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Neutrosophic Sets and Systems, Vol. 75, 2025
University of New Mexico
Mohammad Shafiq bin Mohammad Kamari, Zahari Bin Md. Rodzi, Deciphering the Geometric Bonferroni Mean
Operator in Pythagorean Neutrosophic Sets Framework
Deciphering the Geometric Bonferroni Mean Operator in
Pythagorean Neutrosophic Sets Framework
Mohammad Shafiq bin Mohammad Kamari 1, Zahari Bin Md. Rodzi 2,3,*, R.H. Al-Obaidi4, Faisal Al-
Sharqi5, Ashraf Al-Quran6, Rawan A. shlaka7
1 College of Computing, Informatics and Mathematics Studies, MARA University of Technology Negeri Sembilan Branch,
Seremban Campus, Seremban, Negeri Sembilan, 70300, Malaysia; shafiqkamari922@gmail.com
2 College of Computing, Informatics and Mathematics Studies, MARA University of Technology Negeri Sembilan Branch,
Seremban Campus, Seremban, Negeri Sembilan, 70300, Malaysia; zahari@uitm.edu.my
3 Faculty of Science and Technology, National University of Malaysia, Bandar Baru Bangi, Selangor, 43600, Malaysia;
zahari@uitm.edu.my
4 Fuel and Energy Techniques Engineering Department, College of Engineering and Technologies, Al-mustaqbal
University, 51001, Babylon, Iraq, raidh.h.salman@uomus.edu.iq
5 Department of Mathematics, Faculty of Education for Pure Sciences, University of Anbar, Ramadi, Anbar, Iraq,
faisal.ghazi@uoanbar.edu.iq
6 Department of Basic Sciences, Preparatory Year, King Faisal University, Al-Ahsa 31982, Saudi Arabia, aalquran@kfu.edu.sa
7 College of Pharmacy, National University of Science and Technology, Dhi Qar, Iraq, rawan-a.shlaka@nust.edu.iq
* Correspondence: zahari@uitm.edu.my
Abstract: The Geometric Bonferroni Mean (GBM), is an extension of The Bonferroni mean (BM), that
combines both BM and the geometric mean, allowing for the representation of correlations among
the combined factors while acknowledging the inherent uncertainty within the decision-making
process. Within the framework of Pythagorean neutrosophic set (PNS) that encompasses truth,
indeterminacy, and falsity-membership degrees, each criterion can be integrated into a unified PNS
value, portraying the overall evaluation of that criterion by employing the Geometric Bonferroni
mean. This study aims to enhance decision-making in Pythagorean neutrosophic framework by
introducing an aggregation operator to PNS using the Geometric Bonferroni Mean. Additionally, it
proposes a normalized approach to resolve decision-making quandaries within the realm of PNS,
striving for improved solutions. The novel Pythagorean Neutrosophic Normalized Weighted
Geometric Bonferroni Mean (PNNWGBM) aggregating operator has been tested in a case of multi-
criteria decision-making (MCDM) problem involving the selection of Halal products suppliers with
several criteria. The result shows that this aggregating operator is offering dependable and
pragmatic method for intricate decision-making challenges and able to effectively tackle uncertainty
and ambiguity in MCDM problem.
Keywords: aggregating operator; Bonferroni Mean (BM); Geometric Bonferroni Mean (GBM);
Pythagorean neutrosophic set (PNS); multi-criteria decision-making (MCDM).
1. Introduction
Zadeh [1] proposed the concept of fuzzy sets as a category of entities characterized by a spectrum
of membership grades. He broadened the concepts of inclusion, union, intersection, complement,
relation, convexity, and others to apply to these sets, and elucidated several properties associated
with these concepts within the framework of fuzzy sets. Intuitionistic Fuzzy Sets (IFS), first
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Operator in Pythagorean Neutrosophic Sets Framework
introduced by Atanassov [2], assigns both membership and non-membership functions to elements
of a universe, ensuring their combined sum is less than or equal to one. Consequently, IFS offers a
more precise and definitive description compared to fuzzy sets. However, it's limited to handling
incomplete and uncertain information, unable to address the indeterminate and inconsistent
information frequently encountered in real-world scenarios. Hence, as an expansion of fuzzy sets and
IFS, Smarandache introduced neutrosophic sets (NS) in 1995 and published his results in 1998 [3].
Smarandache's definition outlines that a NS, denoted as A within a universal set , is distinguished
by three distinct functions: a truth-membership function, denoted as , an indeterminacy-
membership function, denoted as , and a falsity-membership function, denoted as . The
main strength of the neutrosophic set lies in its enhancement of fuzzy set theory by integrating
membership, non-membership, and indeterminacy parameters, which are crucial for effectively
managing uncertainty in the decision-making process. Smarandache [4] and H. Wang et al. [5]
additionally introduced the concept of a single-valued neutrosophic set (SVNS) through adjustments
to the established conditions such that , and and
, which are better suited for addressing scientific and engineering problems.
Throughout the years, numerous extensions of neutrosophic sets have been developed by other
researchers because of their broad range of descriptive scenarios frequently encountered in various
real-life situations. Example of neutrosophic set extensions are interval neutrosophic set [6],
simplified neutrosophic set [7], neutrosophic soft set [8], multi-valued neutrosophic set [9] and rough
neutrosophic set [10]. Over the past decade, a wealth of intriguing studies on neutrosophic sets has
emerged across diverse domains within multi-criteria decision-making (MCDM), demonstrating
their relevance and impact [11]-[17].
Another novel expansion of neutrosophic set is the Pythagorean neutrosophic sets (PNS). Jansi et
al. [18] extends the theory of correlation coefficient from neutrosophic sets (NS) to Pythagorean
neutrosophic sets (PNS), where 'T' and 'F' represent dependent neutrosophic components. This
extension relaxes the constraint condition requiring the square sum of membership, non-membership,
and indeterminacy to be less than two. The Pythagorean constraint helps in better modeling and
representing complex situations where the interplay between truth, indeterminacy, and falsity is
more intricate. This can be particularly useful in scenarios with high degrees of uncertainty or where
traditional neutrosophic sets might be too rigid. Hence, PNS can provide more accurate and refined
decision-making capabilities. They allow for more sophisticated aggregation and comparison
techniques, leading to potentially better outcomes in decision-making processes involving uncertain
or vague information. This makes PNS a valuable tool in fields where robust decision-making under
uncertainty is essential [19].
In multi-criteria decision-making (MCDM), an aggregating operator is a mathematical function
or method used to combine multiple criteria or attributes into a single composite score or decision
value. The purpose of these operators is to synthesize the diverse information provided by the
different criteria to facilitate decision-making. Originally introduced as an enhancement of the
arithmetic mean, the Bonferroni Mean (BM) is an aggregating operator celebrated for its unique
ability to factor in the significance and interplay between element pairs during aggregation [20]. The
geometric mean serves as an aggregation operator in various fields, particularly in situations where
multiplication or compounding of values is relevant. As an aggregation operator, it combines
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multiple values into a single representative value that retains essential information from the original
dataset. Xia et al. [21] introduces Geometric Bonferroni Mean (GBM), an integration of geometric
mean with BM. The Geometric Bonferroni Mean (GBM) extends the concept of BM by applying
geometric aggregation functions, offering a more flexible approach to handling multiplicative
interactions among variables. The Geometric Bonferroni Mean (GBM) specifically aggregates
multiple criteria in decision-making by capturing their interrelationships, enhancing the accuracy
and robustness of evaluations, particularly in complex scenarios with interdependent attributes.
Moreover, within the realm of Pythagorean fuzzy environments, extensions like the Pythagorean
fuzzy GBM operator have emerged to delineate the connections between parameters and explore
their unique characteristics [22]. These advancements underscore the flexibility and versatility of the
GBM concept across different decision-making contexts. To increase its adaptability and usefulness,
the GBM has been expanded through various modifications. For example, the introduction of
weighted GBM (WGBM) operators assign weights to indicate the relative importance of certain
criteria or attributes, enhancing the aggregation precision. Additionally, the GBM aggregating
operator harbors vast potential for application across a multitude of domains within the realm of
fuzzy and neutrosophic sets [23]-[27]. In addition, many research works have addressed the
significance of Pythagorean fuzzy environments with some mathematical technicals, for example,
Edalatpanah [32-34] discussed the impact of some applications of Pythagorean fuzzy in feature
selection. Dirik and others [35-39] used Pythagorean fuzzy to develop decision models. Mohammed
et al. [40,41] combined the above model with developed topological concepts in topological spaces.
Al-sharqi et al [42] introduced the MCDM method for decision-making based on multi-mathematical
structures like complex fuzzy structure [43,44], fuzzy graph structure [45,46], and some algebraic
structures [47,48]. Al-Quran et al and other [49-52] proposed approaches for the selection of MCDM
technology by using the fuzzy set and its extension method,
The motivation for this study arises from the capabilities of the Pythagorean neutrosophic set,
which has garnered significant attention from researchers in MCDM techniques due to its impressive
performance. A new aggregating operator that combines the strengths of the Geometric Bonferroni
Mean with the versatile capabilities of Pythagorean neutrosophic sets would be a valuable addition
to the field of multi-criteria decision-making. To the best of the authors' knowledge, such an
aggregating operator has not yet been explored, thereby addressing a current gap in the research.
The main objectives of this study are (i) To develop Pythagorean Neutrosophic Geometric
Bonferroni Mean (PNGBM), a novel aggregating operator that integrates PNS methodology with the
classic GBM aggregating operator. (ii) To develop PNNWGBM, a normalized weighted Geometric
Bonferroni Mean aggregating operator within the PNS framework. (iii) To test the applicability of the
developed aggregating operator in a multi-criteria decision-making (MCDM) problem involving the
selection of Halal product suppliers.
The paper is structured as follows: Section 2 introduces fundamental PNS theories. Section 3
thoroughly explains the proposed methodology. Section 4 demonstrates the application of the
suggested methodology through a case study focusing on the selection of Halal products suppliers
with several criteria for the decision makers. Lastly, Section 5 functions as the concluding remarks.
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2. Experimental Section
2.1 Preliminaries
This section covers the fundamental theories essential for the development of PNNWGBM.
Definition 1. [20] Let , and be a collection of positive real numbers ,
then BM is defined as
(1)
Definition 2. [21] Let , and be a set of non-negative numbers . Then:
(2)
Equation (2) is called Geometric Bonferroni Mean (GBM). The GBM possesses the following
characteristics:
1.
2. , for all .
3. is monotonic, if for all .
4. .
Definition 3. [28] Let , and be a set of non-negative numbers .
indicates the importance degree of , satisfying , , for ,
indicates the importance degree of for , satisfying , .
Then we call
(3)
the weighted Geometric Bonferroni Mean (WGBM).
Definition 4. [29] Let be a non-empty set or a universe. We define a Pythagorean fuzzy set
A as
(4)
Where indicate the truth membership and false membership respectively for
each element to the set , and for each . The indeterminacy
membership is given by .
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Definition 5. [5] Let be a universe or non-empty set. A single valued neutrosophic set in
is given by:
(5)
Where and with no limitations on the sum of the components where
.
Definition 6. [30] Let be a universe or non-empty set. A Pythagorean neutrosophic set with
and are dependent neutrosophic components that is given by:
(6)
Where represent the degree of membership, represent the degree of indeterminacy and
represent the degree of non-membership respectively such that and
satisfying
(7)
(8)
Definition 7. [31] Let , and are any two
PNSs, then the following definitions apply to the operating rules for PNSs, which include addition,
multiplication, scalar multiplication, and power operations:
i.
(9)
ii.
(10)
iii.
where and
(11)
iv.
where and .
(12)
2.2 Proposed Method
In this section, our goal is to enhance the capabilities of the GBM operator to accommodate
situations where PNS are employed as input parameters. Therefore, our study involves
implementing the PNGBM operator within the Pythagorean neutrosophic framework.
Definition 8. Let and be a set of non-negative numbers
. Then:
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Operator in Pythagorean Neutrosophic Sets Framework
(13)
Equation (13) was formed based on the operating rules of PNS as mentioned in Definition 7.
Proposition 1. Let with where a PNS set consist of
. For any :
(14)
Proof Let two PNS sets and .
Using the operating rules (iii) in Definition 7, we have
By commencing the operating rules i) based on Definition 7, we get:
Thus, Proposition 1 holds.
Proposition 2. Let and where a PNS set consist of
For any
(15)
Proof By referring to the result from Proposition 1, we get:
and
By commencing the operating rules (ii) based on Definition 7, we get:
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Therefore, Proposition 2 is valid.
Proposition 3. Let and where a PNS set consist of
Provided the value of at which , we have:
(16)
Proof Referring to Equation (14) in Proposition 1, when , we obtain:
Then,
By referring Equation (15), let to obtain the following general form:
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Thus, Equation (16) as stated in Proposition 3 is applicable when .
Next, let
This proof also applies to the case where . Thus, Proposition 3 remains valid.
Consequently, we can infer the following Proposition 4 directly from Proposition 3 as stated earlier.
Proposition 4. Let and where a PNS set consist of
Provided the value of at which , we have:
(17)
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Proposition 5. Let and where a PNS set consist of
For any
(18)
Proof Proposition 5 can be proven as follows by combining the results from Propositions 2 to 4.
By using Equation (15) and letting , we get:
and if , the equation becomes:
Next, let
where
can be referred from Proposition 3, which is
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and
can be referred from Proposition 4, which is
Therefore, operations are performed for
:
Given that Equation (18) remains applicable for . Thus, Proposition 5 holds.
Proposition 6. Let and where a PNS set consist of
For any
(19)
Proof Referring to Proposition 5, we possess the following:
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By commencing the operating rules iii) based on Definition 6, we get:
Hence, Proposition 6 holds. Subsequently, through the application of PNGBM as outlined in
Definition 8, we embark on deducing the PNNWGBM operator in the following steps:
Definition 9. Let and where a PNS set consist of
, then PNNWGBM (Pythagorean Neutrosophic Normalized Weighted Geometric
Bonferroni Mean) is defined for all sets of PNS as follows:
(20)
where the weight vector, signifies the level of importance of for
with the condition
and
Theorem 1. Let and where a PNS set consist of
, the value computed by the PNNWGBM operator in Equation (20) represents a
Pythagorean neutrosophic number with the following components:
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Proof Theorem 1 can be established by using the derived equations from Propositions 1 to 6:
We possess , then
Next, we get:
Therefore,
Furthermore, we can simplify the equation to get:
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(21)
satisfying Equation (7) and (8). Hence, this validates Theorem 1. Building on the established Theorem
1, we can additionally deduce the important properties of the , namely reducibility,
commutativity, idempotency, monotonicity, and boundedness.
Theorem 2. Reducibility: Let then
(22)
Proof Considering , then in line with Definition 9, it follows that
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thus, completing the proof of Theorem 2.
Theorem 3. Idempotency: Let where then
(23)
Proof Given that for every , it follows that
thus, concluding the proof of Theorem 3.
Theorem 4. Commutativity: is any permutation of Pythagorean neutrosophic
numbers . For any
(24)
Proof Let be any permutation of Pythagorean neutrosophic numbers .
Then
thus, completing the proof of Theorem 4.
Theorem 5. Monotonicity: Let where and
where be two collections of PNSs. For any , ,
and , then
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(25)
Proof For the degree of truth, we have for all and . Thus,
and . Following that
For the degree of indeterminacy, since for all and , then we get
and . Therefore,
Likewise, for the degree of falsity, we can observe that:
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Hence, considering that , and , we managed to conclude
that , thereby concluding the proof of Theorem 5.
Theorem 6. Boundedness: Let where a PNS set consist of
and , , then
(26)
Proof Since , using Theorem 3 and 5 as a basis, we obtain:
In a similar manner, we can derive:
Thus, boundedness is obtained, and Theorem 6 proof has been concluded.
3. Results and Discussion
This section may be divided by subheadings. It should provide a concise and precise description
of the experimental results, their interpretation as well as the experimental conclusions that can be
drawn.
3.1. The Multi-Criteria Decision-Making Method Based on PNNWGBM Operator
In this section, we introduce a Multiple Criteria Decision-Making (MCDM) problem within a
Pythagorean neutrosophic setting. Let as the list of suppliers and
as the list of criteria. Then is the weight assigned to criterion for
such that and
. We propose the PNNWGBM operator to
consolidate the overall criteria for each supplier into a singular, aggregated preference. The
computational steps for this method are detailed below.
Step 1. Construct the decision matrix in the form of a Pythagorean neutrosophic set (PNS),
denoted as . A direct-relation matrix, incorporating the criteria
score for each supplier, is formed. Afterward, each criterion is converted to Pythagorean
neutrosophic numbers. The rating scale for PNS numbers is determined by using seven linguistic
scores ranging from negligible to exceptionally significant effect, employing Pythagorean
neutrosophic linguistic variables, as detailed in Table 1.
Table 1. The new Pythagorean neutrosophic linguistic variable [31].
Score
Linguistic Variable
Rating Scale in Pythagorean
Neutrosophic Set
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1
No Effect
2
Low Effect
3
Medium Low Effect
4
Medium Effect
5
Medium High Effect
6
High Effect
7
Very High Effect
Step 2. Each criterion for each supplier is aggregated into a unified value by using PNNWGBM
aggregating operator and the weight vector that represents the importance
of each criterion with varying from 1 to . The weight vector satisfies the condition
and
Step 3. The aggregated values should adhere to the PNS number conditions outlined in Definition 6
such that
Step 4. Deneutrosophicate the PNS numbers, into a crisp value by using
the following formula:
Step 5. According to the crisp value, each supplier is ranked from highest to lowest value which
represents the most ideal supplier for the decision makers to the least ideal.
3.2. Illustrative Example
To illustrate the feasibility of the proposed method, we present an example of a multi-criteria
decision-making problem. A company wants to choose a Halal products supplier for their business.
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Then there are four different suppliers to choose from. Each of these suppliers is going to be evaluated
based on four criteria which are, Quality of Products, Product Variety, Cost and
Pricing, Customer Service and Support and Location and Delivery Options.
Step 1. By using the five criteria and four potential Halal products suppliers
, the initial direct-relation matrix, is obtained Table 2. These values then are
transformed into Pythagorean neutrosophic numbers, which encompass the degrees of truth,
indeterminacy, and falsity.
Table 2. Initial direct-relation matrix, .
C1
C2
C3
C4
C5
S1
6
3
1
7
7
S2
5
3
6
6
1
S3
5
4
5
6
1
S4
2
7
3
4
2
Step 2. The aggregated value was computed using the PNNWGBM operator from Equation (21)
to depict the criterion selection for each supplier. The decision-makers employ a weighting vector
and the result obtained is shown in Table 3.
Table 3. The aggregated value using PNNWGBM operator.
PNNWGBM
S1
(0.9867,0.0091,0.055)
S2
(0.9872,0.0082,0.0417)
S3
(0.9856,0.008,0.0377)
S4
(0.9712,0.0165,0.0901)
Step 3. The aggregated PNS set has been verified and it satisfies the conditions outlined in
Definition 6 where and as
shown in Table 4.
Table 4. PNS number verification.
S1
0.98673
0.00909
0.05501
0.97666
0.97674
S2
0.98718
0.00823
0.04171
0.97626
0.97633
S3
0.98557
0.00795
0.03768
0.97277
0.97284
S4
0.97118
0.01652
0.09007
0.9513
0.95157
Step 4. The aggregated PNS set has been deneutrosophicated into a crisp value to represent the
overall criterion for each supplier.
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Table 5. Crisp value.
S1
0.3503
S2
0.3457
S3
0.3437
S4
0.3593
Step 5. Table 6 presents the ranking of the suppliers according to the crisp value that has been
aggregated by PNNWGBM operator. Therefore, Supplier 4 is the most recommended alternative for
the company to choose.
Table 6. The rankings of suppliers.
PNNWGBM
Rank
S1
0.3503
2
S2
0.3457
3
S3
0.3437
4
S4
0.3593
1
4. Conclusions
This paper has discussed the application of Geometric Bonferroni Mean (GBM) operator to
Pythagorean neutrosophic set framework. The primary aim was to introduce and verify a novel
normalized weighted Geometric Bonferroni Mean, termed PNNWGBM tailored for Pythagorean
neutrosophic sets. The integration of GBM into PNS setting provides a new reliable means for
decision makers in the MCDM problems, offering a nuanced approach to capturing interactions
among variables in decision-making processes. The proposed PNNWGBM method exhibited
promising results, validated through the illustrative example of case study pertaining to Halal
products supplier selection. Moreover, this study contributes to existing literature by broadening the
applications of the Geometric Bonferroni Mean operator and assessing its efficacy in Pythagorean
neutrosophic settings. The findings underscore the potential of PNNWGBM to be applied to existing
MCDM methodologies such as TOPSIS, AHP, PROMETHEE and DEMATEL.
The advantages of this newly developed PNNWGBM aggregating operator compared to
existing operators lie in Pythagorean neutrosophic sets, which provide a more flexible framework for
representing uncertainty than traditional fuzzy sets. By incorporating Pythagorean neutrosophic sets,
the new operator can better capture and process degrees of truth, indeterminacy, and falsity, leading
to more accurate decision-making. Moreover, the combination of PNS methodology with the GBM
aggregating operator enables effective aggregation of information while considering
interdependencies and correlations between different criteria, resulting in a more comprehensive
assessment. Lastly, this new operator can be applied to a wide range of multi-criteria decision-making
(MCDM) problems in fields such as engineering, economics, and social sciences. Its ability to handle
complex and uncertain information makes it suitable for real-world applications where traditional
methods may fall short.
Future research for the GBM operator in Pythagorean neutrosophic set theory encompasses
algorithm development, decision-making applications, uncertainty modelling, integration with other
operators, real-world applications, extensions to fuzzy and neutrosophic sets, theoretical analysis,
robustness and sensitivity analysis, machine learning integration, and comparative studies. These
investigations aim to enhance computational efficiency, explore practical applications, understand
theoretical properties, and assess performance in uncertain environments, ultimately advancing the
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utility and understanding of this operator. Overall, this study lays the groundwork for utilizing the
Geometric Bonferroni Mean aggregation operator in Pythagorean neutrosophic decision-making.
The integration of the Geometric Bonferroni Mean (GBM) with Pythagorean neutrosophic set,
despite its advantages, has limitations including increased computational complexity, sensitivity to
parameter selection, and high data quality requirements. It can also suffer from reduced
interpretability and scalability issues as the number of criteria and interactions increase. The
subjectivity in defining neutrosophic membership functions introduces potential biases, and there
are limited practical applications and case studies validating this approach. Additionally, validating
the outcomes can be challenging due to the abstract and complex nature of the method.
Funding: We would like to acknowledge the Ministry of Higher Education Malaysia for their sponsorship of the
Fundamental Research Grant Scheme (Project Code: FRGS/1/2023/STG06/UITM/02/5). This financial support has
been crucial in advancing our research efforts, and we are grateful for their assistance.
Conflicts of Interest: The authors declare no conflict of interest.
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Received: June 27, 2024. Accepted: Oct 12, 2024
