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\begin{document}

\title{A commentary on ``An improved score function for ranking neutrosophic
sets and its application to decision-making process''}



\author[1]{Akanksha Singh}%
\author[1]{Shahid  Bhat}%
\affil[1]{Thapar Institute of Engineering and Technology}%


\vspace{-1em}



  
  \date{May 17, 2024}


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\selectlanguage{english}
\begin{abstract}
This work aims to study and observe all the existing score functions
that help to rank the single-valued neutrosophic set (SVNS) as well as
interval-valued neutrosophic set (IVNS) to make a better choice among
all the available alternatives in multi-criteria decision-making (MCDM)
problems. An intensive study about all these existing score functions
reveals that there holds some limitations in the method of ranking order
which is misleading the results in decision-making problems. These
observations about the existing score functions of the SVNS and IVNS
have been claimed with the help of well-defined examples, illustrating
an inefficiency of all these existing score functions. Thus, to propose
a valid score function for ranking SVNS and IVNS for making a better
selection among all the other available alternatives in MCDM problems is
still an open challenging research problem.%
\end{abstract}%



\sloppy


\textbf{A commentary on ``An improved score function for ranking
neutrosophic sets and its application to decision-making process''}

Akanksha Singh\textsuperscript{a}11\textsuperscript{2}Corresponding
author \& Current Address: Lecturer, School of Sciences, Baddi
University of Emerging Sciences and Technologies, Makhnumajra, Baddi
district, Solan Baddi, Himachal Pradesh, IN 173205 Email id:
akanksha.singh@baddiuniv.ac.in ORCID ID: 0000-0003-2189-4974 Phone no.:
+91-9888414611\textsuperscript{2}, Shahid Ahmad Bhat\textsuperscript{a3}

\textsuperscript{a}School of Mathematics,

Thapar Institute of Engineering \& Technology (Deemed to be University)

P.O. Box 32, Patiala, Pin -147004, Punjab, India

asingh3\_phd16@thapar.edu\textsuperscript{1},
bhatshahid444@gmail.com\textsuperscript{3}

Abstract: This work aims to study and observe all the existing score
functions that help to rank the single-valued neutrosophic set (SVNS) as
well as interval-valued neutrosophic set (IVNS) to make a better choice
among all the available alternatives in multi-criteria decision-making
(MCDM) problems. An intensive study about all these existing score
functions reveals that there holds some limitations in the method of
ranking order which is misleading the results in decision-making
problems. These observations about the existing score functions of the
SVNS and IVNS have been claimed with the help of well-defined examples,
illustrating an inefficiency of all these existing score functions.
Thus, to propose a valid score function for ranking SVNS and IVNS for
making a better selection among all the other available alternatives in
MCDM problems is still an open challenging research problem.

KEYWORDS - IVNS, MCDM, score function, SVNS

1. INTRODUCTION

In the real-life uncertainty is the only thing which is certain in life,
so the information available in the real-world cannot be crisp always.
This theory was incorporated by a deep thinker Zadeh who proposed a new
theory of sets i.e., fuzzy sets {[}27{]} which brought a huge revolution
in the area of new thinking world and mathematics. Fuzzy sets holds the
idea that in practical life the information available is not always
certain or crisp but beholds the hand of uncertainty together and the
study of this uncertainty would help a lot in the process of decision
making {[}28,29{]}. Later with the time some intriguing extensions of
fuzzy sets were developed like- intuitionistic fuzzy set (IFS) {[}2{]},
interval-valued intuitionistic fuzzy set (IVIFS) {[}3{]}, Pythagorean
fuzzy set (PFS) {[}24-26{]}, interval-valued Pythagorean fuzzy set
(IVPFS) {[}32{]}, neutrosophic set {[}18-20{]}, SVNS {[}23{]} and IVNS
{[}22{]} etc. In this note, a deep study have been made to analyze the
ranking order of some of the extensions of fuzzy sets like, IFS
{[}8,9,30{]}, PFS {[}32{]}, IVPFS {[}4-7,11,12,16,17,31{]}, SVNS
{[}1,10,13-15,21{]} and IVNS {[}10,13{]}. After a rigorous study it has
been observed that there exist some restrictions in the existing methods
{[}10,13{]} for comparing SVNS and IVNS. Some well-defined
counter-examples are chosen where the uncertainty in the data is
expressed in the form of SVNS and IVNS to claim that the existing score
function defined to rank the SVNS and IVNS results incorrectly. The aim
of this note is to make researchers aware that, the shortcomings pointed
out by Nancy and Garg {[}10{]} in the existing methods {[}13{]} is also
occurring in the methods proposed by Nancy and Garg {[}10{]}. Therefore,
to propose the valid methods for the same is still an open challenging
research problem.

\textbf{2. A brief review of existing score functions}

Nancy and Garg {[}10{]} pointed out the shortcomings of the existing
methods {[}13{]} for the ranking of SVNS as well for the ranking of
IVNS. Also, to resolve these limitations, Nancy and Garg proposed new
methods for the same.

\textbf{2.1. Existing score function}

Sahin {[}13{]} proposed the following method for the ranking of two
SVNS, \(A_{1}=\left\langle a_{1},b_{1},c_{1}\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle a_{2},b_{2},c_{2}\right\rangle\). Find
\(K\left(A_{1}\right)=\frac{1+a_{1}-2b_{1}-c_{1}}{2}\)and \(K\left(A_{2}\right)=\frac{1+a_{2}-2b_{2}-c_{2}}{2}\), and check that
\(K\left(A_{1}\right)>K\left(A_{2}\right)\) or\(K\left(A_{1}\right)<K\left(A_{2}\right)\) or\(K\left(A_{1}\right)=K\left(A_{2}\right)\).

\begin{enumerate}
\tightlist
\item
  If \(K\left(A_{1}\right)>K\left(A_{2}\right)\) then\(A_{1}>A_{2}\).
\item
  If \(K\left(A_{1}\right)<K\left(A_{2}\right)\) then\(A_{1}<A_{2}\).
\item
  If \(K\left(A_{1}\right)=K\left(A_{2}\right)\) then\(A_{1}=A_{2}\).
\end{enumerate}

Sahin {[}13{]} also proposed the following method for the ranking of two
IVNS,\(A_{1}=\left\langle\left[a_{1}^{L}{,a}_{1}^{U}\right],\left[b_{1}^{L}{,b}_{1}^{U}\right],\left[c_{1}^{L}{,c}_{1}^{U}\right]\right\rangle\)and\(A_{2}=\left\langle\left[a_{2}^{L}{,a}_{2}^{U}\right],\left[b_{2}^{L}{,b}_{2}^{U}\right],\left[c_{2}^{L},c_{2}^{U}\right]\right\rangle\).
Find\(L\left(A_{1}\right)=\frac{2+a_{1}^{L}+a_{1}^{U}-2b_{1}^{L}-2b_{1}^{U}-c_{1}^{L}-c_{1}^{U}}{4}\)and\(L\left(A_{2}\right)=\frac{2+a_{2}^{L}+a_{2}^{U}-2b_{2}^{L}-2b_{2}^{U}-c_{2}^{L}-c_{2}^{U}}{4}\), and check that
\(L\left(A_{1}\right)>L\left(A_{2}\right)\) or\(L\left(A_{1}\right)<L\left(A_{2}\right)\) or\(L\left(A_{1}\right)=L\left(A_{2}\right)\).

\begin{enumerate}
\tightlist
\item
  If \(L\left(A_{1}\right)>L\left(A_{2}\right)\) then\(A_{1}>A_{2}\).
\item
  If \(L\left(A_{1}\right)<L\left(A_{2}\right)\) then\(A_{1}<A_{2}\).
\item
  If \(L\left(A_{1}\right)=L\left(A_{2}\right)\)then\(\text{\ \ A}_{1}=A_{2}\).
\end{enumerate}

Nancy and Garg {[}10, Section 2, Def. 2.6, Ex. 2.1, pp. 379{]}
considered two different SVNS,\(A_{1}=\left\langle 0.5,0.2,0.6\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle 0.2,0.2,0.3\right\rangle\)
and showed that on considering the existing method {[}13{]}, the
relation\(\text{\ \ A}_{1}=A_{2}\) is obtained. While, it is obvious that
\(A_{1}\neq A_{2}\). On the basis of this numerical example, Nancy and
Garg {[}10, Section 2, Def. 2.6, Ex. 2.1, pp. 379{]} claimed that the
existing method {[}13{]} for the ranking of SVNS is not valid.

It is pertinent to mention that the
SVNS,\(A_{1}=\left\langle 0.5,0.2,0.6\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle 0.2,0.2,0.3\right\rangle\) can also be represented
as IVNS,\(A_{1}=\left\langle\left[0.5,0.5\right],\left[0.2,0.2\right],\left[0.6,0.6\right]\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle\left[0.2,0.2\right],\left[0.2,0.2\right],\left[0.3,0.3\right]\right\rangle\). It can be verified
that on considering the existing method {[}13{]}, the
relation\(\text{\ \ A}_{1}=A_{2}\) is obtained. While, it is obvious that
\(A_{1}\neq A_{2}\). Hence, the existing method {[}13{]} for the ranking
of IVNS is also not valid.

\textbf{2.2. Proposed score function}

To resolve the shortcoming discussed in Section 2.1 of the existing
methods {[}13{]}, Nancy and Garg {[}10, Section 3, Def. 3.1, pp. 379{]}
proposed the following method for the ranking of two
SVNS,\(A_{1}=\left\langle a_{1},b_{1},c_{1}\right\rangle\) and\(A_{2}=\left\langle a_{2},b_{2},c_{2}\right\rangle\).
Find\(N\left(A_{1}\right)=\frac{1+(a_{1}-2b_{1}-c_{1})({2-a}_{1}-c_{1})}{2}\)and\(N\left(A_{2}\right)=\frac{1+(a_{2}-2b_{2}-c_{2})({2-a}_{2}-c_{2})}{2}\), and check that
\(N\left(A_{1}\right)>N\left(A_{2}\right)\) or\(N\left(A_{1}\right)<N\left(A_{2}\right)\) or\(N\left(A_{1}\right)=N\left(A_{2}\right)\).

\begin{enumerate}
\tightlist
\item
  If \(N\left(A_{1}\right)>N\left(A_{2}\right)\) then\(A_{1}>A_{2}\).
\item
  If \(N\left(A_{1}\right)<N\left(A_{2}\right)\) then\(A_{1}<A_{2}\).
\item
  If \(N\left(A_{1}\right)=N\left(A_{2}\right)\) then\(A_{1}=A_{2}\).
\end{enumerate}

Furthermore, Nancy and Garg {[}10, Section 3, Def. 3.2, pp. 381{]}
proposed the following method for the ranking of two
IVNS,\(A_{1}=\left\langle\left[a_{1}^{L}{,a}_{1}^{U}\right],\left[b_{1}^{L}{,b}_{1}^{U}\right],\left[c_{1}^{L}c_{1}^{U}\right]\right\rangle\)and\(A_{2}=\left\langle\left[a_{2}^{L}{,a}_{2}^{U}\right],\left[b_{2}^{L}{,b}_{2}^{U}\right],\left[c_{2}^{L}c_{2}^{U}\right]\right\rangle\).
Find\(M\left(A_{1}\right)=\frac{4+{(a}_{1}^{L}+a_{1}^{U}-c_{1}^{L}-c_{1}^{U}-2b_{1}^{L}-2b_{1}^{U})(4-a_{1}^{L}{-a}_{1}^{U}-c_{1}^{L}{-c}_{1}^{U})}{8}\)and

\(M\left(A_{2}\right)=\frac{4+{(a}_{2}^{L}+a_{2}^{U}-c_{2}^{L}-c_{2}^{U}-2b_{2}^{L}-2b_{2}^{U})(4-a_{2}^{L}{-a}_{2}^{U}-c_{2}^{L}{-c}_{2}^{U})}{8}\), and check that \(M\left(A_{1}\right)>M\left(A_{2}\right)\)
or\(M\left(A_{1}\right)<M\left(A_{2}\right)\) or\(M\left(A_{1}\right)=M\left(A_{2}\right)\).

\begin{enumerate}
\tightlist
\item
  If \(M\left(A_{1}\right)>M\left(A_{2}\right)\) then\(A_{1}>A_{2}\).
\item
  If \(M\left(A_{1}\right)<M\left(A_{2}\right)\) then\(A_{1}<A_{2}\).
\item
  If \(M\left(A_{1}\right)=M\left(A_{2}\right)\)then\(\text{\ \ A}_{1}=A_{2}\).
\end{enumerate}

In this note, it is shown that, there exist two different
SVNS\(\text{\ \ A}_{1}\) and \(A_{2}\ \)such
that\(N{(A}_{1})=N{(A}_{2})\) as well as two different IVNS\(\text{\ \ A}_{1}\)
and \(A_{2}\ \)such that\(M{(A}_{1})=M{(A}_{2})\) i.e., the
shortcomings, pointed out by Nancy and Garg {[}10, Section 2, Def. 2.6,
Ex. 2.1, Ex. 2.2, pp. 379{]} in the existing methods {[}13{]}, is also
occurring in the methods proposed by Nancy and Garg {[}10, Section 2,
Def. 3.1, pp. 379; Def. 3.2, pp. 381{]}. Hence, to propose the valid
methods for the ranking of two SVNS as well as the ranking of two IVNS
is still an open challenging research problem.

\textbf{3. Limitations of proposed method for ranking of two SVNS}

Let \(\text{\ \ A}_{1}=\left\langle 0.8,0.1,0.6\right\rangle\) and\(\text{\ \ A}_{2}=\left\langle 0.8,0.2,0.4\right\rangle\) be two SVNS, then according
to the proposed method {[}10, Section 3, Def. 3.1, pp. 379{]}, discussed
in Section 2.2, \(N{(A}_{1})=N{(A}_{2})=0.5\). Therefore, according to the proposed
method {[}10, Section 3, Def. 3.1, pp. 379{]}, \(A_{1}=A_{2}\).
While, it is obvious that\(\text{\ \ A}_{1}\neq A_{2}\). Hence, the proposed method
{[}10, Section 3, Def. 3.1, pp. 379{]} for the ranking of two SVNS is
not valid.

Similarly, let\(\text{\ \ A}_{1}=\left\langle 0.1,0.0,0.1\right\rangle\) and\(\text{\ \ A}_{2}=\left\langle 0.3,0.0,0.3\right\rangle\) be two SVNS, then
according to the proposed method {[}10, Section 3, Def. 3.1, pp. 379{]},
discussed in Section 2.2, \(N{(A}_{1})=N{(A}_{2})=0.5\). Therefore, according to
the proposed method {[}10, Section 3, Def. 3.1, pp. 379{]},
\(\text{\ A}_{1}=A_{2}\). While, it is obvious that\(\text{\ \ A}_{1}\neq A_{2}\). Hence,
the proposed method {[}10, Section 3, Def. 3.1, pp. 379{]} for the
ranking of two SVNS is not valid.

\textbf{4. Limitations of proposed method for ranking of two IVNS}

Let\(A_{1}=\left\langle\left[0.1,0.7\right],\left[0.05,0.15\right],\left[0.1,0.3\right]\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle\left[0.2,0.8\right],\left[0.05,0.15\right],\left[0.2,0.4\right]\right\rangle\)be any two IVNS, then
according to the proposed method {[}10, Section 3, Def. 3.2, pp. 381{]},
discussed in Section 2.2,\(M{(A}_{1})=M{(A}_{2})=0.5\). Therefore, according to the
proposed method {[}10, Section 3, Def. 3.2, pp. 381{]},
\(A_{1}=A_{2}\). While, it is obvious that\(\text{\ \ A}_{1}\neq A_{2}\). Hence,
the proposed method {[}10, Section 3, Def. 3.2, pp. 381{]} for the
ranking of two IVNS is not valid.

Similarly, let\(A_{1}=\left\langle\left[0.1,0.7\right],\left[0.1,0.1\right],\left[0.1,0.3\right]\right\rangle\)and\(\text{\ \ A}_{2}=\left\langle\left[0.2,0.8\right],\left[0.1,0.1\right],\left[0.2,0.4\right]\right\rangle\)be any two IVNS,
then according to the proposed method {[}10, Section 3, Def. 3.2, pp.
381{]}, discussed in Section 2.2,\(M{(A}_{1})=M{(A}_{2})=0.5\). Therefore,
according to the proposed method {[}10, Section 3, Def. 3.2, pp. 381{]},
\(A_{1}=A_{2}\). While, it is obvious that\(\text{\ \ A}_{1}\neq A_{2}\). Hence,
the proposed method {[}10, Section 3, Def. 3.2, pp. 381{]} for the
ranking of two IVNS is not valid.

It is observed that several such examples are occurring like this, i.e.,
let\(A_{1}=\left\langle\left[0.1,0.7\right],\left[0.0,0.2\right],\left[0.1,0.3\right]\right\rangle\)and\(A_{2}=\left\langle\left[0.2,0.8\right],\left[0.0,0.2\right],\left[0.2,0.4\right]\right\rangle\)also, let another example
be,\(A_{1}=\left\langle\left[0.1,0.2\right],\left[0.0,0.0\right],\left[0.1,0.2\right]\right\rangle\)and\(A_{2}=\left\langle\left[0.4,0.5\right],\left[0.0,0.0\right],\left[0.4,0.5\right]\right\rangle\)and we observe that each
example according to the proposed method {[}10, Section 3, Def. 3.2, pp.
381{]}, discussed in Section 2.2, results into\(M{(A}_{1})=M{(A}_{2})=0.5\)
declaring that \(A_{1}=A_{2}\). While, it is obvious
that\(\text{\ \ A}_{1}\neq A_{2}\). Hence, the proposed method {[}10, Section 3,
Def. 3.2, pp. 381{]} for the ranking of two IVNS is not valid in its
present form.

\textbf{5. Conclusion}

Thus with an intensive analysis, it is clearly observed that Nancy and
Garg's proposed method {[}10, Section 3, Def. 3.1, pp. 379{]} for the
ranking of two SVNS as well as the Nancy and Garg's proposed method
{[}10, Section 3, Def. 3.2, pp. 381{]} for the ranking of two IVNS fails
to find the correct ranking order and hence stands invalid. To propose
the valid methods for the same may be considered as a challenging future
research problem.

\textbf{ACKNOWLEDGMENTS}

The authors would like to thank Dr. Amit Kumar (Associate Professor,
School of Mathematics Thapar Institute of Engineering \& Technology,
Patiala, India) and Dr. S.S. Appadoo (Associate Professor, Department of
Supply Chain Management, University of Manitoba, Canada) for everything.

\textbf{COMPLIANCE WITH ETHICAL STANDARDS}

\textbf{CONFLICT OF INTEREST.}

The authors declare that they do not have any financial or associative
interest indicating a conflict of interest in about submitted work.

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