# A Monotonous Intuitionistic Fuzzy TOPSIS Method under General Linear Orders
via Admissible Distance Measures

Xinxing Wu, Zhiyi Zhu, Chuan Chen, Guanrong Chen, Peide Liu Manuscript
received xx 00, 201x; revised xx 00, 201x.X. Wu is with (1) the School of
Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China; (2)
the Institute for Artificial Intelligence, Southwest Petroleum University,
Chengdu, Sichuan 610500, China. (3) the Zhuhai College of Jilin University,
Zhuhai, Guangdong 519041, China. e-mail: (wuxinxing5201314@163.com).Z. Zhu is
with the School of Sciences, Southwest Petroleum University, Chengdu, Sichuan
610500, China. e-mail: (zhuzhiyi2019@163.com).C. Chen is with the School of
Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China.
e-mail: (chenchuan975201314@163.com).G. Chen is with the Department of
Electrical Engineering, City University of Hong Kong, Hong Kong SAR, China.
e-mail: (eegchen@cityu.edu.hk).P. Liu is with a School of Management Science
and Engineering, Shandong University of Finance and Economics, Jinan, Shandong
250014, China. e-mail: (peide.liu@gmail.com).All correspondences should be
addressed to X. Wu and P. Liu.This work was supported by the National Natural
Science Foundation of China (Nos. 11601449 and 71771140), the Key Natural
Science Foundation of Universities in Guangdong Province (No. 2019KZDXM027),
and the Natural Science Foundation of Sichuan Province (No. 2022NSFSC1821).

###### Abstract

All intuitionistic fuzzy TOPSIS methods contain two key elements: (1) the
order structure, which can affect the choices of positive ideal-points and
negative ideal-points, and construction of admissible distance/similarity
measures; (2) the distance/similarity measure, which is closely related to the
values of the relative closeness degrees and determines the accuracy and
rationality of decision-making. For the order structure, many efforts are
devoted to constructing some score functions, which can strictly distinguish
different intuitionistic fuzzy values (IFVs) and preserve the natural partial
order for IFVs. This paper proves that such a score function does not exist,
namely the application of a single monotonous and continuous function does not
distinguish all IFVs. For the distance or similarity measure, some examples
are given to show that classical similarity measures based on the normalized
Euclidean distance and normalized Minkowski distance do not meet the axiomatic
definition of intuitionistic fuzzy similarity measures. Moreover, some
illustrative examples are given to show that classical intuitionistic fuzzy
TOPSIS methods do not ensure the monotonicity with the natural partial order
or linear orders, which may yield some counter-intuitive results. To overcome
the limitation of non-monotonicity, we propose a novel intuitionistic fuzzy
TOPSIS method, using three new admissible distances with the linear orders
measured by a score degree/similarity function and accuracy degree, or two
aggregation functions, and prove that the proposed TOPSIS method is monotonous
under these three linear orders. This is the first result with a strict
mathematical proof on the monotonicity with the linear orders for the
intuitionistic fuzzy TOPSIS method. Finally, we show two practical examples
and comparative analysis with other decision-making methods to illustrate the
efficiency of the developed TOPSIS method.

###### Index Terms:

Intuitionistic fuzzy set, Distance measure, Similarity measure, TOPSIS, Multi-
attribute decision making.

##  I Introduction

Zadeh (1965) [1] established the fuzzy set (FS) theory by applying membership
degree to measure the importance of a fuzzy element, which generalized the
concept of crisp set, characterized by a characteristic function taking value
$0$ or $1$, by taking any value in the unit interval $[0,1]$. However, due to
the limitation of a membership function that only indicates two (supporting
and opposing) opposite states of fuzziness, the fuzzy set cannot express the
neutral state of “this and also that”. According to this, Atanassov (1986) [2]
generalized Zadeh’s fuzzy set by proposing the concept of intuitionistic fuzzy
sets (IFSs) (see also [3]), characterized by a membership function and a non-
membership function meeting the condition that the sum of the membership
degree and the non-membership degree at every point is less than or equal to
$1$. Every pair of membership and non-membership degrees for IFSs was called
an intuitionistic fuzzy value (IFV) by Xu [4]. Thereafter, various multi-
attribute decision making (MADM) methods under the intuitionistic fuzzy
framework were developed and widely applied. This paper focuses on the
intuitionistic fuzzy TOPSIS method.

Being one of the most well-known MADM methods, TOPSIS was first proposed by
Hwang and Yoon (1981) [5]. The main idea of the TOPSIS is that the most
desirable alternative should be nearest from the positive ideal-point and
meanwhile furthest from the negative ideal-point. Noting that the total order
structure ‘$\leq$’ and the absolute distance ‘$|\cdot|$’ on the real line
$\mathbb{R}$ are admissible (the bigger the real number, the nearer from the
maximum, and the further from the minimum). This can naturally guarantee the
establishment of the TOPSIS and further ensure its monotonicity. Due to the
complex two-dimensional structure of the space of all IFVs, all existing IF
distance measurements for IFVs are not admissible under linear orders on IFVs.
Therefore, we [6] introduced the concept of admissible distance measure with
the linear order ‘$\leq_{{}_{\mathrm{XY}}}$’ of Xu and Yager [7] and
constructed an admissible distance measure $\varrho$ with the linear order
‘$\leq_{{}_{\mathrm{XY}}}$’.

Through careful analysis of the TOPSIS method, it is not difficult to find
that the TOPSIS method contains two key elements: (1) the order structure,
which can affect the choices of positive ideal-points and negative ideal-
points; (2) the distance/similarity measure, which is closely related to the
relative closeness degrees and determines the accuracy and rationality of
decision-making. Therefore, essentially speaking, all improvements on IF
TOPSIS method provide certain improvements on the order structure and the
distance/similarity measures.

To rank IFVs, score function is a useful tool. Xu and Yager [7] introduced the
linear order ‘$\leq_{{}_{\text{XY}}}$’ for IFVs by applying a score function
and an accuracy function. According to the TOPSIS idea [5], Zhang and Xu [8]
proposed another linear order ‘$\leq_{{}_{\text{ZX}}}$’ for IFVs by applying a
similarity function and an accuracy function. Recently, Xing et al. [9]
defined a linear order for IFVs by using a score function expressed by the
Euclidean distance from the maximum point $\langle 1,0\rangle$. Bustince et
al. [10] suggested a general construction of linear orders for intervals
contained in $[0,1]$ by means of aggregation functions. Based on this, De
Miguel et al. [11, 12] developed a general method for constructing linear
orders between pairs of intervals based on aggregation functions, which was
successfully applied to construct linear orders for interval-valued IFSs. Wang
et al. [13] classified the existing score functions of IFVs into two types,
one type consists of score functions without abstention group influence [14,
15, 16, 17, 18, 19], and the other consists of score functions with abstention
group influence [20]. Zeng et al. [21] illustrated that these existing score
functions, the existing accuracy functions in [22, 23], and the measure
methods in [24, 25, 8] for ranking IFVs have a drawback that they may cause
some unreasonable raking results or they cannot distinguish some different
IFVs. To overcome this drawback, Zeng et al. [21] proposed a new score
function $S_{{}_{CK}}$ for IFVs, which was a monotonically increasing
injective with Atanassov’s partial order $\subset$ (see [21, Theorems 3.1 and
3.2]). However, we constructed an example to show that this inference in [21]
does not hold and proved that such a score function does not exist, i.e.,
there is no any continuous injective from the space of all IFVs to
$\mathbb{R}$ that is increasing with Atanassov’s partial order $\subset$. This
means that the score function $S_{{}_{CK}}$ has the same drawback. This
trouble mainly arises from the fact that the space of all IFVs has a two-
dimensional structure, which is not homeomorphic to any closed interval on the
real line $\mathbb{R}$. In fact, we proved that the application of a single
monotonous and continuous function does not distinguish all IFVs.

Being a pair of dual concepts, the normalized distance measures and similarity
measures have been widely studied. Similarly to the axiomatic definition of
similarity measure for fuzzy sets [26, 27], Li and Cheng [28] gave an
axiomatic definition of similarity measures for IFSs by using normalization
(S1), symmetry (S3), and compatibility with Atanassov’s partial order (S4),
i.e., the condition $I_{1}\subset I_{2}\subset I_{3}$ implies that the
similarity measure between $I_{1}$ and $I_{3}$ is smaller than that between
$I_{1}$ and $I_{2}$ and between $I_{2}$ and $I_{3}$. Xu [29] introduced some
new IF similarity measures and applied them to some practical MADM problems.
Xu and Chen [30] presented a comprehensive overview of distance and similarity
measures of IFSs and proposed some continuous distance and similarity measures
for IFSs using the weighted Hamming distance, weighted Euclidean distance, and
weighted Hausdorff distance. Iancu [31] defined some IF similarity measures
using Frank t-norms $T_{\gamma}^{\mathbf{F}}$. Szmidt and Kacprzyk [32]
pointed out that the third parameter (indeterminacy degree) should be
considered when calculating distances for IFSs. Because of the duality between
distance and similarity measures, various three-dimensional IF distance and
similarity measures including indeterminacy degrees were introduced in [33,
34, 35, 36]. However, Atanassov’s partial order only reveals the magnitudes of
membership degrees and non-membership degrees between two IFSs, and thus many
three-dimensional IF similarity measures considering indeterminacy degrees
might not meet the axiomatic condition (S4). In fact, we constructed three
examples to show that Euclidean similarity measure in [34, 37, 35], Minkowski
similarity measure in [29, 35, 34, 37], and a modified similarity measure in
[30] based on the idea of the above TOPSIS does not satisfy the axiomatic
condition (S4). In particular, it should be pointed out that some existing IF
distance and similarity measures are unreasonable for dealing with some
practical decision-making problems. For example, Mitchell [38] showed that Li
and Cheng’s similarity measure [28] may lead to counter-intuitive situations
in some cases. Chen et al. [39] showed some counterexamples to illustrate that
the similarity measures in [40, 41, 42, 38, 23, 43] may produce unreasonable
results in some cases. As noted above, the application of a single monotonous
and continuous function does not distinguish all IFVs. This means that all
continuous IF similarity measures are unable to distinguish every pair of
different IFSs. For example, when we apply a continuous IF similarity measure
for pattern recognition, we always encounter the case that we cannot determine
the classification result. Therefore, the comparative analysis on the
indistinguishability is meaningless ([44, Tables 2–5], [40, Tables 1–2], [39,
Tables 1, 2, 5, 6], [33, Table 2]), because all continuous IF similarity
measures will also encounter the same indistinguishability problem. On the
other hand, all existing distance and similarity measures are only admissible
with Atanassov’s partial order, and thus decision-making results obtained by
these distance or similarity measures can only guarantee the monotonicity with
the partial order. Therefore, to obtain monotonous decision-making methods, we
need to develop new admissible distance and similarity measures with linear
orders.

During the past decade, various generalized IF TOPSIS methods have been
developed. For example, Boran et al. [45] first extended the TOPSIS method to
IF group decision-making with IFV weights. Then, some similar IF TOPSIS
methods with IFV/linguistic weights were developed [46, 37, 47] and widely
applied to practical decision-making problems [46, 48, 49, 50, 51]. Our
Examples 6 and 7 below in this paper show that the TOPSIS methods in [45, 48,
37, 47, 50] may yield unreasonable results even when dealing with the simplest
decision-making problems. Chen et al. [52] proposed a MADM method with crisp
numerical weights based on the TOPSIS method and a new similarity measure for
the IF situation. Zeng et al. [21] pointed out that the MADM method in [52]
cannot distinguish the alternatives in some special situations. Furthermore,
our Example 8 below in this paper demonstrates that the MADM method in [52] is
not monotonous with the linear order ‘$\leq_{{}_{\mathrm{XY}}}$’ of Xu and
Yager [7]. Based on a new distance measure, Shen et al. [33] developed an
extended IF TOPSIS method and applied it to credit risk evaluation. Here, only
some works closely related to this paper are mentioned. For more results on
TOPSIS under the IF or interval-valued IF framework, interested readers are
referred to [53, 54, 55, 56, 57, 58, 59].

Inspired by the above discussions, this paper establishes a monotonous IF
TOPSIS under three popular linear orders, ‘$\leq_{{}_{\text{XY}}}$’ in [7],
‘$\leq_{{}_{\text{ZX}}}$’ in [8], and ‘$\leq_{{}_{A,B}}$’ in ([11, 12, 10]).
More precisely, the main contributions of this paper are as follows:

(1) We construct some counterexamples to illustrate that Euclidean similarity
measure ([37, 34]), Minkowski similarity measure ([29]), and modified
Euclidean similarity measure ([30]) do not satisfy the axiomatic definition of
IF similarity measures (see Examples 1–3).

(2) We prove that there is no any continuous and injective function from the
space of all IFVs to the real line $\mathbb{R}$ that is increasing with
Atanassov’s partial order ‘$\subset$’. This indicates the nonexistence of
continuous similarity measure distinguishing between every pair of IFVs.
Therefore, the comparative analysis on the indistinguishability for IF
similarity measures is meaningless, but unfortunately this is a common problem
for all continuous IF similarity measures.

(3) We construct three simple examples to show that some classical IF TOPSIS
methods in [37, 52, 48, 45, 47, 50] are not monotonous with Atanassov’s
partial order ‘$\subset$’ or the linear order ‘$\leq_{{}_{\text{XY}}}$’ (see
Examples 6–8), which may yield counter-intuitive results. To overcome this
limitation, by proposing three new admissible distances with the linear order
‘$\leq_{{}_{\text{XY}}}$’ or ‘$\leq_{{}_{\textrm{ZX}}}$’ or
‘$\leq_{{}_{A,B}}$’, we develop a novel IF TOPSIS method and prove that it is
monotonically increasing with these three linear orders.

(4) We provide two practical examples and a comparative analysis with other
MADM methods to illustrate the efficiency of our proposed TOPSIS method.

The paper is organized as follows: Section II presents some basic definitions
related to the IFSs, including IFSs, orders for IFSs, and IF distance and
similarity measure. Section III provides some examples to illustrate that
classical similarity measures in [34, 29, 30] based on the normalized
Euclidean distance and normalized Minkowski distance do not meet the axiomatic
definition of IF similarity measures. Section IV proves that the application
of a single monotonous and continuous function does not distinguish all IFVs.
Section V applies three examples to demonstrate that the IF TOPSIS methods in
[37, 52, 48, 45, 47, 50] are not monotonous with Atanassov’s partial order
‘$\subset$’ or the linear order ‘$\leq_{{}_{\text{XY}}}$’ of Xu and Yager [7].
To overcome this limitation, by constructing three new admissible distances
with the linear order ‘$\leq_{{}_{\text{XY}}}$’, ‘$\leq_{{}_{\textrm{ZX}}}$’
or ‘$\leq_{{}_{A,B}}$’, Section VI develops a new IF TOPSIS method and proves
that it is monotonically increasing with these three linear orders. Section
VII presents two practical examples to demonstrate the efficiency of the
proposed TOPSIS method developed in Section VI. Section VIII concludes the
paper with a future research outlook.

##  II Preliminaries

###  II-A Intuitionistic fuzzy set (IFS)

######  Definition II.1 ([3, Definition 1.1])

Let $X$ be the universe of discourse. An intuitionistic fuzzy set (IFS) $I$ in
$X$ is defined as an object in the following form:

$I=\left\\{\langle x;\mu_{{}_{I}}(x),\nu_{{}_{I}}(x)\rangle\mid x\in
X\right\\},$

where the functions $\mu_{{}_{I}}:X\rightarrow[0,1]$ and
$\nu_{{}_{I}}:X\rightarrow[0,1]$ define the degree of membership and the
degree of non-membership of an element $x\in X$ to the set $I$, respectively,
and for every $x\in X$, $\mu_{{}_{I}}(x)+\nu_{{}_{I}}(x)\leq 1.$

Let $\mathrm{IFS}(X)$ denote the set of all IFSs in $X$. For
$I\in\mathrm{IFS}(X)$, the indeterminacy degree $\pi_{{}_{I}}(x)$ of an
element $x$ belonging to $I$ is defined by
$\pi_{{}_{I}}(x)=1-\mu_{{}_{I}}(x)-\nu_{{}_{I}}(x)$. In [4, 35], the pair
$\langle\mu_{{}_{I}}(x),\nu_{{}_{I}}(x)\rangle$ is called an  intuitionistic
fuzzy value (IFV) or an intuitionistic fuzzy number (IFN). For convenience,
use $\alpha=\langle\mu_{\alpha},\nu_{\alpha}\rangle$ to represent an IFV
$\alpha$, which satisfies $\mu_{\alpha}\in[0,1]$, $\nu_{\alpha}\in[0,1]$, and
$0\leq\mu_{\alpha}+\nu_{\alpha}\leq 1$. Additionally,
$s(\alpha)=\mu_{\alpha}-\nu_{\alpha}$ and
$h(\alpha)=\mu_{\alpha}+\nu_{\alpha}$ are called the score degree and the
accuracy degree of $\alpha$, respectively. Let $\tilde{\mathbb{I}}$ denote the
set of all IFVs, i.e.,
${\tilde{\mathbb{I}}}=\\{\langle\mu,\nu\rangle\in[0,1]^{2}\mid\mu+\nu\leq
1\\}$.

Motivated by the basic operations on IFSs, Xu et al. [35, 7] introduced the
following basic operational laws for IFVs.

######  Definition II.2 ([35, Definition 1.2.2])

Let $\alpha=\langle\mu_{\alpha},\nu_{\alpha}\rangle$,
$\beta=\langle\mu_{\beta},\nu_{\beta}\rangle\in\tilde{\mathbb{I}}$. Define

  1. (i)

$\alpha^{\complement}=\langle\nu_{\alpha},\mu_{\alpha}\rangle$.

  2. (ii)

$\alpha\cap\beta=\langle\min\\{\mu_{\alpha},\mu_{\beta}\\},\max\\{\nu_{\alpha},\nu_{\beta}\\}\rangle$.

  3. (iii)

$\alpha\cup\beta=\langle\max\\{\mu_{\alpha},\mu_{\beta}\\},\min\\{\nu_{\alpha},\nu_{\beta}\\}\rangle$.

  4. (iv)

$\alpha\oplus\beta=\langle\mu_{\alpha}+\mu_{\beta}-\mu_{\alpha}\mu_{\beta},\nu_{\alpha}\nu_{\beta}\rangle$.

  5. (v)

$\alpha\otimes\beta=\langle\mu_{\alpha}\mu_{\beta},\nu_{\alpha}+\nu_{\beta}-\nu_{\alpha}\nu_{\beta}\rangle$.

  6. (vi)

$\lambda\alpha=\langle
1-(1-\mu_{\alpha})^{\lambda},(\nu_{\alpha})^{\lambda}\rangle$, $\lambda>0$.

  7. (vii)

$\alpha^{\lambda}=\langle(\mu_{\alpha})^{\lambda},1-(1-\nu_{\alpha})^{\lambda}\rangle$,
$\lambda>0$.

###  II-B Orders for IFSs

Atanassov’s order ‘$\subset$’ [3], defined by that $\alpha\subset\beta$ if and
only if $\alpha\cap\beta=\alpha$, is a partial order on $\tilde{\mathbb{I}}$.

######  Definition II.3 ([11, Definition 4.1])

An order $\leq$ on $\tilde{\mathbb{I}}$ is said to be an IF-admissible order
if it is a linear order and refines Atanassov’s order $\subset$.

To compare any two IFVs, Xu and Yager [7] introduced the following linear
order ‘$\leq_{{}_{\text{XY}}}$’ (see also [4, Definition 3.1] and [35,
Definition 1.1.3]):

######  Definition II.4 ([7, Definition 1])

Let $\alpha_{1}$ and $\alpha_{2}$ be two IFVs.

  * •

If $s(\alpha_{1})<s(\alpha_{2})$, then $\alpha_{1}$ is smaller than
$\alpha_{2}$, denoted by $\alpha_{1}<_{{}_{\text{XY}}}\alpha_{2}$.

  * •

If $s(\alpha_{1})=s(\alpha_{2})$, then

    * –

if $h(\alpha_{1})=h(\alpha_{2})$, then $\alpha_{1}=\alpha_{2}$;

    * –

if $h(\alpha_{1})<h(\alpha_{2})$, then
$\alpha_{1}<_{{}_{\text{XY}}}\alpha_{2}$.

If $\alpha_{1}<_{{}_{\text{XY}}}\alpha_{2}$ or $\alpha_{1}=\alpha_{2}$, then
denote it by $\alpha_{1}\leq_{{}_{\text{XY}}}\alpha_{2}$.

Alongside Xu and Yager’s order ‘$\leq_{{}_{\text{XY}}}$’ in Definition II.4,
Szmidt and Kacprzyk [25] proposed another comparison function,
$\rho(\alpha)=\frac{1}{2}(1+\pi(\alpha))(1-\mu(\alpha))$ for IFVs, which is a
partial order. However, it sometimes cannot distinguish between two IFVs.
Although Xu’s method [4] constructs a linear order for ranking any two IFVs,
its procedure has the following disadvantages: (1) It may result in that the
less we know, the better the IFV, which is not reasonable. (2) It is sensitive
to a slight change of the parameters. (3) It is not preserved under
multiplication by a scalar, namely, $\alpha\leq_{{}_{\text{XY}}}\beta$ might
not imply $\lambda\alpha<_{{}_{\text{XY}}}\lambda\beta$, where $\lambda$ is a
scalar (see [60, Example 1]). To overcome such shortcomings of the above two
ranking methods, Zhang and Xu [8] improved Szmidt and Kacprzyk’s method [25],
according to Hwang and Yoon’s idea [5] and technique for preference order by
similarity to an ideal point. They also introduced a similarity function
$L(\alpha)$, called the $L$-value in [8], for any IFV
$\alpha=\langle\mu_{\alpha},\nu_{\alpha}\rangle$, as follows:

$L(\alpha)=\frac{1-\nu_{\alpha}}{(1-\mu_{\alpha})+(1-\nu_{\alpha})}=\frac{1-\nu_{\alpha}}{1+\pi_{\alpha}}.$
(1)

In particular, if $\nu_{\alpha}<1$, then
$L(\alpha)=\frac{1}{\tfrac{1-\mu_{\alpha}}{1-\nu_{\alpha}}+1}.$ Furthermore,
Zhang and Xu [8] introduced the following order ‘$\leq_{{}_{\textrm{ZX}}}$’
for IFVs by applying the similarity function $L(\\_)$.

######  Definition II.5 ([8])

Let $\alpha_{1}$ and $\alpha_{2}$ be two IFVs.

  * •

If $L(\alpha_{1})<L(\alpha_{2})$, then
$\alpha_{1}<_{{}_{\textrm{ZX}}}\alpha_{2}$;

  * •

If $L(\alpha_{1})=L(\alpha_{2})$, then

    * –

if $h(\alpha_{1})=h(\alpha_{2})$, then $\alpha_{1}=\alpha_{2}$;

    * –

if $h(\alpha_{1})<h(\alpha_{2})$, then
$\alpha_{1}<_{{}_{\textrm{ZX}}}\alpha_{2}$.

If $\alpha_{1}<_{{}_{\textrm{ZX}}}\alpha_{2}$ or $\alpha_{1}=\alpha_{2}$, then
denote it by $\alpha_{1}\leq_{{}_{\textrm{ZX}}}\alpha_{2}$.

######  Definition II.6 ([61, Definition 1.1])

An aggregation function in $[0,1]^{n}$ is a function
$A:[0,1]^{n}\rightarrow[0,1]$ that

  1. (i)

is nondecreasing in each variable;

  2. (ii)

fulfills the boundary conditions

$A(0,\ldots,0)=0\text{ and }A(1,\ldots,1)=1.$

By an equivalent transformation between intervals and IFVs and [10,
Proposition 3.2], the following general construction of linear orders is
proposed for IFVs.

###### Proposition II.1

Let $A$, $B:[0,1]^{2}\rightarrow[0,1]$ be two continuous aggregation functions
satisfying that, for $(x_{1},y_{1})$, $(x_{2},y_{2})\in[0,1]^{2}$, the
identities $A(x_{1},y_{1})=A(x_{2},y_{2})$ and $B(x_{1},y_{1})=B(x_{2},y_{2})$
can hold only if $x_{1}=x_{2}$ and $y_{1}=y_{2}$. Define the order
$\leq_{{}_{A,B}}$ on $\tilde{\mathbb{I}}$ as follows:
$\langle\mu_{1},\nu_{1}\rangle\leq_{{}_{A,B}}\langle\mu_{2},\nu_{2}\rangle$ if
and only if $A(\mu_{1},1-\nu_{1})<A(\mu_{2},1-\nu_{2})$ or
$(A(\mu_{1},1-\nu_{1})=A(\mu_{2},1-\nu_{2})$ and $B(\mu_{1},1-\nu_{1})\leq
B(\mu_{2},1-\nu_{2}))$. Then, $\leq_{{}_{A,B}}$ is an admissible order on
$\tilde{\mathbb{I}}$.

For $\gamma\in[0,1]$, consider an aggregation function
$K_{\gamma}:[0,1]^{2}\rightarrow[0,1]$ defined by
$K_{\gamma}(x,y)=x+\gamma(y-x).$ For $\gamma_{1}$, $\gamma_{2}\in[0,1]$ with
$\gamma_{1}\neq\gamma_{2}$, according to Proposition II.1, it follows that the
order $\leq_{{}_{\gamma_{1},\gamma_{2}}}$ on $\tilde{\mathbb{I}}$ defined by
$\alpha\leq_{{}_{\gamma_{1},\gamma_{2}}}\beta$ if and only if
$K_{\gamma_{1}}(\mu_{\alpha},1-\nu_{\alpha})<K_{\gamma_{1}}(\mu_{\beta},1-\nu_{\beta})$
or
($K_{\gamma_{1}}(\mu_{\alpha},1-\nu_{\alpha})=K_{\gamma_{1}}(\mu_{\beta},1-\nu_{\beta})$
and $K_{\gamma_{2}}(\mu_{\alpha},1-\nu_{\alpha})\leq
K_{\gamma_{2}}(\mu_{\beta},1-\nu_{\beta})$) is an admissible order on
$\tilde{\mathbb{I}}$.

###  II-C IF distance and similarity measure

Li and Cheng [28] introduced an axiomatic definition of similarity measure for
IFSs, which was then improved by Mitchell [62] as follows. More results on the
similarity measure can be found in [34, 63].

######  Definition II.7 ([62])

Let $X$ be the universe of discourse. A mapping
$\mathbf{S}:\mathrm{IFS}(X)\times\mathrm{IFS}(X)\rightarrow[0,1]$ is called an
admissible similarity measure with the order $\subset$ on $\mathrm{IFS}(X)$ if
it satisfies the following conditions: for any $I_{1}$, $I_{2}$,
$I_{3}\in\mathrm{IFS}(X)$,

  1. (S1)

$0\leq\mathbf{S}(I_{1},I_{2})\leq 1$.

  2. (S2)

$\mathbf{S}(I_{1},I_{2})=1$ if and only if $I_{1}=I_{2}$.

  3. (S3)

$\mathbf{S}(I_{1},I_{2})=\mathbf{S}(I_{2},I_{1})$.

  4. (S4)

If $I_{1}\subset I_{2}\subset I_{3}$, then
$\mathbf{S}(I_{1},I_{3})\leq\mathbf{S}(I_{1},I_{2})$ and
$\mathbf{S}(I_{1},I_{3})\leq\mathbf{S}(I_{2},I_{3})$.

###### Remark 1

The admissible similarity measure with the order $\subset$ was also called
similarity measure by Hung and Yang [64] and Szmidt [34]. When no ambiguity is
possible, we simply call it similarity measure.

###### Definition II.8

Let $X$ be the universe of discourse and $I_{1}$, $I_{2}\in\mathrm{IFS}(X)$.
If
$\langle\mu_{{}_{I_{1}}}(x),\nu_{{}_{I_{1}}}(x)\rangle\leq_{{}_{\text{XY}}}\langle\mu_{{}_{I_{2}}}(x),\nu_{{}_{I_{2}}}(x)\rangle$
holds for all $x\in X$, then we say that $I_{1}$ is smaller than or equal to
$I_{2}$ under the linear order $\leq_{{}_{\text{XY}}}$, denoted by
$I_{1}\leq_{{}_{\text{XY}}}I_{2}$.

Based on Definition II.8, we introduce the improved similarity measure
definition for IFSs below:

###### Definition II.9

Let $X$ be the universe of discourse. A mapping
$\mathbf{S}:\mathrm{IFS}(X)\times\mathrm{IFS}(X)\rightarrow[0,1]$ is called an
admissible similarity measure with the order $\leq_{{}_{\text{XY}}}$ on
$\mathrm{IFS}(X)$ if it satisfies the conditions (S1)–(S3) in Definition II.7,
and the following one (S4′):

  1. (S4′)

For any $I_{1}$, $I_{2}$, $I_{3}\in\mathrm{IFS}(X)$, if
$I_{1}\leq_{{}_{\text{XY}}}I_{2}\leq_{{}_{\text{XY}}}I_{3}$, then
$\mathbf{S}(I_{1},I_{3})\leq\mathbf{S}(I_{1},I_{2})$ and
$\mathbf{S}(I_{1},I_{3})\leq\mathbf{S}(I_{2},I_{3})$.

Now, we recall some classical distances and similarity measures for IFSs.

The normalized Hamming distance in [32] is:

$d_{\mathrm{Ha}}(I_{1},I_{2})=\frac{1}{2n}\sum_{j=1}^{n}H_{j},$ (2)

where
$H_{j}=|\mu_{I_{1}}(x_{j})-\mu_{I_{2}}(x_{j})|+|\nu_{I_{1}}(x_{j})-\nu_{I_{2}}(x_{j})|+|\pi_{I_{1}}(x_{j})-\pi_{I_{2}}(x_{j})|$.

The normalized Euclidean distance in [32] is:

$d_{\mathrm{Eu}}(I_{1},I_{2})=\sqrt{\frac{1}{2n}\sum_{j=1}^{n}E_{j}},$ (3)

where
$E_{j}=|\mu_{I_{1}}(x_{j})-\mu_{I_{2}}(x_{j})|^{2}+|\nu_{I_{1}}(x_{j})-\nu_{I_{2}}(x_{j})|^{2}+|\pi_{I_{1}}(x_{j})-\pi_{I_{2}}(x_{j})|^{2}$.

The normalized Minkowski distance in [32, 34] is:

$d_{{}_{\mathrm{M}}}^{(\alpha)}(I_{1},I_{2})=\sqrt[\alpha]{\frac{1}{2n}\sum_{j=1}^{n}M_{j}},$
(4)

where
$M_{j}=|\mu_{I_{1}}(x_{j})-\mu_{I_{2}}(x_{j})|^{\alpha}+|\nu_{I_{1}}(x_{j})-\nu_{I_{2}}(x_{j})|^{\alpha}+|\pi_{I_{1}}(x_{j})-\pi_{I_{2}}(x_{j})|^{\alpha}$
and $\alpha\geq 1$.

By using the normalized Hamming distance, Szmidt and Kacprzyk [65] introduced
the following similarity measure $\mathbf{S}_{\mathrm{SK}}^{1}$ for two IFSs
$I_{1}$ and $I_{2}$:

$\mathbf{S}_{{}_{\mathrm{SK}}}^{1}(I_{1},I_{2})=\frac{d_{\mathrm{Ha}}(I_{1},I_{2})}{d_{\mathrm{Ha}}(I_{1},I_{2}^{\complement})},$
(5)

where $I_{2}^{\complement}$ is the complement of $I_{2}$. If we replace the
normalized Hamming distance with the normalized Euclidean distance, we can
obtain the following “similarity measure” $\mathbf{S}_{\mathrm{SK}}^{1}$ for
two IFSs $I_{1}$ and $I_{2}$:

$\mathbf{S}_{{}_{\mathrm{SK}}}^{2}(I_{1},I_{2})=\frac{d_{\mathrm{Eu}}(I_{1},I_{2})}{d_{\mathrm{Eu}}(I_{1},I_{2}^{\complement})}.$
(6)

Clearly, both $\mathbf{S}_{\mathrm{SK}}^{1}$ and
$\mathbf{S}_{\mathrm{SK}}^{2}$ are not similarity measures, because their
values may exceed $1$. Then, the following similarity measures were introduced
by Szmidt [34] using the normalized Hamming and Euclidean distances:

$\mathbf{Sim}_{{}_{\text{H}}}^{1}(I_{1},I_{2})=1-d_{\mathrm{Ha}}(I_{1},I_{2}),$
(7)

and

$\mathbf{Sim}_{{}_{\text{E}}}^{2}(I_{1},I_{2})=1-d_{\mathrm{Eu}}(I_{1},I_{2}).$
(8)

Based on the normalized Minkowski distance $d_{{}_{\mathrm{M}}}^{(\alpha)}$,
Xu [29] introduced the following Minkowski similarity measure (see also [37,
30]): for $\alpha>0$,

$\mathbf{S}_{{}_{\text{Xu}}}^{(\alpha)}(I_{1},I_{2})=1-d_{{}_{\mathrm{M}}}^{(\alpha)}(I_{1},I_{2}).$
(9)

Clearly, $\mathbf{S}_{{}_{\text{Xu}}}^{(1)}=\mathbf{Sim}_{{}_{\text{H}}}^{1}$
and $\mathbf{S}_{{}_{\text{Xu}}}^{(2)}=\mathbf{Sim}_{{}_{\text{E}}}^{2}$.

Motivated by the idea of the TOPSIS of Hwang and Yoon [5], Xu and Chen [30]
modified Eqs. (5) and (6) as follows:

$\mathbf{S}_{{}_{\text{XC}}}^{1}(I_{1},I_{2})=\frac{d_{\mathrm{Ha}}(I_{1},I_{2}^{\complement})}{d_{\mathrm{Ha}}(I_{1},I_{2})+d_{\mathrm{Ha}}(I_{1},I_{2}^{\complement})},$
(10)
$\mathbf{S}_{{}_{\text{XC}}}^{2}(I_{1},I_{2})=\frac{d_{\mathrm{Eu}}(I_{1},I_{2}^{\complement})}{d_{\mathrm{Eu}}(I_{1},I_{2})+d_{\mathrm{Eu}}(I_{1},I_{2}^{\complement})}.$
(11)

##  III The drawbacks of some existing similarity measures

This section illustrates that the similarity measures defined by Eqs. (8),
(9), and (11) do not meet the property (S4) in the axiomatic definition of
intuitionistic fuzzy similarity measures.

###### Example 1

Let the universe of discourse $X=\\{x_{1}\\}$, and
$I_{1}=\left\\{\frac{\langle 0,1\rangle}{x_{1}}\right\\}$,
$I_{2}=\left\\{\frac{\langle 0.1,0\rangle}{x_{1}}\right\\}$, and
$I_{3}=\left\\{\frac{\langle 0.4,0\rangle}{x_{1}}\right\\}$. Clearly,
$I_{1}\subset I_{2}\subset I_{3}$. By direct calculation, we have

$\displaystyle\mathbf{Sim}_{{}_{\text{E}}}^{2}(I_{1},I_{2})=$ $\displaystyle
1-\sqrt{\frac{|0-0.1|^{2}+|1-0|^{2}+|0-0.9|^{2}}{2}}$ $\displaystyle=$
$\displaystyle 1-\sqrt{0.91},$

and

$\displaystyle\mathbf{Sim}_{{}_{\text{E}}}^{2}(I_{1},I_{3})=$ $\displaystyle
1-\sqrt{\frac{|0-0.4|^{2}+|1-0|^{2}+|0-0.6|^{2}}{2}}$ $\displaystyle=$
$\displaystyle 1-\sqrt{0.76},$

and thus
$\mathbf{Sim}_{{}_{\text{E}}}^{2}(I_{1},I_{2})<\mathbf{Sim}_{{}_{\text{E}}}^{2}(I_{1},I_{3})$.
This, together with $I_{1}\subset I_{2}\subset I_{3}$, implies that the
formula $\mathbf{Sim}_{{}_{\text{H}}}^{1}$ defined by Eq. (8) is not a
similarity measures on IFSs.

###### Example 2

Let the universe of discourse $X=\\{x_{1}\\}$ and $I_{1}=\left\\{\frac{\langle
0,1\rangle}{x_{1}}\right\\}$,
$I_{2}=\left\\{\frac{\langle\mu_{2},0\rangle}{x_{1}}\right\\}$, and
$I_{3}=\left\\{\frac{\langle\mu_{3},0\rangle}{x_{1}}\right\\}$ be three IFSs
on $X$ such that $0<\mu_{2}<\mu_{3}<0.5$. Clearly, $I_{1}\subset I_{2}\subset
I_{3}$. Fix any $\alpha>1$. By direct calculation, we have

$\mathbf{S}_{{}_{\text{M}}}^{(\alpha)}(I_{1},I_{2})=1-\sqrt[\alpha]{\frac{(\mu_{2})^{\alpha}+1+(1-\mu_{2})^{\alpha}}{2}},$

and

$\mathbf{S}_{{}_{\text{M}}}^{(\alpha)}(I_{1},I_{3})=1-\sqrt[\alpha]{\frac{(\mu_{3})^{\alpha}+1+(1-\mu_{3})^{\alpha}}{2}}.$

Let $\Gamma(x)=1-\sqrt[\alpha]{\frac{x^{\alpha}+1+(1-x)^{\alpha}}{2}}$
($x\in(0,0.5)$). Noting that $\alpha>1$ and $x\in(0,0.5)$, by direct
calculation, we get
$\Gamma^{\prime}(x)=-\frac{1}{2}(\frac{x^{\alpha}+1+(1-x)^{\alpha}}{2})^{\frac{1}{\alpha}-1}\cdot(x^{\alpha-1}-(1-x)^{\alpha-1})>0$,
and thus the function $\Gamma$ is strictly increasing on $(0,0.5)$. This,
together with $0<\mu_{2}<\mu_{3}<0.5$, implies that
$\mathbf{S}_{{}_{\text{M}}}^{(\alpha)}(I_{1},I_{2})=\Gamma(\mu_{2})<\Gamma(\mu_{3})=\mathbf{S}_{{}_{\text{M}}}^{(\alpha)}(I_{1},I_{3})$.
Therefore, the formula $\mathbf{S}_{{}_{\text{M}}}^{(\alpha)}$ defined by Eq.
(9) is not a similarity measures on IFSs for any $\alpha>1$.

###### Example 3

Let the universe of discourse $X=\\{x_{1}\\}$, and
$I_{1}=\left\\{\frac{\langle 0,1\rangle}{x_{1}}\right\\}$,
$I_{2}=\left\\{\frac{\langle 0.9,0.01\rangle}{x_{1}}\right\\}$, and
$I_{3}=\left\\{\frac{\langle 0.901,0.007\rangle}{x_{1}}\right\\}$. Clearly,
$I_{1}\subset I_{2}\subset I_{3}$. By direct calculation, we have

$\displaystyle\mathbf{S}_{{}_{\text{XC}}}^{2}(I_{1},I_{2})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\frac{|0-0.01|^{2}+|1-0.9|^{2}+|0-0.09|^{2}}{2}}}{\sqrt{\frac{|0-0.9|^{2}+|1-0.01|^{2}+|0-0.09|^{2}}{2}}+\sqrt{\frac{|0-0.01|^{2}+|1-0.9|^{2}+|0-0.09|^{2}}{2}}}$
$\displaystyle\approx$ $\displaystyle 0.09141,$

and

$\displaystyle\mathbf{S}_{{}_{\text{XC}}}^{2}(I_{1},I_{3})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\frac{|0.007|^{2}+|1-0.901|^{2}+|0.092|^{2}}{2}}}{\sqrt{\frac{|0.901|^{2}+|1-0.007|^{2}+|0.092|^{2}}{2}}+\sqrt{\frac{|0.007|^{2}+|1-0.901|^{2}+|0.092|^{2}}{2}}}$
$\displaystyle\approx$ $\displaystyle 0.09148,$

and thus
$\mathbf{S}_{{}_{\text{XC}}}^{2}(I_{1},I_{2})<\mathbf{S}_{{}_{\text{XC}}}^{2}(I_{1},I_{3})$.
This, together with $I_{1}\subset I_{2}\subset I_{3}$, implies that the
formula $\mathbf{S}_{{}_{\text{XC}}}^{2}$ defined by Eq. (11) is not a
similarity measures on IFSs.

##  IV A remark on score functions for IFVs

Zeng et al. [21] introduced the following score value $S_{{}_{CK}}(\\_)$ for
IFV $\alpha=\langle\mu_{\alpha},\nu_{\alpha}\rangle$:

$S_{{}_{CK}}(\alpha)=(\mu_{\alpha}-\nu_{\alpha})-(1-\mu_{\alpha}-\nu_{\alpha})\times\frac{\log_{2}(2-\mu_{\alpha}-\nu_{\alpha})}{100}.$
(12)

Then, they proved the following basic properties for the score function
$S_{{}_{CK}}(\\_)$.

######  Theorem IV.1 ([21, Theorem 3.1])

Assume that $\alpha$ and $\beta$ are two IFVs. If $\alpha\neq\beta$, then
$S_{{}_{CK}}(\alpha)\neq S_{{}_{CK}}(\beta)$.

######  Theorem IV.2 ([21, Theorem 3.2])

Assume that $\alpha$ and $\beta$ are two IFVs. If $\alpha\supset\beta$, then
$S_{{}_{CK}}(\alpha)>S_{{}_{CK}}(\beta)$.

However, the following example shows that Theorem IV.1 does not hold.

###### Example 4

Choose $\alpha=\langle 0,0\rangle$ and
$\beta=\langle\frac{99}{200},\frac{101}{200}\rangle$. By direct calculation,
we have
$S_{{}_{CK}}(\alpha)=(0-0)-(1-0-0)\times\frac{\log_{2}2}{100}=-\frac{1}{100}$
and
$S_{{}_{CK}}(\beta)=(\frac{99}{200}-\frac{101}{200})-(1-\frac{99}{200}-\frac{101}{200})\times\frac{\log_{2}(2-1)}{100}=-\frac{1}{100}.$
This implies that Theorem IV.1 does not hold since $\alpha\neq\beta$.

In fact, we can prove that there is no any continuous function from
$\tilde{\mathbb{I}}$ to $\mathbb{R}$ simultaneously meeting the conditions in
Theorems IV.1 and IV.2, which indicates that the two-dimensional structure of
IFVs is too complex to distinguish all IFVs with only a single monotonous and
continuous function, where the monotonicity is under Atanassov’s order
‘$\subset$’, and the continuity is under the topology of subset of
$\mathbb{R}^{2}$.

###### Theorem IV.3

There is no any continuous function
$f:\tilde{\mathbb{I}}\rightarrow\mathbb{R}$ satisfying the following two
conditions:

  1. (1)

$f$ is injective, i.e., for any $\alpha$, $\beta\in\tilde{\mathbb{I}}$ with
$\alpha\neq\beta$, $f(\alpha)\neq f(\beta)$;

  2. (2)

$f$ is increasing under the partial order $\subset$, i.e., for any $\alpha$,
$\beta\in\tilde{\mathbb{I}}$ with $\alpha\subset\beta$, $f(\alpha)\leq
f(\beta)$.

###### Proof:

Suppose on the contrary that there exists a continuous function
$f:\tilde{\mathbb{I}}\rightarrow\mathbb{R}$ simultaneously satisfying the
conditions (1) and (2).

(i) Let $\varphi(\nu)=f(\langle 0.25,\nu\rangle)$ ($\nu\in[0.25,0.75]$).
Clearly, $\varphi$ is continuous since $f$ is continuous on
$\tilde{\mathbb{I}}$. For any $0.25\leq\nu_{1}\leq\nu_{2}\leq 0.75$, by
$\langle 0.25,\nu_{1}\rangle\supset\langle 0.25,\nu_{2}\rangle$ and condition
(2), one has $\varphi(\nu_{1})=f(\langle 0.25,\nu_{1}\rangle)\geq f(\langle
0.25,\nu_{2}\rangle)=\varphi(\nu_{2})$. This, together with condition (1),
implies that $\varphi(\\_)$ is strictly decreasing on $[0.25,0.75]$. Thus,
$\varphi((0.25,0.75])=[\varphi(0.75),\varphi(0.25))=[f(\langle
0.25,0.75\rangle),f(\langle 0.25,0.25\rangle))$ by the intermediate value
theorem.

(ii) Let $\psi(\nu)=f(\langle\mu,0.25\rangle)$ ($\mu\in[0,0.25]$). Clearly,
$\psi$ is continuous since $f$ is continuous on $\tilde{\mathbb{I}}$. For any
$0\leq\mu_{1}\leq\mu_{2}\leq 0.25$, by
$\langle\mu_{1},0.25\rangle\subset\langle\mu_{2},0.25\rangle$ and condition
(2), one has $\psi(\mu_{1})=f(\langle\mu_{1},0.25\rangle)\leq
f(\langle\mu_{2},0.25\rangle)=\psi(\mu_{2})$. This, together with condition
(1), implies that $\psi(\\_)$ is strictly increasing on $[0,0.25]$. Thus,
$\psi([0,0.25))=[\psi(0),\psi(0.25))=[f(\langle 0,0.25\rangle),f(\langle
0.25,0.25\rangle))$ by the intermediate value theorem.

Summing (i) and (ii), one can easily verify that
$\Lambda=\varphi((0.25,0.75])\cap\psi([0,0.25))=[\max\\{f(\langle
0,0.25\rangle),f(\langle 0.25,0.75\rangle)\\},f(\langle 0.25,0.25\rangle))$ is
a non-degenerate interval, i.e.,

$\max\\{f(\langle 0,0.25\rangle),f(\langle 0.25,0.75\rangle)\\}<f(\langle
0.25,0.25\rangle),$

implying that, for any $\xi\in\Lambda$, there exist $\nu\in(0.25,0.75]$ and
$\mu\in[0,0.25)$ such that $\varphi(\nu)=\xi$ and $\psi(\mu)=\xi$, and thus
there exist $0<\mu<0.25<\nu\leq 0.75$ such that $\varphi(\nu)=f(\langle
0.25,\nu\rangle)=\xi=f(\langle\mu,0.25\rangle)=\psi(\mu)$. This contradicts
with condition (1). ∎

Shen et al. [33] pointed out that many existing distance measures cannot
determine the classification results for some pattern recognition problems
(see [33, Table 2]), i.e., their dual similarity measures cannot distinguish
between some pair of IFVs. To overcome this drawback, they proposed a new
distance measure $d_{{}_{\mathrm{Sh}}}$ as follows: for
$\alpha=\langle\mu_{\alpha},\nu_{\alpha}\rangle$,
$\beta=\langle\mu_{\beta},\nu_{\beta}\rangle\in\tilde{\mathbb{I}}$,

$\displaystyle
d_{{}_{\mathrm{Sh}}}(\alpha,\beta)=\sqrt{\frac{(\tilde{\mu}_{\alpha}-\tilde{\mu}_{\beta})^{2}+(\tilde{\nu}_{\alpha}-\tilde{\nu}_{\beta})^{2}}{2}},$

where
$\tilde{\mu}_{\alpha}=\mu_{\alpha}(1+\frac{2}{3}\pi_{\alpha}(1+\pi_{\alpha}))$,
$\tilde{\nu}_{\alpha}=\nu_{\alpha}(1+\frac{2}{3}\pi_{\alpha}(1+\pi_{\alpha}))$,
$\tilde{\mu}_{\beta}=\mu_{\beta}(1+\frac{2}{3}\pi_{\beta}(1+\pi_{\beta}))$,
and
$\tilde{\nu}_{\beta}=\nu_{\beta}(1+\frac{2}{3}\pi_{\beta}(1+\pi_{\beta}))$.

Fix $\beta\in\tilde{\mathbb{I}}$ and define
$\mathcal{G}(\alpha)=1-d_{{}_{\mathrm{Sh}}}(\alpha,\beta)$ for
$\alpha=\langle\mu_{\alpha},\nu_{\alpha}\rangle\in\tilde{\mathbb{I}}$. From
[33, Theorem 1], it follows that, for $\alpha_{1}$
$\alpha_{2}\in\tilde{\mathbb{I}}$ with $\alpha_{1}\subset\alpha_{2}$, one has
$\mathcal{G}(\alpha_{1})\leq\mathcal{G}(\alpha_{2})$, i.e., the function
$\mathcal{G}$ satisfies the condition (2) of Theorem IV.3. This, together with
Theorem IV.3 and the continuity of $\mathcal{G}$, implies that $\mathcal{G}$
is not injective, and thus there exist two different IFVs $\alpha_{1}$ and
$\alpha_{2}\in\tilde{\mathbb{I}}$ such that
$\mathcal{G}(\alpha_{1})=\mathcal{G}(\alpha_{2})$, implying that
$d_{{}_{\mathrm{Sh}}}(\alpha_{1},\beta)=d_{{}_{\mathrm{Sh}}}(\alpha_{2},\beta)$.
Therefore, the distance measure $d_{{}_{\mathrm{Sh}}}$ has the same drawback
(see Example 5). In fact, by Theorem IV.3, we conclude that there is no any
continuous distance measure that can overcome the above drawback. Therefore,
the comparative analysis in [44, Tables 2–5], [40, Tables 1–2], [39, Tables 1,
2, 5, 6], and [33, Table 2] on the indistinguishability is meaningless.

###### Example 5

Let $\beta=\langle 0,0\rangle$ and $\alpha=\langle
x,y\rangle\in\tilde{\mathbb{I}}$. Then,

$\displaystyle d_{{}_{\mathrm{Sh}}}(\alpha,\beta)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{[1+\frac{2}{3}(1-x-y)(2-x-y)]^{2}\times(x^{2}+y^{2})}{2}}=0.5,$

i.e.,

$\left[1+\frac{2}{3}(1-x-y)(2-x-y)\right]^{2}\times(x^{2}+y^{2})=0.5.$ (13)

Clearly, Eq. (13) has infinitely many solutions. This means that there exist
infinitely many IFVs, whose distances from $\beta$ are all equal to $0.5$.

##  V A monotonous IF TOPSIS method with the linear orders
$\leq_{{}_{\text{XY}}}$ and $\leq_{{}_{\text{ZX}}}$

Suppose that there are $n$ alternatives ${A}_{i}$ ($i=1,2,\ldots,n$) evaluated
with respect to $m$ attributes $\mathscr{O}_{j}$ ($j=1,2,\ldots,m$). The sets
of the alternatives and attributes are denoted by
$A=\\{A_{1},A_{2},\ldots,A_{n}\\}$ and
$\mathscr{O}=\\{\mathscr{O}_{1},\mathscr{O}_{2},\ldots,\mathscr{O}_{m}\\}$,
respectively. The rating (or evaluation) of each alternative $A_{i}\in A$
($i=1,2,\ldots,n$) on each attribute $\mathscr{O}_{j}$ ($j=1,2,\ldots,m$) is
expressed with an IFS
$r_{ij}=\left\\{\frac{\langle\mu_{ij},\nu_{ij}\rangle}{(A_{i},\mathscr{O}_{j})}\right\\}$,
denoted by $r_{ij}=\langle\mu_{ij},\nu_{ij}\rangle$ for short, where
$\mu_{ij}\in[0,1]$ and $\nu_{ij}\in[0,1]$ are respectively the satisfaction
(or membership) degree and dissatisfaction (or non-membership) degree of the
alternative $A_{i}\in A$ on the attribute $\mathscr{O}_{j}$ satisfying the
condition $0\leq\mu_{ij}+\nu_{ij}\leq 1$. A multi-attribute decision-making
(MADM) problem with IFSs is expressed in matrix form shown in Table I.

TABLE I: IF decision matrix $R=(r_{ij})_{n\times m}$

| $\mathscr{O}_{1}$ | $\mathscr{O}_{2}$ | $\ldots$ | $\mathscr{O}_{m}$
---|---|---|---|---
$A_{1}$ | $\langle\mu_{11},\nu_{11}\rangle$ | $\langle\mu_{12},\nu_{12}\rangle$ | $\ldots$ | $\langle\mu_{1m},\nu_{1m}\rangle$
$A_{2}$ | $\langle\mu_{21},\nu_{21}\rangle$ | $\langle\mu_{22},\nu_{22}\rangle$ | $\ldots$ | $\langle\mu_{2m},\nu_{2m}\rangle$
$\vdots$ | $\vdots$ | $\vdots$ | $\ddots$ | $\vdots$
$A_{n}$ | $\langle\mu_{n1},\nu_{n1}\rangle$ | $\langle\mu_{n2},\nu_{n2}\rangle$ | $\ldots$ | $\langle\mu_{nm},\nu_{nm}\rangle$

To follow the common sense, a good method for MADM must guarantee the
monotonicity, i.e., the higher the score of each attribute of the alternative
is, the higher the ranking is. Meanwhile, decision-making results obtained by
this good method must be consistent with our intuitive judgment, when dealing
with the simplest problems, for which the decision making results can be
obtained by direct observation and comparison (see Examples 6 and 7). In
analyzing the TOPSIS method with crisp score values in $[0,1]$, we can find
that the unit interval $[0,1]$ has excellent algebraic and topological
structures as follows: (1) The unit interval $[0,1]$ has a natural linear
order structure $\leq$ and a natural metric structure $|\cdot|$; (2) This
natural metric structure $|\cdot|$ can ensure that the smaller the value in
$[0,1]$ is, the farther away from $1$, and the closer away from $0$; (3) The
order topology induced by the linear order $\leq$ is consistent with the
topology induced by the metric $|\cdot|$. These good structures of $[0,1]$ can
ensure that the TOPSIS method of Hwang and Yoon [5] is monotonous. Recently,
we [6] proved that the space $\tilde{\mathbb{I}}$ of all IFVs with the order
topology induced by the linear order $<_{{}_{\text{XY}}}$, defined in
Definition II.4, is not metrizable, i.e., there is no such good distance for
$\tilde{\mathbb{I}}$ with the linear order $<_{{}_{\text{XY}}}$. Nevertheless,
we still construct an admissible distance with the order
$\leq_{{}_{\text{XY}}}$ in [6]. In the following, we will show that this
distance is very important for our proposed monotonous IF TOPSIS method.

###  V-A Limitation in TOPSIS method of Li [37]

First, we recall a fundamental IF TOPSIS method from [37] and use two examples
show that the TOPSIS method in [37] does not have the basic monotonicity with
the partial order $\subset$, which may lead the decision-making result to be
unreasonable and inconsistent with the actual situation, when dealing with
some even simplest decision-making problems.

The main process of IF TOPSIS method in [37, Section 3.3] is summarized as
follows:

Step 1: Determine the alternatives $A=\\{A_{1},A_{2},\ldots,A_{n}\\}$ and
attributes
$\mathscr{O}=\\{\mathscr{O}_{1},\mathscr{O}_{2},\ldots,\mathscr{O}_{m}\\}$,
respectively;

Step 2: Construct the IF decision matrix
$R=(r_{ij}=\langle\mu_{ij},\nu_{ij}\rangle)_{m\times n}$, as shown in Table I;

Step 3: Determine the weights of the attributes expressed with the IF weight
vector $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{m})^{\top}$, where
$\omega_{j}=\langle\rho_{j},\vartheta_{j}\rangle\in\tilde{\mathbb{I}}$;

Step 4: Compute the weighted IF decision matrix
$\overline{R}=(\overline{r}_{ij}=\omega_{j}\otimes r_{ij})$ by Definition II.2
(v), i.e.,
$\overline{r}_{ij}=\langle\rho_{j}\mu_{ij},\vartheta_{j}+\nu_{ij}-\vartheta_{j}\nu_{ij}\rangle$;

Step 5: Determine the IF positive ideal-point
$\mathbf{A}^{+}=(\langle\mu^{+}_{1},\nu^{+}_{1}\rangle,\langle\mu^{+}_{2},\nu^{+}_{2}\rangle,\ldots,\langle\mu^{+}_{m},\nu^{+}_{m}\rangle)^{\top}$
and the IF negative ideal-point
$\mathbf{A}^{-}=(\langle\mu^{-}_{1},\nu^{-}_{1}\rangle,\langle\mu^{-}_{2},\nu^{-}_{2}\rangle,\ldots,\langle\mu^{-}_{m},\nu^{-}_{m}\rangle)^{\top}$
as follows:

$\displaystyle\mu^{+}_{j}=\max_{1\leq i\leq
n}\\{\bar{\mu}_{ij}\\},\quad\nu^{+}_{j}=\min_{1\leq i\leq
n}\\{\bar{\nu}_{ij}\\},$ $\displaystyle\mu^{-}_{j}=\min_{1\leq i\leq
n}\\{\bar{\mu}_{ij}\\},\quad\nu^{-}_{j}=\max_{1\leq i\leq
n}\\{\bar{\nu}_{ij}\\};$

Step 6: Compute the Euclidean distances
$d_{\mathrm{Eu}}(A_{i},\mathbf{A}^{+})$ and
$d_{\mathrm{Eu}}(A_{i},\mathbf{A}^{-})$ of the alternatives $A_{i}$
($i=1,2,\ldots,n$) from $\mathbf{A}^{+}$ and $\mathbf{A}^{-}$ by using formula
(3);

Step 7: Calculate the relative closeness degrees $\mathscr{C}_{i}$ of the
alternatives $A_{i}$ ($i=1,2,\ldots,n$) to the IF positive ideal-point
$\mathbf{A}^{+}$ by the following formula:

$\mathscr{C}_{i}=\frac{d_{\mathrm{Eu}}(A_{i},\mathbf{A}^{-})}{d_{\mathrm{Eu}}(A_{i},\mathbf{A}^{+})+d_{\mathrm{Eu}}(A_{i},\mathbf{A}^{-})};$

Step 8: Rank the alternatives $A_{i}$ ($i=1,2,\ldots,n$) according to the
nonincreasing order of the relative closeness degrees $\mathscr{C}_{i}$ and
select the most desirable alternative.

###### Example 6

Suppose that there exist $4$ alternatives $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$
evaluated with respect to $2$ benefit attributes $\mathscr{O}_{1}$,
$\mathscr{O}_{2}$. The sets of the alternatives and attributes are denoted by
$\\{A_{1},A_{2},A_{3},A_{4}\\}$ and $\\{\mathscr{O}_{1},\mathscr{O}_{2}\\}$,
respectively. Assume that the IF weight vector of $\mathscr{O}_{1}$ and
$\mathscr{O}_{2}$ is $\omega=(\omega_{1},\omega_{2})^{\top}=(\langle
1,0\rangle,\langle 1,0\rangle)^{\top}$. The IF decision-making matrix is
expressed as shown in Table II.

TABLE II: IF decision matrix $R=(r_{ij})_{4\times 2}$

| $\mathscr{O}_{1}$ | $\mathscr{O}_{2}$
---|---|---
$A_{1}$ | $\langle 0,1\rangle$ | $\langle 0,1\rangle$
$A_{2}$ | $\langle 0.9,0.01\rangle$ | $\langle 0.9,0.01\rangle$
$A_{3}$ | $\langle 0.901,0.007\rangle$ | $\langle 0.901,0.007\rangle$
$A_{4}$ | $\langle 1,0\rangle$ | $\langle 1,0\rangle$

If we use the above TOPSIS method [37, Section 3.3], by direct calculation, it
can be verified that the weighted IF decision matrix is given as shown in
Table III.

TABLE III: Weighted IF decision matrix $\overline{R}=(\omega_{j}\otimes
r_{ij})_{4\times 2}$

| $\mathscr{O}_{1}$ | $\mathscr{O}_{2}$
---|---|---
$A_{1}$ | $\langle 0,1\rangle$ | $\langle 0,1\rangle$
$A_{2}$ | $\langle 0.9,0.01\rangle$ | $\langle 0.9,0.01\rangle$
$A_{3}$ | $\langle 0.901,0.007\rangle$ | $\langle 0.901,0.007\rangle$
$A_{4}$ | $\langle 1,0\rangle$ | $\langle 1,0\rangle$

The IF positive ideal-point $\mathbf{A}^{+}$ and the IF negative ideal-point
$\mathbf{A}^{-}$ are obtained as follows:

$\mathbf{A}^{+}=(\langle 1,0\rangle,\langle 1,0\rangle)\text{ and
}\mathbf{A}^{-}=(\langle 0,1\rangle,\langle 0,1\rangle),$

respectively. According to the Euclidean distance of the alternatives $A_{1}$,
$A_{2}$, $A_{3}$, and $A_{4}$ from $\mathbf{A}^{+}$ and $\mathbf{A}^{-}$
obtained by [37, Eqs. (3.27) and (3.28)], the relative closeness degrees
$\mathscr{C}_{j}$ of the alternatives $A_{1}$, $A_{2}$, $A_{3}$, and $A_{4}$
to the IF positive ideal-point can be calculated as follows:

$\mathscr{C}_{1}=\frac{d_{\mathrm{Eu}}(A_{1},\mathbf{A}^{-})}{d_{\mathrm{Eu}}(A_{1},\mathbf{A}^{+})+d_{\mathrm{Eu}}(A_{1},\mathbf{A}^{-})}=0,$
$\mathscr{C}_{2}=\frac{d_{\mathrm{Eu}}(A_{2},\mathbf{A}^{-})}{d_{\mathrm{Eu}}(A_{2},\mathbf{A}^{+})+d_{\mathrm{Eu}}(A_{2},\mathbf{A}^{-})}=0.9085917,$
$\mathscr{C}_{3}=\frac{d_{\mathrm{Eu}}(A_{3},\mathbf{A}^{-})}{d_{\mathrm{Eu}}(A_{3},\mathbf{A}^{+})+d_{\mathrm{Eu}}(A_{3},\mathbf{A}^{-})}=0.9085194,$

and

$\mathscr{C}_{4}=\frac{d_{\mathrm{Eu}}(A_{4},\mathbf{A}^{-})}{d_{\mathrm{Eu}}(A_{4},\mathbf{A}^{+})+d_{\mathrm{Eu}}(A_{4},\mathbf{A}^{-})}=1,$

respectively. Therefore, the ranking order of $A_{1}$, $A_{2}$, $A_{3}$, and
$A_{4}$ is: $A_{4}\succ A_{2}\succ A_{3}\succ A_{1}$. However, $A_{4}\succ
A_{3}\succ A_{2}\succ A_{1}$ by a direct observation with $\langle
1,0\rangle\supset\langle 0.901,0.007\rangle\supset\langle
0.9,0.01\rangle\supset\langle 0,1\rangle$. This means that the ranking order
obtained by the IF TOPSIS method in [37] is not consistent with the real
situation.

The following example demonstrates that the above IF TOPSIS may yield some
unreasonable decision-making results, even if we restrict the normalized
weight vector $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{m})^{\top}$ in
Step 3 to be positive real numbers, i.e., $\omega_{j}\in(0,1]$ and
$\sum_{j=1}^{m}\omega_{i}=1$.

###### Example 7

Suppose that there exist $4$ alternatives $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$
evaluated with respect to $2$ benefit attributes $\mathscr{O}_{1}$,
$\mathscr{O}_{2}$. The sets of the alternatives and attributes are denoted by
$\\{A_{1},A_{2},A_{3},A_{4}\\}$ and $\\{\mathscr{O}_{1},\mathscr{O}_{2}\\}$,
respectively. Assume that the weight vector of $\mathscr{O}_{1}$ and
$\mathscr{O}_{2}$ is
$\omega=(\omega_{1},\omega_{2})^{\top}=(0.5,0.5)^{\top}$.The IF decision-
making matrix is expressed as shown in Table IV.

TABLE IV: IF decision-making matrix $R=(r_{ij})_{4\times 2}$

| $\mathscr{O}_{1}$ | $\mathscr{O}_{2}$
---|---|---
$A_{1}$ | $\langle 0,1\rangle$ | $\langle 0,1\rangle$
$A_{2}$ | $\langle 0.99,0.0001\rangle$ | $\langle 0.99,0.0001\rangle$
$A_{3}$ | $\langle 0.990199,0.49\times 10^{-4}\rangle$ | $\langle 0.990199,0.49\times 10^{-4}\rangle$
$A_{4}$ | $\langle 1,0\rangle$ | $\langle 1,0\rangle$

By direct calculation, it can be verified that the weighted IF decision matrix
is given as shown in Table V.

TABLE V: Weighted IF decision matrix $\overline{R}=(\omega_{j}\cdot
r_{ij})_{4\times 2}$

| $\mathscr{O}_{1}$ | $\mathscr{O}_{2}$
---|---|---
$A_{1}$ | $\langle 0,1\rangle$ | $\langle 0,1\rangle$
$A_{2}$ | $\langle 0.9,0.01\rangle$ | $\langle 0.9,0.01\rangle$
$A_{3}$ | $\langle 0.901,0.007\rangle$ | $\langle 0.901,0.007\rangle$
$A_{4}$ | $\langle 1,0\rangle$ | $\langle 1,0\rangle$

If we use the TOPSIS method in [37], by Example 6, we know that the ranking
order of $A_{1}$, $A_{2}$, $A_{3}$, and $A_{4}$ is: $A_{4}\succ A_{2}\succ
A_{3}\succ A_{1}$. This is also an unreasonable decision-making result.

###### Remark 2

(1) Examples 6 and 7 show that, for some weight vector with either IFVs or
real numbers, even for some simple decision-making problems, the TOPSIS method
in [37] may lead to some unreasonable decision-making results.

(2) Careful readers can verify that by applying the TOPSIS methods in [45, 48,
47, 50] to Examples 6, the same result can be obtained. This means that the
TOPSIS methods in [45, 48, 47, 50] may produce unreasonable results when
dealing with the simplest decision-making problems.

###  V-B Limitation in TOPSIS method of Chen et al. [52]

The above two examples show that the TOPSIS method in [37] is not monotonous
with Atanassov’s partial order $\subset$. Recently, Chen et al. [52] developed
a monotonous TOPSIS method with the partial order $\subset$ based on a new
similarity measure. However, the following example shows that the TOPSIS
methods in [52] is not monotonous with the linear order
$\leq_{{}_{\text{XY}}}$.

The main process of IF TOPSIS method in [52] is summarized as follows:

Step 1: Determine the alternatives $A=\\{A_{1},A_{2},\ldots,A_{n}\\}$ and
attributes
$\mathscr{O}=\\{\mathscr{O}_{1},\mathscr{O}_{2},\ldots,\mathscr{O}_{m}\\}$,
respectively, and construct the IF decision matrix
$R=(r_{ij}=\langle\mu_{ij},\nu_{ij}\rangle)_{m\times n}$, as shown in Table I;

Step 2: Determine the IF positive ideal-point
$\mathbf{A}^{+}=(\langle\mu^{+}_{1},\nu^{+}_{1}\rangle,\langle\mu^{+}_{2},\nu^{+}_{2}\rangle,\ldots,\langle\mu^{+}_{m},\nu^{+}_{m}\rangle)^{\top}$
and the IF negative ideal-point
$\mathbf{A}^{-}=(\langle\mu^{-}_{1},\nu^{-}_{1}\rangle,\langle\mu^{-}_{2},\nu^{-}_{2}\rangle,\ldots,\langle\mu^{-}_{m},\nu^{-}_{m}\rangle)^{\top}$
as follows:

$\langle\mu^{+}_{j},\nu^{+}_{j}\rangle=\begin{cases}\langle\max\limits_{1\leq
i\leq n}\\{{\mu}_{ij}\\},\min\limits_{1\leq i\leq
n}\\{{\nu}_{ij}\\}\rangle,&\mathscr{O}_{j}\in\mathscr{O}^{+},\\\
\langle\min\limits_{1\leq i\leq n}\\{{\mu}_{ij}\\},\max\limits_{1\leq i\leq
n}\\{{\nu}_{ij}\\}\rangle,&\mathscr{O}_{j}\in\mathscr{O}^{-},\end{cases}$

and

$\langle\mu^{-}_{j},\nu^{-}_{j}\rangle=\begin{cases}\langle\min\limits_{1\leq
i\leq n}\\{{\mu}_{ij}\\},\max\limits_{1\leq i\leq
n}\\{{\nu}_{ij}\\}\rangle,&\mathscr{O}_{j}\in\mathscr{O}^{+},\\\
\langle\max\limits_{1\leq i\leq n}\\{{\mu}_{ij}\\},\min\limits_{1\leq i\leq
n}\\{{\nu}_{ij}\\}\rangle,&\mathscr{O}_{j}\in\mathscr{O}^{-},\end{cases}$

where $\mathscr{O}^{+}$ is the set of benefit attributes and $\mathscr{O}^{-}$
is the set of cost attributes;

Step 3: Compute the degree of indeterminacy
$\pi_{j}^{+}=1-\mu^{+}_{j}-\nu^{+}_{j}$ of the positive ideal-point
$\langle\mu^{+}_{j},\nu^{+}_{j}\rangle$ for each attribute $\mathscr{O}_{j}$
($j=1,2,\ldots,n$);

Step 4: Compute the degree of indeterminacy
$\pi_{j}^{-}=1-\mu^{-}_{j}-\nu^{-}_{j}$ of the negative ideal-point
$\langle\mu^{-}_{j},\nu^{-}_{j}\rangle$ for each attribute $\mathscr{O}_{j}$
($j=1,2,\ldots,n$);

Step 5: Compute the degree of similarity $g_{ij}^{+}$ between the evaluating
IFV $r_{ij}$ of the alternative $A_{i}$ with respect to the attribute
$\mathscr{O}_{j}$ and the positive ideal-point
$\langle\mu^{+}_{j},\nu^{+}_{j}\rangle$ of the attribute $\mathscr{O}_{j}$ to
construct the positive similarity matrix $G^{+}=(g_{ij}^{+})_{m\times n}$,
where
$g_{ij}^{+}=1-\frac{|2(\mu_{j}^{+}-\mu_{ij})-(\nu_{j}^{+}-\nu_{ij})|}{3}\times(1-\frac{\pi_{j}^{+}+\pi_{ij}}{2})-\frac{|2(\nu_{j}^{+}-\nu_{ij})-(\mu_{j}^{+}-\mu_{ij})|}{3}\times\frac{\pi_{j}^{+}+\pi_{ij}}{2};$

Step 6: Compute the degree of similarity $g_{ij}^{-}$ between the evaluating
IFV $r_{ij}$ of the alternative $A_{i}$ with respect to the attribute
$\mathscr{O}_{j}$ and the negative ideal-point
$\langle\mu^{-}_{j},\nu^{-}_{j}\rangle$ of the attribute $\mathscr{O}_{j}$ to
construct the negative similarity matrix $G^{-}=(g_{ij}^{-})_{m\times n}$,
where
$g_{ij}^{-}=1-\frac{|2(\mu_{j}^{-}-\mu_{ij})-(\nu_{j}^{-}-\nu_{ij})|}{3}\times(1-\frac{\pi_{j}^{-}+\pi_{ij}}{2})-\frac{|2(\nu_{j}^{-}-\nu_{ij})-(\mu_{j}^{-}-\mu_{ij})|}{3}\times\frac{\pi_{j}^{-}+\pi_{ij}}{2};$

Step 7: Compute the weighted positive score
$S_{i}^{+}=\sum_{j=1}^{m}\omega_{j}g_{ij}^{+}$ and the weighted negative score
$S_{i}^{-}=\sum_{j=1}^{m}\omega_{j}g_{ij}^{-}$ of each alternative $A_{i}$
($i=1,2,\ldots,n$), where $\omega_{j}$ is the weight of criterion
$\mathscr{O}_{j}$ such that $\omega_{j}\in(0,1]$ and
$\sum_{j=1}^{m}\omega_{j}=1$;

Step 8: Compute the relative degree of closeness
$T(A_{i})=\frac{S_{i}^{+}}{S_{i}^{+}+S_{i}^{-}}$ of each alternative $A_{i}$.
The larger the value of $T(A_{i})$, the better the preference order of
alternative $A_{i}$. Then, rank the alternatives $A_{i}$ ($i=1,2,\ldots,n$)
according to the nonincreasing order of the relative closeness degrees
$T(A_{1})$, $T(A_{2})$, $\ldots$, $T(A_{n})$.

###### Example 8

Suppose that there exist $4$ alternatives $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$
evaluated with respect to $2$ benefit attributes $\mathscr{O}_{1}$,
$\mathscr{O}_{2}$. The sets of the alternatives and attributes are denoted by
$\\{A_{1},A_{2},A_{3},A_{4}\\}$ and $\\{\mathscr{O}_{1},\mathscr{O}_{2}\\}$,
respectively. Assume that the weight vector of $\mathscr{O}_{1}$ and
$\mathscr{O}_{2}$ is $\omega=(0.5,0.5)^{\top}$. The IF decision-making matrix
is expressed as shown in Table VI.

TABLE VI: IF decision matrix $R$

| $\mathscr{O}_{1}$ | $\mathscr{O}_{2}$
---|---|---
$A_{1}$ | $\langle 0,1\rangle$ | $\langle 0,1\rangle$
$A_{2}$ | $\langle 0.3,0\rangle$ | $\langle 0.3,0\rangle$
$A_{3}$ | $\langle 0.64,0.36\rangle$ | $\langle 0.64,0.36\rangle$
$A_{4}$ | $\langle 1,0\rangle$ | $\langle 1,0\rangle$

The IF positive ideal-point $\mathbf{A}^{+}$ and the IF negative ideal-point
$\mathbf{A}^{-}$ are obtained as follows: $\mathbf{A}^{+}=(\langle
1,0\rangle,\langle 1,0\rangle)$ and $\mathbf{A}^{-}=(\langle
0,1\rangle,\langle 0,1\rangle),$ respectively. By using the TOPSIS method in
[52], we obtain the positive similarity matrix $G^{+}$ and the negative
similarity matrix $G^{-}$ as follows:

$G^{+}=(g_{ij}^{+})_{4\times 2}=\begin{bmatrix}0&0\\\ 0.615&0.615\\\
0.64&0.64\\\ 1&1\\\ \end{bmatrix},$

and

$G^{-}=(g_{ij}^{-})_{4\times 2}=\begin{bmatrix}1&1\\\ 0.385&0.385\\\
0.36&0.36\\\ 0&0\\\ \end{bmatrix}.$

Then, the weighted positive scores
$S^{+}_{i}=\omega_{1}g_{i1}^{+}+\omega_{2}g_{i2}^{+}$ ($i=1,2,3,4$) and the
weighted negative scores
${S_{i}^{-}}=\omega_{1}g_{i1}^{-}+\omega_{2}g_{i2}^{-}$ ($i=1,2,3,4$) of the
alternatives $A_{1}$, $A_{2}$, $A_{3}$, and $A_{4}$ can be calculated as
follows:

$S_{1}^{+}=0,\ S_{2}^{+}=0.615,\ S_{3}^{+}=0.64,\ S_{4}^{+}=1,$

and

$S_{1}^{-}=1,\ S_{2}^{+}=0.385,\ S_{3}^{+}=0.36,\ S_{4}^{-}=0.$

Therefore, the relative degree of closeness
$T(A_{i})=\frac{S_{i}^{+}}{S_{i}^{+}+S_{i}^{-}}$ ($i=1,2,3,4$) of the
alternatives $A_{1}$, $A_{2}$, $A_{3}$, and $A_{4}$ are given as follows:

$T(A_{1})=0,\ T(A_{2})=0.615,\ T(A_{3})=0.64,\ T(A_{4})=1,$

and thus the ranking order of $A_{1}$, $A_{2}$, $A_{3}$, and $A_{4}$ is:
$A_{4}\succ A_{3}\succ A_{2}\succ A_{1}$. However, it can be verified that
$A_{4}\succ A_{2}\succ A_{3}\succ A_{1}$ by a direct observation with $\langle
1,0\rangle\geq_{{}_{\text{XY}}}\langle
0.3,0\rangle\geq_{{}_{\text{XY}}}\langle
0.64,0.36\rangle\geq_{{}_{\text{XY}}}\langle 0,1\rangle$.

Summing up Examples 6–8, an interesting question is whether there exists an IF
TOPSIS method that is monotonous with the linear order $\leq_{{}_{\text{XY}}}$
or $\leq_{{}_{\text{ZX}}}$? In the following section, we will establish an IF
TOPSIS method that is monotonous with the linear order $\leq_{{}_{\text{XY}}}$
or $\leq_{{}_{\text{ZX}}}$.

##  VI A monotonous IF TOPSIS method

The fundamental cause for counterintuitive decision-making results in Examples
6–8 lies in the structure of metrics for IFVs. In [6], we defined a metric
$\varrho$ in $\tilde{\mathbb{I}}$ as follows: for $\alpha$,
$\beta\in\tilde{\mathbb{I}}$,

$\varrho(\alpha,\beta)=\begin{cases}\frac{1}{3}(1+|s(\alpha)-s(\beta)|),&s(\alpha)\neq
s(\beta),\\\
\frac{1}{3}(|h(\alpha)-h(\beta)|),&s(\alpha)=s(\beta),\end{cases}$

where $s(\alpha)$ and $h(\alpha)$ are the score degree and the accuracy degree
of $\alpha$, respectively. Furthermore, we [6] proved the following basic
properties of $\varrho$.

######  Theorem VI.1 ([6])

  1. (1)

$\varrho(\alpha,\beta)\in[0,1]$ and $\varrho(\alpha,\beta)=0$ if and only if
$\alpha=\beta$.

  2. (2)

$\varrho(\alpha,\beta)=1$ if and only if ($\alpha=\langle 0,1\rangle$ and
$\beta=\langle 1,0\rangle$) or ($\alpha=\langle 1,0\rangle$ and $\beta=\langle
0,1\rangle$).

  3. (3)

$\varrho(\alpha,\beta)=\varrho(\beta,\alpha)$.

  4. (4)

For any $\alpha$, $\beta$, $\gamma\in\tilde{\mathbb{I}}$,
$\varrho(\alpha,\beta)+\varrho(\beta,\gamma)\geq\varrho(\alpha,\gamma)$.

  5. (5)

For any $\alpha$, $\beta$, $\gamma\in\tilde{\mathbb{I}}$, if
$\alpha\leq_{{}_{\text{XY}}}\beta\leq_{{}_{\text{XY}}}\gamma$, then
$\varrho(\alpha,\beta)\leq\varrho(\alpha,\gamma)$ and
$\varrho(\beta,\gamma)\leq\varrho(\alpha,\gamma)$.

Based on the similarity function $L(\alpha)$, similarly to the metric
$\varrho$, define the parametric metrics $\varrho^{(\lambda)}$ and
$\tilde{\varrho}^{(\lambda)}$ in $\tilde{\mathbb{I}}$ as follows: for
$\alpha$, $\beta\in\tilde{\mathbb{I}}$,

$\begin{split}&\varrho^{(\lambda)}(\alpha,\beta)\\\
=&\begin{cases}\frac{1}{1+2\lambda}(1+\lambda\cdot|s(\alpha)-s(\beta)|),&s(\alpha)\neq
s(\beta),\\\
\frac{1}{1+2\lambda}(|h(\alpha)-h(\beta)|),&s(\alpha)=s(\beta),\end{cases}\end{split}$
(14)

and

$\begin{split}&\tilde{\varrho}^{(\lambda)}(\alpha,\beta)\\\
=&\begin{cases}\frac{1}{1+\lambda}(1+\lambda\cdot|L(\alpha)-L(\beta)|),&L(\alpha)\neq
L(\beta),\\\
\frac{1}{1+\lambda}(|h(\alpha)-h(\beta)|),&L(\alpha)=L(\beta),\end{cases}\end{split}$
(15)

where $\lambda\geq 1$ is a parameter, and $L(\alpha)$ and $h(\alpha)$ are the
similarity function and the accuracy degree of $\alpha$, respectively.

Let $A$ and $B$ be two aggregation functions satisfying the condition in
Proposition II.1. Based on Proposition II.1, define another parametric metric
$\varrho_{{}_{A,B}}^{(\lambda)}$ in $\tilde{\mathbb{I}}$ as follows: for
$\alpha$, $\beta\in\tilde{\mathbb{I}}$,

$\begin{split}&\varrho_{{}_{A,B}}^{(\lambda)}(\alpha,\beta)\\\
=&\begin{cases}\frac{1}{2}(1+|\overline{A}(\alpha)-\overline{A}(\beta)|),&\overline{A}(\alpha)\neq\overline{A}(\beta),\\\
\frac{1}{2}(|\overline{B}(\alpha)-\overline{B}(\beta)|),&\overline{A}(\alpha)=\overline{A}(\beta),\end{cases}\end{split}$
(16)

where $\lambda\geq 1$ is a parameter,
$\overline{A}(\alpha)=A(\mu_{\alpha},1-\nu_{\alpha})$, and
$\overline{B}(\alpha)=B(\mu_{\alpha},1-\nu_{\alpha})$. In particular, by
taking $A=K_{\gamma_{1}}$ and $B=K_{\gamma_{2}}$ with
$\gamma_{1}\neq\gamma_{2}$ and direct calculation, we have

$\begin{split}&\varrho_{{}_{K_{\gamma_{1}},K_{\gamma_{2}}}}^{(\lambda)}(\alpha,\beta)\\\
=&\begin{cases}\frac{1}{1+\lambda}(1+\lambda\cdot
E_{\gamma_{1}}(\alpha,\beta)),&E_{\gamma_{1}}(\alpha,\beta)\neq 0,\\\
\frac{1}{1+\lambda}E_{\gamma_{2}}(\alpha,\beta),&E_{\gamma_{1}}(\alpha,\beta)=0,\end{cases}\end{split}$
(17)

where
$E_{\gamma}(\alpha,\beta)=|\overline{K}_{\gamma}(\alpha)-\overline{K}_{\gamma}(\beta)|=|(1-\gamma)(\mu_{\alpha}-\mu_{\beta})-\gamma(\nu_{\alpha}-\nu_{\beta})|$.

###### Remark 3

The parameter $\gamma_{1}$ in Eq. (17) can be regarded as the preference for
decision-makers to choose the membership and non-membership:

(1) If $\gamma_{1}>0.5$, then the decision-makers prefer non-membership to
membership, i.e., the decision-makers are pessimistic.

(2) If $\gamma_{1}<0.5$, then the decision-makers prefer membership to non-
membership, i.e., the decision-makers are optimistic.

(3) If $\gamma_{1}=0.5$, then the decision-makers have no preference for
membership and non-membership, i.e., the decision-makers are neutral.

Similarly to the proof of Theorem VI.1 in [6], we can prove that the metrics
$\varrho^{(\lambda)}$, $\tilde{\varrho}^{(\lambda)}$, and
$\varrho_{{}_{A,B}}^{(\lambda)}$ have the following basic properties.

###### Theorem VI.2

Let $\lambda\geq 1$ and
$\rho\in\\{\varrho^{(\lambda)},\tilde{\varrho}^{(\lambda)},\varrho_{{}_{A,B}}^{(\lambda)}\\}$.
Then,

  1. (1)

$\rho(\alpha,\beta)\in[0,1]$ and $\rho(\alpha,\beta)=0$ if and only if
$\alpha=\beta$.

  2. (2)

$\rho(\alpha,\beta)=1$ if and only if ($\alpha=\langle 0,1\rangle$ and
$\beta=\langle 1,0\rangle$) or ($\alpha=\langle 1,0\rangle$ and $\beta=\langle
0,1\rangle$).

  3. (3)

$\rho(\alpha,\beta)=\varrho(\beta,\alpha)$.

  4. (4)

For any $\alpha$, $\beta$, $\gamma\in\tilde{\mathbb{I}}$,
$\rho(\alpha,\beta)+\rho(\beta,\gamma)\geq\rho(\alpha,\gamma)$.

  5. (5)

For any $\alpha$, $\beta$, $\gamma\in\tilde{\mathbb{I}}$, if
$\alpha\leq\beta\leq\gamma$, then $\rho(\alpha,\beta)\leq\rho(\alpha,\gamma)$
and $\rho(\beta,\gamma)\leq\rho(\alpha,\gamma)$.

For the MADM problem with IFSs, by using the three metrics defined by Eqs.
(14)–(16), we propose a new IF TOPSIS method as follows:

Step 1: (Construct the decision matrix) Supposing that the decision-maker gave
the rating (or evaluation) of each alternative $A_{i}\in A$ ($i=1,2,\ldots,n$)
on each attribute $\mathscr{O}_{j}$ ($j=1,2,\ldots,m$) in the form of IFNs
$r_{ij}=\langle\mu_{ij},\nu_{ij}\rangle$, construct an IF decision matrix
$R=(r_{ij})_{n\times m}$ as shown in Table I.

Step 2: (Normalize the decision matrix) Transform the IF decision matrix
$R=(r_{ij})_{n\times m}$ to the normalized IF decision matrix
$\overline{R}=(\bar{r}_{ij})_{n\times
m}=(\langle\bar{\mu}_{ij},\bar{\nu}_{ij}\rangle)_{n\times m}$ as follows:

$\bar{r}_{ij}=\begin{cases}r_{ij},&\text{for benefit attribute
}\mathscr{O}_{j},\\\ r_{ij}^{\complement},&\text{for cost attribute
}\mathscr{O}_{j},\end{cases}$

where $r_{ij}^{\complement}$ is the complement of $r_{ij}$.

Step 3: (Determine the positive and negative ideal-points) Determine the IF
positive ideal-point
$\mathbf{A}^{+}=(\langle\mu^{+}_{1},\nu^{+}_{1}\rangle,\langle\mu^{+}_{2},\nu^{+}_{2}\rangle,\ldots,\langle\mu^{+}_{m},\nu^{+}_{m}\rangle)^{\top}$
and IF negative ideal-point
$\mathbf{A}^{-}=(\langle\mu^{-}_{1},\nu^{-}_{1}\rangle,\langle\mu^{-}_{2},\nu^{-}_{2}\rangle,\ldots,\langle\mu^{-}_{m},\nu^{-}_{m}\rangle)^{\top}$
as follows:

$\displaystyle\mu^{+}_{j}=\max_{1\leq i\leq
n}\\{\bar{\mu}_{ij}\\},\quad\nu^{+}_{j}=\min_{1\leq i\leq
n}\\{\bar{\nu}_{ij}\\},$ $\displaystyle\mu^{-}_{j}=\min_{1\leq i\leq
n}\\{\bar{\mu}_{ij}\\},\quad\nu^{-}_{j}=\max_{1\leq i\leq
n}\\{\bar{\nu}_{ij}\\}.$

Step 4: (Compute the weighted similarity measures) Choose $\lambda\geq 1$ and
compute the weighted similarity measures between the alternatives $A_{i}$
($i=1,2,\ldots,n$) and the IF positive ideal-point $\mathbf{A}^{+}$ and
between the alternatives $A_{i}$ ($i=1,2,\ldots,n$) and the IF negative ideal-
point $\mathbf{A}^{-}$ by using the following formulas:

$\displaystyle\mathbf{S}(A_{i},\mathbf{A}^{+})=1-\sum_{j=1}^{m}\omega_{j}\cdot\varrho^{(\lambda)}(\langle\bar{\mu}_{ij},\bar{\nu}_{ij}\rangle,\langle\mu^{+}_{j},\nu^{+}_{j}\rangle),$
(18) $\displaystyle(\text{resp., }$
$\displaystyle\mathbf{S}(A_{i},\mathbf{A}^{+})=1-\sum_{j=1}^{m}\omega_{j}\cdot\tilde{\varrho}^{(\lambda)}(\langle\bar{\mu}_{ij},\bar{\nu}_{ij}\rangle,\langle\mu^{+}_{j},\nu^{+}_{j}\rangle),$
(19)
$\displaystyle\mathbf{S}(A_{i},\mathbf{A}^{+})=1-\sum_{j=1}^{m}\omega_{j}\cdot\varrho_{{}_{A,B}}^{(\lambda)}(\langle\bar{\mu}_{ij},\bar{\nu}_{ij}\rangle,\langle\mu^{+}_{j},\nu^{+}_{j}\rangle)),$
(20)

and

$\displaystyle\mathbf{S}(A_{i},\mathbf{A}^{-})=1-\sum_{j=1}^{m}\omega_{j}\cdot\varrho^{(\lambda)}(\langle\bar{\mu}_{ij},\bar{\nu}_{ij}\rangle,\langle\mu^{-}_{j},\nu^{-}_{j}\rangle),$
(21) $\displaystyle(\text{resp., }$
$\displaystyle\mathbf{S}(A_{i},\mathbf{A}^{-})=1-\sum_{j=1}^{m}\omega_{j}\cdot\tilde{\varrho}^{(\lambda)}(\langle\bar{\mu}_{ij},\bar{\nu}_{ij}\rangle,\langle\mu^{-}_{j},\nu^{-}_{j}\rangle),$
(22)
$\displaystyle\mathbf{S}(A_{i},\mathbf{A}^{-})=1-\sum_{j=1}^{m}\omega_{j}\cdot\varrho_{{}_{A,B}}^{(\lambda)}(\langle\bar{\mu}_{ij},\bar{\nu}_{ij}\rangle,\langle\mu^{-}_{j},\nu^{-}_{j}\rangle)).$
(23)

By Theorems VI.1 and VI.2, it is easy to see that the similarity measures
obtained by Eqs. (18)–(20) are admissible similarity measures with the orders
$\leq_{{}_{\text{XY}}}$, $\leq_{{}_{\text{ZX}}}$, and $\leq_{{}_{A,B}}$,
respectively.

Step 5: (Compute the relative closeness degrees) Calculate the relative
closeness degrees $\mathscr{C}_{i}$ of the alternatives $A_{i}$
($i=1,2,\ldots,n$) to the IF positive ideal-point $\mathbf{A}^{+}$ by using
the following formula:

$\mathscr{C}_{i}=\frac{\mathbf{S}(A_{i},\mathbf{A}^{+})}{\mathbf{S}(A_{i},\mathbf{A}^{+})+\mathbf{S}(A_{i},\mathbf{A}^{-})}.$
(24)

Step 6: (Rank the alternatives) Rank the alternatives $A_{i}$
($i=1,2,\ldots,n$) according to the nonincreasing order of the relative
closeness degrees $\mathscr{C}_{i}$ and select the most desirable alternative.

###### Remark 4

(1) By Theorems VI.1 and VI.2, it is easy to see that
$\mathbf{S}(A_{i},\mathbf{A}^{+})+\mathbf{S}(A_{i},\mathbf{A}^{-})$ in Eq.
(24) is always nonzero. This overcomes the limitation that many TOPSIS method
may lead to the situation that the denominator is equal to $0$ when computing
the relative closeness degrees.

(2) For practical MADM problems, in order to eliminate the effect of the
constant term $1$ in formulas (14)–(17) as much as possible, the parameter
$\lambda$ should be chosen as large as possible.

######  Theorem VI.3 (Monotonicity)

Using Eqs. (18) and (21), the above proposed method is increasing with the
linear order $\leq_{{}_{\text{XY}}}$, i.e., for the MADM problem expressed in
Table I, if there exist $1\leq i_{1},i_{2}\leq n$ such that
$\bar{r}_{i_{1}j}\leq_{{}_{\text{XY}}}\bar{r}_{i_{2}j}$ holds for all $1\leq
j\leq m$, then $\mathscr{C}_{i_{1}}\leq\mathscr{C}_{i_{2}}$, i.e., $A_{i_{2}}$
is better than $A_{i_{1}}$ ranked by the proposed method. In particular, the
proposed method is increasing with Atanassov’s order ‘$\subset$’.

###### Proof:

Fix $\lambda\geq 1$. Let
$\mathbf{A}^{+}=(\langle\mu^{+}_{1},\nu^{+}_{1}\rangle,\langle\mu^{+}_{2},\nu^{+}_{2}\rangle,\ldots,$
$\langle\mu^{+}_{m},\nu^{+}_{m}\rangle)^{\top}$ and
$\mathbf{A}^{-}=(\langle\mu^{-}_{1},\nu^{-}_{1}\rangle,\langle\mu^{-}_{2},\nu^{-}_{2}\rangle,\ldots,\langle\mu^{-}_{m},\nu^{-}_{m}\rangle)^{\top}$
be the IF positive ideal-point and the IF negative ideal-point obtained by
Step 3, respectively. Clearly,
$\langle\mu^{-}_{j},\nu^{-}_{j}\rangle\leq_{{}_{\text{XY}}}\bar{r}_{i_{1}j}\leq_{{}_{\text{XY}}}\bar{r}_{i_{2}j}\leq_{{}_{\text{XY}}}\langle\mu^{+}_{j},\nu^{+}_{j}\rangle$.
By Theorem VI.2, we have

$\varrho^{(\lambda)}(\langle\mu^{-}_{j},\nu^{-}_{j}\rangle,\bar{r}_{i_{1}j})\leq\varrho^{(\lambda)}(\langle\mu^{-}_{j},\nu^{-}_{j}\rangle,\bar{r}_{i_{2}j}),$

and

$\varrho^{(\lambda)}(\langle\mu^{+}_{j},\nu^{+}_{j}\rangle,\bar{r}_{i_{2}j})\leq\varrho^{(\lambda)}(\langle\mu^{+}_{j},\nu^{+}_{j}\rangle,\bar{r}_{i_{1}j}),$

and thus,

$\mathbf{S}(A_{i_{1}},\mathbf{A}^{-})\geq\mathbf{S}(A_{i_{2}},\mathbf{A}^{-}),$

and

$\mathbf{S}(A_{i_{1}},\mathbf{A}^{+})\leq\mathbf{S}(A_{i_{2}},\mathbf{A}^{+})\text{
by Eqs.~{}\eqref{Eq-1} and \eqref{Eq-2}}.$

This, together with Eq. (24), implies that

(1) if $\mathbf{S}(A_{i_{1}},\mathbf{A}^{+})=0$, then

$\mathscr{C}_{i_{1}}=\frac{\mathbf{S}(A_{i_{1}},\mathbf{A}^{+})}{\mathbf{S}(A_{i_{1}},\mathbf{A}^{+})+\mathbf{S}(A_{i_{1}},\mathbf{A}^{-})}=0\leq\mathscr{C}_{i_{2}};$

(2) if $\mathbf{S}(A_{i_{1}},\mathbf{A}^{+})>0$, then
$\mathbf{S}(A_{i_{2}},\mathbf{A}^{+})\geq\mathbf{S}(A_{i_{1}},\mathbf{A}^{+})>0$,
and thus

$\displaystyle\mathscr{C}_{i_{1}}=$
$\displaystyle\frac{\mathbf{S}(A_{i_{1}},\mathbf{A}^{+})}{\mathbf{S}(A_{i_{1}},\mathbf{A}^{+})+\mathbf{S}(A_{i_{1}},\mathbf{A}^{-})}=\frac{1}{1+\frac{\mathbf{S}(A_{i_{1}},\mathbf{A}^{-})}{\mathbf{S}(A_{i_{1}},\mathbf{A}^{+})}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{1+\frac{\mathbf{S}(A_{i_{2}},\mathbf{A}^{-})}{\mathbf{S}(A_{i_{2}},\mathbf{A}^{+})}}=\frac{\mathbf{S}(A_{i_{2}},\mathbf{A}^{+})}{\mathbf{S}(A_{i_{2}},\mathbf{A}^{+})+\mathbf{S}(A_{i_{2}},\mathbf{A}^{-})}=\mathscr{C}_{i_{2}}.$

Therefore, $\mathscr{C}_{i_{1}}\leq\mathscr{C}_{i_{2}}$. ∎

By Theorem VI.2, similarly to the proof of Theorem VI.3, it is not difficult
to check that the following result holds.

######  Theorem VI.4 (Monotonicity)

Using Eqs. (19) and (22) (resp., Eqs. (20) and (23)), the above proposed
method is increasing with the linear order $\leq_{{}_{\text{ZX}}}$ (resp.,
$\leq_{{}_{A,B}}$).

######  Example 9 (Continuation of Example 7)

Consider the MADM problem described in Example 7. If the proposed TOPSIS
method in this section is used based on Eq. (14) with $\lambda=1$, by direct
calculation, it can be verified that $\mathscr{C}_{1}=0$,
$\mathscr{C}_{2}=0.99495$, $\mathscr{C}_{3}=0.995075$, $\mathscr{C}_{4}=1$.
Thus, the ranking order of the alternatives $A_{1}$, $A_{2}$, $A_{3}$, and
$A_{4}$ is: $A_{4}\succ A_{3}\succ A_{2}\succ A_{1}$, which is consistent with
the result obtained by directly observing in Example 7.

######  Example 10 (Continuation of Example 8)

Consider the MADM problem described in Example 8. If the proposed TOPSIS
method in this section is used based on Eq. (14) with $\lambda=1$, by direct
calculation, it can be verified that $\mathscr{C}_{1}=0$,
$\mathscr{C}_{2}=0.65$, $\mathscr{C}_{3}=0.64$, $\mathscr{C}_{4}=1$. Thus, the
ranking order of the alternatives $A_{1}$, $A_{2}$, $A_{3}$, and $A_{4}$ is:
$A_{4}\succ A_{2}\succ A_{3}\succ A_{1}$, which is consistent with the result
obtained by directly observing in Example 8.

###### Remark 5

Observing from Examples 9 and 10, it can be seen that the proposed TOPSIS
method can effectively overcome the limitations of the TOPSIS methods in [37,
52], which is consistent with the result proved in Theorem VI.3. Furthermore,
this shows that the proposed TOPSIS method is superior to those in [37, 52].

##  VII Illustrative examples

This section provides two practical examples to illustrate the efficiency of
the above proposed TOPSIS method. One is an IF MADM problem on the choice of
suppliers in the supply chain management (see Example 11). The ranking order
obtained by the proposed TOPSIS method is slightly different from the results
obtained by those TOPSIS methods in [46, 49, 51, 33]. However, the most
desirable alternatives are consistent. The other is an IF MADM problem on the
choice of project managers (see Example 12). The ranking order, obtained by
the proposed TOPSIS method under the case that the decision-maker is neutral
or pessimistic, is consistent with those results obtained by the TOPSIS
methods in [52, 66, 46, 49, 51].

######  Example 11 ([52, Example 5.1])

Assume that there are five alternatives $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$,
and $A_{5}$ of suppliers and four attributes $\mathscr{O}_{1}$,
$\mathscr{O}_{2}$, $\mathscr{O}_{3}$, and $\mathscr{O}_{4}$ to assess these
five alternatives, so as to choose the best supplier among these five
alternatives in the supply chain management, where $\mathscr{O}_{1}$ is the
“Product Quality”, $\mathscr{O}_{2}$ is the “Service”, $\mathscr{O}_{3}$ is
the “Delivery”, $\mathscr{O}_{4}$ is the “Sustainability” and
$\mathscr{O}_{1}$, $\mathscr{O}_{2}$, $\mathscr{O}_{3}$, and $\mathscr{O}_{4}$
are benefit attributes, with weight vector
$\omega=(0.25,0.4,0.2,0.15)^{\top}$.

Step 1: (Construct the decision matrix) The decision matrix
$R=(r_{ij})_{5\times 4}$ given by the decision maker is listed in Table VII.

TABLE VII: The decision matrix $R$

| $\mathscr{O}_{1}$ | $\mathscr{O}_{2}$ | $\mathscr{O}_{3}$ | $\mathscr{O}_{4}$
---|---|---|---|---
$A_{1}$ | $\langle 0.6,0.3\rangle$ | $\langle 0.5,0.2\rangle$ | $\langle 0.2,0.5\rangle$ | $\langle 0.1,0.6\rangle$
$A_{2}$ | $\langle 0.8,0.2\rangle$ | $\langle 0.8,0.1\rangle$ | $\langle 0.6,0.1\rangle$ | $\langle 0.3,0.4\rangle$
$A_{3}$ | $\langle 0.6,0.3\rangle$ | $\langle 0.4,0.3\rangle$ | $\langle 0.4,0.2\rangle$ | $\langle 0.5,0.2\rangle$
$A_{4}$ | $\langle 0.9,0.1\rangle$ | $\langle 0.5,0.2\rangle$ | $\langle 0.2,0.3\rangle$ | $\langle 0.1,0.5\rangle$
$A_{5}$ | $\langle 0.7,0.1\rangle$ | $\langle 0.3,0.2\rangle$ | $\langle 0.6,0.2\rangle$ | $\langle 0.4,0.2\rangle$

Step 2: (Normalize the decision matrix) Since $\mathscr{O}_{1}$,
$\mathscr{O}_{2}$, $\mathscr{O}_{3}$, and $\mathscr{O}_{4}$ are all benefit
attributes, we have $\overline{R}=(\bar{r}_{ij})_{5\times 4}=R$.

Step 3: (Determine the positive and negative ideal-points) The IF positive
ideal-point is

$\mathbf{A}^{+}=(\langle 0.9,0.1\rangle,\langle 0.8,0.1\rangle,\langle
0.6,0.1\rangle,\langle 0.5,0.2\rangle)^{\top},$

and IF negative ideal-point is

$\mathbf{A}^{-}=(\langle 0.6,0.3\rangle,\langle 0.3,0.3\rangle,\langle
0.2,0.5\rangle,\langle 0.1,0.6\rangle)^{\top}.$

Steps 4 and 5: (Compute the relative closeness degrees) Choose $\lambda=100$
and calculate the relative closeness degrees $\mathscr{C}_{i}$ of the
alternatives $A_{i}$ ($i=1,2,3,4,5$) to the IF positive ideal-point
$\mathbf{A}^{+}$ by Eqs. (18), (21), and (24): $\mathscr{C}_{1}=0.4321$,
$\mathscr{C}_{2}=0.5709$, $\mathscr{C}_{3}=0.4750$, $\mathscr{C}_{4}=0.4876$,
$\mathscr{C}_{5}=0.5053.$

Step 6: (Rank the alternative) Because
$\mathscr{C}_{2}>\mathscr{C}_{5}>\mathscr{C}_{4}>\mathscr{C}_{3}>\mathscr{C}_{1}$,
the ranking order of the alternatives $A_{i}$ ($i=1,2,3,4,5$) is: $A_{2}\succ
A_{5}\succ A_{4}\succ A_{3}\succ A_{1}.$

Repeating Steps 1–3, by applying Eqs. (19) and (22), we obtain the following
result:

Step 4 and 5: (Compute the relative closeness degrees) Choose $\lambda=100$
and calculate the relative closeness degrees $\mathscr{C}_{i}$ of the
alternatives $A_{i}$ ($i=1,2,3,4,5$) to the IF positive ideal-point
$\mathbf{A}^{+}$ by Eqs. (19), (22), and (24): $\mathscr{C}_{1}=0.4371$,
$\mathscr{C}_{2}=0.5618$, $\mathscr{C}_{3}=0.4694$, $\mathscr{C}_{4}=0.4922$,
$\mathscr{C}_{5}=0.4925.$

Step 6: (Rank the alternatives) Because
$\mathscr{C}_{2}>\mathscr{C}_{5}>\mathscr{C}_{4}>\mathscr{C}_{3}>\mathscr{C}_{1}$,
the ranking order of the alternatives $A_{i}$ ($i=1,2,3,4,5$) is: $A_{2}\succ
A_{5}\succ A_{4}\succ A_{3}\succ A_{1}.$

### Comparative analysis

From Table VIII, which shows a comparison of the ranking orders of the
alternatives in Example 11 for different MADM methods, it can be observed that
(1) our results are exactly the same, which are consistent with the result
$A_{2}\succ A_{5}\succ A_{4}\succ A_{3}\succ A_{1}$ in [52, 66, 21]; (2) the
result obtained by Xu’s IFWA operator in [4] is different from the results
obtained by all other methods; (3) since the TOPSIS method of Büyüközkan and
Güleryüz in [49] is based on the normalized Euclidean distance defined by Eq.
(3), which does not satisfy the axiomatic definition of IF distance measure
(see Example 1), the ranking result $A_{2}\succ A_{5}\succ A_{3}\succ
A_{4}\succ A_{1}$ may be unreasonable; (4) the best choice is always $A_{2}$.

TABLE VIII: A comparison of the ranking orders of the alternatives in Example
11 for different MADM methods

Methods | Ranking orders
---|---
Chen et al.’s TOPSIS method in [52] | $A_{2}\succ A_{5}\succ A_{4}\succ A_{3}\succ A_{1}$
Wang and Wei’s TOPSIS method in [66] | $A_{2}\succ A_{5}\succ A_{4}\succ A_{3}\succ A_{1}$
Altan Koyuncu et al.’s TOPSIS method in [46] | $A_{2}\succ A_{5}\succ A_{3}\succ A_{4}\succ A_{1}$
Büyüközkan and Güleryüz’s TOPSIS method in [49] | $A_{2}\succ A_{5}\succ A_{3}\succ A_{4}\succ A_{1}$
Zhang et al.’s TOPSIS method in [51] | $A_{2}\succ A_{5}\succ A_{3}\succ A_{4}\succ A_{1}$
Zeng et al.’s VIKOR method in [21] | $A_{2}\succ A_{5}\succ A_{4}\succ A_{3}\succ A_{1}$
Shen et al.’s TOPSIS method in [33] | $A_{2}\succ A_{5}\succ A_{3}\succ A_{4}\succ A_{1}$
Xu’s IFWA operator method in [4] | $A_{2}\succ A_{4}\succ A_{5}\succ A_{3}\succ A_{1}$
Our TOPSIS based on $\varrho^{(100)}$ | $A_{2}\succ A_{5}\succ A_{4}\succ A_{3}\succ A_{1}$
Our TOPSIS based on $\tilde{\varrho}^{(100)}$ | $A_{2}\succ A_{5}\succ A_{4}\succ A_{3}\succ A_{1}$
Our TOPSIS based on $\varrho_{{}_{K_{0.2},K_{0.4}}}^{(100)}$ | $A_{2}\succ A_{5}\succ A_{4}\succ A_{3}\succ A_{1}$
Our TOPSIS based on $\varrho_{{}_{K_{0.5},K_{0.4}}}^{(100)}$ | $A_{2}\succ A_{5}\succ A_{4}\succ A_{3}\succ A_{1}$
Our TOPSIS based on $\varrho_{{}_{K_{0.6},K_{0.4}}}^{(100)}$ | $A_{2}\succ A_{5}\succ A_{4}\succ A_{3}\succ A_{1}$

To illustrate the detailed influence of the parameters $\lambda$ and
$\gamma_{1}$ on the decision-making results in Example 11 by using metrics
$\varrho^{(\lambda)}$, $\tilde{\varrho}^{(\lambda)}$, and
$\varrho_{{}_{K_{\gamma_{1}},K_{\gamma_{2}}}}^{(\lambda)}$, the relative
closeness degrees $\mathscr{C}_{i}$ of each alternative $A_{i}$ obtained by
$\varrho^{(\lambda)}$, $\tilde{\varrho}^{(\lambda)}$, and
$\varrho_{{}_{K_{\gamma_{1}},K_{\gamma_{2}}}}^{(\lambda)}$ are shown in Figs.
1 (a), (b), and (c)–(d), respectively. As can be seen from Fig. 1, the ranking
orders of the alternatives using different values of parameters $\lambda$ and
$\gamma_{1}$ remain the same and are stabilized , when the parameter $\lambda$
is large enough, and thus the preferences of decision makers do not affect the
ranking results in this example. This indicates that our method is effective
and stable.

(a) Relative closeness degrees of $A_{1}$–$A_{5}$ obtained by
$\varrho^{(\lambda)}$

(b) Relative closeness degrees of $A_{1}$–$A_{5}$ obtained by
$\tilde{\varrho}^{(\lambda)}$

(c) Relative closeness degrees of $A_{1}$–$A_{5}$ obtained by
$\varrho_{{}_{K_{\gamma_{1}},K_{\gamma_{2}}}}^{(\lambda)}$
($0\leq\gamma_{1}<0.5$, $\gamma_{2}=1$)

(d) Relative closeness degrees of $A_{1}$–$A_{5}$ obtained by
$\varrho_{{}_{K_{\gamma_{1}},K_{\gamma_{2}}}}^{(\lambda)}$
($0.5<\gamma_{1}<1$, $\gamma_{2}=1$)

Figure 1: Relative closeness degrees of $A_{1}$–$A_{5}$ in Example 11

In summary, our proposed TOPSIS method has the following advantages:

  1. (1)

It is monotonous under the linear order ‘$\leq_{{}_{\text{XY}}}$’ or
‘$\leq_{{}_{\textrm{ZX}}}$’ or ‘$\leq_{{}_{A,B}}$’. This can overcome the
limitation of non-monotonicity for some classical IF TOPSIS methods in [48,
45, 52, 37, 47, 50]. In addition, other IF TOPSIS methods can at most
guarantee the monotonicity under Atanassov’s partial order $\subset$.

  2. (2)

Based on the admissible distances with linear orders, our method is more in
line with the essential features of the original TOPSIS introduced by Hwang
and Yoon [5].

  3. (3)

As can be seen from Example 11, the ranking orders of the alternatives using
different MADM methods are slightly different. However, the best choice is the
same. Moreover, our preference order is stable when the parameter $\lambda$ is
large enough. This indicates that our method is effective and stable.

  4. (4)

Compared to the TOPSIS methods in [33, 52], our method requires less
computation and fewer steps.

######  Example 12 ([21, Example 5.2])

Assume that there is a committee of a company, which decides to choose a
project manager from five alternatives $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$, and
$A_{5}$ with four attributes $\mathscr{O}_{1}$, $\mathscr{O}_{2}$,
$\mathscr{O}_{3}$, and $\mathscr{O}_{4}$, where $\mathscr{O}_{1}$ is “Self-
Confidence”, $\mathscr{O}_{2}$ is “Personality”, $\mathscr{O}_{3}$ is “Past
Experience”, $\mathscr{O}_{4}$ is the “Proficiency in Project Management” and
$\mathscr{O}_{1}$, $\mathscr{O}_{2}$, $\mathscr{O}_{3}$, and $\mathscr{O}_{4}$
are all benefit attributes, with weight vector
$\omega=(0.1,0.2,0.3,0.4)^{\top}$.

Assume the decision matrix $R=(r_{ij})_{5\times 4}$ given by the committee is
as listed in Table IX.

TABLE IX: The decision matrix $R$

| $\mathscr{O}_{1}$ | $\mathscr{O}_{2}$ | $\mathscr{O}_{3}$ | $\mathscr{O}_{4}$
---|---|---|---|---
$A_{1}$ | $\langle 0.4,0.5\rangle$ | $\langle 0.3,0.6\rangle$ | $\langle 0.4,0.4\rangle$ | $\langle 0.5,0.3\rangle$
$A_{2}$ | $\langle 0.4,0.4\rangle$ | $\langle 0.5,0.4\rangle$ | $\langle 0.3,0.5\rangle$ | $\langle 0.3,0.4\rangle$
$A_{3}$ | $\langle 0.4,0.6\rangle$ | $\langle 0.5,0.5\rangle$ | $\langle 0.4,0.6\rangle$ | $\langle 0.4,0.6\rangle$
$A_{4}$ | $\langle 0.3,0.4\rangle$ | $\langle 0.2,0.6\rangle$ | $\langle 0.1,0.9\rangle$ | $\langle 0.4,0.4\rangle$
$A_{5}$ | $\langle 0.5,0.4\rangle$ | $\langle 0.3,0.6\rangle$ | $\langle 0.3,0.5\rangle$ | $\langle 0.47,0.5\rangle$

Step 1: (Normalize the decision matrix) Since $\mathscr{O}_{1}$,
$\mathscr{O}_{2}$, $\mathscr{O}_{3}$, and $\mathscr{O}_{4}$ are all benefit
attributes, we have $\overline{R}=(\bar{r}_{ij})_{5\times 4}=R$.

Step 2: (Determine the positive and negative ideal-points) The IF positive
ideal-point is

$\mathbf{A}^{+}=(\langle 0.5,0.4\rangle,\langle 0.5,0.4\rangle,\langle
0.4,0.4\rangle,\langle 0.5,0.3\rangle)^{\top},$

and IF negative ideal-point is

$\mathbf{A}^{-}=(\langle 0.3,0.6\rangle,\langle 0.2,0.6\rangle,\langle
0.1,0.9\rangle,\langle 0.3,0.6\rangle)^{\top}.$

Step 3: (Compute the relative closeness degrees) Choose $\lambda=100$ and
calculate the relative closeness degrees $\mathscr{C}_{i}$ of the alternatives
$A_{i}$ ($i=1,2,3,4,5$) to the IF positive ideal-point $\mathbf{A}^{+}$ by
Eqs. (18), (21), and (24): $\mathscr{C}_{1}=0.5565$, $\mathscr{C}_{2}=0.5295$,
$\mathscr{C}_{3}=0.5058$, $\mathscr{C}_{4}=0.4555$, $\mathscr{C}_{5}=0.5171.$

Step 4: (Rank the alternatives) Because
$\mathscr{C}_{1}>\mathscr{C}_{2}>\mathscr{C}_{5}>\mathscr{C}_{3}>\mathscr{C}_{4}$,
the ranking order of the alternatives $A_{i}$ ($i=1,2,3,4,5$) is: $A_{1}\succ
A_{2}\succ A_{5}\succ A_{3}\succ A_{4}.$

Repeating Steps 1–2, by applying Eqs. (19) and (22), we obtain the following
result:

Step 3: (Compute the relative closeness degrees) Calculate the relative
closeness degrees $\mathscr{C}_{i}$ of the alternatives $A_{i}$
($i=1,2,3,4,5$) to the IF positive ideal-point $\mathbf{A}^{+}$ by Eqs. (19),
(22), and (24): $\mathscr{C}_{1}=0.5531$, $\mathscr{C}_{2}=0.5329$,
$\mathscr{C}_{3}=0.5042$, $\mathscr{C}_{4}=0.4583$, $\mathscr{C}_{5}=0.5198.$

Step 4: (Rank the alternative) Because
$\mathscr{C}_{1}>\mathscr{C}_{2}>\mathscr{C}_{5}>\mathscr{C}_{3}>\mathscr{C}_{4}$,
the ranking order of the alternatives $A_{i}$ ($i=1,2,3,4,5$) is: $A_{1}\succ
A_{2}\succ A_{5}\succ A_{3}\succ A_{4}.$

From Table X, which shows a comparison of the ranking orders of the
alternatives in Example 12 for different MADM methods, it can be observed that
our results based on the metrics $\varrho^{(100)}$, $\tilde{\varrho}^{(100)}$,
$\varrho_{{}_{K_{0.5},K_{0.4}}}^{(100)}$, and
$\varrho_{{}_{K_{0.6},K_{0.4}}}^{(100)}$ are consistent with the ranking
orders obtained by the MADM methods in [52, 66, 46, 49, 51, 21].

TABLE X: A comparison of the ranking orders of the alternatives in Example 12
for different MADM methods

Methods | Ranking orders
---|---
Chen et al.’s TOPSIS method in [52] | $A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$
Wang and Wei’s TOPSIS method in [66] | $A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$
Altan Koyuncu et al.’s TOPSIS method in [46] | $A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$
Büyüközkan and Güleryüz’s TOPSIS method in [49] | $A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$
Zhang et al.’s TOPSIS method in [51] | $A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$
Zeng et al.’s VIKOR method in [21] | $A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$
Our TOPSIS based on $\varrho^{(100)}$ | $A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$
Our TOPSIS based on $\tilde{\varrho}^{(100)}$ | $A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$
Our TOPSIS based on $\varrho_{{}_{K_{0.2},K_{0.4}}}^{(100)}$ | $A_{1}\succ A_{3}\succ A_{5}\succ A_{2}\succ A_{4}$
Our TOPSIS based on $\varrho_{{}_{K_{0.5},K_{0.4}}}^{(100)}$ | $A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$
Our TOPSIS based on $\varrho_{{}_{K_{0.6},K_{0.4}}}^{(100)}$ | $A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$

To illustrate the detailed influence of the parameters $\lambda$ and
$\gamma_{1}$ on the decision-making results in Example 12 by using the metrics
$\varrho^{(\lambda)}$, $\tilde{\varrho}^{(\lambda)}$, and
$\varrho_{{}_{K_{\gamma_{1}},K_{\gamma_{2}}}}^{(\lambda)}$, the relative
closeness degrees $\mathscr{C}_{i}$ of each alternative $A_{i}$ obtained by
$\varrho^{(\lambda)}$, $\tilde{\varrho}^{(\lambda)}$, and
$\varrho_{{}_{K_{\gamma_{1}},K_{\gamma_{2}}}}^{(\lambda)}$ are shown in Fig. 2
(a), (b), and (c)–(d), respectively. Compared to Example 11, the preferences
of decision makers greatly affect the ranking results in this example: (1) For
$0\leq\gamma_{1}<0.5$, the ranking order is: $A_{1}\succ A_{3}\succ A_{5}\succ
A_{2}\succ A_{4}$. (2) For $0.5\leq\gamma_{1}<1$, the ranking order is:
$A_{1}\succ A_{2}\succ A_{5}\succ A_{3}\succ A_{4}$.

(a) Relative closeness degrees of $A_{1}$–$A_{5}$ obtained by
$\varrho^{(\lambda)}$

(b) Relative closeness degrees of $A_{1}$–$A_{5}$ obtained by
$\tilde{\varrho}^{(\lambda)}$

(c) Relative closeness degrees of $A_{1}$–$A_{5}$ obtained by
$\varrho_{{}_{K_{\gamma_{1}},K_{\gamma_{2}}}}^{(\lambda)}$
($0\leq\gamma_{1}<0.5$, $\gamma_{2}=1$)

(d) Relative closeness degrees of $A_{1}$–$A_{5}$ obtained by
$\varrho_{{}_{K_{\gamma_{1}},K_{\gamma_{2}}}}^{(\lambda)}$
($0.5<\gamma_{1}<1$, $\gamma_{2}=1$)

Figure 2: Relative closeness degrees of $A_{1}$–$A_{5}$ in Example 12

##  VIII Conclusions

This paper is devoted to establishing a monotonous IF TOPSIS method with three
typical linear orders, ‘$\leq_{{}_{\text{XY}}}$’ in [7],
‘$\leq_{{}_{\textrm{ZX}}}$’ in [8], and ‘$\leq_{{}_{A,B}}$’ in [11, 12].
Noting that the TOPSIS method is closely related to the order structure and
the metric/similarity measure, we first discuss some examples to show that
some classical similarity measures in [37, 34, 29, 30], including Euclidean
similarity measure, Minkowski similarity measure, and modified Euclidean
similarity measure, do not satisfy the axiomatic definition of IF similarity
measures. Then, we prove the nonexistence of a continuous function that can
distinguish IFV by a real number and is increasing with Atanassov’s order
‘$\subset$’. As a direct corollary, we prove that there is no any continuous
similarity measure that can distinguish between each pair of IFVs. Moreover,
we show some illustrative examples to demonstrate that some classical IF
TOPSIS methods in [48, 45, 52, 37, 47, 50] are not monotonous with Atanassov’s
partial order ‘$\subset$’ or the linear order ‘$\leq_{{}_{\text{XY}}}$’, which
may yield counter-intuitive results. To overcome this limitation, by using
three new parametric admissible distances with the linear order
‘$\leq_{{}_{\text{XY}}}$’ or ‘$\leq_{{}_{\textrm{ZX}}}$’ or
‘$\leq_{{}_{A,B}}$’, we develop a novel IF TOPSIS method and prove that it is
monotonically increasing with these two linear orders. Finally, we show two
practical examples with comparative analysis to other MADM methods to
illustrate the efficiency of our TOPSIS method.

Because the proposed TOPSIS method depends on the choice of the linear orders,
choosing an appropriate order for a given problem is very important for
practical decision-making. Meanwhile, because the construction method of
admissible distances with linear orders presented in this paper is relatively
rough, which fails to capture all properties of the the corresponding linear
order, this may cause inaccurate decision-making results in some cases. In the
future, therefore, we will further study the general construction of linear
orders and admissible distance/similarity measures for IFVs, which will be
useful for building more effective IF TOPSIS methods.

## References

  * [1] L. A. Zadeh, “Fuzzy sets,” _Inf. Control_ , vol. 8, no. 3, pp. 338–353, 1965\.
  * [2] K. T. Atanassov, “Intuitionistic fuzzy set,” _Fuzzy Sets Syst._ , vol. 20, pp. 87–96, 1986.
  * [3] ——, _Intuitionistic Fuzzy Sets: Theory and Applications_ , ser. Studies in Fuzziness and Soft Computing. Springer, Berlin, Heidelberg, 1999, vol. 35.
  * [4] Z. Xu, “Intuitionistic fuzzy aggregation operators,” _IEEE Trans. Fuzzy Syst._ , vol. 15, no. 6, pp. 1179–1187, 2007.
  * [5] C.-L. Hwang and K. Yoon, _Multiple Attribute Decision Making: Methods and Applications_ , ser. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, Heidelberg, 1981, vol. 186.
  * [6] X. Wu, T. Wang, Q. Liu, P. Liu, G. Chen, and X. Zhang, “Topological and algebraic structures of Atanassov’s intuitionistic fuzzy-values space,” https://arxiv.org/abs/2111.12677.
  * [7] Z. Xu and R. R. Yager, “Some geometric aggregation operators based on intuitionistic fuzzy sets,” _Int. J. Gen. Syst._ , vol. 35, pp. 417–433, 2006.
  * [8] X. Zhang and Z. Xu, “A new method for ranking intuitionistic fuzzy values and its application in multi-attribute decision making,” _Fuzzy Optim. Decis. Making_ , vol. 11, pp. 135–146, 2012.
  * [9] Z. Xing, W. Xiong, and H. Liu, “A Euclidean approach for ranking intuitionistic fuzzy values,” _IEEE Trans. Fuzzy Syst._ , vol. 26, no. 1, pp. 353–365, 2018.
  * [10] H. Bustince, J. Fernandez, A. Kolesárová, and R. Mesiar, “Generation of linear orders for intervals by means of aggregation functions,” _Fuzzy Sets Syst._ , vol. 220, pp. 69–77, 2013.
  * [11] L. De Miguel, H. Bustince, J. Fernandez, E. Induráin, A. Kolesárová, and R. Mesiar, “Construction of admissible linear orders for interval-valued Atanassov intuitionistic fuzzy sets with an application to decision making,” _Inf. Fusion_ , vol. 27, pp. 189–197, 2016\.
  * [12] L. De Miguel, H. Bustince, B. Pekala, U. Bentkowska, I. Da Silva, B. Bedregal, R. Mesiar, and G. Ochoa, “Interval-valued Atanassov intuitionistic OWA aggregations using admissible linear orders and their application to decision making,” _IEEE Trans. Fuzzy Syst._ , vol. 24, no. 6, pp. 1586–1597, 2016\.
  * [13] J. Wang, J. Zhang, and S.-Y. Liu, “A new score function for fuzzy MCDM based on vague set theory,” _Int. J. Comput. Cognit._ , vol. 4, no. 1, 2006.
  * [14] S.-M. Chen and J.-M. Tan, “Handling multicriteria fuzzy decision-making problems based on vague set theory,” _Fuzzy Set Syst._ , vol. 67, no. 2, pp. 163–172, 1994.
  * [15] A. Kharal, “Homeopathic drug selection using intuitionistic fuzzy sets,” _Homeopathy_ , vol. 98, no. 1, pp. 35–39, 2009.
  * [16] L. Lin, X.-H. Yuan, and Z.-Q. Xia, “Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets,” _J. Comput. Syst. Sci._ , vol. 73, no. 1, pp. 84–88, 2007.
  * [17] H.-W. Liu and G.-J. Wang, “Multi-criteria decision-making methods based on intuitionistic fuzzy sets,” _Eur. J. Oper. Res._ , vol. 179, no. 1, pp. 220–233, 2007.
  * [18] J. Wang and J. Li, “Multi-criteria fuzzy decision-making method based on cross entropy and score functions,” _Expert Syst. Appl._ , vol. 38, no. 1, pp. 1032–1038, 2011.
  * [19] J. Ye, “Improved method of multicriteria fuzzy decision-making based on vague sets,” _Comput. Aided Des._ , vol. 39, no. 2, pp. 164–169, 2007.
  * [20] J. Wang, K. Li, and H. Zhang, “Interval-valued intuitionistic fuzzy multi-criteria decision-making approach based on prospect score function,” _Knowl.-Based Syst._ , vol. 27, pp. 119–125, 2012.
  * [21] S. Zeng, S.-M. Chen, and L.-W. Kuo, “Multiattribute decision making based on novel score function of intuitionistic fuzzy values and modified VIKOR method,” _Inf. Sci._ , vol. 488, pp. 76–92, 2019.
  * [22] D. H. Hong and C.-H. Choi, “Multicriteria fuzzy decision-making problems based on vague set theory,” _Fuzzy Sets Syst._ , vol. 114, no. 1, pp. 103–113, 2000.
  * [23] J. Ye, “Using an improved measure function of vague sets for multicriteria fuzzy decision-making,” _Expert Syst. Appl._ , vol. 37, no. 6, pp. 4706–4709, 2010.
  * [24] K. Guo, “Amount of information and attitudinal-based method for ranking Atanassov’s intuitionistic fuzzy values,” _IEEE Trans. Fuzzy Syst._ , vol. 22, no. 1, pp. 177–188, 2013.
  * [25] E. Szmidt and J. Kacprzyk, “Amount of information and its reliability in the ranking of Atanassov’s intuitionistic fuzzy alternatives,” in _Recent Advances in Decision Making_ , E. Rakus-Andersson, R. R. Yager, N. Ichalkaranje, and L. C. Jain, Eds. Springer Berlin Heidelberg, 2009, pp. 7–19.
  * [26] X. Liu, “Entropy, distance measure and similarity measure of fuzzy sets and their relations,” _Fuzzy Sets Syst._ , vol. 52, no. 3, pp. 305–318, 1992\.
  * [27] P. Wang, _Theory of Fuzzy Sets and Their Applications_. Shanghai Science and Technology Publishing House, Shanghai, 1982.
  * [28] D.-F. Li and C. Cheng, “New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions,” _Pattern Recognit. Lett._ , vol. 23, pp. 221–225, 2002.
  * [29] Z. Xu, “Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making,” _Fuzzy Optim. Decis. Making_ , vol. 6, no. 2, pp. 109–121, 2007.
  * [30] Z. Xu and J. Chen, “An overview of distance and similarity measures of intuitionistic fuzzy sets,” _Int. J. Uncertain. Fuzziness. Knowl.-Based Syst._ , vol. 16, no. 04, pp. 529–555, 2008.
  * [31] I. Iancu, “Intuitionistic fuzzy similarity measures based on frank t-norms family,” _Pattern Recognit. Lett._ , vol. 42, pp. 128–136, 2014.
  * [32] E. Szmidt and J. Kacprzyk, “Distances between intuitionistic fuzzy sets,” _Fuzzy Sets Syst._ , vol. 114, no. 3, pp. 505–518, 2000.
  * [33] F. Shen, X. Ma, Z. Li, Z. Xu, and D. Cai, “An extended intuitionistic fuzzy TOPSIS method based on a new distance measure with an application to credit risk evaluation,” _Inf. Sci._ , vol. 428, pp. 105–119, 2018.
  * [34] E. Szmidt, _Distances and Similarities in Intuitionistic Fuzzy Sets_ , ser. Studies in Fuzziness and Soft Computing. Springer, Berlin, Heidelberg, 2014, vol. 307.
  * [35] Z. Xu and X. Cai, _Intuitionistic Fuzzy Information Aggregation: Theory and Applications_ , ser. Mathematics Monograph Series. Science Press, 2012, vol. 20.
  * [36] Y. Yang and F. Chiclana, “Consistency of 2D and 3D distances of intuitionistic fuzzy sets,” _Expert Syst. Appl._ , vol. 39, no. 10, pp. 8665–8670, 2012.
  * [37] D.-F. Li, _Decision and Game Theory in Management with Intuitionistic Fuzzy Sets_ , ser. Studies in Fuzziness and Soft Computing. Springer-Verlag Berlin Heidelberg, 2014, vol. 308.
  * [38] H. B. Mitchell, “On the Dengfeng-Chuntian similarity measure and its application to pattern recognition,” _Pattern Recognit. Lett._ , vol. 24, no. 16, pp. 3101–3104, 2003.
  * [39] S.-M. Chen, S.-H. Cheng, and T.-C. Lan, “A novel similarity measure between intuitionistic fuzzy sets based on the centroid points of transformed fuzzy numbers with applications to pattern recognition,” _Inf. Sci._ , vol. 343–344, pp. 15–40, 2016.
  * [40] S.-M. Chen and C.-H. Chang, “A novel similarity measure between Atanassov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition,” _Inf. Sci._ , vol. 291, pp. 96–114, 2015.
  * [41] W.-L. Hung and M.-S. Yang, “Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance,” _Pattern Recognit. Lett._ , vol. 25, no. 14, pp. 1603–1611, 2004.
  * [42] Z. Liang and P. Shi, “Similarity measures on intuitionistic fuzzy sets,” _Pattern Recognit. Lett._ , vol. 24, no. 15, pp. 2687–2693, 2003.
  * [43] H. Zhang and L. Yu, “New distance measures between intuitionistic fuzzy sets and interval-valued fuzzy sets,” _Inf. Sci._ , vol. 245, pp. 181–196, 2013\.
  * [44] F. E. Boran and D. Akay, “A biparametric similarity measure on intuitionistic fuzzy sets with applications to pattern recognition,” _Inf. Sci._ , vol. 255, pp. 45–57, 2014.
  * [45] F. E. Boran, S. Genç, M. Kurt, and D. Akay, “A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method,” _Expert Syst. Appl._ , vol. 36, no. 8, pp. 11 363–11 368, 2009.
  * [46] C. Altan Koyuncu, E. Aydemir, and A. C. Başarır, “Selection industry 4.0 maturity model using fuzzy and intuitionistic fuzzy TOPSIS methods for a solar cell manufacturing company,” _Soft Comput._ , vol. 25, no. 15, pp. 10 335–10 349, 2021.
  * [47] A. Memari, A. Dargi, M. R. A. Jokar, R. Ahmad, and A. R. A. Rahim, “Sustainable supplier selection: a multi-criteria intuitionistic fuzzy TOPSIS method,” _J. Manuf. Syst._ , vol. 50, pp. 9–24, 2019.
  * [48] F. Boran, K. Boran, and T. Menlik, “The evaluation of renewable energy technologies for electricity generation in Turkey using intuitionistic fuzzy TOPSIS,” _Energy Sources, Part B_ , vol. 7, no. 1, pp. 81–90, 2012\.
  * [49] G. Büyüközkan and S. Güleryüz, “Multi criteria group decision making approach for smart phone selection using intuitionistic fuzzy TOPSIS,” _Int. J. Comput. Intell. Syst._ , vol. 9, no. 4, pp. 709–725, 2016.
  * [50] B. D. Rouyendegh, A. Yildizbasi, and P. Üstünyer, “Intuitionistic fuzzy TOPSIS method for green supplier selection problem,” _Soft Comput._ , vol. 24, no. 3, pp. 2215–2228, 2020.
  * [51] L. Zhang, J. Zhan, and Y. Yao, “Intuitionistic fuzzy TOPSIS method based on CVPIFRS models: an application to biomedical problems,” _Inf. Sci._ , vol. 517, pp. 315–339, 2020.
  * [52] S.-M. Chen, S.-H. Cheng, and T.-C. Lan, “Multicriteria decision making based on the TOPSIS method and similarity measures between intuitionistic fuzzy values,” _Inf. Sci._ , vol. 367, pp. 279–295, 2016.
  * [53] D. A. Wood, “Supplier selection for development of petroleum industry facilities, applying multi-criteria decision making techniques including fuzzy and intuitionistic fuzzy TOPSIS with flexible entropy weighting,” _J. Nat. Gas Sci. Eng._ , vol. 28, pp. 594–612, 2016.
  * [54] Z. Yue, “TOPSIS-based group decision-making methodology in intuitionistic fuzzy setting,” _Inf. Sci._ , vol. 277, pp. 141–153, 2014.
  * [55] Y. Liu, J.-W. Bi, and Z.-P. Fan, “A method for ranking products through online reviews based on sentiment classification and interval-valued intuitionistic fuzzy TOPSIS,” _Int. J. Inf. Technol. Decis. Making_ , vol. 16, no. 06, pp. 1497–1522, 2017.
  * [56] C.-Y. Wang and S.-M. Chen, “Multiple attribute decision making based on interval-valued intuitionistic fuzzy sets, linear programming methodology, and the extended TOPSIS method,” _Inf. Sci._ , vol. 397, pp. 155–167, 2017\.
  * [57] M. Yazdi, “Risk assessment based on novel intuitionistic fuzzy-hybrid-modified TOPSIS approach,” _Safety Sci._ , vol. 110, pp. 438–448, 2018.
  * [58] S. Zeng, S.-M. Chen, and K.-Y. Fan, “Interval-valued intuitionistic fuzzy multiple attribute decision making based on nonlinear programming methodology and TOPSIS method,” _Inf. Sci._ , vol. 506, pp. 424–442, 2020.
  * [59] M. Singh, R. Rathi, J. Antony, and J. A. Garza-Reyes, “Lean six sigma project selection in a manufacturing environment using hybrid methodology based on intuitionistic fuzzy madm approach,” _IEEE Trans. Eng. Manage._ , 2021, doi:10.1109/TEM.2021.3049877.
  * [60] G. Beliakov, H. Bustince, D. P. Goswami, U. K. Mukherjee, and N. R. Pal, “On averaging operators for Atanassov’s intuitionistic fuzzy sets,” _Inf. Sci._ , vol. 181, no. 6, pp. 1116–1124, 2011.
  * [61] M. Grabisch, J.-L. Marichal, R. Mesiar, and E. Pap, _Aggregation Functions_ , ser. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2009, vol. 127.
  * [62] H. B. Mitchell, “On the Dengfeng-Chuntian similarity measure and its application to pattern recognition,” _Pattern Recognit. Lett._ , vol. 24, pp. 3101–3104, 2003.
  * [63] P. Burillo and H. Bustince, “Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets,” _Fuzzy Sets Syst._ , vol. 78, no. 3, pp. 305–316, 1996.
  * [64] W.-L. Hung and M.-S. Yang, “On similarity measures between intuitionistic fuzzy sets,” _Int. J. Intell. Syst._ , vol. 23, no. 3, pp. 364–383, 2008\.
  * [65] E. Szmidt and J. Kacprzyk, “A similarity measure for intuitionistic fuzzy sets and its application in supporting medical diagnostic reasoning,” in _Rutkowski L., Siekmann J.H., Tadeusiewicz R., Zadeh L.A. (eds) Artificial Intelligence and Soft Computing–ICAISC 2004. ICAISC 2004._ , ser. Lecture Notes in Computer Science, vol. 3070. Springer, Berlin, Heidelberg, 2004, pp. 388–393.
  * [66] H. Wang and G. Wei, “An effective supplier selection method with intuitionistic fuzzy information,” in _Proceedings of the 2008 International Conference on Wireless Communications, Networking and Mobile Computing, Dalian, China_. IEEE, 2008, pp. 1–4.