A Class of Population Mean Estimators in Stratified Random Sampling:

A Case Study on Fine Particulate Matter in the North of Thailand

NUANPAN LAWSON1*, NATTHAPAT THONGSAK2

1Department of Applied Statistics, Faculty of Applied Science,

King Mongkut’s University of Technology North Bangkok,

1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800,

THAILAND

2State Audit Office of the Kingdom of Thailand,

Bangkok, 10400,

THAILAND

Abstract: - Residents of Thailand’s upper northern have been facing hazardous air quality with the amount of

fine particulate matter rising several times higher than the standards of the World Health Organization for many

years which is classified as a level that severely affects public health. The dust problem is an urgent issue in

Thailand that needs to be solved. Assessment of pollution data in advance can help the Thai government in

planning to abolish and prevent ongoing dust problems for Thai citizens. A new class of population mean

estimators is proposed under stratified random sampling. The bias and mean square error of the proposed

estimators are studied using a Taylor series approximation. A simulation study and an application to air

pollution data in the north of Thailand to investigate the performance of the estimators. The results from the air

pollution data in the north of Thailand present that the proposed estimators offer the highest efficiency

concerning others.

Key-Words: - Ratio estimator, Fine particulate matter, Mean square error, Stratified random sampling,

Population mean, Thailand.

Received: November 3, 2023. Revised: November 19, 2023. Accepted: December 15, 2023. Published: March 8, 2024.

1 Introduction

Thailand’s air pollution has become an increasing

concern at present due to the amount of dust

exceeding Thailand’s and the World Health

Organization’s guidelines. Multiple provinces in

the north where a 24-hour average of fine particulate

matter with a diameter less than 2.5 microns

(PM2.5) and fine particulate matter with a diameter

less than 10 microns (PM10) on a 24-hour average

exceed the standard of 50 micrograms per cubic

meter and the air quality index (AQI) exceeds the

standard value of 100 which is classified as levels

that severely affect human health. In northern

Thailand, there is an abundance of provinces

tackling pollution from dust, due to seasonal

agricultural practices and geography consisting of

mountainous terrain accounting for the trapping of

pollution that passes into the area. There is a

significant correlation between pollution and

citizens’ health and quality of life, as shown by an

increased incidence of upper respiratory symptoms

and lung cancer, a worrying problem. Lung cancer

due to PM2.5 pollution affects genetically

susceptible people at a rapid and unprecedented rate,

which is an increasing concern amongst the

population. The Thai government needs to monitor

the air quality data and assess the situation closely

including implementing measures to control burning

in forest areas, farmland, and areas along the

highway strictly. Nevertheless, the government

must enact a strict policy requesting people's

cooperation to refrain from burning waste and

agricultural waste in preparation for farming to

prevent forest fires which can lead to dust issues in

the north of Thailand.

Possessing air pollution data can assist in policy

planning and preventing harm to life. Thailand's air

pollution data have been investigated by many

researchers. Carbon monoxide is used to estimate

the average PM2.5 in Dindang, Thailand using the

known prior information in simple random sampling

without replacement (SRSWOR), [1]. The best

estimators with the smallest mean square error

(MSE) that were suggested by [1], are based on the

known median of the auxiliary variable and quartile

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.19

Nuanpan Lawson, Natthapat Thongsak

E-ISSN: 2224-2880

160

Volume 23, 2024

average. The [1], estimators assist in saving budget

and time with a small sampling fraction. The

transformation technique to create combined

estimators was called upon to estimate the average

PM2.5 via nitrogen dioxide in Chiang Rai under

double sampling which results in improving the

performance of the combined estimators concerning

the single ones, [2]. The air pollution from vehicles

in Selangor, Malaysia, an area with heavy traffic

congestion has been studied in [3]. They found that

PM10 and ozone are the key factors contributing to

air pollution in Selangor although the air pollution

there seemed to be reduced in the period of their

study, [4], [5].

In sample surveys the ratio and regression

estimators for estimating population mean, an

average of a specified characteristic of a group, are

well known in assisting to heighten the efficiency of

estimators when an available auxiliary variable has

a positive relation to a study variable, a ratio

estimator was introduced for estimating the study

variable population mean under SRSWOR when the

population mean of the auxiliary variable is present,

[6]. To increase the precision of the population

mean estimator, some parameters of an auxiliary

variable such as coefficient of variation, correlation,

and kurtosis are applied to modify the usual ratio

estimator, [7], [8], [9]. Five ratio estimators using a

regression estimator and known parameters to

estimate the population mean were proposed by

[10], [11], [12], [13].

Stratified random sampling is one of the

sampling techniques that is suitable for use when the

units in the population are homogenous within the

same stratum and heterogeneous between different

strata. In the technique of stratified sampling, a

population comprising N units is subdivided into

distinct subpopulations, for example, it could be

divided into geographical characteristics, age group,

and gender. The subpopulations are called strata.

Dividing strata is useful when we want to find the

size of each stratum. The samples in each stratum

are selected via simple random sampling. Under

stratified random sampling, there are two types of

ratio estimator; a separate and combined one. Ratio

estimators were adopted by [14], under SRSWOR

that were suggested by [7], [8], [9], which resulted

in four separate ratio estimators under stratified

random sampling and found that the proposed

estimators are more efficient than the usual separate

ratio estimator. More works that proposed to modify

ratio estimator under stratified random sampling can

be seen in [15], [16], [17], [18].

In the prevailing study, we aim to propose a

new class of separate ratio estimators utilizing some

parameters of an auxiliary variable under stratified

random sampling. We acquired the formula of bias

and mean square error of the proposed class of

estimators up to the first-degree Taylor

approximation. A simulation study and an

application to air pollution in the north of Thailand

are considered to demonstrate the capacity of the

proposed class of estimators.

2 Existing Estimators

2.1 The Usual Separate Ratio Estimator

The usual separate ratio estimator for the population

mean is

RS

1

ˆ,

Lh

hh

X

Y W y x











(1)

where

1

h

N

h i h

i

X x N





is the population mean of an

auxiliary variable in stratum

; 1,2,3,...,h h L

,

1

h

n

h i h

i

x x n





and

1

h

n

h i h

i

y y n





are the sample

means of the auxiliary and study variables in

stratum

h

based on a sample of size

h

n

,

respectively,

h

N

WN



is the stratum weight, and

h

N

is the population size in stratum

h

, such that

1

.

L

h

NN





The bias and MSE of

RS

ˆ

Y

are



 

2

RS

1

ˆ,

L

h h h xh h xh yh

h

Bias Y W Y C C C











 

2 2 2 2

RS

1

ˆ2,

L

h h h yh xh h xh yh

h

MSE Y W Y C C C C





  



where

xh xh h

C S X

is the auxiliary variable’s

population coefficient of variation in stratum

h

,

xyh

h

xh yh

S

SS





is the population correlation

coefficient between the auxiliary and study variables

in stratum

h

,

 

2

1

h

N

xh i h

i

h

S x X

N





,

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.19

Nuanpan Lawson, Natthapat Thongsak

E-ISSN: 2224-2880

161

Volume 23, 2024

 

2

1

h

N

yh i h

i

h

S y Y

N





, and

  

1

h

N

xyh i h i h

i

h

S x X y Y

N

  



2.2 The Adjusted Ratio, [14], Estimators

Ratio estimators proposed by [8], [9], were adopted

under SRSWOR to stratified random sampling, [14].

The [14], estimators are:

1

TL

ˆ,

Lh xh

hh

hh xh

XC

Y W y xC















(2)

 

2

TL2

1

ˆ,

Lhh

hh

hhh

Xx

Y W y xx



















(3)

 

2

12

TL3

ˆ,

Lh h xh

hh

hh h xh

x X C

Y W y x x C



















(4)

 

2

1

T

2

L4

ˆ,

Lxh h h

hh

hxh h h

C X x

Y W y C x x



















(5)

where

 

4

2

1

24

( 1) 3( 1)

( 1)( 2)( 3) ( 2)( 3)

h

N

h h i

ih

h

h h h xh h h

N N x X N

xN N N S N N





 



    



is the population coefficient of kurtosis of the

auxiliary variable in stratum

h

.

The biases and MSEs of the [14], estimators are:



2

1

TL

ˆ,

Lhh

h h h xh h xh yh

hh xh h xh

XX

Bias Y W Y C C C

X C X C







   





   





   







   

TL2

2

122

ˆ,

Lhh

h h h xh h xh yh

hh h h h

XX

Bias Y W Y C C C

X x X x









   





   





   







 

   

 

2

22

2

22

TL3

1

ˆ,

Lh h h h

h h h xh h xh yh

hh h xh h h xh

x X x X

Bias Y W Y C C C

x X C x X C











   





   





   







   

T

122

4

2

L

ˆ,

Lxh h xh h

h h h xh h xh yh

hxh h h xh h h

C X C X

Bias Y W Y C C C

C X x C X x









   





   





   







2

2 2 2 2

1TL

1

ˆ2,

Lhh

h h h yh xh h xh yh

hh xh h xh

XX

MSE Y W Y C C C C

X C X C







   



  

   





   







   

2

TL2

2 2 2

122

ˆ2,

Lhh

h h h yh xh h xh yh

hh h h h

XX

MSE Y W Y C C C C

X x X x









   



  

   





   







 

   

 

2

22

2 2 2 2

122

TL3

ˆ2,

Lh h h h

h h h yh xh h xh yh

hh h xh h h xh

x X x X

MSE Y W Y C C C C

x X C x X C











   



  

   





   







   

2

2 2 2 2

2

L4

12

T

ˆ2.

Lxh h xh h

h h h yh xh h xh yh

hxh h h xh h h

C X C X

MSE Y W Y C C C C

C X x C X x









   



  

   





   





The usual separate ratio in (1) and [14], estimators

in (2) to (5) in a general form are:

R

1

ˆ,

Lh h h

hh

hh h h

A X D

Y W y A x D















(6)

The bias and MSE of the

R

ˆ

Y

in general forms are"



 

22

R

1

ˆ,

L

h h h h xh h h xh yh

h

Bias Y W Y C C C

   







(7)



 

2 2 2 2 2

R

1

ˆ2.

L

h h h yh h xh h h xh yh

h

MSE Y W Y C C C C

   



  



(8)

where

0 and

hh

AD

are constants or functions of

the auxiliary variable in stratum

h

, and

hh

h

h h h

AX

A X D





.

3 Proposed Estimator

A new group of separate ratio estimators is shown

under stratified random sampling inspired by the

work of [10], under SRS.

 

Reg

1

ˆ,

Lh h h

h h h h h

hh h h

A X D

Y W y b X x A x D









   









(9)

where

h

b

is a sample regression coefficient of

h



in

stratum

h

.

Let

0

hh

h

yY

Y







then

 

0

1

h h h

yY





, let

1

hh

h

xX

X







then

 

1

h h h

xX





, such that

   

 

2 2 2 2

0 1 0 1

01

0, , ,

.

h h y x

xy

E E E C E C

E C C

     

  

   



Rewriting (9) in terms of

01

and

hh



,

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.19

Nuanpan Lawson, Natthapat Thongsak

E-ISSN: 2224-2880

162

Volume 23, 2024

   

 



Reg 0 1

11

1

0 1 1

1

22

0 1 1 1

1

0 1 1

1

22

0 1 1 1

ˆ11

1

Lh h h

h h h h h h

hh h h h

L

h h h h h h h h h

h

L

h h h h h h h h h h h

h

L

h h h h h h h h h h

h

h h h h h h h h h h

A X D

Y W Y b X A X D

W Y Y b X

W Y Y b X Y

Y b X

 

   

     

   

      













   











   





    



   

  





2h

Y

The estimation error of

Reg

ˆ

Y

is:





Reg 0 1 1

1

2 2 2

0 1 1 1

ˆL

h h h h h h h h h

h

h h h h h h h h h h hh

Y Y W Y b X Y

Y b X Y

   

      



   

  



The

Reg

ˆ

Y

bias



Reg Reg

ˆˆ

Bias Y E Y Y

 

2

1

,

L

h h h h h h h xh h xh yh

h

W Y K C C C

    





  





(10)

The

Reg

ˆ

Y

MSE



2

Reg Reg

ˆˆ

MSE Y E Y Y

 





2

0 1 1

1

2

0 1 1

1

2 2 2 2 2 2 2 2 2

0 1 1 0 1

1

22

0 1 1

2

22

L

h h h h h h h h h

h

L

h h h h h h h h h

h

L

h h h h h h h h h h h h h h

h

h h h h h h h h h

E W Y b X Y

W E Y b X Y

W E Y b X Y b X Y

Y b X Y

   

     

    





  





  

   





   

 

2

2 2 2 2

1

2

L

h h h yh h h h xh h h h h xh yh

h

W Y C K C K C C

     



    



(11)

Some members of

Reg

ˆ

Y

are represented in Table 1.

Table 1. Some of the proposed estimators,

Reg

ˆ, 1,2,...,5

i

Yi

4 Theoretical Comparison

The effectiveness of the suggested estimator is

evaluated in comparison to the currently employed

estimators; the usual separate ratio, and Tailor and

Lone estimators. The MSE is used as a criterion in

comparison with the new estimator in (11) with the

existing ones in the general form in (8) as follows.

The suggested

Reg

ˆ

Y

exhibits superior performance

to the existing

R

ˆ

Y

if:



Reg R

ˆˆ

MSE Y MSE Y

 

2

22

2 2 2 2 2 2 2

11

22

2 2 2 2 2 2

11

2

2 2 2 2

2

LL

yh h h h xh

h h h h h h yh h xh h h xh yh

hh

h h h h xh yh

LL

h h h xh

h h h h h h h xh h h xh yh

hh

h h h h xh yh

h h h h h h xh h

C K C

W Y W Y C C C C

K C C

KC

W Y W Y C C C

K C C

W Y K C



    

  



    

  

   









  

























 

2 2 2

11

22

LL

xh h h h h h h h xh yh h h xh yh

hh

C W Y K C C C C

     



  



 

2

2 2 2 2 2 2

11

2

LL

h h h xh h h h h h h h h xh yh h h h h

hh

W Y C K W Y C C K

        



    



5 A Simulation Study

The simulation studies illustrate and compare the

effectiveness of the suggested estimators against

those already in existence. A population of size

2,000N

is divided into 3 strata and variable

( , )XY

are generated via bivariate normal

distribution for each stratum. The strata parameters

are:

1ststratum:

1 1 1 1 1 1

1,000, 400, 500, 0.3, 1.4, 0.8

xy

N X Y C C



     

2nd stratum:

2 2 2 2 2 2

600, 550, 700, 1.0, 1.6, 0.6

xy

N X Y C C



     

3rd stratum:

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.19

Nuanpan Lawson, Natthapat Thongsak

E-ISSN: 2224-2880

163

Volume 23, 2024

3 3 3 3 3 3

400, 550, 350, 1.1, 0.9, 0.4

xy

N X Y C C



     

A sample of sizes

100, 200, 400, and 600n

are allocated from the population

2,000N

using

SRSWOR. We allocate the sample sizes n to each

stratum using proportional allocation. For

100,n

the allocated sample sizes for the 1st, 2nd, and 3rd

strata are

1 2 3

50, 30, 20n n n  

, for

200,n

the

allocated sample sizes for the 1st, 2nd and 3rd strata

are

1 2 3

100, 60, 40n n n  

, for

400n

the

allocated sample sizes for the 1st, 2nd and 3rd strata

are

1 2 3

200, 120, 80n n n  

and for

600n

the

allocated sample sizes for the 1st, 2nd and 3rd

stratums are

1 2 3

300, 180, 120n n n  

,

respectively. R program, [19], is applied to repeat

the simulation 10,000 times.

The new and existing biases and MSEs are

calculated by:



10,000

1

ˆˆ

,

10,000 i

i

Bias Y Y Y









10,000 2

1

ˆˆ

.

10,000 i

i

MSE Y Y Y







The biases and MSEs are represented in Table 2.

The results in Table 2 showed that the

introduced estimators worked well because they

gave both less bias and MSE than the existing ones.

All proposed calculations assisting with unique

known parameters of the auxiliary variable gave

similar results for both biases and MSEs. The larger

sample sizes gave smaller biases and MSEs

concerning smaller sample sizes. We can see that

the introduced estimators utilizing the sample

regression estimator are more effective than the

usual separate ratio and [14], ones under stratified

random sampling.

Table 2. Biases and MSEs of the estimators

6 An Application to Air Pollution

Data in Northern Thailand

Northern Thailand air pollution data from 2003-

2020 are applied to assess the effectiveness of the

newest estimators, [20]. The monthly average

density of nitrogen dioxide NO2 (mg per square

metre) is to be the auxiliary variable and the

concentration of PM2.5 (micron per cubic metre) is

considered as the study variable with a population

1,728N

units. The population parameters are

summarized below.

 

2

39.34, 2.48, 1.23, 0.30, 3.36,

yx

Y X C C x



    

0.68.





We considered eight provinces in northern

Thailand as the strata to study the average amount

of PM2.5. The provinces that are included in the

study are Chiang Mai, Chiang Rai, Lamphun,

Lampang, Phrae, Phayao, Nan, and Mae Hong Son

(

216, 1,2,...,8

h

Nh

). A sample

520n

is

randomly picked from a population

1,728N

using

SRSWOR. The samples of sizes

65

h

n

based on

proportional allocation are randomly taken from

each strata using SRSWOR. The parameters in each

stratum are showcased in Table 3.

Table 3. Population parameters for each province in

northern Thailand

The estimated PM2.5 and percentage relative

efficiencies (PREs) of the estimators against the

usual separate ratio estimator are depicted in Table

4.

The new group of estimators gave a better

performance which produced higher PREs

compared to all existing estimators. The proposed

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.19

Nuanpan Lawson, Natthapat Thongsak

E-ISSN: 2224-2880

164

Volume 23, 2024

estimator

Reg5

ˆ

Y

that used the benefit of both

xh

C

and

 

2hx



gave the highest PRE in this situation and

gave a closer estimated PM2.5 to the population.

The results revealed that the recommended

estimators achieved much more than the existing

ones based on the air pollution dataset.

Table 4. Estimated values of PM2.5 and PREs of the

estimators

7 Conclusion

A new class of separate ratio estimators for

predicting population mean are investigated through

this study under stratified random sampling. Some

available insights on the auxiliary variable have

been implemented to increase the efficiency of the

population mean estimator. The outcomes of the

simulation study and the application to air pollution

data in northern Thailand indicated that the

suggested estimators performed more effectively

than the typical separate ratio approach and [14]

estimators under stratified random sampling. As

expected, larger sample sizes resulted in smaller

MSEs for all situations. The top-performing

estimator outperformed all other existing estimators,

delivering nearly thrice the efficiency. In subsequent

research, additional established auxiliary variables

may be employed to predict the population means of

the variable under study, and the new estimators can

be formulated to suit more intricate survey

frameworks. e.g., cluster sampling and stratified

single-stage cluster sampling. Nevertheless, the

proposed estimators can help estimate other

application data in many areas of interest.

Acknowledgement:

We appreciate all unknown referees for their

valuable comments.

References:

[1] Lawson, N., Improved ratio type estimators

using some prior information in sample

surveys: a case study of fine particulate matter

in Thailand, WSEAS Transactions on

Systems, Vol. 22, 2023. pp. 538-542.

[2] Thongsak, N. and Lawson, N., A combined

family of dual to ratio estimators using a

transformed auxiliary variable, Lobachevskii

Journal of Mathematics, Vol.43, No.9, 2022,

pp. 2621-2633.

[3] Fadzil, A., Shuhaili, A., Ihsan, S. Z. and Faris,

W. F., Air pollution study of vehicles

emission in high volume traffic: Selangor,

Malaysia as a case study, WSEAS

Transactions on Systems, Vol. 12, 2013. pp.

67-84.

[4] Austin, W., Carattini, S, Gomez-Mahecha, J.

and Pesko, M.F., The effects of

contemporaneous air pollution on COVID-19

morbidity and mortality, Journal of

Environmental Economics and

Management, Vol. 119, 2023, pp. 102815.

[5] Beloconi, A. and Vounatsou, P, Long-term air

pollution exposure and COVID-19 case-

severity: An analysis of individual-level data

from Switzerland. Environmental Research,

Vol. 216, 2023, pp. 114481.

[6] Cochran, W.G. Sampling Techniques. 3rd

edition. India: Wiley Eastern Limited,

1940.

[7] Sisodia, B. V. S., and Dwivedi ,V. K., A

modified ratio estimator using coefficient of

variation of auxiliary variable, Journal of

the Indian Society of Agricultural Statistics,

Vol.33, No.1, 1981, pp.13-18.

[8] Upadhyaya, L. N., and Singh, H. P., Use of

transformed auxiliary variable in estimating

the finite Population Mean, Biometrical

Journal, Vol.41, No.5, 1999, pp. 627-636.

[9] Singh, H. P., Tailor, R., Tailor, R., and

Kakran, M. S., An improved estimator of

population mean using power transformation,

Journal of the Indian Society of Agricultural

Statistics, Vol. 58, No. 2, 2004, pp. 223-230.

[10] Kadilar, C. and Cingi, H., Ratio estimators in

simple random sampling, Applied

Mathematics and Computation, 2004, Vol.

151, pp. 893-902.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.19

Nuanpan Lawson, Natthapat Thongsak

E-ISSN: 2224-2880

165

Volume 23, 2024

[11] Kadilar, C. and Cingi, H., An improvement in

estimating the population mean by using the

correlation coefficient, Hacettepe Journal of

Mathematics and Statistics, 2006, Vol. 35, pp.

103 -109.

[12] Nangsue, N., Adjusted ratio and regression

type estimators for estimation of population

mean when some observations are missing,

International Scholarly and Scientific

Research & Innovation, Vo.3, No.5,

2009, pp. 334-337.

[13] Koç, T. and Koç, H. A new class of

quantileregression ratio-type estimators for

finite population mean in stratified

random sampling, Axioms, Vol. 12, No.

7, 2023, pp. 713.

[14] Tailor, R. and Lone, H. A., Separate ratio-type

estimators of population mean in stratified

random sampling, Journal of Modern Applied

Statistical Methods, 2014, Vol. 13, No.1,

pp.223-233.

[15] Kadilar, C. and Cingi, H., Ratio estimators in

stratified random sampling, Journal of

Modern Applied Statistical Methods, 2003,

Vol.45, No.2, pp. 218-225.

[16] Kadilar, C. and Cingi, H., A new ratio

estimator in stratified random sampling,

Communications in Statistics-Theory and

Methods, 2005, Vol. 34, No.3, pp. 597-602.

[17] Singh, R.V.K, and Ahmed, A., Ratio-type

estimators in stratified random sampling using

auxiliary attribute, Proceedings of the

International MultiConference of Engineers

and Computer Scientists 2014 Vol I, IMECS

2014, March 12 - 14, 2014, Hong Kong.

[18] Sharma, V. and Kumar, S., Simulation study

of ratio type estimators in stratified

randomsampling using multi-auxiliary

information, Thailand Statistician, 2020, Vol.

18, No. 3, pp.281-289.

[19] R Core Team, R: A language and environment

for statistical computing. R Foundation for

Statistical Computing, Vienna, Austria, 2021,

[Online], https://www.R-project.org/

(Accessed Date: November 5, 2023)

[20] Inness, A., Ades, M., Agustí-Panareda, A.,

Barré, J., Benedictow, A., Blechschmidt, A.-

M., Dominguez, J. J., Engelen, R., Eskes, H.,

Flemming, J., Huijnen, V., Jones, L., Kipling,

Z., Massart, S., Parrington, M., Peuch, V.-H.,

Razinger, M., Remy, S., Schulz, M., and

Suttie, M. The CAMS Reanalysis of

Atmospheric Composition. Atmospheric

Chemistry and Physics, 2019, Vol. 19. No. 6,

pp. 3515-3556.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of funding for research presented in a

scientific article or scientific article itself

This work was funded by King Mongkut’s

University of Technology North Bangkok, Thailand.

Contract number 671159.

Conflict of Interest

The author has no conflicts 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

_US

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.19

Nuanpan Lawson, Natthapat Thongsak

E-ISSN: 2224-2880

166

Volume 23, 2024