy

Let

 

1

1t

w u kY k y





    

 



11... p t p

t

k Y Y

   



    

11 1

... r

q t q i it

tiX

    











    

By changing the integral variable, we can obtain

the following integral equation:

     

1

0

1

1lt

wu

kY

F u F w f

k k k



  











  

  



11 1

... r

q t q i it

tiX dw

     



















     

. (11)

If

 

~

t

Y Exp



and

 

1y

f y e







;

0y

, then

   

 

1

11

1

0

1

1...

11

1

t

tr

qtq i it

i

l

wu

kY

kk

X

F u F w e dw

k





  

  



































 









 





(12)

Let

 



111 1

11... r

tqtq i it

ti

u kY X

k

M u e

     















     





,

then we obtain

   

 



0

1 ; 0

lwk

Mu

F u F w e dw u l

k







   



.

Let

 

0

lwk

g F w e dw









, then

 

1.

Mu g

k

Fu



  



Consequently, we obtain

 

11

1

...

11

1

( ) 1

tr

t

q t q i it

i

u kY x

k

F u e g

k

  



  

















 



  





(13)

By solving for constant

g

, we obtain

g

 

0

lwk

F w e dw









   

 

0

1

lwk

gM w e dw

k





















 

111 1

00

11... r

tq t q i it

ti

ll

u kY x

wk

k

wk

ge

e dw k

e dw

     















     









 







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 

   

11

1

...

1

11

tr

t

q t q i it

i

lk

kY x

klk

ke

g

ee



  

  



 































  



















.

(14)

By substituting constant

g

Eq. (14) into Eq. (13),

we arrive at

 



 

111 1

11...

1

r

tqtq i it

ti

u kY X

k

e

Fu k

     















     



 



 

   

11

1

...

1

tr

t

q t q i it

i

lk

kY l

x

kk

ke

ee



  



  



 

 

































  































. (15)

Using the Fredholm integral equation of the

second kind, the explicit one-sided formulations for

the ARL of a MAX(q,r) process operating on a

modified EWMA control chart can be derived.

Fredholm integral equations are encountered in

various fields because they are essential for

analyzing and solving problems that involve

functions and integrals. They offer a robust

mathematical framework that facilitates the

modeling and comprehension of complex

phenomena. When the process is in a state of

control with exponential parameters

0





, we

obtain the following explicit solution for

0

ARL

:

 

   

00

11

1

0

010

0

1

...

1

tr

t

q t q i it

i

ul

kk

kY l

X

kk

ee

ARL

ee



   

  



  



 















































.

(16)

Similarly, the explicit solution for can be

expressed as when the process is in the out-of-

control state with an exponential parameter

1





.

 

   

11

1

111

1

...

1

tr

t

q t q i it

i

ul

kk

kY l

x

kk

ee

ARL

ee



   

  



  



 















































.

(17)

The Existence and Uniqueness of the Explicit

Formulas. Here, we show the existence and

uniqueness of the solution to the integral equation

in Eq. (12). First, we define

 



1

11

1

0

1

1...

11

( ( )) 1

t

qtq

t

r

i it

i

l

wu

kY

kk

X

T F u F w e dw

k





    





















 



    



 





(18)

Theorem 1.

Banach’s fixed-point theorem, [24].

Let

[0, ]Nl

be a set of all of the continuous

functions on complete metric

 

,,Xd

and assume

that

:T X X

is a contraction mapping with

contraction constant

01s

; i.e.,

12

( ) ( )T F T F

1 2 1 2

,s F F F F X   

. Subsequently,

(.)FX

is

unique at

( ( ))T F u

()Fu

; i.e., it has a unique fixed

point in

.X

Proof: To show that

T

defined in (13) is a

contraction mapping for

12

, [0, ]F F C l

, we use the

inequality

1 2 1 2 1

( ) ( ) ,T F T F s F F F   

2(0, )F N l

with

01s

. Consider Eq.(8) and Eq.(13), then

12

( ) ( )T F T F 



 

12

0

[0, ]

()

sup ( ( ) ( ))

lwk

ul

Mu F w F w e dw

k













 

12

[0, ]

sup ( ) 1

lk

ul

F F M u e















  





 

12 [0, ]

1 sup ( )

lk

ul

F F e M u











  

12

s F F 



,

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E-ISSN: 2415-1521

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Volume 11, 2023

where

 

[0, ]

1 sup ( )

lk

ul

s e M u











and

 

111 1

11... r

tq t q i it

ti

u kY X

k

M u e

     















     





;

0 1.s

The solution exists and is unique, as

demonstrated by the application of Banach's fixed-

point theorem.

3.2 The NIE for the ARL of an MAX(q,r)

Process on a Modified EWMA Control

Chart

The NIE approach is widely used for evaluating the

ARL. It can be based on one of several quadrature

rules (midpoint, trapezoidal, Simpson’s rule, and

Gauss-Legendre), all of which give ARLs that are

very close to each other, [25]. In the present study,

we use the Gauss-Legendre rule to evaluate the

ARL. An integral equation of the second kind for

the ARL on the modified EWMA control chart for

the MAX(q, r) process in (15) can be approximated

by using the quadrature formula. The Gauss-

Legendre quadrature rule is applied as follows:

Given that

 

111

1

...

ji

t

jt

r

q t q i it

i

aa

k

kY

f a f k

x



  



  















    















(18)

The approximation for the integral is in the form

   

 

1

0

lm

jj

j

F w f w dw w f a







(19)

where

1

2

jb

aj

m









and

; 1,2,...,

jb

w j m

m



.

Using the Gauss-Legendre quadrature formula,

numerical approximation

()Fu

for the integral

equations can be found as the solution to the

following linear equations:

 

1

mji

i j j

j

aa

F a w F a f

kk



















 



 

111 1

... r

tq t q i it

ti

kY X

k

     













     



and

 

1

mjm

m j j

j

aa

F a w F a f

kk













 









 

111 1

... r

tt q t q i it

i

kY X

k

     













      







.

This set of

m

equations with

m

unknowns can be

rewritten in matrix form. The column vector

of

 

i

Fa

is

     

 

1 1 2

, ,...,

mm

F a F a F a



L

. Since

1(1,1,...,1)

m

1

is a column vector of ones and

mm

R

is a matrix, we can define m to the mth element of

matrix

R

as follows:

 

   

1

11 1

1

1,

...

ji

t

ij j r

t q t q i it

i

aa

kY

kk

R w f

kx





    







   













 

   







and

 

1,1,...,1

mdiagI

as a unit matrix of order

m

.

If

 

1

IR

exists, the numerical approximation for

the integral equation in terms of the matrix can be

written as

 

1

11m m m m m



  

G I R 1

. Finally, by

substituting

i

a

with

u

in

 

i

Fa

, the numerical

integration equation for function

 

Fu

can be

derived as

 

1

mj

jj

j

au

F u w F a f

kk













 









 

111 1

... r

tt q t q i it

i

kY X

k

     













      







. (20)

Here, we compare the results for ARL0 and

ARL1 derived by using explicit formulas and the

NIE method for a MAX(q,r) process with

exponential white noise running on a modified

EWMA chart. The numerical results were

computed by using MATHEMATICA with the

number of division points set as 1,000. The

performances are reported as the absolute

percentage relative error, which is derived as

WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2023.11.41

Sittikorn Khamrod, Yupaporn Areepong,

Saowanit Sukparungsee, Rapin Sunthornwat

E-ISSN: 2415-1521

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Volume 11, 2023

(%) 100

Explicit Formula NIE

Explicit Formula

ARL ARL

APRE ARL





For comparison, the performance measure for

the ARL of a MAX(q,r) process with exponential

white noise on the EWMA and modified EWMA

control chart is the RMI, which is computed as

,,

1,

[]

1

[]

nshift i shift i

ishift i

ARL Min ARL

RMI n Min ARL















where

,shift i

ARL

is the ARL of the control chart

when a shift in the process mean is detected and

,

[]

shift i

Min ARL

is the minimum value of the ARL at

the same level.

4 Numerical Results

The results from Table 1 (Appendix) and Table 2

(Appendix) allocated for upper control limit (l)

which is related to the implementation process

running on modified EWMA control charts as

reported for MAX(2,1) and MAX(3,2) respectively,

also compare both Table 1 (Appendix) and Table 2

(Appendix) reveal different results of

i



. The

outcomes for the one-sided ARL when using the

explicit formulas and the NIE method to verify

Table 3 (Appendix) and Table 4 (Appendix) define

ARL0 = 370, and



= 0.05, 0.10, 0.15, and 0.20

described in

Table 3 (Appendix) was

1



=2.5 and Table 4

(Appendix) was

1



=1 and

2



=3 are especially in

conditions that reveal different results

i



.

Moreover, the CPU time for the explicit formulas

was minuscule spending less than 1 second while

that for the NIE method was around 9 seconds, and

the central processing unit (CPU) time (System:

AMD Ryzen 7 5700U with Radeon

Graphics@1.8GHz. Processor, 16GB RAM. 64-bit

Operating System).

According to a comparison of the ARL values

between EWMA and modified EWMA control

charts processes for a MAX(2,3) wherever ARL0 =

370,

11





,

22





,

33





,

1 2 3 1X X X  

,

1



=0.10, and

2



=0.20 found that

0.05





the

accomplishment of EWMA control chart it’s better

than modified EWMA control chart for k=0.5 when

shift size more than or equal to initiative 0.3

Furthermore, k=1, k=5 and k=10 found that

performance of EWMA greater than modified

EWMA at shift size was 0.4. For

0.10





modified

EWMA reveal that performance was better than

EWMA in all of the shift size from Table 5

(Appendix). According to the data presented in

Figure 1, the investigation of ARL values for

EWMA and modified EWMA control charts shows

that the modified EWMA performs more efficiently

than the standard EWMA as the parameter k

increases.

Application: Example 1

At this moment, we allocated the closing price of

natural gas with the crude oil WTI price as the

exogenous variable from 1 July to 31 August 2022,

as summarized in Table 6 (Appendix). The

performance of the modified EWMA was better

than that of the EWMA control chart except for

shift size = 0.4 for all k. Moreover, the results show

that the modified EWMA control chart less than

RMI values was under the EWMA control chart.

From Figure 2, the ARL of the EWMA and

modified EWMA control charts observation

research found that the efficiency of modified

EWMA was better than EWMA when k increased.

Finally, the ARL and the RMI values trended to

decrease when k increases from Figure 3.

Fig. 1: The ARL of the EWMA and modified EWMA control charts simulation data for (a)

0.05





and (b)

0.10





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E-ISSN: 2415-1521

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Volume 11, 2023

Fig. 2: The ARL of the EWMA and modified EWMA control charts for example 1

when (a)

0.05





and (b)

0.10





(a) (b)

Fig. 3: The RMI values of the EWMA and modified EWMA control charts for example 1

when (a)

0.05





and (b)

0.10





(a) (b)

Fig. 4: The RMI values of the EWMA and modified EWMA control charts for example 2

when (a)

0.05





and (b)

0.10





Application: Example 2

At this time, we spend the closing stock price for

KTB Public Company Limited with the USD/JPY

and EUR/USD are daily foreign exchange rates as

the exogenous variable from 1 August to 15

September 2022, as performed in Table 7

(Appendix). The accomplishment

of the EWMA was better than that of the modified

EWMA control chart for all shift sizes except for

k=

4





. Furthermore, the results show that the

modified EWMA control chart revealed RMI larger

values than the EWMA control chart except result

for k=

4





from Figure 4.

5 Conclusion

The Explicit formulas were verified for the ARL of

a MAX(q,r) process with exponential white noise

running on a modified EWMA control chart. The

results from notifying the upper control limit which

is related to implementing process running on

modified EWMA control charts as performed for a

MAX(q,r). The precision of the proposed explicit

formulas was suggested as the absolute in terms of

percentage difference deviation when compared

with the NIE method. Acknowledgeable, they were

using code and fasting to calculate the way the

CPU time to point out was less than the NIE

method. The practical applies to real data for the

MAX(q,r) process which results show that a

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DOI: 10.37394/232018.2023.11.41

Sittikorn Khamrod, Yupaporn Areepong,

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E-ISSN: 2415-1521

457

Volume 11, 2023

modified EWMA control chart is good for notating

small shifts in the process mean. Particularly, in

some cases could be chosen some specific k=

4





for modified EWMA control chart performed

outcomes greater than the EWMA control chart.

In this specific study, we chose k values of 0.5,

1, 5, and 10. It is clear that higher values of k result

in improved detection performance. However, in

real-world data applications, certain lower k values,

especially k=1, demonstrate efficiency comparable

to their larger counterparts. Furthermore, even

when



values vary for different shift sizes, the

ARL remains consistent.

Acknowledgement:

This research was funded by Thailand Science

Research and Innovation Fund (NSRF), and King

Mongkut’s University of Technology North

Bangkok with Contract no. KMUTNB-FF-66-04.

References:

[1] M. Kovarik, P. Klimek, The usage of time

series control charts for financial process

analysis, Journal of Competitiveness. Vol 4,

No. 3, 2012, pp.29-45.

[2] A. Koetsier, S. N. Van Der Veer, K. J. Jager,

N. Peek, N. F. De Keizer, Control charts in

healthcare quality improvement, Methods of

information in medicine, Vol 51, No. 03,

2012, pp.189-198.

[3] W. A. Shewhart, Economic control of quality

of manufactured product, D. Van Nostrand

Company, 1931.

[4] E. S. Page, Continuous inspection schemes,

Biometrika, Vol 41, No. 1/2, 1954, pp.100-

115.

[5] S. W. Roberts, Control chart tests based

on geometric moving average,

Technometrics, Vol 1, No. 3, 1959, pp.239-

250.

[6] A. K. Patel, J. Divecha, Modified

exponentially weighted moving average

(EWMA) control chart for an analytical

process data, Journal of Chemical

Engineering and Materials Science, Vol 2,

No. 1, 2011, pp.12-20.

[7] N. Khan, M. Aslam, C. H. Jun, Design of a

control chart using a modified EWMA

statistic, Quality and Reliability Engineering

International, Vol 33, No. 5, 2017, pp.1095-

1104.

[8] Y. Areepong, A. Novikov, Martingale

approach to EWMA control charts for

changes in exponential distribution, Journal

of Quality Measurement and Analysis, Vol 4,

No. 1, 2008, pp.197-203.

[9] W. Suriyakat, Y. Areepong, S.

Sukparungsee, G. Mititelu, (2012). On

EWMA procedure for AR (1) observations

with exponential white noise, International

Journal of Pure and Applied Mathematics,

Vol 77, No. 1, 2012, pp.73-83.

[10] Y. Supharakonsakun, Y. Areepong, S.

Sukparungsee, Numerical approximation of

ARL on modified EWMA control chart for

Ma (1) process, Lecture Notes in Engineering

and Computer Science, Vol. 2239, No. 1,

2019, pp.272-275.

[11] C. W. Champ, S. E. Rigdon, A comparison

of the Markov chain and the integral

equation approaches for evaluating the run

length distribution of quality control charts,

Communications in Statistics-Simulation and

Computation, Vol 20, No. 1, 1991, pp. 191-

204.

[12] K. Petcharat, The effectiveness of CUSUM

control chart for trend stationary seasonal

autocorrelated data, Thailand

Statistician, Vol 20, No. 2, 2022, pp. 475-

488.

[13] S. Phanyaem, Explicit formulas and

numerical integral equation of ARL for

SARX (P,r)L model based on CUSUM

chart, Mathematics and Statistics, Vol 10,

No. 1, 2022, pp. 88-99.

[14] W. Suriyakat and K. Petcharat, Exact run

length computation on EWMA control chart

for stationary moving average process with

exogenous variables, Mathematics and

Statistics, Vol 10, No. 3, 2022, pp. 624-635.

[15] C. Chananet and S. Phanyaem, Improving

CUSUM control chart for monitoring a

change in processes based on seasonal ARX

model, IAENG International Journal of

Applied Mathematics, Vol 52, No. 3, 2022,

pp.1-8.

[16] Petcharat K., Designing the Performance of

EWMA Control Chart for Seasonal Moving

Average Process with Exogenous

Variables. IAENG International Journal of

Applied Mathematics, 53(2), 2023, pp.1-9.

[17] J. Andel, On AR (1) processes with

exponential white noise, Communications in

Statistics-Theory and Methods, Vol. 17, No.

5, 1988, pp.1481–1495.

[18] F. J. Girón, E. Caro, J. I. Domínguez,

A conjugate family for AR (1) processes with

exponential errors, Communications in

WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2023.11.41

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Saowanit Sukparungsee, Rapin Sunthornwat

E-ISSN: 2415-1521

458

Volume 11, 2023

Statistics-Theory and Methods, Vol. 23, No.

6, 1994, pp.1771–1784.

[19] M. Ibazizen, H. Fellag, Bayesian estimation

of an AR (1) process with exponential white

noise, Statistics, Vol. 37, No. 5, 2003,

pp.365–372.

[20] P. Paichit, An integral equation procedure for

average run length of control chart of ARX

(p) processes, Far East Journal of

Mathematical Sciences, Vol 99, No. 3, 2016,

pp. 359-381.

[21] K.Petcharat, The performance of EWMA

control chart for MAX (1, r) process,

In: Proceedings of The International

MultiConference of Engineers and Computer

Scientists (IMECS 2021), October 2021.

[22] S. Phanyaem, Average run length of

CUSUM chart based on SARX (P, r)L model,

In: Proceedings of The International

MultiConference of Engineers and Computer

Scientists (IMECS 2021), October 2021.

[23] G. Mititelu, Y. Areepong, S. Sukparungsee,

A. Novikov, Explicit analytical solutions for

the average run length of CUSUM and

EWMA charts. East-West Journal of

Mathematics, 2010.

[24] R. S. Palais, (2007). A simple proof of the

Banach contraction principle, Journal of

Fixed Point Theory and Applications, Vol 2,

2007, pp.221-223.

[25] P. Phanthuna, Y. Areepong, S.

Sukparungsee, Numerical integral equation

methods of average run length on modified

EWMA control chart for exponential AR (1)

process, In: Proceedings of the International

MultiConference of Engineers and Computer

Scientists 2018, Hong Kong, March 2018.

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E-ISSN: 2415-1521

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Volume 11, 2023

APPENDIX

Table 1. Upper control limit values (l) for a MAX(2,1) process running on modified EWMA control charts

when ARL0 = 370,

1



=2.5 and

X

=1

Parameter



Modified EWMA

k = 0.5

k = 1

k = 5

k = 10

1



= 0.30

0.05

0.24913371

0.500416482

2.50745388

5.0162278

2



= 0.50

0.10

0.25503099

0.507821090

2.53911469

5.0799058

0.15

0.26185780

0.515851454

2.57146878

5.1452603

0.20

0.26942969

0.524444580

2.60454198

5.2123718

1



= 0.30

0.05

0.09084219

0.182566560

0.91485655

1.8301810

2



= -0.50

0.10

0.09245095

0.183978770

0.91910885

1.8385789

0.15

0.09441432

0.185652263

0.92341818

1.8470413

0.20

0.09665296

0.187550392

0.92778394

1.8555688

1



= 0.20

0.05

0.18389680

0.36945555

1.8512819

3.7035240

2



= 0.30

0.10

0.18779194

0.37383657

1.8685122

3.7380478

0.15

0.19238343

0.37869642

1.8860113

3.7731850

0.20

0.19752500

0.38397799

1.9037848

3.8089550

1



= -0.20

0.05

0.12284740

0.24685785

1.2370013

2.4746377

2



= 0.30

0.10

0.12516834

0.24911515

1.2447209

2.4899964

0.15

0.12796327

0.25171206

1.2525365

2.5055183

0.20

0.13112825

0.25460318

1.2604481

2.5212067

Table 2. Upper control limit values (l) for a MAX(3,2) process running on modified EWMA control charts

when ARL0 = 370,

1



=1,

2



=3,

1

X

=1 and

2

X

=1

Parameter



Modified EWMA

k = 0.5

k = 1

k = 5

k = 10

1



= 0.10

0.05

0.05498546

0.11052074

0.55384312

1.10796860

2



= 0.20

0.10

0.05588707

0.11120248

0.55543597

1.11105809

3



= 0.50

0.15

0.05700636

0.11205043

0.55706280

1.11416699

0.20

0.05829371

0.11304018

0.55872267

1.11729532

1



= -0.10

0.05

0.011073177

0.022261258

0.11156007

0.22317691

2



= -0.20

0.10

0.011237178

0.022356223

0.11164074

0.22331113

3



= -0.50

0.15

0.011445845

0.022486974

0.11173058

0.22344968

0.20

0.011688782

0.022648007

0.11182928

0.22359248

1



= 0.70

0.05

0.1110828

0.22322707

1.11859545

2.23776429

2



= 0.30

0.10

0.1131330

0.22515245

1.12491999

2.25031930

3



= 0.50

0.15

0.1156136

0.22738921

1.13132329

2.26299299

0.20

0.1184295

0.22989500

1.13780499

2.27578710

1



= -0.70

0.05

0.00549699

0.011051296

0.055382745

0.11079367

2



= -0.30

0.10

0.00557731

0.011095784

0.055407725

0.11082969

3



= -0.50

0.15

0.00567985

0.011158195

0.055437504

0.11086808

0.20

0.00579944

0.011235759

0.055471902

0.11090879

WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2023.11.41

Sittikorn Khamrod, Yupaporn Areepong,

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E-ISSN: 2415-1521

460

Volume 11, 2023

Table 3. The one-sided ARL for a MAX(2,1) process running on the modified EWMA chart when ARL0 = 370,

1



=2.5,

X

=1, and

k

=1

ARL

NIE

(time: s.)

APRE



l

i



Shift

size

Explicit

(time: <0.001)

NIE

0.05

0.500416482

=0.30

0.00

370.00004

370.00003

(9.109)

0.00000138

=0.50

0.001

279.55999

279.55998

(9.141)

0.00000127

0.003

187.77302

(9.125)

0.00000116

0.005

141.36583

(9.266)

0.00000110

0.007

113.35555

(9.141)

0.00000106

0.01

87.38934

(9.187)

0.00000102

0.03

34.61916

(9.297)

0.00000090

0.05

21.62573

(9.203)

0.00000084

0.07

15.75544

(9.125)

0.00000079

0.10

11.23569

(9.187)

0.00000072

0.30

4.08730

(9.297)

0.00000043

0.50

2.70231

(9.016)

0.00000027

0.70

2.13800

(9.140)

0.00000017

1.00

1.73869

(9.313)

0.00000010

0.10

0.18397877

= 0.30

0.00

370.00001

(9.226)

0.00000032

= –0.50

0.001

248.30778

(9.047)

0.00000025

0.003

149.78596

(9.094)

0.00000020

0.005

107.24114

(9.141)

0.00000017

0.007

83.52199

(9.234)

0.00000016

0.01

62.71985

(9.046)

0.00000015

0.03

23.61055

(9.079)

0.00000012

0.05

14.58250

(9.203)

0.00000011

0.07

10.58087

(9.312)

0.00000010

0.10

7.53611

(9.063)

0.00000009

0.30

2.82179

(9.109)

0.00000004

0.50

1.94746

(9.406)

0.00000002

0.70

1.60449

(9.203)

0.00000001

1.00

1.37145

(9.203)

0.00000001

0.15

0.37869642

= 0.20

0.00

370.00003

370.00157

(8.907)

0.00041399

= 0.30

0.001

259.39509

259.39584

(9.219)

0.00029015

0.003

162.38681

162.38710

(9.046)

0.00018153

0.005

118.22208

118.22224

(9.109)

0.00013208

0.007

92.96597

92.96607

(9.141)

0.00010380

0.01

70.42923

70.42928

(9.188)

0.00007856

0.03

27.04125

27.04126

(9.328)

0.00002997

0.05

16.82440

(9.093)

0.00001851

0.07

12.26643

(9.094)

0.00001340

0.10

8.78158

(9.125)

0.00000947

0.30

3.31637

(9.250)

0.00000321

0.50

2.26670

(9.219)

0.00000193

0.70

1.84116

(9.047)

0.00000136

1.00

1.54163

(9.282)

0.00000093

0.20

0.2546031792

= –0.20

0.00

370.00004

370.00003

(9.110)

0.00000173

= 0.30

0.001

242.53585

242.53584

(9.093)

0.00000119

0.003

143.66407

(9.141)

0.00000078

0.005

102.10439

(9.250)

0.00000060

0.007

79.22260

(9.140)

0.00000050

0.01

59.32160

(9.094)

0.00000042

0.03

22.31806

(9.203)

0.00000025

0.05

13.84747

(9.234)

0.00000020

0.07

10.09879

(9.109)

0.00000018

0.10

7.24638

(9.360)

0.00000015

0.30

2.80812

(9.125)

0.00000007

0.50

1.96840

(9.062)

0.00000004

0.70

1.63237

(9.156)

0.00000002

1.00

1.39916

(9.282)

0.00000001

WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2023.11.41

Sittikorn Khamrod, Yupaporn Areepong,

Saowanit Sukparungsee, Rapin Sunthornwat

E-ISSN: 2415-1521

461

Volume 11, 2023

Table 4. The one-sided ARL for a MAX(3,2) process running on the modified EWMA chart

when ARL0 = 370,

11





,

23





,

11X

,

21X

and

ARL

NIE

(time: s.)



l

i



Shift

size

Explicit

(time: <0.001 s.)

NIE

APRE

0.05

0.11052074

=0.10

0.00

370.00018

(9.141)

0.000000064

=0.20

0.001

244.71619

(9.125)

0.000000058

=0.50

0.003

145.86586

(9.250)

0.000000053

0.005

103.87154

(9.421)

0.000000050

0.007

80.63748

(9.156)

0.000000049

0.01

60.36506

(9.250)

0.000000047

0.03

22.51801

(9.219)

0.000000043

0.05

13.83562

(9.172)

0.000000039

0.07

9.99715

(9.171)

0.000000037

0.10

7.08390

(9.281)

0.000000032

0.30

2.61368

(9.125)

0.000000019

0.50

1.80732

(9.141)

0.000000006

0.70

1.49929

(9.250)

0.000000007

1.00

1.29579

(9.328)

0.000000000

0.10

0.022356223

= –0.10

0.00

370.00055

(8.969)

0.000000005

= –0.20

0.001

204.74900

(9.141)

0.000000003

= –0.50

0.003

108.12396

(9.094)

0.000000002

0.005

73.44573

(9.110)

0.000000002

0.007

55.60454

(9.250)

0.000000002

0.01

40.75057

(9.281)

0.000000002

0.03

14.65928

(9.109)

0.000000002

0.05

8.96310

(9.110)

0.000000001

0.07

6.48300

(9.156)

0.000000000

0.10

4.62131

(9.250)

0.000000002

0.30

1.84333

(9.156)

0.000000000

0.50

1.37951

(9.125)

0.000000000

0.70

1.21500

(9.141)

0.000000000

1.00

1.11464

(9.219)

0.000000000

0.15

0.22738921

=0.7

0.00

370.00029

370.00028

(9.078)

0.000000862

=0.3

0.001

246.28581

(9.219)

0.000000626

=0.5

0.003

147.62694

(9.203)

0.000000437

0.005

105.43034

(9.140)

0.000000355

0.007

82.01068

(9.219)

0.000000309

0.01

61.53276

(9.016)

0.000000269

0.03

23.18543

(9.172)

0.000000185

0.05

14.35886

(9.265)

0.000000159

0.07

10.44791

(9.266)

0.000000143

0.10

7.47111

(9.109)

0.000000126

0.30

2.84641

(9.172)

0.000000063

0.50

1.97790

(9.219)

0.000000035

0.70

1.63298

(9.219)

0.000000024

1.00

1.39553

(9.093)

0.000000014

0.20

0.0112357586

= –0.7

0.00

370.00057

(8.953)

0.000000003

= –0.3

0.001

177.84872

(9.204)

0.000000002

= –0.5

0.003

87.28694

(9.187)

0.000000001

0.005

57.86667

(9.156)

0.000000001

0.007

43.29746

(9.125)

0.000000001

0.01

31.44761

(9.156)

0.000000000

0.03

11.22574

(9.047)

0.000000000

0.05

6.91007

(9.141)

0.000000000

0.07

5.04318

(9.187)

0.000000000

0.10

3.64789

(9.250)

0.000000000

0.30

1.58725

(9.047)

0.000000000

0.50

1.25331

(9.141)

0.000000000

0.70

1.13841

(9.125)

0.000000000

1.00

1.07063

(9.266)

0.000000000

1k

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DOI: 10.37394/232018.2023.11.41

Sittikorn Khamrod, Yupaporn Areepong,

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E-ISSN: 2415-1521

462

Volume 11, 2023

Table 5. Comparison of the ARL values for a MAX(2,3) process running on EWMA and

modified EWMA control

Shift

size

EWMA

Modified EWMA

k=0.5

k=1

k=5

k=10

h=  

l =0.033308641

l =0.0669569

l =0.335539028

l =0.671248923

0.05

0.00

370

0.01

292.80788

101.16734

54.32554

29.66845

27.35029

0.02

232.80528

57.67087

29.25695

15.75359

14.52804

0.03

185.94795

39.90579

20.00313

10.86156

10.04007

0.04

149.19079

30.27319

15.19340

8.36644

7.75483

0.05

120.23014

24.24248

12.25001

6.85416

6.37090

0.1

43.59987

11.68273

6.25939

3.80200

3.57955

0.2

7.99463

5.38844

3.28070

2.27957

2.18619

0.3

2.50677

3.44416

2.33687

1.78539

1.73250

0.4

1.40197

2.56320

1.89449

1.54697

1.51283

0.5

1.12728

2.08603

1.64657

1.40944

1.38565

RMI

8.422293

1.886133

0.662234

0.081343

0.028025

Shift

size

h=0.0001077388

l =0.033828632

l =0.067306441

l =0.336146642

l =0.672394345

0.10

0.00

370

0.01

322.51351

84.86241

50.19911

29.35727

27.24036

0.02

281.89360

47.30034

26.92598

15.58778

14.46976

0.03

247.04728

32.50870

18.40651

10.74975

10.00081

0.04

217.06999

24.61424

13.99326

8.28274

7.72544

0.05

191.21124

19.71478

11.29728

6.78767

6.34754

0.1

105.07854

9.60684

5.81994

3.77059

3.56846

0.2

37.13064

4.57126

3.09828

2.26568

2.18124

0.3

15.69653

3.00955

2.23433

1.77710

1.72953

0.4

7.76816

2.29701

1.82841

1.54134

1.51079

0.5

4.44225

1.90831

1.60038

1.40531

1.38414

RMI

15.285597

1.395811

0.531361

0.048156

0.000000

Table 6. Comparison of the ARL values for a MAX(2,1) process running on EWMA and modified EWMA

control charts when ARL0 = 370,

10.097





,

11X

,

1



= –1.223 and

2



= –0.699

Shift size

EWMA

Modified EWMA

k=0.5

k=1

k=5

k=10

h=0.000001869

l=0.000928768

l =0.005070046

l =0.0300315284

l =0.06096242

0.05

0.00

370

0.01

196.318682

9.879934

7.893869

6.525167

6.367563

0.02

111.368549

5.156209

4.258706

3.635864

3.563844

0.03

66.583664

3.580090

3.045727

2.671272

2.627740

0.04

41.597636

2.803836

2.446131

2.192959

2.163366

0.05

27.011222

2.348224

2.092422

1.909602

1.888120

0.10

12.631543

1.847012

1.700065

1.593096

1.580402

0.20

5.120406

1.498224

1.422805

1.366535

1.359769

0.30

1.334549

1.158702

1.143858

1.132074

1.130610

0.40

1.063653

1.077259

1.073194

1.069828

1.069400

0.50

1.019385

1.046045

1.045018

1.044139

1.044025

RMI

8.992296

0.162988

0.074903

0.012883

0.005645

Shift size

h=0.000014193

l =0.001181441

l =0.0046159374

l =0.0292793555

l =0.0601950977

0.10

0.00

370

0.01

55.41693832

8.767793

7.475860

10.501405

8.069057

0.02

24.81047391

4.621807

4.053835

5.469681

4.351070

0.03

14.16833753

3.241983

2.913299

3.787001

3.109271

0.04

9.131643368

2.563354

2.349964

2.956700

2.494832

0.05

6.367520148

2.165516

2.017906

2.468416

2.131976

0.10

3.654008003

1.728601

1.650002

1.929678

1.728795

0.20

2.124412983

1.425515

1.390603

1.552740

1.442986

0.30

1.153776272

1.132785

1.131058

1.181233

1.153324

0.40

1.042066983

1.063647

1.065998

1.090083

1.078967

0.50

1.016593125

1.037492

1.040280

1.054521

1.049006

RMI

1.585831

0.052834

0.006216

0.143323

0.040219



WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2023.11.41

Sittikorn Khamrod, Yupaporn Areepong,

Saowanit Sukparungsee, Rapin Sunthornwat

E-ISSN: 2415-1521

463

Volume 11, 2023

Table 7. Comparison of the ARL values for a MAX(2,2) process running on EWMA and modified EWMA

control charts when ARL0 = 370,

10.078





,

25.663





135X

,

21X

,

1



= –0.831 and

2



= –0.837

Shift

size

EWMA

Modified EWMA

4

k





k=0.5

k=1

k=5

k=10

h =

1.39570210-12

l =

8.2808810-14

l =

1.5524510-8

l =

4.1203410-8

l =

2.6406110-7

l =

5.4558310-7

0.05

0.000

370

0.0001

55.918932

53.406181

67.259655

68.038452

68.740983

68.833544

0.0002

30.439245

28.922156

37.217006

37.689487

38.116837

38.173492

0.0003

21.007965

19.919012

25.833354

26.171546

26.477718

26.518395

0.0004

16.090669

15.264063

19.844924

20.108433

20.347094

20.378836

0.0005

13.075530

12.391242

16.151187

16.367230

16.562943

16.588986

0.0010

6.892126

6.522190

8.527914

8.642525

8.746367

8.760201

0.0020

3.732005

3.532678

4.599306

4.659564

4.714135

4.721406

0.0030

2.677330

2.537740

3.277312

3.318731

3.356223

3.361219

0.0040

2.155613

2.047267

2.617535

2.649298

2.678041

2.681870

0.0050

1.847935

1.759412

2.224267

2.250101

2.273474

2.276588

RMI

0.048731

0.001852

0.262755

0.277648

0.291120

0.292905

Shift

size

h=

2.7920610-12

l =

1.6566610-13

l =

9.5152710-9

l =

3.1049710-8

l =

2.4768610-7

l =

5.2812410-7

0.10

0.000

370

0.0001

52.47239

48.914789

65.291415

66.884843

68.478415

68.702274

0.0002

28.45630

26.394065

36.031611

36.992373

37.956722

38.092651

0.0003

19.62371

18.191344

24.988924

25.674280

26.363133

26.460350

0.0004

15.03460

13.935765

19.189626

19.722242

20.257953

20.333604

0.0005

12.22358

11.324846

15.615910

16.051605

16.490011

16.551944

0.0010

6.47125

6.002218

8.248586

8.477580

8.708160

8.740756

0.0020

3.53672

3.293741

4.456621

4.575165

4.694545

4.711425

0.0030

2.55796

2.393403

3.181705

3.262097

3.343061

3.354509

0.0040

2.07385

1.949563

2.545875

2.606789

2.668148

2.676826

0.0050

1.78832

1.689018

2.167173

2.216185

2.265572

2.272558

RMI

0.066618

0.000442

0.315924

0.347874

0.380027

0.384565

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

- Sittikorn Khamrod carried out the simulation of

Section 4.

- Yupaporn Areepong has organized the

conceptualization and validation

- Saowanit Sukparungsee has implemented the

methodology and software.

- Rapin Sunthornwat was responsible for

Mathematics and Statistics.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

This research was funded by Thailand Science

Research and Innovation Fund (NSRF), and King

Mongkut’s University of Technology North

Bangkok with Contract no. KMUTNB-FF-66-04.

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



WSEAS TRANSACTIONS on COMPUTER RESEARCH

DOI: 10.37394/232018.2023.11.41

Sittikorn Khamrod, Yupaporn Areepong,

Saowanit Sukparungsee, Rapin Sunthornwat

E-ISSN: 2415-1521

464

Volume 11, 2023