https://doi.org/10.31449/inf.v42i4.1303 Informatica 42 (2018) 617–627 617 
Entropy, Distance and Similarity Measures under Interval-Valued 
Intuitionistic Fuzzy Environment 
Pratiksha Tiwari and Priti Gupta 
Delhi Institute of Advanced Studies, Plot No. 6, Sector- 25, Rohini, Delhi, India 
E-mail: parth12003@yahoo.co.in 
 
Keywords: entropy measure, distance, similarity measure, interval valued intuitionistic fuzzy sets 
Received: July 21, 2016 
 
This paper presents new axiomatic definitions of entropy measure using concept of probability and 
distance for interval-valued intuitionistic fuzzy sets (IvIFSs) by considering degree of hesitancy which is 
consistent with the definition of entropy given by De Luca and Termini. Thereafter, we propose some 
entropy measures and also derived relation between distance, entropy and similarity measures for 
IvIFSs. Further, we checked the performance of proposed entropy and similarity measures on the basis 
of intuition and compared with the existing entropy and similarity measures using numerical examples. 
Lastly, proposed similarity measures are used to solve problems in the field of pattern recognition and 
medical diagnoses.  
Povzetek: V prispevku so predstavljene nove aksiomatske definicije entropijske mere za intervalno 
intuicionistične mehke množice. 
1 Introduction 
Fuzzy set theory (Zadeh, 1965) is tool that can handle 
uncertainty and imprecision effortlessly. Interval-valued, 
intuitionistic, interval-valued intuitionistic fuzzy sets 
(Zadeh (1975), Atanassov (1986), Atanassov & Gargov 
(1989)), vague sets (Gau & Buehrer, 1993) and R-fuzzy 
sets (Yang, Hinde, 2010) are various generalizations of 
Fuzzy sets (FSs). From all these generalizations IvIFSs 
and intuitionistic fuzzy sets (IFSs) are two conventional 
extensions of FSs. IvIFSs are more practical and flexible 
than IFSs as they are characterized by membership and 
non-membership degree range instead of real numbers. It 
makes IvIFSs more useful in dealing with real world 
complexities which arises due to insufficient information, 
lack of data, imprecise knowledge and human nature 
wherein range is provided instead of real numbers. 
Distance, entropy and similarity measures are the central 
arenas that are investigated by various researchers under 
intuitionistic and interval-valued fuzzy environment (IFE 
and IvFE). These measures identify the similarity or 
dissimilarity between two FSs. Till date, vivid entropy, 
distance or similarity measures are presented by various 
investigators. Some of these research findings are 
mentioned as follows: Xu (2007 a, b) introduced the 
concept of similarity between IvIFSs along with some 
distance measure. Zang et al. (2009) defined a entropy 
axiomatically for interval-valued fuzzy sets (IvFSs) and 
discussed relation between entropy and similarity 
measures. Xu and Yager (2009) studied preference 
relation and defined similarity measure under IvFE and 
interval-valued intuitionistic environment (IvIFE).  Wei 
et al (2011) derived a generalized measure of entropy for 
IvIFSs. Cosine similarity measures for IvIFSs are defined 
by both Ye (2012) and Singh (2012). Sun & Liu (2012), 
Hu & Li (2013), Zhang et al. (2014) proposed entropy 
and similarity measure along with their relationship for 
IvIFSs. Applications of the aforesaid entropy, distance 
and similarity measures are for recognition of patterns, 
medical diagnoses, and decision making with multiple 
criteria and expert systems problems. However, most of 
these distance, similarity or entropy measures do not 
consider hesitancy index between IvIFSs. Hesitance 
index play a very important role when membership and 
non-membership degree do not differ much for two 
IvIFSs but their hesitant index does. Some of the authors, 
Xu (2007), Xu & Xia (2011), Wu et al. (2014) 
considered hesitancy into the measure of distance, 
similarity and entropy developed by them. Since 
hesitancy index also has a vital role in any decision 
making as it outclasses the existing methods and deals 
with decision process in a better way. Dammak et al. 
(2016) studies some possibility measures in multi-criteria 
decision making under 𝐼𝑣 𝐼 𝐹𝐸 . Tiwari & Gupta(in press) 
proposed generalized entropy and similarity measure for 
𝐼𝑣𝐼𝐹𝑆 with application in decision making. Zang et al. 
(2016) defined some operations on 𝐼𝑣𝐼𝐹𝑆𝑠 and proposed 
some aggregation operators for 𝐼𝑣𝐼𝐹𝑆𝑠 w.r.t. the 
restricted interval Shapley function with application in  
multi-criteria decision making. In this paper we have 
developed some of the distance, entropy and similarity 
measures by taking all the three degrees in account and 
applied it to pattern recognition and medical diagnoses 
under 𝐼𝑣𝐼𝐹 𝐸 .  
This work is organized in various sections. Section 2 
has basic definition and operations on 𝐼𝑣𝐼𝐹𝑆𝑠 . Section 3, 
presents the relationship between distance and entropy 
measures along with example to check the performance 
of entropy measures on the basis of intuition. A relation 
between measure of entropy and similarity measure is 
proposed in Section 4. Further, comparison of new 
similarity measures with the few existing one in done. 

618 Informatica 42 (2018) 617–627 P. Tiwari et al.  
 
Thereafter in section 5 we applied new similarity 
measures to recognition of patterns and medical 
diagnoses. Lastly conclusion is drawn in Section 6. 
2 IvIFSs along with its distance and 
similarity measures 
This section has definitions and concepts for IvIFSs. In 
this paper Ω={𝑥 1
,….,𝑥 𝑛 } denotes the universe of 
discourse;ℂ(Ω) and IvIFSs(Ω) denote all crisp sets and 
𝐼𝑣𝐼𝐹𝑆𝑠 respectively in Ω. 
Definition 1 ( Atanassov & Gargov,1989): An IvIFS  A 
in the finite universe Ω is defined by a triplet 
〈𝑥 𝑖 ,𝑀𝑉
𝐴 (𝑥 𝑖 ),𝑁𝑉
𝐴 (𝑥 𝑖 ),𝐻𝑉
𝐴 (𝑥 𝑖 )〉 as 𝑥 𝑖 ∈Ω where 
𝑀𝑉
𝐴 (𝑥 𝑖 )=[𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )] is called membership 
value interval ,𝑁𝑉
𝐴 (𝑥 𝑖 ) =[𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )] is called 
non-membership value interval and 𝐻𝑉
𝐴 (𝑥 𝑖 )=
[𝐻𝑉
𝐴𝐿
(𝑥 𝑖 ),𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )],𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )=1−𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−
𝑁𝑉
𝐴 𝑈 (𝑥 𝑖 ) , 𝐻 𝐴𝑈
(𝑥 𝑖 )=1−𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ) such 
that 0 ≤𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )+𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )≤1 for each 𝑥 𝑖 ∈𝐴 . 
Liu (1992) defined distance and similarity measures 
for IvIFSs axiomatically which are given as follows:  
Definition 2: For any two IvIFSs A and B, a real valued 
function D:IvIFSs(Ω)×IvIFSs(Ω)⟶[0,1] is termed as 
a distance measure of IvIFSs on Ω, if it satisfies the 
below mentioned axioms: 
1. For any crisp set A , we have D(A,A
̅
 )=1. 
2. Distance between any two IvIFSs A and B is 
zero  iff A=B. 
3. Distance measure is symmetrical w.r.t to any 
two IvIFSs A and B. 
4. For any three IvIFSs A,B and C such that A⊆
B⊆C, we have D(A,C)≥D(A,B) 
and D(A,C)≥D(B,C). 
Distance between FSs was presented by (Kacprzyk, 
1997). Then its extension was proposed by Atanassov in 
1999 as two dimensional distances whereas third 
parameter hesitancy degree in distance was introduced by 
Szmidt and Kacprzyk (2000) for intuitionistic fuzzy sets. 
Yang & Chiclana (2012) proved three dimensional 
distance consistency over two dimensional distances. 
Grzegorzewski (2004) and Park et al. (2007) gave 
distance measure for IvFSs and IvIFSs respectively. Here 
we extend the distance measures by considering 
hesitancy degree for IvIFSs. For any two IvIFSs A and B, 
we define the following measures of distance: 
1) Normalized Euclidean Distance  
D
1
(A,B)={
1
12𝑛 ∑ [(𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−
𝑛 𝑖 =1
𝑀𝑉
𝐵𝐿
(𝑥 𝑖 ))
2
+(𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 ))
2
+
(𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 ))
2
+(𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−
𝑁𝑉
𝐵𝑈
(𝑥 𝑖 ))
2
+(𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 ))
2
+
(𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 ))
2
]}
1
2
⁄
  
    …(1) 
2) Normalized Hamming Distance  
𝐃 𝟐 (𝐀 ,𝐁 )=
𝟏 𝟖𝐧
∑ [|𝑴𝑽
𝑨𝑳
(𝒙 𝒊 )−𝑴𝑽
𝑩𝑳
(𝒙 𝒊 )|+
𝒏 𝒊 =𝟏 |𝑴𝑽
𝑨𝑼
(𝒙 𝒊 )−𝑴𝑽
𝑩𝑼
(𝒙 𝒊 )|+|𝑵𝑽
𝑨𝑳
(𝒙 𝒊 )−
𝑵𝑽
𝑩𝑳
(𝒙 𝒊 )|+|𝑵𝑽
𝑨𝑼
(𝒙 𝒊 )−𝑵𝑽
𝑩𝑼
(𝒙 𝒊 )|+
|𝑯𝑽
𝑨𝑳
(𝒙 𝒊 )−𝑯𝑽
𝑩𝑳
(𝒙 𝒊 )|+|𝑯𝑽
𝑨𝑼
(𝒙 𝒊 )−
𝑯𝑽
𝑩𝑼
(𝒙 𝒊 )|]    
    …(2)  
3) Hamming Hausdorff Normalized Distance 
D
3
(A,B)=
1
4n
∑ [|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
𝑛 𝑖 =1
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|+|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−
𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|+
|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−
𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|]    
    …(3) 
4) Hausdorff  Normalized Hamming Distance 
D
4
(A,B)=
1
4n
∑ 𝑚𝑎𝑥 {
 
 
 
 
|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|
2
,
|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|
2
,
|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|
2
}
 
 
 
 
𝑛 𝑖 =1
    …(4) 
5) Averaged fifth Distance Measure  
D
5
(A,B)=
1
2n
∑
{
 
 
 
 
[
|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|
+|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|
+|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|
]
8
+
𝑛 𝑖 =1
𝑚𝑎𝑥 (
|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|,
|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|,
|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|
)
4
}
 
 
 
 
  
     …(5) 
6) Generalized Measure of Distance , for p≥2, 
D
6
(A,B)={
1
12n
∑ (|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
𝑛 𝑖 =1
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝑝 +(|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−
𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝑝 +
(|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−
𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝑝 }
1/p
    
    …(6) 
Definition 3: Let A and B be any two IvIFSs, a real 
valued function S:IvIFSs(Ω)×IvIFSs(Ω)⟶[0,1] is 
defined as a measure of similarity for IvIFSs on Ω, if it 
satisfies axioms mentioned below: 
1. For any crisp set A , we have S(A,A
̅
 )=0  
2. Measure of similarity between any two IvIFSs is 
1 iff A=B. 
3. Measure of similarity is symmetric w.r.t. any 
two IvIFSs. 
4. For any three IvIFSs A,B and C such that A⊆
B⊆C. We have  S(A,C)≤S(A,B) and 
S(A,C)≤S(B,C) . 

Entropy, Distance and Similarity Measures under... Informatica 42 (2018) 617–627 619 
From axiomatic definition of distance and similarity 
measures it is clear that S (A,B)=1− D(A,B) where  A 
and B are IvIFSs , D  and S are distance and similarity 
measure for IvIFSs respectively. 
2.1 Entropy measure for IvIFSs 
In 1972, De Luca and Termini defined measure of 
entropy for FSs. Hung & Yang (2006) extended 
definition for IFSs considering hesitancy degree. The 
following definition for entropy is an extension of 
definition of entropy proposed by Hung & Yang (2006) 
for IvIFSs. 
Definition 4: A real valued function E:IvIFSs(Ω)→
[0,1] is termed as measure of entropy under IvIFE, if 
below mentioned axioms are satisfied: 
1. E(A)=0, ∀A∈ ℂ(Ω) ; 
2. E(A)=1, iff  𝑀 𝐴 (𝑥 𝑖 )=𝑁 𝐴 (𝑥 𝑖 )=𝐻 𝐴 (𝑥 𝑖 )=
[
1
3
,
1
3
], ∀ 𝑥 𝑖 ∈Ω; 
3. E(A)≤E(B) , if A is less fuzzy than B; 
4. E(A)=E(A
̅
) , where A
̅
 is complement of  A , 
where A,B ∈IvIFSs(Ω) . 
Above definition is steady with description of measure of 
entropy given by De Luca & Termini (1972). As it is 
known that complete description of an IvIFS A∈Ω has 
three degrees membership, non-membership and 
hesitancy with 𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )+𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )+𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )=1 
and 𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )+𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )+𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )=1 with  
0≤𝑀𝑉
𝐴𝑈
(𝑥 𝑖 ),𝑁𝑉
𝐴𝑈
(𝑥 𝑖 ),𝐻𝑉
𝐴𝐿
(𝑥 𝑖 ), 
𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ),𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )≤1. By taking all the 
three in to consideration we may assume them as 
probability measure. Therefore the entropy is maximum 
when all the variables are equal (i.e. 𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )=
𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )= 𝐻𝑉
𝐴 𝐿 (𝑥 𝑖 )=
1
3
 and 𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )=𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )=
 𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )=
1
3
) and zero (minimum) when only one 
variable is exists (i.e. 𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )=𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )=
1,𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )=𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )= 0,𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )=𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )=0 
or  𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )=𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )=0,𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )=𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )=
 1,𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )=𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )=1 
or  𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )=𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )=0,𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )=
𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )= 0,𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )=𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )=1).  
Again we extend the definition of entropy given by Zang 
et al. (2014) based on distance for IvIFSs. In following 
definition we have considered degree of hesitation, 
which is not considered by other definitions of entropy. 
Definition 5: A real-valued function E:IvIFSs(Ω)→
[0,1] is termed as measure of entropy under IvIFE, if the 
following axioms are satisfied: 
 
1. E(A)=0, ∀A∈ ℂ(Ω) ; 
2. E(A)=1, iff all the three description of IvIFSs 
intervals satifies   𝑀𝑉
𝐴 (𝑥 𝑖 )=𝑁𝑉
𝐴 (𝑥 𝑖 )=
𝐻𝑉
𝐴 (𝑥 𝑖 )=[
1
3
,
1
3
], ∀ 𝑥 𝑖 ∈Ω 
3. If D(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)≥
D(B,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) ,then E(A)≤E(B),
∀A,B∈IvIFSs(Ω) , where D is measure of 
distance. 
4.  E(A)=E(A
̅
) , where A
̅
 is complement of  A , 
where A,B ∈IvIFSs(Ω) . 
In the next section, we derive a relation which relates 
measure of distance and entropy for IvIFSs, which 
satisfies all the axioms of the definition of entropy. 
3 Relation between measure of 
distance and entropy 
Here, we develop a technique which obtains entropy 
measure for IvIFSs which satisfies the aforementioned 
properties. 
Theorem 2: Let D
j
, j=1,…,6 be the above-mentioned 
six distance measure equations (1)-(6) between IvIFSs, 
then, E
j
(A)=1−3D
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) , j=
1,…,6  for any A ∈IvIFSs(Ω) are measure of entropy of 
IvIFSs. 
Proof: We prove that E
𝑗 (A) , for j=1,…,6  satisfies 
conditions given by definition 5. 
Property1): If A∈ ℂ(Ω)⇒𝐴 (𝑥 𝑖 )=〈[1,1],[0,0],[0,0]〉 
or 𝐴 (𝑥 𝑖 )=〈[0,0],[1,1],[0,0]〉, ∀ 𝑥 𝑖 ∈𝛺 , then for j=
1,…,6 
D
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)=
1
3
. 
Thus, E
j
(A)=0 
Property 2): For all j=1,…,6,E
j
(A)=1 
⟺1−3D
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)=1 
⟺3D
𝑗 (A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)=0 
⟺ A=〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉 
Property3): Let A and B be any two IvIFSs and 
D
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)≥
D
j
(B,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) then 
1−3D
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)
≥1−3D
j
(B,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) 
⟹E
j
(A)≤E
j
(B) ,  for all j=1,…,6 
Property 4) : Let A be any 𝐼𝑣𝐼𝐹𝑆 then A
̅
=
{〈𝑥 𝑖 ,[𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )],[𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )]〉 𝑥 𝑖 ∈Ω ⁄ } 
⟹D
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)
=D
𝑗 (A
̅
,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) 
Thus, E
𝑗 (A)=E
j
(A
̅
) , for all j=1,…,6. ∎ 
From theorem 2 and various distance formulas’ 
mentioned (equation (1) to (6)), we get corresponding 
entropy formulas as follows: 

620 Informatica 42 (2018) 617–627 P. Tiwari et al.  
 
𝐸 1
(𝐴 )=1−3{
1
12𝑛 ∑ [(𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
)
2
+
𝑛 𝑖 =1
(𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
)
2
+(𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
)
2
+
(𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
)
2
+(𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
)
2
+
(𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
)
2
]}
1
2
⁄
   
     
𝐸 2
(𝐴 )=1−
3
8𝑛 ∑ [|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
|+
𝑛 𝑖 =1
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
|+|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
|+
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
|+|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
|+
|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
|]    
      
𝐸 3
(𝐴 )=1−
3
4𝑛 ∑ [|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
|∨
𝑛 𝑖 =1
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
|+|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
|∨
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
|+|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
|∨
|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
|]    
 
𝐸 4
(𝐴 )=1−
3
2𝑛 ∑ 𝑚𝑎𝑥 {
 
 
 
 
|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−1/3|+|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−1/3|
2
,
|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−1/3|+|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−1/3|
2
,
|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−1/3|+|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−1/3|
2
}
 
 
 
 
𝑛 𝑖 =1
   
E
5
(A,B)
=1
−
3
2n
∑
{
 
 
 
 
 
 
[
|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−1/3|+|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−1/3|
+|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−1/3|+|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−1/3|
+|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−1/3|+|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−1/3|
]
8
n
i=1
+
max(
|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−1/3|+|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−1/3|,
|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−1/3|+|𝑁 𝑉 𝐴𝑈
(𝑥 𝑖 )−1/3|,
|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−1/3|+|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−1/3|
)
2
}
 
 
 
 
 
 
 
 
𝐸 6
(𝐴 ,𝐵 )=1−3∑ [
1
12𝑛 [(|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
|∨
𝑛 𝑖 =1
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
|)
𝑝 +(|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
|∨
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
|)
𝑝 +(|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−
1
3
|∨
|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−
1
3
|)
𝑝 ]]
1
𝑝 ⁄
    
To check the consistency of proposed entropy measures 
with the intuitionist beliefs we have used the following 
example.  
 
Example: Consider two IvIFSs 𝐴 =
{𝑥 ,〈[0.2,0.2],[0.2,0.3],[0.5,0.6]〉,𝑥 ∈𝛺 } and 𝐵 =
{𝑥 ,〈[0.2,0.3],[0.4,0.6],[0.1,0.4]〉,𝑥 ∈𝛺 }, clearly we can 
see that A is more fuzzy than B. Then E
𝑗 (A) and E
𝑗 (B) 
are given in Table 1: 
Entropies A B 
E
1
 0.5570 0.65866 
E
2
 0.675 0.7 
E
3
 0.6 0.525 
E
4
 0.675 0.75 
E
5
 0.5125 0.6 
E
6
 0.7171 0.672 
Table 1: Entropies. 
Since E
3
(A)>E
3
(B) and E
6
(A)>E
6
(B) which 
indicates that E
3
and E
6
are consistent with the intuition. 
3.1 Comparison of existing entropy 
measure with proposed entropy 
measures 
We compared performance of existing entropy measures 
with the proposed measures with the help of an example. 
Let A be an IvIFS, then 
𝐸 𝑍𝐽
(𝐴 )=
1
𝑛 ∑
𝑚𝑖𝑛 (𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ))+𝑚𝑖 𝑛 (𝑀𝑉
𝐴𝑈
(𝑥 𝑖 ),𝑁𝑉
𝐴𝑈
(𝑥 𝑖 ))
𝑚𝑎𝑥 (𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ))+𝑚𝑎𝑥 (𝑀𝑉
𝐴𝑈
(𝑥 𝑖 ),𝑁𝑉
𝐴𝑈
(𝑥 𝑖 ))
𝑛 =1
, 
 
𝐸 𝑊𝑊
(𝐴 )=
1
n
∑
[
min(MV
AL
(x
i
),NV
AL
(x
i
))
+min(MV
AU
(x
i
),NV
AU
(x
i
))+HV
AL
(x
i
)+HV
AU
(x
i
)
]
[
max(MV
AL
(x
i
),NV
AL
(x
i
))
+max(MV
AU
(x
i
),NV
AU
(x
i
))++HV
AL
(x
i
)+HV
AU
(x
i
)
]
n
=1
, 
 
𝐸 𝑍𝑀
(𝐴 )=
1
𝑛 ∑ [1−(𝑀𝑉
̅̅̅̅̅
𝐴 (𝑥 𝑖 )+
𝑛 𝑖 =1
𝑁𝑉
̅̅̅̅
𝐴 (𝑥 𝑖 ))𝑒 1−(𝑀𝑉
̅̅̅̅̅
𝐴 (𝑥 𝑖 )+𝑁𝑉
̅̅̅̅
𝐴 (𝑥 𝑖 ))
], 
 
where 𝑀𝑉
̅̅̅̅̅
𝐴 (𝑥 𝑖 )=𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )+𝜏 ∆𝑀𝑉
𝐴 (𝑥 𝑖 ) , 𝑁𝑉
̅̅̅̅
𝐴 (𝑥 𝑖 )=
𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )+𝜏 ∆𝑁𝑉
𝐴 (𝑥 𝑖 ) and ∆𝑀 𝑉 𝐴 (𝑥 𝑖 )=𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−
𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ) and ∆𝑁𝑉
𝐴 (𝑥 𝑖 )=𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ) , 𝜏 ∈
[0,1]  
are the measures of entropies given by Zang et al. (2010), 
Wei et al.(2011) and Zang et al. (2011) respectively. 
Example: Consider the example from Sun & Liu (2012) 
to review the entropies for an IvIFSs 𝐴 𝑖 which are as 
follows: 
 
“𝐴 1
={𝑥 ,〈[0.1,0.2],[0.2,0.4],[0.4,0.7]〉,𝑥 ∈𝛺 },  
𝐴 2
={𝑥 ,〈[0.2,0.2],[0.3,0.5],[0.3,0.5]〉,𝑥 ∈𝛺 }, 
𝐴 3
={𝑥 ,〈[0.2,0.4],[0,0],[0.6,0.8]〉,𝑥 ∈𝛺 }, 
𝐴 4
={𝑥 ,〈[0.3,0.4],[0,0.142857],[0.4571,0.7]〉,𝑥 ∈𝛺 }, 
𝐴 5
={𝑥 ,〈[0.1,0.1],[0,0.2],[0.6,0.9]〉,𝑥 ∈𝛺 },  
𝐴 6
={𝑥 ,〈[0,0.2],[0,0.2],[0.4,1.0]〉,𝑥 ∈𝛺 }” 

Entropy, Distance and Similarity Measures under... Informatica 42 (2018) 617–627 621 
The values of  𝐸 𝑍𝐽
(𝐴 1
)=0.5=𝐸 𝑍𝐽
(𝐴 2
) , 
𝐸 𝑊𝑊
(𝐴 3
)=0.7=𝐸 𝑊𝑊
(𝐴 4
) and 𝐸 𝑍𝑀
(𝐴 5
)=0.5=
𝐸 𝑍𝑀
(𝐴 6
) . Thus, the entropies 𝐸 𝑍𝐽
( 𝐴 𝑖 ) , 𝐸 𝑊𝑊
( 𝐴 𝑖 ) and 
𝐸 𝑍𝑀
( 𝐴 𝑖 ) are unreasonable. Proposed entropies E
j
, j=
1,2,..6 can discriminate the fuzziness of all the IvIFSs 
𝐴 𝑖 ,𝑖 =1,…,6 given as follows and give the reasonable 
results given in Table 2. 
Here we have proposed some entropy measures and 
evaluated its performance on the basis of intuitionistic 
belief and comparison with existing measures. In next 
section we propose relation between measures of entropy 
and similarity under IvIFE. Also, we have defined new 
measures of similarity and have evaluated their 
performance by comparing them with some existing 
measures. 
4 Relations between measure of 
entropy and similarity together 
with new similarity measures 
In this section contains definition of new measures of 
similarity under IvIFE and determined an important 
relation between entropy and similarity measure IvIFSs 
which is discussed as follows. 
Theorem 3: Let 𝑆 𝑗 , for 𝑗 =1,..,6 be measure of 
similarity of IvIFSs w.r.t. the measure of distance 𝐷 𝑗 , for 
𝑗 =1,..,6 respectively, and A be any IvIFS. Then 
𝐸 𝑗 (𝐴 )=3𝑆 𝑗 (𝐴 ,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)−2, for 𝑗 =
1,…,6 are measures of entropy for IvIFSs. 
 
Proof: We prove that E
j
(A) , for j=1,…,6  satisfies 
conditions given by definition 5. 
 
Property 1): If A∈ ℂ(Ω)⇒𝐴 (𝑥 𝑖 )=〈[1,1],[0,0],[0,0]〉 
or 𝐴 (𝑥 𝑖 )=〈[0,0],[1,1],[0,0]〉, ∀ 𝑥 𝑖 ∈𝛺 , then for j=
1,…,6 
S
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)
=1−D
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)
=
2
3
. 
Thus, E
j
(A)=0 
 
Property 2): For all 𝑗 =1,…,6,𝐸 𝑗 (𝐴 )=1 
⟺3S
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)−2=1 
⟺S
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)=1 
⟺ A=〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉 
Property3): Let A and B be any two IvIFSs and 
D
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)≥
D
j
(B,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) then 
1−D
𝑗 (A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)
≤1−D
𝑗 (B,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) 
⟺S
j
(A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)
≤S
𝑗 (B,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) 
⟺E
𝑗 (A)≤E
j
(B) ,  for all 𝑗 =1,…,6 
Property 4) : Let A be any IvIFS then A
̅
=
{〈𝑥 𝑖 ,[𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )],[𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )]〉 𝑥 𝑖 ∈Ω ⁄ } 
⟹𝐷 𝑗 (A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)
=D
j
(A
̅
,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) 
⟹S
𝑗 (A,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)
=S
𝑗 (A
̅
,〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) 
Thus, E
j
(A)=E
j
(A
̅
) , for all j=1,…,6. ∎ 
Next, we present a conversion technique to define 
similarity measures established by entropy measure for 
IvIFSs. 
 
Definition 6  : For any two IvIFSs A and B in Ω, such 
that both A and B are defined by the triplet 
〈𝑥 𝑖 ,[𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )],[𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ),𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )]〉  and  
〈𝑥 𝑖 ,[𝑀𝑉
𝐵𝐿
(𝑥 𝑖 ),𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )],[𝑁𝑉
𝐵𝐿
(𝑥 𝑖 ),𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )]〉 respec
tively. we define an IvIFSs  ∅(A,B) using A and B as 
given below: 
𝑀𝑉
∅(𝐴 ,𝐵 )𝐿 (𝑥 𝑖 ) = 
1
3
{1−[𝑚𝑎𝑥 (|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)]
1/2
}; 
𝑀𝑉
∅(𝐴 ,𝐵 )𝑈 (𝑥 𝑖 ) = 
1
3
{1−[𝑚𝑎𝑥 (|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑀 𝑉 𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑁 𝐴𝑈
(𝑥 𝑖 )−𝑁 𝐵𝑈
(𝑥 𝑖 )|,|𝐻 𝐴𝐿
(𝑥 𝑖 )−𝐻 𝐵𝐿
(𝑥 𝑖 )|∨
|𝐻 𝐴𝑈
(𝑥 𝑖 )−𝐻 𝐵𝑈
(𝑥 𝑖 )|)]}; 
𝑁 𝑉 ∅(𝐴 ,𝐵 )𝐿 (𝑥 𝑖 ) = 
1
3
{1+[𝑚𝑖𝑛 (|𝑀 𝑉 𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
 𝐸 1
(𝐴 𝑖 ) 𝐸 2
(𝐴 𝑖 ) 𝐸 3
(𝐴 𝑖 ) 𝐸 4
(𝐴 𝑖 ) 𝐸 5
(𝐴 𝑖 ) 𝐸 6
(𝐴 𝑖 ) 
A
1
 0.58167 0.625 0.45 0.675 0.4875 0.6063 
A
2
 0.735425 0.75 0.65 0.8 0.675 0.765479 
A
3
 0.367544 0.4 0.3 0.45 0.15 0.490098 
A
4
 0.523512 0.582143 0.425 0.607143 0.398214 0.566987 
A
5
 0.27889 0.3 0.15 0.3 0.05 0.395848 
A
6
 0.271989 0.375 0.01 0.45 0.1375 0.292893 
Table 2: Comparison of entropies. 

622 Informatica 42 (2018) 617–627 P. Tiwari et al.  
 
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)]
2
}; 
𝑁𝑉
∅(𝐴 ,𝐵 )𝑈 (𝑥 𝑖 ) = 
1
3
{1+[𝑚𝑖𝑛 (|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻 𝑉 𝐵𝐿
(𝑥 𝑖 )|∨
|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)]}. 
Theorem 4: E(∅(A,B) ) be a similarity measure for 
IvIFSs A and B, where E is an entropy. 
 
Proof: To prove that E(∅(A,B) ) is a measure of 
similarity, we need to prove property given by definition 
3 holds . 
 
Property 1): If 𝐴 ∈ ℂ(𝛺 )⇒𝐴 (𝑥 𝑖 )=〈[1,1],[0,0],[0,0]〉 
or 𝐴 (𝑥 𝑖 )=〈[0,0],[1,1],[0,0]〉, for any 𝑥 𝑖 ∈𝛺 , then 
𝑀𝑉
∅(𝐴 ,𝐴 ̅ )𝐿 (𝑥 𝑖 )=0=𝑀𝑉
∅(𝐴 ,𝐴 ̅ )𝑈 (𝑥 𝑖 ) ; 
And 𝑁𝑉
∅(𝐴 ,𝐴 ̅ )𝐿 (𝑥 𝑖 )=1=𝑁𝑉
∅(𝐴 ,𝐴 ̅ )𝑈 (𝑥 𝑖 ) 
Thus, ∅(𝐴 ,𝐴 ̅
)={〈𝑥 𝑖 ,[0,0],[1,1],[0,0]〉 𝑥 𝑖 ∈Ω ⁄ } 
⟹ S(A,A
̅
)=E(∅(A,A
̅
) ) = 0 
 
Property 2): Assume that S(A,B)=1⇒E(∅(A,B))=1 
⟺𝑀𝑉
∅(𝐴 ,𝐵 )
(𝑥 𝑖 )=𝑁𝑉
∅(𝐴 ,𝐵 )
(𝑥 𝑖 )=𝐻𝑉
∅(𝐴 ,𝐵 )
(𝑥 𝑖 )=[
1
3
,
1
3
] 
⟺ 𝑚𝑎𝑥 (|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|
∨|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )
−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|
∨|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )
−𝐻 𝑉 𝐵𝐿
(𝑥 𝑖 )|
∨|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)=0 
and 𝑚𝑖𝑛 (|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−
𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−
𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|,|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−
𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)=0 
⟺|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|
=0, 
|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|=0, 
and |𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|=
0. 
⟺𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )=𝑀𝑉
𝐵𝐿
(𝑥 𝑖 ),𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )=𝑀𝑉
𝐵𝑈
(𝑥 𝑖 ), 
𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )=𝑁𝑉
𝐵𝐿
(𝑥 𝑖 ),𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )=𝑁𝑉
𝐵𝑈
(𝑥 𝑖 ) 
and 𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )=𝐻𝑉
𝐵 𝐿 (𝑥 𝑖 ),𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )=𝐻𝑉
𝐵𝑈
(𝑥 𝑖 ) . 
⟺A=B. 
 
Property 3): ∅(𝐴 ,𝐵 )=∅(𝐵 ,𝐴 ) by definition of 
𝑀𝑉
∅(𝐴 ,𝐵 )𝐿 (𝑥 𝑖 ),𝑀𝑉
∅(𝐴 ,𝐵 )𝑈 (𝑥 𝑖 ) ,𝑁𝑉
∅(𝐴 ,𝐵 )𝐿 (𝑥 𝑖 ),
𝑁𝑉
∅(𝐴 ,𝐵 )𝑈 (𝑥 𝑖 ) for any 𝑥 𝑖 ∈𝛺 
⟹𝐸 (∅(𝐴 ,𝐵 ))=𝐸 (∅(𝐵 ,𝐴 )) 
⟺𝑆 (𝐴 ,𝐵 )=𝑆 (𝐵 ,𝐴 ) 
 
Property 4): Let A,B and C be any three IvIFSs such that 
A⊆B⊆C for any 𝑥 𝑖 ∈𝛺 , we have 𝑀𝑉
𝐴 (𝑥 𝑖 )≤
𝑀𝑉
𝐵 (𝑥 𝑖 )≤𝑀𝑉
𝐶 (𝑥 𝑖 ) ,𝑁𝑉
𝐴 (𝑥 𝑖 )≥𝑁𝑉
𝐵 (𝑥 𝑖 )≥𝑁𝑉
𝐶 (𝑥 𝑖 ) or 
𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )≤𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )≤𝑀𝑉
𝐶𝐿
(𝑥 𝑖 ) ,𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )≥
𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )≥𝑁𝑉
𝐶𝐿
(𝑥 𝑖 ) and  𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )≤𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )≤
𝑀𝑉
𝐶𝑈
(𝑥 𝑖 ) ,𝑁 𝑉 𝐴𝑈
(𝑥 𝑖 )≥𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )≥𝑁𝑉
𝐶𝑈
(𝑥 𝑖 ) . 
⟹|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐶 𝐿 (𝑥 𝑖 )|≥|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|, 
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐶𝑈
(𝑥 𝑖 )|≥|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|; 
|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐶𝐿
(𝑥 𝑖 )|≥|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|, 
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐶𝑈
(𝑥 𝑖 )|≥|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|; 
and |𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐶𝐿
(𝑥 𝑖 )|=|2(𝑀𝑉
𝐶𝐿
(𝑥 𝑖 )−
𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ))+2(𝑁𝑉
𝐶𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ))|≥
|2(𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ))+2(𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )−
𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ))|=|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|,  
Similarly, we have |𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐶𝑈
(𝑥 𝑖 )|≥
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )| 
So, we have  
𝑀𝑉
∅(𝐴 ,𝐵 )
(𝑥 𝑖 )≤𝑀𝑉
∅(𝐴 ,𝐶 )
(𝑥 𝑖 )≤[
1
3
,
1
3
] and 𝑁𝑉
∅(𝐴 ,𝐵 )
(𝑥 𝑖 )≥
𝑁𝑉
∅(𝐴 ,𝐶 )
(𝑥 𝑖 )≥[
1
3
,
1
3
] for any 𝑥 𝑖 ∈𝛺 . ⟹∅(A,C)⊆
∅(A,B)⊆〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉 
Similarly, we have ⟹∅(A,C)⊆∅(B,C)⊆
〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉. Thus 
D(∅(A,B),〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)≤
 D(∅(A,C),〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) 
and D(∅(B,C),〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)≤
 D(∅(A,C),〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) .  
So form definition of entropy corresponding to distance 
function, we get E(∅(A,C))≤E(∅(A,B)) and 
E(∅(A,C))≤E(∅(B,C)) 
or S(∅(A,C))≤S(∅(A,B)) and S(∅(A,C))≤
S(∅(B,C)) .      ∎ 
 
Corollary 1: Let E be an entropy measure for IvIFSs and 
∅(A,B) be an IvIFS defined on two IvIFSs A and 
B acaccording to definition 6, then  E(∅(A,B)
̅̅̅̅̅̅̅̅̅
) measure 
of similarity for 𝐴 ,𝐵 ∈𝐼𝑣𝐼𝐹𝑆𝑠 (𝛺 ) . 
 
Proof: Proof followed from the definition of complement 
interval-valued intuitionistic fuzzy sets and theorem 4.
      ∎ 
Definition 7: Let 𝐴 ,𝐵 ∈𝐼𝑣𝐼 𝐹 𝑆𝑠 (𝛺 )  , we can define 
𝐼𝑣𝐼𝐹𝑆 η(A,B) using A,B as follows: 
𝑀𝑉
𝜂 (𝐴 ,𝐵 )𝐿 (𝑥 𝑖 ) = 
1
3
{1+[𝑚𝑖𝑛 ((|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|
𝛼 ),(|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁 𝑉 𝐵𝑈
(𝑥 𝑖 )|)
𝛼 ,(|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 )]
2
}; 
𝑀𝑉
𝜂 (𝐴 ,𝐵 )𝑈 (𝑥 𝑖 ) = 
1
3
{1+[𝑚𝑖𝑛 ((|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|
𝛼 ),(|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 ,(|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 )]}; 
𝑁𝑉
𝜂 (𝐴 ,𝐵 )𝐿 (𝑥 𝑖 ) = 
1
3
{1−[𝑚𝑎𝑥 ((|𝑀 𝑉 𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑀𝑉
𝐴 𝑈 (𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|
𝛼 ),(|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 ,(|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻 𝑉 𝐵𝑈
(𝑥 𝑖 )|)
𝛼 )]
1/2
}; 
𝑁𝑉
𝜂 (𝐴 ,𝐵 )𝑈 (𝑥 𝑖 ) = 
1
3
{1−[𝑚𝑎𝑥 ((|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|
𝛼 ),(|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁 𝑉 𝐵𝑈
(𝑥 𝑖 )|)
𝛼 ,(|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨
|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 )]}, 
Where 𝛼 ∈[1,∞[ and 𝑥 𝑖 ∈𝛺 . 

Entropy, Distance and Similarity Measures under... Informatica 42 (2018) 617–627 623 
Theorem 5: For any two 𝐼𝑣𝐼𝐹𝑆𝑠 A and B,E(η(A,B) ) is a 
measure of similarity, where E is an entropy measure. 
 
Proof: To prove that E(η(A,B) ) is a measure of 
similarity, we need to prove property given by definition 
3 holds . 
 
Property 1): If 𝐴 ∈ ℂ(𝛺 )⇒𝐴 (𝑥 𝑖 )=〈[1,1],[0,0],[0,0]〉 
or 𝐴 (𝑥 𝑖 )=〈[0,0],[1,1],[0,0]〉, for any 𝑥 𝑖 ∈𝛺 , then 
𝑀𝑉
∅(𝐴 ,𝐴 ̅ )𝐿 (𝑥 𝑖 )=1=𝑀𝑉
∅(𝐴 ,𝐴 ̅ )𝑈 (𝑥 𝑖 ) ; 
and 𝑁𝑉
∅(𝐴 ,𝐴 ̅ )𝐿 (𝑥 𝑖 )=0=𝑁𝑉
∅(𝐴 ,𝐴 ̅ )𝑈 (𝑥 𝑖 ) 
Thus, ∅(A,A
̅
)={〈𝑥 𝑖 ,[1,1],[0,0],[0,0]〉 𝑥 𝑖 ∈Ω ⁄ } 
⟹ S(A,A
̅
)=E(∅(A,A
̅
) ) = 0 
 
Property 2): Assume that S(A,B)=1 
Then E(η(A,B))=1 
⟺𝑀𝑉
𝜂 (𝐴 ,𝐵 )
(𝑥 𝑖 )=[
1
3
,
1
3
]=𝑁𝑉
𝜂 (𝐴 ,𝐵 )
(𝑥 𝑖 )
= 𝐻𝑉
𝜂 (𝐴 ,𝐵 )
(𝑥 𝑖 ) 
 
⟺(|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|
∨|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 =0, 
(|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵 𝑈 (𝑥 𝑖 )|)
𝛼 =0, 
and (|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−
𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 =0. 
⟺𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )=𝑀𝑉
𝐵𝐿
(𝑥 𝑖 ),𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )
=𝑀𝑉
𝐵𝑈
(𝑥 𝑖 ),𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )
=𝑁𝑉
𝐵𝐿
(𝑥 𝑖 ),𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )=𝑁𝑉
𝐵𝑈
(𝑥 𝑖 ) 
and 𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )=𝐻𝑉
𝐵𝐿
(𝑥 𝑖 ),𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )=𝐻𝑉
𝐵𝑈
(𝑥 𝑖 ) . 
⟺A=B. 
 
Property 3): η(A,B)=η(B,A) by definition of 
𝑀𝑉
𝜂 (𝐴 ,𝐵 )𝐿 (𝑥 𝑖 ),𝑀𝑉
𝜂 (𝐴 ,𝐵 )𝑈 (𝑥 𝑖 ) ,𝑁𝑉
𝜂 (𝐴 ,𝐵 )𝐿 (𝑥 𝑖 ),
𝑁𝑉
𝜂 (𝐴 ,𝐵 )𝑈 (𝑥 𝑖 ) for any 𝑥 𝑖 ∈𝛺 
⟹E(∅(A,B))=E(∅(B,A)) 
⟺S(A,B)=S(B,A) 
 
Property 4): Let A,B and C be  any three IvIFSs such that 
A⊆B⊆C, then for any 𝑥 𝑖 ∈𝛺 , we have 𝑀𝑉
𝐴 (𝑥 𝑖 )≤
𝑀𝑉
𝐵 (𝑥 𝑖 )≤𝑀𝑉
𝐶 (𝑥 𝑖 ) ,𝑁𝑉
𝐴 (𝑥 𝑖 )≥𝑁𝑉
𝐵 (𝑥 𝑖 )≥𝑁𝑉
𝐶 (𝑥 𝑖 ) or 
𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )≤𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )≤𝑀𝑉
𝐶𝐿
(𝑥 𝑖 ) ,𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )≥
𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )≥𝑁𝑉
𝐶𝐿
(𝑥 𝑖 ) and 𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )≤𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )≤
𝑀𝑉
𝐶𝑈
(𝑥 𝑖 ) , 𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )≥𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )≥𝑁𝑉
𝐶𝑈
(𝑥 𝑖 ) . 
⟹|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐶𝐿
(𝑥 𝑖 )|≥|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|, 
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐶𝑈
(𝑥 𝑖 )|≥|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|; 
|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁 𝑉 𝐶𝐿
(𝑥 𝑖 )|≥|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|, 
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐶𝑈
(𝑥 𝑖 )|≥|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|; 
and |𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐶𝐿
(𝑥 𝑖 )|=|2(𝑀𝑉
𝐶𝐿
(𝑥 𝑖 )−
𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ))+2(𝑁𝑉
𝐶 𝐿 (𝑥 𝑖 )−𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ))|≥
|2(𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐴𝐿
(𝑥 𝑖 ))+2(𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )−
𝑁𝑉
𝐴𝐿
(𝑥 𝑖 ))|=|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|,  
Similarly, we have |𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐶𝑈
(𝑥 𝑖 )|≥
|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )| 
So, we have from definition 7 
𝑀𝑉
𝜂 (𝐴 ,𝐶 )
(𝑥 𝑖 )≥𝑀𝑉
𝜂 (𝐴 ,𝐵 )
(𝑥 𝑖 )≥[
1
3
,
1
3
] and 𝑁𝑉
𝜂 (𝐴 ,𝐶 )
(𝑥 𝑖 )≤
𝑁𝑉
𝜂 (𝐴 ,𝐵 )
(𝑥 𝑖 )≤[
1
3
,
1
3
] for any 𝑥 𝑖 ∈𝛺 . ⟹η(A,C)⊇
η(A,B)⊇〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉 
Similarly, we have ⟹η(A,C)⊇η(B,C)⊇
〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉. Thus  
D(η(A,B),〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)
≤ D(η(A,C),〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) 
and D(η(B,C),〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉)≤
 D(η(A,C),〈[
1
3
,
1
3
],[
1
3
,
1
3
],[
1
3
,
1
3
]〉) .  
So from definition of entropy corresponding to distance 
function, we get E(∅(A,C))≤E(∅(A,B)) and 
E(η(A,C))≤E(η(B,C)) 
or S(η(A,C))≤S(η(A,B)) and S(η(A,C))≤
S(η(B,C)) .     ∎ 
 
Corollary 2: Let E be an entropy for 𝐼𝑣𝐼𝐹𝑆𝑠 and 
η(A,B) be an 𝐼𝑣𝐼𝐹𝑆 defined on 𝐴 ,𝐵 ∈𝐼𝑣𝐼𝐹𝑆𝑠 (𝛺 )  as 
defined in definition 7, then  E(η(A,B)
̅̅̅̅̅̅̅̅̅
) is measure of 
similarity for 𝐴 ,𝐵 ∈𝐼𝑣𝐼𝐹𝑆𝑠 (𝛺 ) . 
Proof: Proof followed from the definition of complement 
of 𝐼 𝑣𝐼𝐹𝑆 𝑠 and theorem 5. ∎ 
4.1 Weighted similarity measure 
Let w=(w
1
,w
2
,…,w
n
)
T
 be the weights provided to 
each element 𝑥 𝑖 ∈𝛺 ,𝑖 =1,2,…,𝑛 . Then the weighted 
similarity measure based on the aforesaid similarity 
measures are defined as 𝑆 (𝐴 ,𝐵 )=
∑ 𝑤 𝑖 𝑆 (𝐴 (𝑥 𝑖 ),𝐵 (𝑥 𝑖 ))
𝑛 𝑖 =1
, where 𝑤 𝑖 ≥0 and ∑ 𝑤 𝑖 =1
𝑛 𝑖 =1
. 
4.2 Comparison with some select measures 
of similarity 
Here, we compare the performance of proposed measure 
of similarity with some of the existing similarity 
measures as follows. 
For any two IvIFSs A and B, then some existing 
similarity measures are given as follows: 
• 𝑆 𝑊 (𝐴 ,𝐵 )=
1
𝑛 ∑
4−(𝑀𝑉
𝐿 (𝑥 𝑖 )+𝑀𝑉
𝑈 (𝑥 𝑖 )+𝑁𝑉
𝐿 (𝑥 𝑖 )+𝑁𝑉
𝑈 (𝑥 𝑖 ))+(𝐻𝑉
𝐿 (𝑥 𝑖 )+𝐻𝑉
𝑈 (𝑥 𝑖 ))
4+(𝑀𝑉
𝐿 (𝑥 𝑖 )+𝑀𝑉
𝑈 (𝑥 𝑖 )+𝑁𝑉
𝐿 (𝑥 𝑖 )+𝑁𝑉
𝑈 (𝑥 𝑖 ))+(𝐻𝑉
𝐿 (𝑥 𝑖 )+𝐻𝑉
𝑈 (𝑥 𝑖 ))
𝑛 𝑖 =1
,  
where 𝑀𝑉
𝐿 (𝑥 𝑖 )=|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|, 𝑀 𝑉 𝑈 (𝑥 𝑖 )=
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|, 𝑁𝑉
𝐿 (𝑥 𝑖 )=|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−
𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|, 𝑁𝑉
𝑈 (𝑥 𝑖 )=|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−
𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|,𝐻𝑉
𝐿 (𝑥 𝑖 )=𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )+𝐻𝑉
𝐵𝐿
(𝑥 𝑖 ) and 
𝐻𝑉
𝑈 (𝑥 𝑖 )=𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )+𝐻𝑉
𝐵𝑈
(𝑥 𝑖 ) is proposed by Wu et 
al.(2014). 
⟺ 𝑚𝑖𝑛 (
(|𝑀 𝑉 𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 ,
(|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 ,
(|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 )
=0 
and 
𝑚𝑎𝑥 (
(|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 ,
(|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 ,
(|𝐻𝑉
𝐴𝐿
(𝑥 𝑖 )−𝐻𝑉
𝐵𝐿
(𝑥 𝑖 )|∨|𝐻𝑉
𝐴𝑈
(𝑥 𝑖 )−𝐻𝑉
𝐵𝑈
(𝑥 𝑖 )|)
𝛼 )
=0 
 

624 Informatica 42 (2018) 617–627 P. Tiwari et al.  
 
• 𝑆 𝐻𝐿
(𝐴 ,𝐵 )=1−
1
4𝑛 ∑ |𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−
𝑛 𝑖 =1
𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|+
|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|+|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|is 
given by Hu and Li (2013). 
 
• 𝑆 𝑆 (𝐴 ,𝐵 )=
1
𝑛 ∑
[
(𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )+𝑀𝑉
𝐴𝑈
(𝑥 𝑖 ))(𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )+𝑀𝑉
𝐵𝑈
(𝑥 𝑖 ))
+(𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )+𝑁𝑉
𝐴𝑈
(𝑥 𝑖 ))(𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )+𝑁𝑉
𝐵𝑈
(𝑥 𝑖 ))
]
[
 
 
 √
(𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )+𝑀𝑉
𝐴𝑈
(𝑥 𝑖 ))
2
+(𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )+𝑁𝑉
𝐴𝑈
(𝑥 𝑖 ))
2
√
(𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )+𝑀𝑉
𝐵𝑈
(𝑥 𝑖 ))
2
+(𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )+𝑁𝑉
𝐵𝑈
(𝑥 𝑖 ))
2
]
 
 
 
𝑛 𝑖 =1
is 
introduced by Singh(2012) 
• 𝑆 𝑆𝑢
(𝐴 ,𝐵 )=
1
𝑛 ∑
|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|⋁|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|
+|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|∨|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|
3−𝑚𝑖 𝑛 {
|𝑀𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑀𝑉
𝐵𝐿
(𝑥 𝑖 )|⋁|𝑁𝑉
𝐴𝐿
(𝑥 𝑖 )−𝑁𝑉
𝐵𝐿
(𝑥 𝑖 )|,
|𝑀𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑀𝑉
𝐵𝑈
(𝑥 𝑖 )|∨|𝑁𝑉
𝐴𝑈
(𝑥 𝑖 )−𝑁𝑉
𝐵𝑈
(𝑥 𝑖 )|
}
𝑛 𝑖 =1
 
 
is proposed by Sun & Liu (2012). 
To review the performance of similarity measures let 
us consider an example. Consider the following IvIFSs 
A={𝑥 𝑖 , 〈[0.5,0.5],[0.5,0.5],[0,0]〉,𝑥 𝑖 ∈Ω},  
B={𝑥 𝑖 , 〈[0.3,0.4],[0.4,0.5],[0.1,0.3]〉,𝑥 𝑖 ∈Ω},  
C={𝑥 𝑖 , 〈[0.3,0.3],[0.3,0.3],[0.4,0.4]〉,𝑥 𝑖 ∈Ω}, 
 D={𝑥 𝑖 , 〈[0.6,0.6],[0.4,0.4],[0,0]〉,𝑥 𝑖 ∈Ω} 
Intuitively, it is clear that A is more similar to D than 
B and C. The result corresponding to measure of 
similarity measures given in Table 3: 
 A B A C A D 
S
W
 0.83333 0.69308 0.8181 
S
HL
 0.9 0.8 0.9 
S
S
 0.99227 0.9 0.98058 
S
Su
 0.89655 0.8571 0.9310 
S
1
(ϕ) 0.7450 0.73722 0.8646 
S
2
(ϕ) 0.78806 0.761886 0.89594 
S
3
(ϕ) 0.72614 0.68377 0.84188 
S
4
(ϕ) 0.78806 0.74188 0.89594 
S
5
(ϕ) 0.68210 0.62283 0.84391 
S
6
(ϕ) for p=2 0.77639 0.77141 0.87090 
S
1
(η) 0.93205 0.89426 0.98708 
S
2
(η) 0.95217 0.92075 0.99184 
S
3
(η) 0.91759 0.85853 0.98418 
S
4
(η) 0.95217 0.92075 0.99184 
S
5
(η) 0.92825 0.88113 0.98776 
S
6
(η) for p=2 0.93292 0.89590 0.98709 
Table 3: Comparison of Similarity Measures. 
From the similarity measures listed in table 3, we can see 
that 
S
W
, S
HL
 and S
S
 are inconsistent with intuition where as 
S
Su
, S
j
(ϕ) and S
j
(η),j=1,…,6. 
In this section we have derived a relation between 
entropy and similarity measure. Then we defined some 
similarity measures, compared its performance with 
existing similarity measures. In section 5 we applied 
proposed similarity measures to draw conclusion in 
pattern recognition and medical diagnoses. 
5 Applications of proposed 
similarity measures 
Here the proposed similarity measures are applied to 
some of the situation that deals with imperfect 
information. 
5.1 Pattern recognition 
Here we use an example of pattern recognition 
considered by Xu (2007) and adapted by Wei et al. 
(2011) and Wu et al.(2014) for classification of building 
material. 
Example: There are four types of building materials 
𝐴 𝑖 ,𝑖 =1,2,3,4 and an anonymous building material B, 
which is characterized by the IvIFSs defined on 𝑋 =
{𝑥 1
,𝑥 2
,…,𝑥 12
} with weighted vector 
 w=(
0.1,0.05,0.08,0.06,0.03,0.07,
 0.09,0.12,0.15,0.07,0.13,0.05
)
T
 
and we have the data given as follows by Xu (2007). 
“𝐴 1
 
=
{
 
 
 
 
 
 (
〈𝑥 1
,[0.1,0.2],[0.5,0.6]〉,〈𝑥 2
,[0.1,0.2],[0.7,0.8]〉,
〈𝑥 3
,[0.5,0.6],[0.3,0.4]〉,〈𝑥 4
,[0.8,0.9],[0.0,0.1]〉,
)
(
〈𝑥 5
,[0.4,0.5],[0.3,0.4]〉,〈𝑥 6
,[0.0,0.1],[0.8,0.9]〉,
〈𝑥 7
,[0.3,0.4],[0.5,0.6]〉,〈𝑥 8
,[1.0,1.0],[0.0,0.0]〉,
)
(
〈𝑥 9
,[0.2,0.3],[0.6,0.7]〉,〈𝑥 10
,[0.4,0.5],[0.4,0.5]〉,
〈𝑥 11
,[0.7,0.8],[0.1,0.2]〉,〈𝑥 12
,[0.4,0.5],[0.4,0.5]〉
)
}
 
 
 
 
 
 
 
𝐴 2
=
{
 
 
 
 
 
 (
〈𝑥 1
,[0.5,0.6],[0.3,0.4]〉,〈𝑥 2
,[0.6,0.7],[0.1,0.2]〉,
〈𝑥 3
,[1.0,1.0],[0.0,0.0]〉,〈𝑥 4
,[0.1,0.2],[0.6,0.7]〉,
)
(
〈𝑥 5
,[0.0,0.1],[0.8,0.9]〉,〈𝑥 6
,[0.7,0.8],[0.1,0.2]〉,
〈𝑥 7
,[0.5,0.6],[0.3,0.4]〉,〈𝑥 8
,[0.6,0.7],[0.2,0.3]〉,
)
(
〈𝑥 9
,[1.0,1.0],[0.0,0.0]〉,〈𝑥 10
,[0.1,0.2],[0.7,0.8]〉,
〈𝑥 11
,[0.0,0.1],[0.8,0.9]〉,〈𝑥 12
,[0.7,0.8],[0.1,0.2]〉
)
}
 
 
 
 
 
 
 
𝐴 3
=
{
 
 
 
 
 
 (
〈𝑥 1
,[0.4,0.5],[0.3,0.4]〉,〈𝑥 2
,[0.6,0.7],[0.2,0.3]〉,
〈𝑥 3
,[0.9,1.0],[0.0,0.0]〉,〈𝑥 4
,[0.0,0.1],[0.8,0.9]〉,
)
(
〈𝑥 5
,[0.0,0.1],[0.8,0.9]〉,〈𝑥 6
,[0.6,0.7],[0.2,0.3]〉,
〈𝑥 7
,[0.1,0.2],[0.7,0.8]〉,〈𝑥 8
,[0.2,0.3],[0.6,0.7]〉,
)
(
〈𝑥 9
,[0.5,0.6],[0.2,0.4]〉,〈𝑥 10
,[1.0,1.0],[0.0,0.0]〉,
〈𝑥 11
,[0.3,0.4],[0.4,0.5]〉,〈𝑥 12
,[0.0,0.1],[0.8,0.9]〉
)
}
 
 
 
 
 
 
 
 
𝐴 4
=
{
 
 
 
 
 
 (
〈𝑥 1
,[1.0,1.0],[0.0,0.0]〉,〈𝑥 2
,[1.0,1.0],[0.0,0.0]〉,
〈𝑥 3
,[0.8,0.9],[0.0,0.1]〉,〈𝑥 4
,[0.7,0.8],[0.1,0.2]〉,
)
(
〈𝑥 5
,[0.0,0.1],[0.7,0.9]〉,〈𝑥 6
,[0.0,0.1],[0.8,0.9]〉,
〈𝑥 7
,[0.1,0.2],[0.7,0.8]〉,〈𝑥 8
,[0.1,0.2],[0.7,0.8]〉,
)
(
〈𝑥 9
,[0.4,0.5],[0.3,0.4]〉,〈𝑥 10
,[1.0,1.0],[0.0,0.0]〉,
〈𝑥 11
,[0.3,0.4],[0.4,0.5]〉,〈𝑥 12
,[0.0,0.1],[0.8,0.9]〉
)
}
 
 
 
 
 
 
 
 
𝐵 =
{
 
 
 
 
 
 (
〈𝑥 1
,[0.9,1.0],[0.0,0.0]〉,〈𝑥 2
,[0.9,1.0],[0.0,0.0]〉,
〈𝑥 3
,[0.7,0.8],[0.1,0.2]〉,〈𝑥 4
,[0.6,0.7],[0.1,0.2]〉,
)
(
〈𝑥 5
,[0.0,0.1],[0.8,0.9]〉,〈𝑥 6
,[0.1,0.2],[0.7,0.8]〉,
〈𝑥 7
,[0.1,0.2],[0.7,0.8]〉,〈𝑥 8
,[0.1,0.2],[0.7,0.8]〉,
)
(
〈𝑥 9
,[0.4,0.5],[0.3,0.4]〉,〈𝑥 10
,[1.0,1.0],[0.0,0.0]〉,
〈𝑥 11
,[0.3,0.4],[0.4,0.5]〉,〈𝑥 12
,[0.0,0.1],[0.7,0.9]〉
)
}
 
 
 
 
 
 
” 

Entropy, Distance and Similarity Measures under... Informatica 42 (2018) 617–627 625 
We need to identify which pattern is most similar to B 
using the maximum degree principle of measures of 
similarity between IvIFSs. Using the anticipated 
similarity measures defined in this paper, we get the 
following results given in table 4: 
 A
1
 B A
2
 B A
3
 B A
4
 B 
S
1
(ϕ) 0.688569 0.803061 0.84671 0.936548 
S
2
(ϕ) 0.725141 0.854747 0.868809 0.95075 
S
3
(ϕ) 0.654357 0.742627 0.818343 0.9265 
S
4
(ϕ) 0.725141 0.861997 0.868809 0.95075 
S
5
(ϕ) 0.587711 0.789371 0.803214 0.926125 
S
6
(ϕ) 
for p=2 0.738413 0.80768 0.863582 0.939988 
S
1
(η) 0.780631 0.773645 0.768612 0.979413 
S
2
(η) 0.816725 0.811175 0.804125 0.98595 
S
3
(η) 0.767465 0.749444 0.76081 0.975 
S
4
(η) 0.816725 0.811175 0.804125 0.98595 
S
5
(η) 0.725088 0.716763 0.706188 0.978925 
S
6
(η) 
for p=2 0.814325 0.801194 0.810194 0.979588 
Table 4: Application to pattern recognition. 
From the above values it is clear that B is most similar to 
A
4
 as the value corresponding to each similarity measure 
is highest for A
4
. So, we can conclude that A
4
building 
material consistent with the specification and this result 
is consistent with the results presented by Wu et 
al.(2014). 
5.2 Medical diagnoses 
Many authors Wei et al. (2011), Wu et al. (2014), Singh 
(2012)  employed IvIFSs to execute medical diagnosis in 
their works. Here we use the data used by Singh(2012) to 
do medical diagnosis using the proposed measure of 
similarity  
 
Example: Let A and B be the set that represent the set of 
diagnoses and symptoms respectively given as A=
{〈A
1
,Viral fever〉,〈A
2
,Malaria 〉,〈A
3
,Typhoid〉} and =
{〈B
1
,Temperature 〉,〈B
2
,Headache〉,〈B
3
,Cough〉} . 
Assume the patient is represented by 
 𝑃 ={
〈𝐵 1
,[0.6,0.8],[0.1,0.2]〉,〈𝐵 2
,[0.3,0.7],[0.2,0.3]〉,
〈𝐵 3
,[0.6,0.8],[0.1,0.2]〉
}  
and the weights corresponding to each attribute is equal 
and each diagnosis is given by the following 𝐼𝑣𝐼𝐹𝑆𝑠 
 
A
1
={
〈B
1
,[0.4,0.5],[0.3,0.4]〉,〈B
2
,[0.4,0.6],[0.2,0.4]〉,
 〈B
3
,[0.4,0.8],[0.1,0.2]〉
} 
A
2
={
〈B
1
,[0.3,0.6],[0.3,0.4]〉,〈B
2
,[0.5,0.6],[0.3,0.4]〉,
〈B
3
,[0.4,0.5],[0.1,0.3]〉
} 
A
3
={
〈B
1
,[0.7,0.8],[0.1,0.2]〉,〈B
2
,[0.6,0.7],[0.1,0.3]〉,
 〈B
3
,[0.3,0.4],[0.1,0.2]〉
} 
Using the proposed similarity measure we classify the 
patient P in one of the diagnoses A
1
,A
2
,A
3
.  The results 
are as follows in Table 5. 
 
 
 A
1
 P A
2
 P A
3
 P 
S
1
(ϕ) 0.80666 0.753483 0.770859 
S
2
(ϕ) 0.840736 0.788069 0.808633 
S
3
(ϕ) 0.766473 0.703639 0.743099 
S
4
(ϕ) 0.840736 0.788069 0.808633 
S
5
(ϕ) 0.761105 0.682104 0.712949 
S
6
(ϕ) for p=2 0.821222 0.776552 0.796418 
S
1
(η) 0.916133 0.872675 0.885256 
S
2
(η) 0.938333 0.9025 0.911667 
S
3
(η) 0.89835 0.847525 0.865842 
S
4
(η) 0.938333 0.9025 0.911667 
S
5
(η) 0.9075 0.85375 0.8675 
S
6
(η) for p=2 0.918319 0.877512 0.891129 
Table 5: Application to Medical diagnoses. 
From the above table 5 it is clear that patient P can be 
diagnosed with viral fever. 
6 Conclusion 
Entropy, distance and similarity measure are 
significant research area in fuzzy information theory as 
they are efficient tools to deal with uncertain and 
insufficient information. Here we have derived new 
definition of entropy based on distance measure by 
considering degree of hesitancy in to account and derived 
relation between distance, entropy and similarity 
measures under IvIFE. Further, we have compared the 
derived similarity measures with some of the existing 
similarity measure and instances are used to show that 
the derived measures are able to draw conclusion when 
existing measures give the same result. Thereafter, 
proposed measures of similarity are applied to 
recognition of patterns and medical diagnoses. 
References 
[1] Atanassov, K.(1986), Intuitionistic fuzzy sets, 
Fuzzy Sets and Systems, 20 (1), 87–96. 
https://doi.org/10.1016/S0165-0114(86)80034-3 
[2] Atanassov, K. T. (1999). Intuitionistic Fuzzy 
Sets. Heidelberg, New York: Physica-Verlag. 
https://doi.org/10.1007/978-3-7908-1870-3 
[3] Atanassov, K. , & Gargov, G.(1989) . Interval-
valued intuitionistic fuzzy sets, Fuzzy Sets and 
Systems, 31 (3) , 343–349. 
https://doi.org/10.1016/0165-0114(89)90205-4 
[4] Dammak, F. Baccour, L and Alimi, A.M.(2016), 
An Exhaustive Study of Possibility Measures of 
Interval-valued Intuitionistic Fuzzy Sets abd 
Application to Multicriteria Decision Making,  
Advances in Fuzzy Systems, Vol 2016, Article 
ID 9185706, 10 pages. 
http://dx.doi.org/10.1155/2016/9185706 
[5] De Luca A.,& Termini S. (1972). A definition 
of a non-probabilistic entropy in setting of fuzzy 
sets. Information and Control, 20, 30. 

626 Informatica 42 (2018) 617–627 P. Tiwari et al.  
 
https://doi.org/10.1016/S0019-9958(72)90199-
41-312. 
[6] Gau, W.L. ,& Buehrer, D.J.(1993) , Vague sets, 
IEEE Transactions on systems, Man and 
Cybernetics, 23 (2), 610–614. 
https://doi.org/10.1109/21.229476 
[7] Grzegorzewski, P. (2004). Distances between 
intuitionistic fuzzy sets and/or interval-valued 
fuzzy sets based on the hausdorff metric. Fuzzy 
Sets and Systems, 48, 319–328. 
https://doi.org/10.1016/j.fss.2003.08.005 
[8] Hu, K. & Li, J. (2013), The entropy and 
similarity measure of interval valued 
intuitionistic fuzzy sets and their relationship, 
International Journal of Fuzzy Systems, Vol. 
15(3), 279-288. 
[9] Hung, W., & Yang, M., (2006), Fuzzy entropy 
in intuitionistic fuzzy sets, International Journal 
of Intelligent Systems, Vol.21, 443-451. 
https://doi.org/10.1002/int.20131 
[10] Kacprzyk, J. (1997). Multistage Fuzzy Control. 
Chichester: Wiley. 
[11] Liu, X.C.(1992), Entropy, distance measure and 
similarity measure of fuzzy sets and their 
relations, Fuzzy Sets and Systems, 52,  305–318. 
https://doi.org/10.1016/0165-0114(92)90239-Z 
[12] Park, J. H., Lin, K.M., Park, J.S. & Kwun, Y.C. 
(2007), Distance between interval-valued 
intuitionistic fuzzy sets, Journal of Physics: 
Conference Series 96. 
https://doi.org/10.1088/1742-6596/96/1/012089 
[13] Singh, P.(2012), A new method on measure of 
similarity between interval-valued intuitionistic 
fuzzy sets for pattern recognition, Journal of 
Applied & Computational Mathematics, 
Vol.1(1), 1-5. 
https://doi.org/10.4172/2168-9679.1000101 
[14] Sun, M. & Liu, J.,(2012), New entropy and 
similarity measures for interval-valued 
intuitionistic fuzzy set, Journal of information & 
Computational Sciences, vol.9(18), 5799-5806. 
[15] Szmidt, E., & Kacprzyk, J. (2000). Distance 
between intuitionistic fuzzy sets. Fuzzy Sets and 
Systems, 114(3), 505–518. 
https://doi.org/10.1016/S0165-0114(98)00244-9 
[16] Tiwari, P. & Gupta, P. (in press), Generalized 
Interval-valued Intuitionistic Fuzzy Entropy 
with some Similarity Measures, International 
Journal of Computing Science and Mathematics. 
[17] Wei, C.P. , Wang, P. , & Zhang, Y.Z. (2011),  
Entropy, similarity measures of interval-valued 
intuitionistic fuzzy sets and their applications, 
Inform. Sci., 181, 4273–4286. 
https://doi.org/10.1016/j.ins.2011.06.001 
[18] Wu, C., Luo, P., Li, Y. & Ren, X. (2014), A 
new similarity measure of interval-valued 
intuitionistic fuzzy sets considering its hesitancy 
degree and applications in expert systems, 
Mathematical Problems in Engineering. 
http://dx.doi.org/10.1155/2014/359214  
[19] Xia, M. &Xu, Z. (2010), Some new similarity 
measures for intuitionistic fuzzy values and their 
application in group decision making , Journal 
of Systems Science and System Engineering, 
Vol. 19(4), 430-452. 
https://doi.org/10.1007/s11518-010-5151-9 
[20] Xu, Z. (2007a), On similarity measures of 
interval-valued intuitionistic fuzzy sets and their 
application to pattern recognitions, Journal of 
Southeast University (English Edition) 23 (1), 
139–143. 
[21] Xu, Z. (2007 b), Some similarity measures of 
intuitionistic fuzzy sets and their application to 
multiple attribute decision making, Fuzzy 
Optimization and Decision Making, Vol. 6(2), 
109-121. 
https://doi.org/10.1007/s10700-007-9004-z 
[22] Xu, Z.S. &Yager, R. R., (2009), Intuitionistic 
and interval valued intuitionistic fuzzy 
preference relations and their measures of 
similarity for the evaluation of agreement within 
a group, Fuzzy Optimization and Decision 
Making, Vol. 8(2), 123-139. 
https://doi.org/10.1007/s10700-009-9056-3 
[23] Yang, Y., & Chiclana, F. (2012), Consistency of 
2D and 3D distances of intuitionistic fuzzy sets, 
Expert Systems with Applications, Vol. 39(10), 
8665–8670. 
https://doi.org/10.1016/j.eswa.2012.01.199 
[24] Yang, Y.J. & Hinde, C. (2010), A new 
extension of fuzzy sets using rough sets: R-
fuzzy sets, Information Sciences, 180, 354–365. 
https://doi.org/10.1016/j.ins.2009.10.004 
[25] Ye, J. (2012), Multicriteria decision making 
method using the Dice similarity measure based 
on the reduct intuitionistic fuzzy sets of interval- 
valued intuitionistic fuzzy sets, Applied 
Mathematical Modelling, vol.36(9), 4466-4472. 
https://doi.org/10.1016/j.apm.2011.11.075 
[26] Zadeh, L. A.(1965), Fuzzy sets, Information and 
Control 8 (3), 338–356. 
https://doi.org/10.1016/S0019-9958(65)90241-
X 
[27] Zadeh, L.A.(1975), The concept of a linguistic 
variable and its application to approximate 
reasoning-I, Information Science, 8, 199–249. 
https://doi.org/10.1016/0020-0255(75)90036-5 
[28] Zang, Q.S., Jiang, S.Y., Jia, B. G., & Luo, S. H., 
(2010), Some information measures for interval- 
valued intuitionistic fuzzy sets, Information 
Sciences, Vol.180, 5130-5145. 
https://doi.org/10.1016/j.ins.2010.08.038 
[29] Zhang, Y.J., Ma, P.J., Su, H.,& Zhang, 
C.P.(2011), Entropy on interval-valued 
intuitionistic fuzzy sets and its application in 
multi-attribute decision making, 2011 
proceedings of 14th International Conference 
on Fusion (FUSION),1-7. 
[30] Zhang, S. Li, X. & Meng, F.(2016), An 
Approach to Multi-criteria Decision Making 
under Interval-valued Intuitionistic Fuzzy Vales 

Entropy, Distance and Similarity Measures under... Informatica 42 (2018) 617–627 627 
and Interval Fuzzy Measures , Journal of 
Industrial and Production Engineering, Vol 
33(4). 
https://doi.org/10.1080/21681015.2016.1146362 
[31] Zhang, Q., Xing, H., Liu, F., Ye, J. & Tang, P. 
(2014), Some entropy measures for interval –
valued intuitionistic fuzzy sets  based on 
distances and their relationships with similarity 
and inclusion measures, Information Sciences, 
283, 55-69. 
https://doi.org/10.1016/j.ins.2014.06.012 
[32] Zhang, H., Zhang, W., & Mei, C. (2009), 
Entropy of interval-valued fuzzy sets based on 
distance and its relationship with similarity 
measure, Knowledge-Based Systems, Vol. 22, 
449-454. 
https://doi.org/10.1016/j.knosys.2009.06.007 
  

628 Informatica 42 (2018) 617–627 P. Tiwari et al.