Novel entropy and distance measures for interval-valued intuitionistic fuzzy sets with application in multi-criteria group decision-making

Abstract

Entropy and distance are the most important information-theoretic measures. These measures have found useful applications in different areas. In the present communication, we study the entropy and distance measures under an interval-valued intuitionistic fuzzy (IVIF) environment using an exponential function. First, it presents the novel exponential entropy and distance measures for interval-valued intuitionistic fuzzy sets (IVIFSs) with proof of their authenticity. A method is offered to solve multi-criteria group decision-making (MCGDM) problems in the IVIF environment based on the weighted exponential entropy measure. The performance of the proposed IVIF MCGDM method is shown by taking two case studies. Second, the success and strength of the proposed IVIF distance measures are demonstrated by comparing them with the existing ones. Further, the paper advances an approach to solve multi-attribute decision-making problems under the IVIF environment. Finally, it considers a real-world example to illustrate the applicability and authenticity of the proposed approach. In doing so, the proposed approach is compared with existing methods to exhibit its advantages. Thus, the proposed IVIF information measures and multi-criteria group decision-making method are more suitable to solve real-life decision-making problems.

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Novel entropy and distance measures for interval-
valued intuitionistic fuzzy sets with application in
multi-criteria group decision-making
Anshu Ohlan
To cite this article: Anshu Ohlan (2022): Novel entropy and distance measures for interval-valued
intuitionistic fuzzy sets with application in multi-criteria group decision-making, International Journal
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INTERNATIONAL JOURNAL OF GENERAL SYSTEMS
https://doi.org/10.1080/03081079.2022.2036138
Novel entropy and distance measures for interval-valued
intuitionistic fuzzy sets with application in multi-criteria
group decision-making
Anshu Ohlan
Department of Education, Government of Haryana, Rohtak, India
ABSTRACT
Entropy and distance are the most important information-theoretic
measures. These measures have found useful applications in dif-
ferent areas. In the present communication, we study the entropy
and distance measures under an interval-valued intuitionistic fuzzy
(IVIF) environment using an exponential function. First, it presents
the novel exponential entropy and distance measures for interval-
valued intuitionistic fuzzy sets (IVIFSs) with proof of their authentic-
ity. A method is offered to solve multi-criteria group decision-making
(MCGDM) problems in the IVIF environment based on the weighted
exponential entropy measure. The performance of the proposed IVIF
MCGDM method is shown by taking two case studies. Second, the
success and strength of the proposed IVIF distance measures are
demonstrated by comparing them with the existing ones. Further,
the paper advances an approach to solve multi-attribute decision-
making problems under the IVIF environment. Finally, it considers a
real-world example to illustrate the applicability and authenticity of
the proposed approach. In doing so, the proposed approach is com-
pared with existing methods to exhibit its advantages. Thus, the pro-
posed IVIF information measures and multi-criteria group decision-
making method are more suitable to solve real-life decision-making
problems.
ARTICLE HISTORY
Received 29 June 2021
Accepted 4 January 2022
KEYWORDS
interval-valued intuitionistic
fuzzy set; entropy measure;
distance measure;
decision-making; VIKOR;
TOPSIS
1. Introduction
Zadeh’s (1965) fuzzy set theory has been widely followed for developing several new
methodologies and concepts to handle vagueness and imprecision. As a generalization of
fuzzy set (FS) theory, Atanassov (1986) suggested the intuitionistic fuzzy set (IFS). After-
ward, several measures of information in IFSs have been studied by dierent scholars from
various aspects (Szmidt and Kacprzyk 2001;ZengandGuo2008; Chen and Randyanto
2013;ChenandChang2015;Yager2015;Ohlan2016; Ohlan and Ohlan 2016;Ohlan2020;
Ohlan and Ohlan 2022; Joshi et al. 2018; Rani, Jain, and Hooda 2018a; Zeng, Chen, and
Kuo 2019; Liu, Chen, and Wang 2020; Meng, Chen, and Yuan 2020;&KumarandChen
2021). Atanassov and Gargov (1989), by oering membership and non-membership inter-
val, broadened IFS theory to interval-valued intuitionistic fuzzy set (IVIFS) theory to deal
CONTACT Anshu Ohlan anshu.gahlawat@yahoo.com
© 2022 Informa UK Limited, trading as Taylor & FrancisGroup

2A. OHLAN
with ambiguous information and models of non-statistical ambiguity. After that, Atanassov
(1994) presented more operations on IVIFSs and considered their basic properties. The
work on IVIFS theory has been further extended by several researchers in useful ways
(Liu, Zheng, and Xiong 2005;Xu2007a;Zhangetal.2010;ChenandChiou2014;Meng
and Chen 2016;WangandChen2017;Xia2018; Wei et al. 2019;ChenandChu2020;
Atanassov 2020; Zeng, Chen, and Fan 2020;Li,Suo,andLi2021). It is well-accepted that
IVIFSs can deal with the large complexity in an uncertain social environment. Accordingly,
it is extremely required to introduce consistent and reliable entropy and distance measures
in the IVIF environment to deal with real-life problems.
Some measures of information have been introduced by several researchers in the
framework of IVIFS with their applications in various elds in the last decades. For exam-
ple, the subsethood and entropy measures from IFSs to IVIFSs were extended by Liu,
Zheng, and Xiong (2005). Zhang and Jiang (2010), Zhang et al. (2010), Wei, Wang, and
Zhang (2011), Zhang et al. (2011), Wei and Zhang (2015), and Singh (2012) introduced
the entropy measures for IVIFSs and showed their application in solving multi-attribute
decision-making problems. Likewise, Zhang et al. (2014) proposed some new measures of
entropy based on distance and presented their relationships with similarity and inclusion
measures in IVIFSs. Ye (2011), Xu (2010), and Düğenci (2016) studied the cross-entropy
and distance measures for IVIFSs to nd their application in group decision-making. Wei
et al. (2019) presented novel generalized exponential IF and IVIF entropy measures with
parameters of knowledge and reliability. Further, Baccour and Alimi (2019)proposedtwo
distance measures between IFSs and generalized them to IVIFSs. Thereafter, an IVIF dis-
tance measure and TOPSIS method were introduced by Garg and Kumar (2020). Che,
Suo, and Li (2021) introduced an approach to construct IVIF entropies using a distance
function.
Multi-criteria decision-making (MCDM) and multi-attribute decision-making
(MADM) techniques are used widely for obtaining the best solution from a set of alterna-
tives in precise situations. Because of the cumulative complexity of the social environment
and lack of precise information, decisions are mostly taken by a team of experts instead
of entities. In sum, the applications of IVIFSs for resolving a variety of decision-making
processes have been gained signicant considerations (Chen, Hsiao, and Yen Y 2011;Li,
Chen, and Huang 2010;Li2011;Parketal.2011;Yue2011;Qietal.2011;ZhangandYu
2012;Chenetal.2012;Ye2013; Nguyen 2016; Wan, Xu, and Dong 2016a,2016b;Chenand
Huang 2017;Wan, Wang, and Dong 2017,2018a,2018b; Wan, Wang, and Dong 2019;Wan,
Xu, and Dong 2020a,2020b; Zeng, Chen, and Fan 2020;Ohlan2021 &Zindani,Maity,and
Bhowmik 2021).
There exist a few entropy and distance measures of IVIFSs to deal with decision prob-
lems. However, investigation of the methods of decision-making employing IVIF entropy
and distance measures for solving problems involving the uncertain socio-economic envi-
ronment is scant. Motivated by the rising importance of fuzzy decision-making methods,
we introduce exponential entropy and distance measures of IVIFSs by using membership
and non-membership intervals of IVIFS. In addition, the corresponding weighted IVIF
exponential entropy measure is proposed.
The central contribution of the current study to the existing literature on measures and
methods of decision-making is summarized as follows:

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 3
1. Novel entropy and distance measures based on exponential function are introduced
for interval-valued intuitionistic fuzzy sets.
2. A method to solve multi-criteria group decision-making (MCGDM) problems in the
IVIF environment based on the weighted exponential entropy measure is discussed.
3. The exibility and authenticity of the proposed IVIF MCGDM method are presented
by taking two case studies.
4. Further, it compares the performance of the proposed distance measure with the
existing distance measures.
5. It proposes an approach to solve multi-attribute decision-making problems using the
proposed IVIF distance measure. The success and strength of the planned method are
authenticated by reasonable examples in the literature.
6. The proposed technique is compared with existing methods to exhibit its advantages.
The rest of the paper is organized as follows. Section 2 briey reviews some fundamental
denitions and concepts relating to IFSs and IVIFSs, including the properties of entropy
and distance measures for IVIFSs used in the study. Section 3 introduces and proves the
exponential entropy and distance measures for IVIFSs. In the same section, we also dene
the corresponding weighted entropy and distance measures for IVIFSs. In Section 4, rstly,
we present the application of IVIF exponential entropy measure in decision-making with
the help of an example. Secondly, we apply the proposed IVIFSs based weighted expo-
nential entropy measure to introduce multi-criteria group decision-making (MCGDM)
methodology. Then, we explore the achievement of the proposed MCGDM method by
solving two case studies. A comparative analysis of the proposed MCGDM method with
theexistingentropiesandmethodsisalsocarriedoutinthesamesection.Thesuperi-
ority and capability of the proposed measure are shown by comparing it with the existing
distance/divergence measures in Section 5. Section 6 illustrates an approach to solve multi-
attribute decision-making problems using the proposed IVIF distance measure. Further,
classical VIKOR and TOPSIS methods of decision-making are also extended to deal with
IVIF data. A comparative analysis with the existing decision-making technique is made to
present the advantage of the proposed methodology. Further, a discussion on the advan-
tages and limitations of the proposed approach is extended in the same section. The nal
sectionpresentstheconclusionderivedfromthestudyinthisprocess.
2. Preliminaries
This section introduces the basic denitions and concepts relating to IFSs, IVIFSs, and the
properties of entropy and distance measures in IVIFSs employed in this study.
Denition 2.1: (Atanassov 1986) An intuitionistic fuzzy set Edened on a nite universe
of discourse Y={y1,y2,...,ym}is expressed as
E=yi,μE(yi),νE(yi)/yi∈Y
where μE:Y→[0, 1], νE:Y→[0, 1] with the condition μE:Y→[0, 1], 0 ≤μE(yi)+
νE(yi)≤1∀yi∈Y.
The numbers μE(yi),νE(yi)∈[0, 1] indicate the degree of membership and non-
membership of yito E,respectively.

4A. OHLAN
For every intuitionistic fuzzy set in Ywe will call πE(yi)=1−μE(yi)−νE(yi),the
intuitionistic index or degree of hesitation of yiin E.Itisapparentthat0≤πE(yi)≤1
for each yi∈Y. For a fuzzy set Ein Y,πE(yi)=0whenνE(yi)=1−μE(yi).
Atanassov and Gargov (1989) established the following concepts of interval-valued
intuitionistic fuzzy sets (IVIFSs):
Denition 2.2: (Atanassov and Gargov 1989)letCdenotes the set of all closed subinter-
vals of the interval [0,1] and Ybe a nite universe of discourse. An interval-valued intu-
itionistic fuzzy set Ein Yis presented in the form E=yi,μE(yi),νE(yi)/yi∈Ywhere
μE:Y→C,νE:Y→Chaving the condition that 0 ≤sup(μE(yi)) +sup(νE(yi)) ≤1.
The intervals μE(yi)and νE(yi)specify the degree of membership and non-membership
of yito E,respectively.HereμE(yi),andνE(yi)are closed intervals instead of the real num-
bers. Their lower and upper boundaries are shown by μEL(yi),μEU (yi),νEL(yi),andνEU (yi),
respectively. So, IVIFS can be expressed as
E=yi,[μEL(yi),μEU (yi)], [νEL (yi),νEU (yi)]/yi∈Ywhere 0 ≤(μEU(yi)+νEU (yi))
≤1, μEL(yi)≥0, νEL (yi)≥0.
we call interval
πE(yi)=1−μE(yi)−νE(yi)=[1 −μEU (yi)−νEU (yi),1−μEL (yi),νEL(yi)]=
[πEL(yi),πEU (yi)]isdegreeofhesitancyyito E.
If μE(yi)=μEL(yi)=μEU (yi)and νE(yi)=νEL (yi)=νEU (yi), then the given IVIFS
can be reduced to an ordinary IFS.
Liu, Zheng, and Xiong (2005) provided the following denition of entropy measure of
IVIFSs analogous to the axiomatic denition of entropy measure for IFSs (Bustince and
Burillo 1995).
Denition 2.3: (Liu, Zheng, and Xiong 2005) A real-valued function H:IVIFS(Y)→
[0, 1] is said to be an entropy measure of IVIFS(Y), if it meets the below conditions:
(H1)H(B)=0ifB=([1, 1], [0, 0])or B=([0, 0], [1, 1])for each yi∈Y;
(H2)H(B)=1ifandonlyif[μBL (yi),μBU (yi)]=[νBL(yi),νBU (yi)]foreachyi∈Y;
(H3)H(B)=H(Bc);
(H4)H(B)≤H(C)if B⊆Cwhen μCL(yi)≤νCL(yi)and μCU (yi)≤νCU (yi)for each
yi∈Y,
or C⊆Bwhen μCL(yi)≥νCL(yi)and μCU (yi)≥νCU (yi)for each yi∈Y.
Denition 2.4: (Düğenci 2016)Amappingd:IVIFS(Y)×IVIFS(Y)→[0, 1] is called
a distance measure between E∈IVIFS(Y)and F∈IVIFS(Y)if d(E,F)meets below-
mentioned properties:
(d1)0≤d(E,F)≤1
(d2)d(E,F)=0ifandonlyifE=F
(d3)d(E,F)=d(F,E)
(d4)IfE⊆F⊆G,E,F,G∈IVIFS(Y)
then d(E,F)≤d(E,G)and d(F,G)≤d(E,G).

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 5
3. Exponential entropy and distance measures for IVIFSs
In the present section, we rst recall Pal and Pal (1991) exponential entropy measure of a
fuzzy set Fas
eH(F)=1
n(√e−1)
n

i=1
[μF(yi)e(1−μF(yi)) +(1−μF(yi))eμF(yi)−1] (1)
Now if we assume Fis an IVIFS, then according to the method introduced by Zhang et al.
(2010), the average possible membership degree of element yito IVIFS Fcan be considered
as
μF(yi)=1
2μFL(yi)+μFU (yi)
2+1−νFL(yi)+νFU (yi)
2
=μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4(2)
ButtheabovemethodfailstoconsiderthehesitancydegreewhiletransforminganIVIFS
into FS. Using the above method of transformation of an FS to IVIFS, we now introduce an
IVIFS entropy measure (Rani, Jain, and Hooda 2018b) and analogous to Pal and Pal (1991)
exponential entropy measure (1) as follows:
Denition 3.1: Let F∈IVIFS(Y),Y={y1,y2,...,yn}, the entropy measure is dened
by
HE(F)=1
n(√e−1)
n

i=1⎧
⎪
⎨
⎪
⎩μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4e1−μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4
+1−μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4eμFL (yi)+μFU (yi)+2−νFL(yi)−νFU (yi)
4−1⎫
⎪
⎬
⎪
⎭
(3)
In the subsequent theorem, we demonstrate the validity of the proposed measure (3).
Theorem 3.1: The function HE(F)dened in (3) is an interval-valued intuitionistic fuzzy
entropy measure.
Proof :In order to prove that HE(F)is an entropy measure of IVIFSs, it only requires to
satisfy (H1)–(H
4) in Denition 2.3.
(H1):LetFbeacrispsetthenwehaveeitherF=([1, 1], [0, 0])or F=([0, 0], [1, 1])
i.e. [μFL(yi),μFU (yi)]=[1, 1] and [νFL (yi),νFU (yi)]=[0, 0]
or [μFL(yi),μFU (yi)]=[0, 0] and [νFL(yi),νFU (yi)]=[1, 1] for each yi∈Y,weget
HE(F)=0.
If we assume μFL(yi)+μFU (yi)+2−νFL(yi)−νFU (yi)
4=ψF(yi)(4)
then HE(F)=1
n(√e−1)
n

i=1
[ψF(yi)e(1−ψF(yi)) +(1−ψF(yi))eψF(yi)−1] (5)

6A. OHLAN
isthesameasPalandPal(1991) exponential entropy (1) which becomes zero if ψF(yi)=0
or 1 for each yi∈Y.
i.e., μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4=0(6)
or μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4=1(7)
for each yi∈Y. Now Equations (6) and (7) hold if either F=([1, 1], [0, 0])or F=
([0, 0], [1, 1])i.e. Fisacrispset.
(H2): If [μFL(yi),μFU (yi)]=[νFL(yi),νFU (yi)]foreachyi∈Y, we evidently get
HE(F)=1 from Equation (3).
On the other hand, if it is assumed that HE(F)=1, we have to show [μFL(yi),μFU (yi)]=
[νFL(yi),νFU (yi)].
From Equation (5), we get HE(F)=1
n
n

i=1
g(ψF(yi))
where g(ψF(yi)) =[ψF(yi)e(1−ψF(yi)) +(1−ψF(yi))eψF(yi)−1]
(√e−1)for each yi∈Y(8)
Now if HE(F)=1implies 1
n
n

i=1
g(ψF(yi)) =1
⇒g(ψF(yi)) =1foreach yi∈Y.(9)
Dierentiating Equation (8) with respect to (w.r.t.) ψF(yi)andequatingtozero,weget
∂g(ψF(yi))
∂(ψF(yi)) =e(1−ψF(yi)) −ψF(yi)e(1−ψF(yi)) −eψF(yi)+(1−ψF(yi))eψF(yi)
√e−1=0
⇒(1−ψF(yi))e(1−ψF(yi)) −ψF(yi)eψF(yi)
√e−1=0
⇒(1−ψF(yi))e(1−ψF(yi)) =ψF(yi)eψF(yi)for each yi∈Y
⇒(1−ψF(yi)) =ψF(yi)
⇒ψF(yi)=0.5 for each yi∈Y.
∂2g(ψF(yi))
∂(ψF(yi))2=−e(1−ψF(yi)) −(1−ψF(yi))e(1−ψF(yi)) −eψF(yi)−ψF(yi)eψF(yi)
√e−1
=(ψF(yi)−2)e(1−ψF(yi))−(1+ψF(yi))eψF(yi)
√e−1<0atψF(yi)=0.5 for each yi∈Y.
So g(ψF(yi)) is maximum at ψF(yi)=0.5 and is a concave function and hence from
Equation (8) it is obtained that HE(F)achieves the maximum at ψF(yi)=0.5 which implies
that [μFL(yi),μFU (yi)]=[νFL(yi),νFU (yi)].
(H3): From Fc=yi,[νFL (yi),νFU (yi)], [μFL(yi),μFU (yi)]/yi∈Yand Equation (3),
we can easily get HE(F)=HE(Fc).

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 7
(H4): If we take α=μFL(yi)+μFU (yi)and β=νFL(yi)+νFU (yi)then from Equation
(3) we have
HE(α,β) =α+2−β
4e1−α+2−β
4+1−α+2−β
4eα+2−β
4−1(10)
where α,β∈[0, 1].
Taking the partial derivative of (10) with respect to αand β,respectively,weget
∂HE
∂α =1
4β+2−α
4e1−α+2−β
4−α+2−β
4eα+2−β
4(11)
and similarly, ∂HE
∂β =1
4α+2−β
4e1−β+2−α
4−β+2−α
4eβ+2−α
4(12)
Put ∂HE
∂α =0and∂HE
∂β =0 to nd the critical points, we get α=β(13)
From (11) and (13), we get
∂HE
∂α ≥0whenα≤βand ∂HE
∂α ≤0whenα≥βfor any α,β∈[0, 1]. Thus, HE(α,β) is
increasing w.r.t. αfor α≤βand decreasing when α≥β.
Similarly, we get ∂HE
∂β ≤0whenα≤βand ∂HE
∂β ≥0whenα≥β.
Now if F⊆Gwith μGL(yi)≤νGL (yi)and μGU (yi)≤νGU (yi)for each yi∈Y.
Then we have μFL(yi)≤μGL(yi)≤νGL (yi)≤νFL(yi),
μFU (yi)≤μGU (yi)≤νGU (yi)≤νFU (yi)
It implies that μFL(yi)≤νFL(yi),andμFU (yi)≤νFU (yi). Thus, from the monotonic nature
of HE(α,β) and Equation (3), we obtained HE(F)≤HE(G).
Likewise, when F⊇Gwith μGL(yi)≥νGL(yi)and μGU (yi)≥νGU (yi)for each yi∈Y,
onecanalsoprovethatHE(F)≤HE(G).
By taking the weight of each element yi∈Y, a weighted exponential entropy measure
of an IVIFS Fis proposed as follows:
HWE(F)=1
n(√e−1)
×
n

i=1
wi⎧
⎪
⎨
⎪
⎩μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4e1−μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4
+1−μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4eμFL (yi)+μFU (yi)+2−νFL(yi)−νFU (yi)
4−1⎫
⎪
⎬
⎪
⎭
(14)
where wi∈[0, 1], i=1, 2, ...,nand n

i=1
wi=1.
If we consider wi=1
n,i=1, 2, ...,n,thenHWE (F)=HE(F).
It can be easily checked that the weighted exponential entropy measure of an IVIFS F
also satises all the properties (H1)–(H
4) of an entropy measure provided in Section 2.

8A. OHLAN
We now turn to evaluate the cross-entropy measure of IVIFSs. In 1999, Fan and Xie
(1999) presented fuzzy information for discrimination of Fagainst Gas
I(F,G)=
n

i=1
[1 −(1−μF(yi))e(μF(yi)−μG(yi)) −μF(yi)e(μG(yi)−μF(yi))] (15)
The fuzzy distance/divergence between F and G is expressed as follows
DE(F,G)=I(F,G)+I(G,F)
=
n

i=12−(1−μF(yi)+μG(yi))e(μF(yi)−μG(yi))
−(1−μG(yi)+μF(yi))e(μG(yi)−μF(yi))(16)
Now by using the above method of transformation of an FS to IVIFS and analogous
to the fuzzy distance/divergence measure (16), we can evidently dene the below distance
measure of IVIFSs.
Denition 3.2: Let F,G∈IVIFS(Y),Y={y1,y2,...,yn}, the IVIFSs based dis-
tance/divergence measure is dened by
IE(F,G)=
n

i=1
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1−1−μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4
eμFL (yi)+μFU (yi)+2−νFL(yi)−νFU (yi)
4−μGL(yi)+μGU (yi)+2−νGL (yi)−νGU (yi)
4
−μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4
eμGL(yi)+μGU (yi)+2−νGL (yi)−νGU (yi)
4−μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(17)
However, IE(F,G)is not symmetric. Accordingly, the symmetric distance measure for
IVIFSs should be as
DE(F,G)=IE(F,G)+IE(G,F). (18)
Theorem 3.2: The function DE(F,G)dened in (18) is a distance measure for IVIFSs.
Proof :In order to show that DE(F,G)is a sensible interval-valued intuitionistic fuzzy
distance measure, we need to satisfy (d1)–(d
4) in Denition 2.4.
(d1) Since in above-dened measure (17) the function is convex, refer to theorem 3.1 of
Fan and Xie (1999). The measure DE(F,G)increases as ρ(F−G)=|(μFL (yi)+μFU (yi))
−(μGL(yi)+μGU (yi))|+|(νFL(yi)+νFU (yi)) −(νGL (yi)+νGU (yi))|increase.
Thus, DE(F,G)achieves its maximum at the following degenerate cases: either F=
([1, 1], [0, 0])and G=([0, 0], [1, 1])or F=([0, 0], [1, 1])and G=([1, 1], [0, 0]).
It provides us that 0 ≤DE(F,G)≤1.
(d2)IfF=Gthen μFL(yi)=μGL(yi),μFU (yi)=μGU (yi),
if νFL(yi)=νGL (yi),νFU (yi)=νGU (yi),thenμFL(yi)−μFL(yi)=0, μFU (yi)−μGU
(yi)=0, νFL(yi)−νFL (yi)=0, νFU (yi)−νGU (yi)=0. Therefore, DE(F,G)=0.

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 9
The property (d3)isalsotriviallysatisedbymeasure(17).
(d4)IfF⊆G⊆H,wehaveρ(F−G)≤ρ(F−H)and ρ(G−H)≤ρ(F−H)
Then it is easy to see that DE(F,G)≤DE(F,H)and DE(G,H)≤DE(F,H).
Thus, it is clear that DE(F,G)is a cross-entropy or distance measure between IVIFSs F
and Gas DE(F,G)satises (d1)–(d
4).
Theorem 3.3: The relation between HE(F)and IE(F,G)is given by the following
HE(F)=1−√e
n√e−1IEF,1
2.
Proof: √eIEF,1
2=√e
n

i=1
(1−(1−OF(yi))eOF(yi)−1
2−OFe1
2−OF(yi)
where OF(yi)=μFL(yi)+μFU (yi)+2−νFL(yi)+νFU (yi)
4
=
n

i=1√e−(1−OF(yi))eOF(yi)−OF(yi)e1−OF(yi)
=−
n

i=1(1−OF(yi))eOF(yi)+OF(yi)e1−OF(yi)−1+1−√e
=−
n

i=1
((1−OF(yi))eOF(yi)+OF(yi)e1−OF(yi)−1)+n√e−1
=−n1−√eHE(F)+n√e−1=n√e−1(1−HE(F))
Thus, we get HE(F)=1−√e
n(√e−1)IEF,1
2.
4. Applications
This section presents the application and consistency of the proposed exponential and
weighted exponential entropy measures of IVIFS. First, we oer an application of IVIF
exponential entropy measure (3) to a decision-making problem. Secondly, we apply the
proposed IVIFSs based weighted exponential entropy measure (14) to a multi-criteria
group decision-making (MCGDM) problem. Thereafter, we will also compare the perfor-
mance of the proposed MCGDM method with the existing methods oered by Düğenci
(2016), and Park et al. (2011).
4.1. Application of exponential entropy measure of IVIFSs
Example 1. Take an example of decision-making provided in Wang, Li, and Wang (2009).
It is assumed that a nancial executive of a corporate entity is evaluating four alterna-
tive investment opportunities O=O1,O2,O3,O4. The rm mandates that a nancial
executive has to assess the following characteristics: risk (x1), growth (x2), socio-political
concerns (x3), and environmental impact (x4). Assume that a nancial executive prefers to

10 A. OHLAN
use interval-valued intuitionistic fuzzy number (IVIFN) to evaluate available alternatives
on each characteristic. The assumed decision matrix is
R=⎡
⎢
⎢
⎣
[0.42, 0.48], [0.4, 0.5]
[0.6, 0.7], [0.05, 0.25]
[0.4, 0.5], [0.2, 0.5]
[0.4, 0.5], [0.4, 0.5]
[0.5, 0.8], [0.1, 0.2]
[0.3, 0.6], [0.3, 0.4]
[0.3, 0.5], [0.4, 0.5]
[0.1, 0.3], [0.2, 0.4]
[0.7, 0.8], [0.1, 0.2]
[0.2, 0.4], [0.4, 0.5]
[0.6, 0.7], [0.2, 0.3]
[0.5, 0.6], [0.2, 0.3]
[0.55, 0.75], [0.15, 0.25]
[0.6, 0.7], [0.1, 0.3]
[0.5, 0.7], [0.1, 0.2]
[0.7, 0.8], [0.1, 0.2]
⎤
⎥
⎥
⎦
By using the proposed entropy measure (1), we can easily compute
HE(O1)=0.8899, HE(O2)=0.9488, HE(O3)=0.8610, HE(O4)=0.8474.
i.e. HE(O4)<HE(O3)<HE(O1)<HE(O2) which coincides with the ranking result
obtainedinWang,Li,andWang(2009).
4.2. A method of group decision-making for IVIFSs based on weighted exponential
entropy measure
We now apply the proposed IVIFSs based weighted exponential entropy measure to multi-
criteria group decision-making (MCGDM) problem. In doing so, we introduce a method
ofMCGDMusingtheIVIFweightedentropymeasureandpresentanillustrativecasestudy
to reveal its application in the real world (Ohlan and Ohlan 2022).
We provide a method of group decision-making to prefer an alternative with IVIFSs
when the weights are partially known. The proposed weighted exponential entropy
measure of IVIFSs to MCGDM is used under the IVIF environment with the known
experts and unknown criteria weights. Let us assume that A={A1,A2,...,An}be set of
nalternatives, D={d1,d2,...,dk}be a set of kexperts with the weighting vector λ=
(λ1,λ2,...,λk)such that λi∈[0, 1] and k

i=1
λi=1, C={c1,c2,...,cm}be a set of mcri-
teria with the weighting vector wj∈[0, 1] and m

j=1
wj=1. The procedural steps for method
of group decision-making with IVIFSs based weighted exponential entropy measure are as
follows:
Step 1. Determine the interval-valued intuitionistic fuzzy decision matrices (IVIFDM).
R(i)=(r(i)
lj )n×mpresents the scores of the applicants provided by the experts.
Step 2. In the process of group decision-making, the scores of available options given
by experts are required to be collected. Based on these scores, develop a collective IVIF
decision matrix having the values denoted by zlj =([μljL,μljU ], [νljL ,νljU ]).Todoso,we
employ the IIFWG operator (Xu 2007b):
zlj =([μljL,μlj U ], [νljL,νljU ])=IVIFWAλ(r(1)
lj ,r(2)
lj ,...,r(k)
lj )
=λ1r(1)
lj ⊕λ2r(2)
lj ⊕...⊕λkr(k)
lj

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 11
=!"1−
k
#
i=1
(1−μljL)λi,1−
k
#
i=1
(1−μljU )λi$,"k
#
i=1
(νljL)λi,
k
#
i=1
(νljU )λi$%. (19)
Step 3. Calculate the weight vector of the criteria.
Here we apply a method for determining the weights of the criteria using Equation (3)
in the IVIF environment when their values are completely unknown:
wj=1−HEj
m−
m

j=1
HEj
,j=1, 2, 3, ...,m(20)
where HEj =1
n
n

i=1
HE(Ai),andHE(Ai)is IVIF entropy measure dened in Equation (3).
Step 4. Calculate the weighted interval-valued information measure (WIVIM) HWE(A)
for each of the alternative Alusing Equation (14).
Step 5. Ranking of alternatives:
After obtaining the numerical values for dierent alternatives, rank the alternatives in
ascending order according to their numerical measurement of information. The leading
Al, with the smallest value of HWE(A), will be the best alternative.
4.2.1. Illustrative example
We now demonstrate a case study of Düğenci (2016)andParketal.(2011) to show the
success of the proposed IVIF MCGDM method utilizing the introduced weighted entropy
measure.
Example 2: Selecting Best Personnel (Düğenci 2016)
A service corporation is recruiting an advertising professional. Six applicants qualied
for preliminary examination and went for further evaluation. In this selection process, a
team of three subject experts has been constituted to assess all six applicants and to conduct
an interview. For further evaluation, ve assessment criteria are presented: communication
skill (C1), uency in an international language (C2), demonstrative stability (C3), earlier
experience (C4), and level of condence (C5).
The best personnel will be selected applying the proposed IVIF MCGDM method as
follows:
Step 1: Determine the interval-valued intuitionistic fuzzy decision matrices (IVIFDM).
The scores of six applicants provided by three experts subject to criteria are presented
in Tables 1–3.
Tab le 1. IVIFDM R(1)by expert 1st for personnel selection problem.
A1A2A3A4A5A6
C1([0.5,0.6],[0.2,0.3]) ([0.6,0.7],[0.2,0.3]) ([0.7,0.8],[0.1,0.2]) ([0.5,0.7],[0.1,0.2]) ([0.4,0.6],[0.1,0.3]) ([0.5,0.8],[0.1,0.2])
C2([0.6,0.7],[0.2,0.3]) ([0.7,0.8],[0.1,0.2]) ([0.5,0.6],[0.2,0.3]) ([0.6,0.7],[0.2,0.3]) ([0.5,0.7],[0.2,0.3]) ([0.4,0.6],[0.2,0.3])
C3([0.4,0.5],[0.2,0.4]) ([0.5,0.7],[0.1,0.3]) ([0.6,0.7],[0.2,0.3]) ([0.5,0.8],[0.1,0.2]) ([0.6,0.7],[0.1,0.2]) ([0.5,0.6],[0.2,0.3])
C4([0.7,0.8],[0.1,0.2]) ([0.8,0.9],[0.0,0.1]) ([0.5,0.7],[0.1,0.2]) ([0.6,0.8],[0.0,0.1]) ([0.7,0.8],[0.1,0.2]) ([0.4,0.8],[0.1,0.2])
C5([0.5,0.7],[0.1,0.2]) ([0.7,0.8],[0.1,0.2]) ([0.6,0.8],[0.1,0.2]) ([0.4,0.5],[0.3,0.4]) ([0.5,0.7],[0.1,0.2]) ([0.5,0.7],[0.1,0.2])

12 A. OHLAN
Tab le 2. IVIFDM R(2)by expert 2nd for personnel selection problem.
A1A2A3A4A5A6
C1([0.6,0.7],[0.2,0.3]) ([0.6,0.7],[0.2,0.3]) ([0.5,0.6],[0.2,0.3]) ([0.4,0.5],[0.3,0.4]) ([0.5,0.6],[0.3,0.4]) ([0.5,0.7],[0.2,0.3])
C2([0.5,0.6],[0.2,0.3]) ([0.7,0.8],[0.1,0.2]) ([0.6,0.8],[0.0,0.1]) ([0.7,0.9],[0.0,0.1]) ([0.7,0.8],[0.1,0.2]) ([0.5,0.6],[0.2,0.3])
C3([0.3,0.4],[0.4,0.6]) ([0.5,0.7],[0.1,0.2]) ([0.5,0.6],[0.2,0.3]) ([0.5,0.7],[0.1,0.2]) ([0.4,0.6],[0.2,0.3]) ([0.3,0.5],[0.4,0.5])
C4([0.7,0.8],[0.1,0.2]) ([0.7,0.9],[0.0,0.1]) ([0.7,0.8],[0.1,0.2]) ([0.6,0.7],[0.1,0.2]) ([0.6,0.8], [0.1,0.2]) ([0.5,0.6],[0.2,0.3])
C5([0.4,0.5],[0.2,0.4]) ([0.5,0.6],[0.2,0.3]) ([0.5,0.7],[0.1,0.2]) ([0.5,0.6],[0.2,0.3]) ([0.5,0.7],[0.1,0.2]) ([0.6,0.7],[0.2,0.3])
Tab le 3. IVIFDM R(3)by expert 3rd for personnel selection problem.
A1A2A3A4A5A6
C1([0.5,0.6],[0.2,0.3]) ([0.6,0.8],[0.1,0.2]) ([0.4,0.5],[0.2,0.4]) ([0.5,0.6],[0.3,0.4]) ([0.5,0.7],[0.1,0.2]) ([0.6,0.8],[0.1,0.2])
C2([0.4,0.5],[0.2,0.4]) ([0.7,0.8],[0.1,0.2]) ([0.7,0.8],[0.1,0.2]) ([0.5,0.7],[0.2,0.3]) ([0.6,0.8],[0.1,0.2]) ([0.5,0.7],[0.1,0.2])
C3([0.6,0.7],[0.2,0.3]) ([0.6,0.7],[0.2,0.3]) ([0.7,0.9],[0.0,0.1]) ([0.6,0.7],[0.0,0.2]) ([0.7,0.9],[0.0,0.1]) ([0.5,0.6],[0.2,0.3])
C4([0.7,0.8],[0.1,0.2]) ([0.7,0.8],[0.1,0.2]) ([0.5,0.6],[0.2,0.3]) ([0.7,0.8],[0.0,0.1]) ([0.5,0.7],[0.0,0.1]) ([0.5,0.7],[0.2,0.3])
C5([0.4,0.6],[0.3,0.4]) ([0.6,0.7],[0.2,0.3]) ([0.4,0.6],[0.2,0.3]) ([0.4,0.7],[0.1,0.2]) ([0.6,0.7],[0.1,0.2]) ([0.5,0.6],[0.2,0.3])
Step 2: We drive a collective IVIF decision matrix having the values denoted by zlj based
on the scores of the alternatives Al,l=1, 2, ...,ngiven by dierent experts for personnel
selection using Equation (19) with the weight vector of experts, λ=(0.3, 0.5, 0.2).The
collective IVIF matrix for personnel selection problem is presented in Table 4.
Step 3: On using Equation (20), we obtained the following weight vector for dierent
criteria w=(0.1259, 0.2449, 0.1624, 0.3249, 0.1398).
Step 4: We c a l cul at e HWE (A)for each of the alternative Alusing Equation (14) and
weights of the criteria w=(0.1259, 0.2449, 0.1624, 0.3249, 0.1398).Theobtainedvalues
are given in Table 5.
Step 5: Rank the applicants.
Based on the calculated values of HWE(A)and ranking order reported in Table 5,we
obtain that A2is the best applicant who is in accord as obtained in Düğenci (2016).
Example 3: Selecting Best Air-Conditioner (Park et al. 2011)
In this example, a group decision-making problem is concerned with a city development
commissioner to build a municipal library who wants to choose the best air-conditioner
according to the intent conguration of the library (Park et al. 2011). The supplier suggests
four possible choices Cl,(l=1, 2, 3, 4),withthevequalities:(1)performance(q
1), (2)
maintainability (q2), (3) exibility (q3), (4) cost (q4), and (5) safety (q5). It is assumed that
Q={Q1,Q2,Q3,Q4,Q5}be a set of qualities (criteria). Now there are four experts E=
e1,e2,e3having weight vector λ=(0.3.0.2, 0.3, 0.2)whoareinvitedtoassessthecore
competencies of the air conditioners. The experts present the characteristics of the choices
Cl,(l=1, 2, 3, 4)by the IVIFNs R(i)=(r(i)
lj )n×mwith respect to the qualities (criteria)Q=
{Q1,Q2,Q3,Q4,Q5}.
Step 1: Interval-valued intuitionistic fuzzy decision matrices (IVIFDM).
R(i)=(rlj(i))n×mpresents the score of the choices given by the experts, which is
scheduled in Tables 1–3(Park et al. 2011).
Step 2: A collective IVIF decision matrix having the values denoted by zlj based on the
scores of the choices Cl,(l=1, 2, 3, 4)by dierent experts for personnel selection using
Equation (19) and experts weight vector λ=(0.3.0.2, 0.3, 0.2)is obtained in Table 6.

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 13
Tab le 4. Collective IVIF decision matrix having the values zlj .
A1A2A3A4A5A6
C1([0.5528,0.6536],
[0.2000,0.3000])
([0.6,0.7234],
[0.1741,0.2766])
([0.5551,0.6603],
[0.1625,0.2814])
([0.4523,0.5898],
[0.2158,0.3249])
([0.4719,0.6224],
[0.1732,0.3194])
([0.5218,0.7551],
[0.1414,0.2449])
C2([0.5150,0.6163],
[0.2000,0.3178])
([0.7000,0.8000],
[0.1000,0.2000])
([0.5962,0.7538],
[0.000,0.1966])
([0.6378,0.8268],
[0.000,0.1732])
([0.6296,0.7741],
[0.1231,0.2259])
([0.4719,0.6224],
[0.1741,0.2766])
C3([0.4024,0.5055],
[0.2828,0.4625])
([0.5218,0.7000],
[0.1149,0.2449])
([0.5778,0.7219],
[0.000,0.2408])
([0.5218,0.7344],
[0.000,0.2000])
([0.5375,0.7219],
[0.000,0.2132])
([0.4084,0.5528],
[0.2828,0.3873])
C4([0.7000,0.8000],
[0.1000,0.2000])
([0.7344,0.8851],
[0.000,0.1149])
([0.6127,0.7405],
[0.1149,0.2169])
([0.6224,0.7551],
[0.000,0.1414])
([0.6163,0.7831],
[0.000,0.1741])
([0.4719,0.6933],
[0.1625,0.2656])
C5([0.4319,0.5898],
[0.1762,0.3249])
([0.5898,0.6933],
[0.1625,0.2656])
([0.5150,0.7186],
[0.1149,0.2169])
([0.4523,0.5962],
[0.1966,0.3016])
([0.5218,0.7000],
[0.1000,0.2000])
([0.5528,0.6822],
[0.1625,0.2656])

14 A. OHLAN
Tab le 5. Calculated values of WIVIM for per-
sonnel selection problem.
HWE(Al)Rank
A10.1675 5
A20.1311 1
A30.1496 4
A40.1433 2
A50.1463 3
A60.1764 6
Tab le 6. Collective IVIF decision matrix having the values zlj .
C1C2C3C4
Q1([0.4762,0.6378],
[0.1625,0.2814])
([0.3519,0.4719],
[0.3249,0.4939])
([0.3459,0.4949],
[0.3000,0.4277])
([0.3000,0.4319],
[0.3249,0.4850])
Q2([0.3000,0.4622],
[0.3464,0.4638])
([0.1312,0.3135],
[0.2670,0.4516])
([0.6331,0.7831],
[0.1000,0.2169])
([0.1000,0.2211],
[0.6284,0.7789])
Q3([0.6224,0.7234],
[0.1231,0.2259])
([0.3316,0.4319],
[0.3837,0.4850])
([0.5486,0.7741],
[0.1000,0.2259])
([0.1000,0.2314],
[0.5348,0.7686])
Q4([0.5000,0.6331],
[0.1000,0.2656])
([0.1712,0.3043],
[0.5486,0.6746])
([0.4319,0.6163],
[0.2169,0.3178])
([0.2211,0.3213],
[0.4277,0.6059])
Q5([0.1654,0.3795],
[0.3622,0.5348])
([0.7000,0.7551],
[0.1231,0.2259])
([0.5324,0.6751],
[0.1762,0.2980])
([0.1931,0.3150],
[0.5281,0.6541])
Tab le 7. Calculated values of WIVIM for
selecting best air-conditioner.
HWE(Cl)Rank
C10.1785 3
C20.1804 4
C30.1628 1
C40.1650 2
Step 3: On using Equation (20), we obtained the following weight vector of criteria w=
(0.0471, 0.2506, 0.2986, 0.1666, 0.2370).
Step 4: WenowcalculateHWE (Cl)for each of the alternative Clusing Equation (14) and
weights of the criteria w=(0.0471, 0.2506, 0.2986, 0.1666, 0.2370). The calculated values
are presented in Table 7.
Step 5: Rank the alternatives of an air-conditioner.
Based on calculated values of HWE(Cl)and ranking order detailed in Table 7,weobtain
that C3isthebestchoicewhichisthesameasachievedinParketal.(2011).
4.2.2. A comparative analysis
In the present subsection, we compare the performance of the proposed MCGDM method
with the existing methods oered by Düğenci (2016)andParketal.(2011). In illustra-
tive Example 2, considering personnel selection problem, it is easy to see that the ranking
order of the applicants obtained employing our proposed methodology is as:A2A4
A5A3A1A6.ThisorderindicatingA2as an appropriate applicant for selection
is in line with the results of Düğenci (2016). Also from our above Example 3 of best air-
conditioner selection problem, we obtained the ranking order:C3C4C1C2,with
thebestchoiceC3.ThisndingaccordswiththeresultsofParketal.(2011). Thus, the

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 15
proposed method is found most suitable for solving MCGDM problems due to its ability
to deal with the imprecise information, and opinions of DMs. The proposed method of
group decision-making is conned to partially known information. However, it may be
generalized to calculate the weightage of the experts.
Moreover, we calculate the performance of the developed MCGDM method utilizing
the entropy measures oered by Zhang et al. (2014), Meng and Chen (2016) and Mishra
et al. (2020) as follows:
Zhang et al. (2014)entropymeasure:
EZ(F)=1−1
2n
n

i=1&&&&
μFL(yi)−1
2&&&&+&&&&
μFU (yi)−1
2&&&&+&&&&
νFL(yi)−1
2&&&&+&&&&
νFU (yi)−1
2&&&&
Meng and Chen (2016)entropymeasure:
EMC(F)=1
n
n

i=1min{μFL(yi),νFL (yi)}+min{μFU (yi),νFU (yi)}
max{μFL(yi),νFL (yi)}+max{μFU (yi),νFU (yi)}
Mishra et al. (2020)entropymeasures:
EM1(F)=1
n
n

i=1
⎛
⎜
⎜
⎝
cot ⎛
⎜
⎜
⎝
π
4+
(|μFL(yi)−νFL (yi)+μFU (yi)−νFU (yi)|)
(1−πFU (yi)+1−πFL(yi))
8π⎞
⎟
⎟
⎠⎞
⎟
⎟
⎠
EM2(F)=1
2n
n

i=1⎛
⎝
sin μFL(yi)+μFU (yi)+2−(νFL (yi)+νFU (yi))
4π
+sin νFL(yi)+νFU (yi)+2−(μFL (yi)+μFU (yi))
4π⎞
⎠.
The calculative performance of the proposed IVIF weighted entropy measure in the
proposed decision-making methodology is compared with the existing entropy measures
utilizing the numerical examples oered by Düğenci (2016)andParketal.(2011). A
comparative study is accessible in Table 8.
Tab le 8. A comparative study.
Author/s Ranking order of alternatives Best alternative
Example 2. Selecting Best Personnel (Düğenci 2016)
Zhang et al. (2014)A2A4A3A5A1A6A2
Meng and Chen (2016)A2A3A5A4A6A1A2
Düğenci (2016)A2A4A5A3A6A1A2
Mishera et al.(2020)A2A3A4A5A1A6A2
and A2A5A4A3A1A6A2
Proposed one A2A4A5A3A1A6A2
Example 3. Selecting Best Air-Conditioner (Park et al. 2011)
Park et al. (2011)C3C2C1C4C3
Zhang et al. (2014)C3C4C1C2C3
Meng and Chen (2016)C3C4C1C2C3
Mishera et al.(2020)C3C4C1C2C3C4
and C4C1C2C3C4
Proposed one C3C4C1C2C3

16 A. OHLAN
It can be easily seen from Table 8that the ranking order of the alternatives obtained
by the proposed MCGDM method and the existing ones are almost similar. Moreover,
the best alternative coincides with the select comparative decision-making methods and
entropy measures.
5. Comparative study of distance measures for IVIFSs
In the present section, the superiority and capability of the proposed measure of IVIF is
illustrated by comparing it with the existing ones.
Xu (2007a), combining with Hausdro metric, introduced four IVIF distance measures
based on Hamming and Euclidean distance. These are as follows:
D1(F,G)=1
4
n

i=1{|μFL(yi)−μGL (yi)|+|μFU (yi)−μGU (yi)|+|νFL(yi)
−νGL(yi)|+|νFU (yi)−νGU (yi)|}
D2(F,G)=-
.
.
.
/
1
4
n

i=1{(μFL(yi)−μGL (yi))2+(μFU (yi)−μGU (yi))2
+(νFL(yi)−νGL (yi))2+(νFU (yi)−νGU (yi))2}
D3(F,G)=
n

i=1
max{|μFL(yi)−μGL (yi)|,|μFU (yi)−μGU (yi)|,|νFL(yi)
−νGL(yi)|,|νFU (yi)−νGU (yi)|}
D4(F,G)=-
.
.
.
/
n

i=1
max{(μFL(yi)−μGL (yi))2,(μFU (yi)−μGU (yi))2,
(νFL(yi)−νGL (yi))2,(νFU (yi)−νGU (yi))2}
Ye (2011) presented the fuzzy cross-entropy of the IVIFSs by analogy with the IF cross-
entropy measure as
D∗
ye(F,G)=Dye (F,G)+Dye(G,F)
where
Dye(F,G)=
n

i=1
μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4
×log μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
1
2[(μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)) +(μGL(yi)
+μGU (yi)+2−νGL(yi)−νGU (yi))]
+
n

i=1
2−μFL(yi)−μFU (yi)+νFL (yi)+νFU (yi)
4
×log 2−μFL(yi)−μFU (yi)+νFL (yi)+νFU (yi)
1
2[(2−μFL(yi)−μFU (yi)+νFL (yi)+νFU (yi))
+(2−μGL(yi)−μGU (yi)+νGL(yi)+νGU (yi))]
.

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 17
Tab le 9. Comparison of IVIF distance/divergence measures.
IVIFSs D1D2D3D4D∗
ye dGK1dGK 2DE
0.0500 0.0707 0.1000 0.1000 0.0025 0.0524 0.0373 0.0025
F1=yi, [0.4, 0.5], [0.3, 0.5]
G1=yi, [0.3, 0.5], [0.4, 0.5]0.0500 0.0707 0.1000 0.1000 −5.5511e-17 0.1063 0.0654 5.5511e-17
F2=yi, [0.4, 0.5], [0.4, 0.5]
G2=yi, [0.3, 0.5], [0.3, 0.5]0.5000 0.5000 0.5000 0.5000 0.0000 0.0000 0.0000 0.0000
F3=yi, [0.5, 0.5], [0.5, 0.5]
G3=yi, [0, 0], [0, 0]0.5000 0.7071 1.0000 1.0000 N/A 0.7698 0.5946 0.2658
F4=yi, [1, 1], [0, 0]
G4=yi, [0, 0], [0, 0]
Note: “N/A” means the division by zero problem, and “Bold” indicates unreasonable outcomes.
Garg and Kumar (2020) introduced two IVIF exponential distance measures using CNs as
follows:
dGK1(F,G) =
1−exp 0−1
3
n

i=1&&1aF(yi)−1aG(yi)&&+&&1bF(yi)−1bG(yi)&&
+&&1cF(yi)−1cG(yi)&&2
1−exp(−n)
dGK2(F,G) =
1−exp 0−1
3
n

i=1&&1aF(yi)−1aG(yi)&&
2+&&1bF(yi)−1bG(yi)&&
2
+&&1cF(yi)−1cG(yi)&&
21
22
1−exp −√n.
Table 9compares IVIF divergence measures using counterintuitive examples. A glance at
the rst two rows of Table 9reveals that the measures D1,D2,D3and D4don’t have the capa-
bility to dierentiate negative dierence from a positive dierence. Moreover, the measure
D∗
ye represents a shortcoming while calculating the dierence between the pair of IVIFSs
(F2,G2), and division by zero problem for the pair (F4,G4).ItisalsoseenthatmeasureD1
behaves the same for dierent pairs of IVIFSs (F3,G3)and (F4,G4)presented in the last
two rows of Table 9which is one of the types of unreasonable outcome. On the other hand,
measure DE(proposed) overcomes the shortcomings of the existing measures and behaves
dierently in dierent cases.
The distance measures dGK1and dGK2given in columns 7 and 8 also performed well
for dierent counterintuitive cases. However, from the point of view of minimizing the
numerical value of a distance measure between IVIFSs, the proposed measure DEis better
than other measures. It is obtained from Column 9 of Table 9that the proposed measure
DEpresents more reliable results than others. Thus, the proposed measure DEis most rea-
sonable and capable of measuring the dierence between two IVIFSs compared to existing
measures.
6. An approach to multi-attribute decision-making
We now present an approach to solve multi-attribute decision-making (MADM) problems
using the proposed IVIF distance measure. In doing so, we use an example to illustrate the

18 A. OHLAN
relevance of our study in solving MADM problems facing in day-to-day life (Ohlan et al.
2022).Inthesamesection,wealsoextendtheclassicalVIKORandTOPSISmethodsof
decision making in the IVIF environment using the proposed IVIF distance measure.
6.1. Problem description and computational steps
Let us suppose that there are malternatives F={F1,F2,...,Fm}to be estimated according
to nattributes A={A1,A2,...,An}.
The working steps of our approach to decision-making are in order:
Step 1: Build an IVIFV decision matrix.
For this, we assume that the estimation of the alternatives Fiw.r.t. the attributes Ajis an
IVIFV. fij =([cijL ,cijU ], [dijL ,dijU ])with i=1, 2, ...,m;j=1, 2, ...,n.Lettheestimated
IVIFV matrix be A=(fij)m×n.
Step 2: Calculate the weight vector of the attributes.
Here we apply the following method for determining the weights of attributes using
above-dened exponential entropy measure in Equation (3) as
wj=1−HEj
n−
n

j=1
HEj
,j=1, 2, 3, ...,n(21)
where HEj =1
m
m

i=1
HE(Fi),andHE(Fi)is IVIF entropy measure dened in Equation (3).
Step 3:NormalizedtheIVIFVdecisionmatrix
This step normalized the attribute values of the decision matrix. If all attributes Aj,j=
1, 2, ...,nareofbenettype,thenattributevaluesneednottobenormalized.Otherwise,
we normalized the decision matrix as:
A=(fij)m×ninto R=(rij)m×nwhere
rij =0fij for benet attribute aj
fc
ij for cos tattributea
j;i=1, 2, ...,m;j=1, 2, ...,n. (22)
where fc
ij is the compliment of fij, i.e. fc
ij =([dijL,dij U ], [cijL,cijU ])and rij =([eijL ,eijU ],
[fijL,fij U ]);i=1,2, .. . ,m;j=1,2, . . . ,n.
Step 4: Determine the IVIFS positive ideal F+and the IVIFS negative ideal F−:
F+=34Aj,([e+
ijL,e+
ijU ], [g+
ijL,g+
ijU ])5/j=1, 2, ...,n6,
F−=34Aj,([e−
ijL,e−
ijU ], [g−
ijL,g−
ijU ])5/j=1, 2, ...,n6, (23)
where ([e+
ijL,e+
ijU ], [g+
ijL,g+
ijU ])= m
max
i=1eijL,m
max
i=1eijU ,m
min
i=1gijL,m
min
i=1gijU ,
([e−
ijL,e−
ijU ], [g−
ijL,g−
ijU ])= m
min
i=1eijL,m
min
i=1eijU ,m
max
i=1gijL,m
max
i=1gijU . (24)
Step 5:CalculateDE
w(Fi,F+)and DE
w(Fi,F−)using DE
w(F,G)=IE
w(F,G)+IE
w(G,F)

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 19
where
IE
w(F,G)
=
n

i=1
wi
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1−1−μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4
eμFL (yi)+μFU (yi)+2−νFL(yi)−νFU (yi)
4−μGL(yi)+μGU (yi)+2−νGL (yi)−νGU (yi)
4
−μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4
eμGL(yi)+μGU (yi)+2−νGL (yi)−νGU (yi)
4−μFL(yi)+μFU (yi)+2−νFL (yi)−νFU (yi)
4
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
(25)
Step 6: Obtained the relative closeness of the ideal solution.
TherelativeclosenessofthealternativesFi,i=1, 2, ...,mw.r.t. the IVIFS F+is
DE(Fi)=DE
w(Fi,F+)
DE
w(Fi,F+)+DE
w(Fi,F−),i=1, 2, ...,m. (26)
Step 7: Rank the alternative in descending order and select the smallest one.
In brief, the concept used in the above-stated decision-making process is based on the
development of the proposed distance measure as a separation measure to overcome the
shortcomings listed in Section 5.
6.1.1. Illustrative example
Let us consider a decision-making problem discussed in (Meng and Chen 2016). A bank
planstoprovidealoantoabusinessrmtoearnmaximumprot.Thefourpotentialalter-
natives are: F1, electric power rm; F2, household rm; F3,transportrm;F4,realestate
rm. The bank has to make a decision based on four attributes: A1,theriskfactor;A
2,
the robustness of growth; A3, recovery period of investment; A4,impactontheenviron-
ment. The four potential alternatives Fi,i=1, 2, 3, 4 are to be assessed with above-stated
attributes Aj,j=1, 2, 3, 4. The goal here is to nd the most protable investment option.
A panel of experts provides the IVIFV decision matrix as below:
Step 1: Construction of an IVIFV decision matrix (Table 10).
Step 2: The attribute weight vector can be calculated using Equation (21) as follows:
w=(0.2481, 0.1880, 0.2920, 0.2719).
Step 3:NormalizedtheIVIFVdecisionmatrix
Since A1,A
3and A4arecostattributes,andA
2is a benet attribute, so the above IVIF
decision matrix given in Table 11 is normalized using Equation (22). Table 11 shows the
normalized decision matrix.
Table 10. IVIFV decision matrix.
A1A2A3A4
F1([0.3,0.5],[0.2,0.4]) ([0.4,0.5],[0.2,0.3]) ([0.3,0.5],[0.3,0.4]) ([0.5,0.7],[0.1,0.4])
F2([0.3,0.4],[0.4,0.6]) ([0.4,0.6],[0.2,0.4]) ([0.3,0.4],[0.5,0.6]) ([0.4,0.5],[0.3,0.5])
F3([0.4,0.6],[0.2,0.3]) ([0.3,0.6],[0.1,0.3]) ([0.5,0.7],[0.1,0.2]) ([0.2,0.6],[0.2,0.3])
F4([0.5,0.7],[0.2,0.3]) ([0.4,0.5],[0.2,0.4]) ([0.2,0.4],[0.3,0.5]) ([0.2,0.4],[0.3,0.5])

20 A. OHLAN
Table 11. Normalized IVIFV decision matrix.
A1A2A3A4
F1([0.2,0.4],[0.3,0.5]) ([0.4,0.5],[0.2,0.3]) ([0.3,0.4],[0.3,0.5]) ([0.1,0.2],[0.5,0.7])
F2([0.4,0.6],[0.3,0.4]) ([0.4,0.6],[0.2,0.4]) ([0.5,0.6],[0.3,0.4]) ([0.3,0.5],[0.4,0.5])
F3([0.2,0.3],[0.4,0.6]) ([0.3,0.6],[0.1,0.3]) ([0.1,0.2],[0.5,0.7]) ([0.2,0.3],[0.2,0.6])
F4([0.2,0.3],[0.5,0.7]) ([0.4,0.5],[0.2,0.4]) ([0.3,0.5],[0.2,0.4]) ([0.3,0.5],[0.2,0.4])
Table 12. Values of DE
w(Fi,F+)and DE
w(Fi,F−).
DE
w(Fi,F+)DE
w(Fi,F−)
F10.0319 0.0157
F20.0022 0.0584
F30.0513 0.0069
F40.0184 0.0018
Table 13. Relative closeness DE(Fi),i=
1, 2, ...,mof the ideal solution.
DE(Fi)Ranking
F10.6702 2
F20.0363 1
F30.8814 3
F40.9109 4
Step 4: From the above normalized IVIFV decision matrix, we obtain the IVIFS positive
ideal F+and the IVIFS negative ideal F−using Equations (23) and (24) as follows:
F+=0A1,([0.4, 0.6], [0.3, 0.4]),A2,([0.4, 0.6], [0.1, 0.3]),
A3,([0.5, 0.6], [0.2, 0.4]),A4,([0.3, 0.5], [0.2, 0.4])2,
F−=0A1,([0.2, 0.3], [0.5, 0.7]),A2,([0.3, 0.5], [0.1, 0.3]),
A3,([0.1, 0.2], [0.5, 0.7]),A4,([0.1, 0.2], [0.5, 0.7])2.
Step 5: The calculated numerical values of DE
w(Fi,F+)and DE
w(Fi,F−)obtained using
Equation (25) are given in Table 12.
Step 6: We get the relative closeness of the ideal solution using Equation (26) as reported
in Table 13.
Step 7: After obtaining the relative closeness of the alternatives, we get that the ranking
order of the alternatives presented in Table 13 is: F2F1F3F4. Likewise, the smallest
one is F2. It indicates that household enterprise (F2) is the best choice. These results are
similar to those obtained in Meng and Chen (2016).
6.2. Extended IVIF VIKOR and TOPSIS methods for MADM: A comparative analysis
This subsection introduces extended VIKOR and TOPSIS methods of MADM in IVIF
situation. To do so, we utilize the proposed IVIF distance measure.
6.2.1. Extended IVIF VIKOR method for MADM
The Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method was estab-
lished by Opricovic and Tzeng (2002) to optimize multi-attributes. In this literature, some

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 21
scholars (e.g. Dong, Yuan, and Wan 2007; Wan, Wang, and Dong 2013; Wan and Zheng
2015) extended the VIKOR methodology in several fuzzy conditions. Based on the classi-
cal VIKOR methodology (Opricovic and Tzeng 2004;2007), the computational procedural
steps of the extended technique in IVIF environment are as follows:
Step 1–5: Refer to steps 1–5 given in Section 6.1.
Step 6:Computethevalues˜
Siand ˜
Rifor i=1, 2, ...,m,forthealternativeFi,usingthe
relations
˜
Si=
n

i=1
DE
w(Fi,F+)
DE
w(F+,F−),Si∈[0, 1] (27)
˜
Ri=max
iDE
w(Fi,F+)
DE
w(F+,F−),Ri∈[0, 1]. (28)
Step 6:Calculatethe ˜
Qivalues for i=1, 2, ...,mwith the relation
˜
Qi=ψ!˜
Si−˜
S+
˜
S−−˜
S+
i%+(1−ψ)!˜
Ri−˜
R+
˜
R−−˜
R+
i%(29)
where ˜
S+=min
i˜
Si,˜
S−=max
i˜
Si,˜
R+=min
i˜
Ri,˜
R−=max
i˜
Ri.
Also, ψand (1−ψ) representtheweightageofthestrategyofmaximumgrouputility
(majority of attribute) and the weightage of the individual regret, respectively.
Step 7:Rankthealternativesbysortingeach˜
Si,˜
Riand ˜
Qivalues in ascending order. The
alternative with a minimum value of ˜
Qiin accordance with ˜
Siand ˜
Riwillbethebestoption.
Now an application of proposed IVIF distance measure with the extended VIKOR
method is demonstrated using an IVIFV decision matrix presented in Table 10.
Table 14 shows the computed values of ˜
Si,˜
Riand ˜
Qiusing Equations (27)–(29). It also
presents the ranking order of dierent alternatives and compromise results for dierent val-
ues of ψ. Thus, corresponding to the priority ranking of dierent alternatives, it is revealed
that F2is the best option. Thus, the results obtained from the extended VIKOR method are
similar to those obtained in our proposed method (above in Section 6.1) and those of Meng
and Chen (2016).
6.2.2. Extended IVIF TOPSIS method for MADM
The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) was devel-
oped by Hwang and Yoon (1981) for MCDM. Further, TOPSIS is extended by numerous
scholars (e.g. Park et al. 2011;ZhangandYu2012;WangandChen2017;GargandKumar
2020; Zeng, Chen, and Fan 2020) in several fuzzy situations. Based on the classical TOPSIS
methodology (Hwang and Yoon 1981), the computational steps of the extended procedure
are as follows:
The computational procedure of the IVIF TOPSIS method is as follows:
Step 1–5: Refer to steps 1–5 in Section 6.1.
Step 6: Obtained the closeness coecient of the ideal solutions

22 A. OHLAN
Table 14. Computed Values of ˜
Si,˜
Riand ˜
Qi.
Item F1F2F3F4Ranking
Compromise solution
(best option)
˜
Si0.0618 0.0297 0.6923 0.2483 F2F1F4F3F2
˜
Ri0.2834 0.0756 0.4977 0.2126 F2F4F1F3F2
˜
Qi(ψ =0)0.4923 0.0000 1.0000 0.3246 F2F4F1F3F2
˜
Qi(ψ =0.1)0.4479 0.0000 1.0000 0.3251 F2F4F1F3F2
˜
Qi(ψ =0.2)0.4035 0.0000 1.0000 0.3256 F2F4F1F3F2
˜
Qi(ψ =0.3)0.3591 0.0000 1.0000 0.3262 F2F4F1F3F2
˜
Qi(ψ =0.4)0.3148 0.0000 1.0000 0.3267 F2F4F1F3F2
˜
Qi(ψ =0.5)0.2704 0.0000 1.0000 0.3272 F2F4F1F3F2
˜
Qi(ψ =0.6)0. 2260 0.0000 1.0000 0.3278 F2F4F1F3F2
˜
Qi(ψ =0.7)0. 1816 0.0000 1.0000 0.3283 F2F4F1F3F2
˜
Qi(ψ =0.8)0.1372 0.0000 1.0000 0.3288 F2F4F1F3F2
˜
Qi(ψ =0.9)0.0928 0.0000 1.0000 0.3294 F2F4F1F3F2
˜
Qi(ψ =1)0.0484 0.0000 1.0000 0.3299 F2F4F1F3F2
Note: ˜
S−=0.6923, ˜
S+=0.0297, ˜
R−=0.4977, ˜
R+=0.0756.
Table 15. Closeness coefficient DE(Fi),i=
1, 2, ...,m.
DE(Fi)Ranking
F10.3298 2
F20.9637 1
F30.1186 3
F40.0891 4
TherelativeclosenessofthealternativesFi,i=1, 2, ...,mw.r.t. the id eal solution is
DE(Fi)=DE
w(Fi,F−)
DE
w(Fi,F+)+DE
w(Fi,F−),i=1, 2, ...,m. (30)
Step 7: Determine the rank of the alternatives by sorting the descending order of close-
ness coecient DE(Fi). An alternative with the maximum value of DE(Fi)will be the best
option.
We now present the application of the extended TOPSIS method using the IVIFV
decision matrix in Table 10.
Table 15 presents the numerical value of the closeness coecient with the corresponding
ranking of dierent alternatives we obtained using Equation (28). Thus, Table 15 depicts
that F2is the best alternative which seems to be in line with the proposed method in Section
6.1. Therefore, the proposed method is reliable.
6.2.3. A comparative study of proposed and existing methods of MADM
This subsection compares the proposed decision-making method with the extended IVIF
TOPSIS method, extended IVIF VIKOR method, and method given by Meng and Chen
(2016), respectively.
The comparative results presented in Table 16 determine that the outcome gured by
the proposed methodology is in accord with the existing ones. This nding validates our
results. It is also noticed that the proposed method is consistent, simple and ecient among
the compared ones.

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 23
Table 16. Comparative analysis.
Method Ranking Optimal alternative
Meng and Chen (2016)F2F4F1F3F2
Extended IVIF VIKOR Method
(Opricovic and Tzeng 2004;2007)
F2F4F1F3F2
Extended IVIF TOPSIS Method
(Hwang and Yoon 1981)
F2F1F3F4F2
Proposed IVIF MADM Method F2F1F3F4F2
6.3. Further discussion
To develop the decision-making method using the WIVIF entropy measure, the weigh-
tage of each expert is considered. In this way, a method of group decision-making using
weighted exponential entropy measure in IVIF situation is developed which involves the
stage of nding the weights of dierent alternatives. However, to get a fair-minded ranking
of alternatives, a MADM method using IVIF exponential distance measure is developed by
assigning equal weights to all the attributes. Thus, the current study oers an appropriate
method of solving decision-making problems in IVIF settings due to its ability to deal with
the imprecise information and opinions of DMs.
7. Concluding remarks
In this paper, we have extended the decision-making methods under the IVIF environ-
ment. To do so, we proposed novel exponential entropy and distance measures for IVIFSs.
The eciency and consistency of the IVIF exponential entropy measure are demonstrated
with the help of an example. A weighted IVIF exponential entropy measure is provided
based on the proposed exponential entropy measure. A method to solve the MCGDM
problem is presented by employing the proposed weighted IVIF exponential entropy mea-
sure. The comparative analysis of illustrations from the extant literature accomplished the
consistency and applicability of the proposed method of MCGDM in solving the personnel
selection problem.
Further, the capability of the proposed IVIF distance measure is shown by comparing
it with existing distance/divergence measures. An approach has been established to obtain
the solution of the MADM problem applying the introduced distance measure. The eec-
tiveness of the method is presented with an example. Finally, a comparative analysis of the
proposed method with the extended VIKOR, TOPSIS, and other similar existing methods
validate our results. It is established that the proposed measures and methods are much
simpler, ecient, and consistent compared with existing measures and methods of solving
decision-making problems. The limitation of the proposed approach is that its applicabil-
ity is limited to the IVIF distance measure in decision-making. But it can be generalized
further with weight vector. Likewise, a group decision-making method can be established
utilizing weighted measure.
In future research, the measures of information can be extended to Pythagorean fuzzy
sets (PFSs), interval-valued PFSs, picture fuzzy sets, and hesitant fuzzy sets. Moreover, the
applications of the proposed information measure and methods of decision-making can
be explored in the real-world problems relating to supplier selection, weather forecasting,
enterprise resource planning selection, and facility location selection.

24 A. OHLAN
Disclosure statement
No potential conict of interest was reported by the author(s).
Notes on contributor
Anshu O hlan attained Ph.D. in Mathematics from Deenbandhu Chhotu Ram
University of Science and Technology, Murthal (Sonepat), Haryana, India, in
2016. She is presently working as a Lecturer of Mathematics in the Department
of Education, Government of Haryana, India. Her areas of research interest
include fuzzy sets and systems, fuzzy information measures, and fuzzy compu-
tational intelligence design and applications. Her sole-authored research work
hasgainedmorethan230citationsinGooglescholar,withh-index10,andi-10
index 10.
ORCID
Anshu Ohlan http://orcid.org/0000-0003-0489-3793
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