# A Theoretical Development of Distance Measure for Intuitionistic Fuzzy Numbers

Authors: Debashree Guha; Debjani Chakraborty
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

The objective of this paper is to introduce a distance measure for intuitionistic fuzzy numbers. Firstly the existing distance measures for intuitionistic fuzzy sets are analyzed and compared with the help of some examples. Then the new distance measure for intuitionistic fuzzy numbers is proposed based on interval difference. Also in particular the type of distance measure for triangle intuitionistic fuzzy numbers is described. The metric properties of the proposed measure are also studied. Some numerical examples are considered for applying the proposed measure and finally the result is compared with the existing ones.


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## Body

## 1. Introduction

The theory of fuzzy set introduced by Zadeh [1] in 1965 has achieved successful applications in various fields. This is because this theory is an extraordinary tool for representing human knowledge, perception, and so forth. Nevertheless, Zadeh himself established in 1973 knowledge which is better represented by means of some generalizations of fuzzy sets. The so-called extensions of fuzzy set theory arise in this way.Two years after the concept of fuzzy set was proposed, it was generalized by Gogeun and L-fuzzy set [2] was developed. There are also some other extensions of fuzzy sets. Out of several higher-order fuzzy sets, the concept of intuitionistic fuzzy sets (IFSs) proposed by Atanassov [3] in 1986 is found to be highly useful to deal with vagueness. The major advantage of IFS over fuzzy set is that IFSs separate the degree of membership (belongingness) and the degree of nonmembership (nonbelongingness) of an element in the set. Then in 1993, Gau and Buehrer [4] introduced the concept of vague sets, which is another generalization of fuzzy sets. Bustince and Burillo [5] pointed out that the notion of vague set is the same as that of IFSs. Another well-known generalization of ordinary fuzzy sets is the concept of interval-valued fuzzy set [6–10]. There is a strong relationship between interval-valued fuzzy sets and IFSs.Among various extensions of fuzzy sets, IFSs have captured the attention of many researchers in the last few decades. This is mainly due to the fact that IFSs are consistent with human behavior, by reflecting and modeling the hesitancy present in real life situations. Therefore in practice, it is realized that human expressions like perception, knowledge, and behavior are better represented by IFSs rather than fuzzy sets. IFS theory is applied to many different fields such as decision making, logic programming, medical diagnosis, and pattern recognition.In the application of fuzzy sets as well as IFSs, similarity measures play a very important role. But the similarity measures bear a relation to distances in many cases. Therefore the study about the distance measures is very much significant. Developing distance measures is one of the fundamental problems of fuzzy set theory. A lot of research has been done to construct the distance measure between fuzzy sets [11]. Recently some researchers have focused their attention to compute the distances between fuzzy numbers [12–18] also. As important contents in fuzzy mathematics, distance measures between IFSs have also attracted many researchers. Several researchers [19–24] focused on computing the distance between IFSs, which we discuss briefly later in Section 2.4.It has been observed that all the papers discussed above considered distances between IFSs on finite universe of discourses only. But construction of distance measures between IFSs for countable and uncountable universe of discourse is also necessary. With this point of view, the concept of intuitionistic fuzzy number (IFN) [25–29] with the universe of discourse as the real line was introduced and studied.Grzegorzewski [28] introduced two families of metrics in a space of IFNs. A method of ranking IFNs based on these metrics was also suggested and investigated in that work. But this distance measure is not effective for some cases. The distance measures proposed by Grzegorzewski [28] compute crisp distance measures for IFNs. But the well-known fact that needs to be remembered here is that [17] “if we are not certain about the numbers themselves, how can we be certain about the distances among them.” This is the reason why fuzzy distance measure for measuring the distance measure between two fuzzy numbers came into the field. For the same reason it is not reasonable to define crisp distance between IFNs. Our intuition says that the uncertainty or hesitation or lack of knowledge presented in defining IFN should inherently be involved in their corresponding distance measures. Now it can be assumed that an IFN is a collection of points with different degrees of membership and corresponding degree of nonmembership. Therefore the distance between two IFNs is nothing but the distances of pairwise membership and nonmembership functions of the respective points. With this point of view in this paper a new approach is introduced to calculate the distance measure between two IFNs. Here the distance measure is proposed based on interval difference. It is worth noting that the proposed distance measures between IFNs are direct generalizations of the results obtained for the classical fuzzy numbers.The paper is organized as follows. Section2 briefly describes the basic definition and notations of IFS, IFN and LR-type IFN. Also some preliminary result is presented in this section. A short review of the existing distance measures is described in Section 2.4. Section 3 introduces the new distance measure for IFNs. The distance measure for IFNs and TIFNs is derived in Section 3.1 and Section 3.2, respectively. The metric properties are studied in Section 3.3. In Section 4, the proposed distance method is illustrated with the help of some numerical examples. The paper is concluded in Section 5.


## 2. Preliminaries

### 2.1. Intuitionistic Fuzzy Sets—Basic Definition and Notation

LetX denote a universe of discourse. Then a fuzzy set A/ in X is defined as a set of ordered pairs:
(2.1)A/={〈x,μA/(x)〉:x∈X},
where μA/:X→[0,1] and μA/(x) is the grade of belongingness of x into A/ [1]. Thus automatically the grade of nonbelongingness of x into A/ is equal to 1-μA/(x). However, while expressing the degree of membership of any given element in a fuzzy set, the corresponding degree of nonmembership is not always expressed as a compliment to 1. The fact is that in real life, the linguistic negation does not always identify with logical negation [21]. Therefore Atanassov [19, 30–33] suggested a generalization of classical fuzzy set, called IFS.An IFSA in Xis given by a set of ordered triples:
(2.2)A={〈x,μA(x),νA(x)〉:x∈X},
where μA,νA:X→[0,1] are functions such that 0≤μA(x)+νA(x)≤1forallx∈X. For eachx the numbers μA(x) and νA(x) represent the degree of membership and degree of nonmembership of the element x∈X to A⊂X, respectively.It is easily seen thatAIFS={〈x,μA(x),1-μA(x)〉:x∈X} is equivalent to (2.1); that is, each fuzzy set is a particular case of the IFS. We will denote a family of fuzzy sets in X by FS(X), while IFS(X) stands for the family of all IFSs in X.For each elementx∈X we can compute, so called, the intuitionistic fuzzy index of x in A defined as follows:
(2.3)πA(x)=1-μA(x)-νA(x).
The value of πA(x) is called the degree of indeterminacy (or hesitation) of the element x∈X to the IFS A. It is seen immediately that πA(x)∈[0,1]. If A∈FS(X), then πA(x)=0forallx∈X.


### 2.2. Intuitionistic Fuzzy Numbers

Different research works [25–29] were done over Intuitionistic Fuzzy Numbers (IFNs). Taking care of those research works in this section the notion of IFNs is studied. IFN is the generalization of fuzzy number and so it can be represented in the following manner.Figure 1
TheLR- representation of intuitionistic fuzzy number ADefinition 2.1 (Intuitionistic fuzzy numbers).
An intuitionistic fuzzy subsetA={〈x,μA(x),νA(x)〉:x∈X} of the real  line R is called an IFN if the following holds. (i)
There existm∈R,μA(m)=1, and νA(m)=0, (m is called the mean value of A).(ii)
μA is a continuous mapping from R to the closed interval [0,1] and forallx∈R the relation 0≤μA(x)+νA(x)≤1 holds.(iii)
The membership and nonmembership function ofAis of the following form:
(2.4)μA(x)={0for-∞<x≤m-α,f1(x)forx∈[m-α,m],1forx=m,h1(x)forx∈[m,m+β],0form+β≤x<∞,
where f1(x) and h1(x) are strictly increasing and decreasing functions in [m-α,m] and [m,m+β], respectively:
(2.5)νA(x)={1for-∞≤x≤m-α′,f2(x)forx∈[m-α′,m];0≤f1(x)+f2(x)≤1,0forx=mh2(x)forx∈[m,m+β′];0≤h1(x)+h2(x)≤1,1form+β′≤x≤∞.
Herem is the mean value of A.α and β are called left and right spreads of membership function μA(x), respectively. α′ and β′ represented left and right spreads of nonmembership function νA(x), respectively. Symbolically the intuitionistic fuzzy number is represented as AIFN=(m;α,β;α′,β′).
It is to be noted here that the IFNA={〈x,μA(x),νA(x)〉:x∈R}, that is, AIFN=(m;α,β;α′,β′) is a conjunction of two fuzzy numbers: A+=(m;α,β) with a membership function μA+(x)=μA(x) and A-=(m;α′,β′) with a membership function μA-(x)=1-νA(x).Definition 2.2 (LR-type Intuitionistic fuzzy number).
An IFNAIFN is LR-type IFN such that for membership and nonmembership functions 0≤μA(x)+νA(x)≤1 holds and may be defined as follows.
Membership function is of the form as follows:(2.6)μA(x)={0for-∞<x≤m-α,L(m-xα)form-α≤x≤m,α>0,1forx=m,R(x-mβ)form≤x≤m+β,β>0,0form+β≤x<∞.
Nonmembership function is of the form:(2.7)νA(x)={1for-∞≤x≤m-α′,1-L(m-xα/)form-α′≤x≤m,α′>0,0forx=m,1-R(x-mβ/)form≤x≤m+β′,β′>0,1form+β′≤x≤∞.
ProvidedL(1)=R(1)=0, L is for left, and R is for right reference, m is the mean value of A.αand β are called left and right spreads of membership functions, respectively. α′ and β′ represented left and right spreads of nonmembership functions, respectively. Symbolically, we write AIFN=(m;α,β;α′,β′)LR. Here for L(x) and R(x) different functions may be chosen. For example, L(x)=R(x)=max(0,1-|x|p),p≥0, and so forth (Figure 1).Definition 2.3 (Triangle Intuitionistic fuzzy number).
An IFNAIFN=(m;α,β;α′,β′) may be defined as a triangle intuitionistic fuzzy number (TIFN) if and only if its membership and nonmembership functions take the following form:
(2.8)μA(x)={0for-∞<x≤m-α,1-m-xαform-α≤x≤m,α>0,1forx=m,1-x-mβform≤x≤m+β,β>0,0form+β≤x<∞,(2.9)νA(x)={1for-∞≤x≤m-α′,m-xα′form-α′≤x≤m,α′>0,0forx=m,x-mβ′form≤x≤m+β′,β′>0,1form+β′≤x≤∞.Now for a TIFN, we can prove the following result.Proposition 2.4.
Let one consider a TIFN of the formATIFN=(m;α,β;α′,β′); then  α′>α and β′>β.Proof.
The membership and nonmembership functions ofATIFN  is given above in (2.8) and (2.9), respectively.
From (2.8) and (2.9), for m-α≤x≤m, we can write
(2.10)vA(x)+1-m-xα≤1;(sinceμA(x)+νA(x)≤1)⇔νA(x)≤m-xα;⇔m-xα'≤m-xα;⇔(m-x)(1α′-1α)≤0.
Since x≤m, therefore the following can be written:
(2.11)(1α′-1α)≤0⇔(α-α′)≤0⇔α≤α′.
Similarly form≤x≤m+β, we can write
(2.12)νA(x)+1-x-mβ≤1;(sinceμA(x)+νA(x)≤1)⇔νA(x)≤x-mβ;⇔x-mβ′≤x-mβ;⇔(x-m)(1β′-1β)≤0.
Since x≥m, therefore the following can be written:
(2.13)(1β′-1β)≤0⇔(β-β′)≤0⇔β≤β′.
Therefore symbolically a TIFN is represented as ATIFN=(m;α,β;α′,β′:α′>α,β′>β).Definition 2.5 (Positive Intuitionistic fuzzy number).
(i) An IFN denoted asAIFN=(m;α,β;α/,β/) is said to be positive if both m-α≥0 and m-α′≥0.
(ii) A TIFN denoted asATIFN=(m;α,β;α′,β′:α′>α,β′>β) is called positive if  m-α′≥0 (follows from Proposition 2.4).


### 2.3.ε-Cut Representation of IFN

Let us consider an IFNAIFN=(m;α,β;α′,β′) defined on the real line R as described before. The ε cut of IFN AIFN is defined by
(2.14)Aε={〈x,μA(x),νA(x)〉∣μA(x)≥ε&νA(x)≤1-ε}∀ε∈[0,1].Theε cut representation of IFN AIFN generates the following pair of intervals and is denoted by
(2.15)[AIFN]ε={[AμL(ε),AμR(ε)];[A1-νL(ε),A1-νR(ε)]},where the interval[AμL(ε),AμR(ε)] is defined in the following way:
(2.16)AμR(ε)={sup{x∣x∈με}ifε>0,sup{x∣x∈[m-α,m+β]}ifε=0,AμL(ε)={inf{x∣x∈με}ifε>0,inf{x∣x∈[m-α,m+β]}ifε=0.
Here με is defined by με={x∣μA(x)≥ε}.And in a similar manner the interval[A1-νL(ε),A1-νR(ε)] can be defined as follows:
(2.17)A1-νL(ε)={inf{x∣x∈vε}ifε>0,inf{x∣x∈[m-α′,m+β′]}ifε=0,A1-νR(ε)={sup{x∣x∈νε}if1-ε>0,sup{x∣x∈[m-α',m+β']}if1-ε=0,whereνε is defined by νε={x∣νA(x)≤1-ε}={x∣1-νA(x)≥ε}.


### 2.4. Overview of the Existing Distance Measures

#### 2.4.1. Notes on the Distance Measures for IFSs

Let us consider twoA,B∈IFS(X) with membership functions μA(x), μB(x) and nonmembership functions νA(x), νB(x), respectively.Atanassov [19] suggested the distance measures as follows.The normalized Hamming distancedA(A,B) is
(2.18)dA(A,B)=12n∑i=1n[|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|].The normalized Euclidean distanceρA(A,B) is
(2.19)ρA(A,B)=12n∑i=1n[(μA(xi)-μB(xi))2+(νA(xi)-νB(xi))2].Then in 2000, it was shown by Szmidt and Kacprzyk [20] that on computing distance for IFSs, all the three parameters, the degree of membership μ, the degree of non membership ν and the hesitation π describing IFSs, should be taken into account. And therefore they modified the concept of distances proposed by Atanassov [19]. The definition of distances presented by Szmidt and Kacprzyk [20] is given as following:The normalized Hamming distancedSK(A,B) is(2.20)dSK(A,B)=12n∑i=1n[|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|+|πA(xi)-πB(xi)|].The normalized Euclidean distanceρSK(A,B) is(2.21)ρSK(A,B)=12n∑i=1n[(μA(xi)-μB(xi))2+(νA(xi)-νB(xi))2]+(πA(xi)-πB(xi))2.Developing the above distance measures, Szmidt and Kacprzyk [20] claim that as their distance measures for IFSs are calculated incorporating all the three parameters describing IFSs, it reflects distances in three-dimensional spaces. On the other hand, the distance measures proposed by Atanassov [19] are the orthogonal projections of the real distances. And in this respect in their opinion their distance measures for IFSs are better than that of Atanassov.But Grzegorzewski [21] was not convinced with the point of view of Szmidt and Kacprzyk [20]. And based on Hausdorff metric, Grzegorzewski [21] proposed another group of distance measures for IFSs as follows.Normalized Hamming distancedG(A,B) is(2.22)dG(A,B)=1n∑i=1nmax{|μA(xi)-μB(xi)|,|νA(xi)-νB(xi)|}.The normalized Euclidean distanceρG(A,B) is(2.23)ρG(A,B)=1n∑i=1nmax{(μA(xi)-μB(xi))2,(νA(xi)-νB(xi))2}.
Obviously, the above distance measures proposed by Grzegorzewski [21] are easy for application. But in reality it may not fit so well. For example, let us consider three IFSs A,B,C∈IFS(X) where X={x1} and using the notation in (2.2) IFSs A, B, and C are of the following form: A={〈x1,1,0〉}, B={〈x1,0,1〉}, and C={〈x1,0,0〉}. If we use the ten-person-voting model to interpret, it would be noted that A={〈x1,1,0〉} represents ten personwhoall vote for a person; B={〈x1,0,1〉} represents ten persons whoall vote against him; whereas C={〈x1,0,0〉} represents ten personswhoall hesitate. So it is quite reasonable for us to think that the difference between A and C is lesser than the difference between A and B. But for the distances defined above, the difference between A and C is just equal to the difference between A and B, which is not so reasonable for us. This is the shortcomings of the distance measures proposed by Grzegorzewski [21].Again in 2005, Wang and Xin [22] first with help of some examples had shown that the distance measure proposed by Szmidt and Kacprzyk [20] is not reasonable for some cases and then developed the following distance measures:
(2.24)dWX(A,B)=1n∑i=1n[|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|4+max(|μA(xi)-μB(xi)|,|νA(xi)-νB(xi)|)2],(2.25)ρWXp(A,B)=1np∑i=1n[|μA(xi)-μB(xi)|2+|νA(xi)-νB(xi)|2]pp.
Though the above distance measures satisfy the properties of a distance measures, but in practice it is realized that the second one is not suitable for some cases. For example, consider three IFSs A,B,C∈IFS(X) where X={x1} and A,B, and C are of the following form: A={〈x1,1,0〉}, B={〈x1,0,0〉}, and C={〈x1,0.5,0.5〉}. If we use the ten-person-voting model to interpret, it would benotedthat A={〈x1,1,0〉} represents ten personswhoall vote for a person; B={〈x1,0,0〉} represents ten person all hesitate; whereas C={〈x1,0.5,0.5〉} represents half of ten person all vote for a person and the rest vote against him; no one is in hesitation. So it is quite reasonable for us to think that the difference between A and C is lesser than the difference between Aand B. But for the second distance defined above, the difference between A and C is just equal to the difference between A and B, which is not so reasonable for us. This is the shortcomings of the distance measure (2.25) proposed by Wang and Xin [22].Now in 2005, Huang et al. [23] suggested several distance measures for IFSs. At first they developed a group of distance measures to unify the distances proposed by Atanassov [19] and Grzegorzewski [21]. After that they proposed the following group of distance measures for IFSs:(2.26)dLW(A,B)=1n∑i=1n2(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|)2+(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|),ρLW(A,B)=2∑i=1n(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|)2n+∑i=1n(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|),ρLW′(A,B)=2∑i=1n(|μA(xi)-μB(xi)|p+|νA(xi)-νB(xi)|p)1/p(2n)1/p+∑i=1n(|μA(xi)-μB(xi)|p+|νA(xi)-νB(xi)|p)1/p.
In 2006, based on Lp metric Hung and Yang [24] defined the following distance measure:
(2.27)dHYp(A,B)=1n∑i=1n(|μA(xi)-μB(xi)|p+|νA(xi)-νB(xi)|p)1/p.
Now after analyzing the above four distance measures we can say that these measures only reflect the difference between μA(x) and νA(x) and their influence to measure the distances; they do not reflect the influence of degree of indeterminacy or hesitation.So after a short review of the existing measures between IFSs, in our opinion, all the measures have some advantages as well as some disadvantages. Therefore we cannot say that one particular distance measure is the best and should replace others. In our opinion all existing distance measures are valuable. From application point of view it can be said that depending on the characteristics of the data and the specific requirements of the problem, we need to decide what measure should be used.However, after reviewing the existing measures it is seen that the distance measures mentioned above calculate distance measures for IFSs of finite universe of discourse. Therefore the problem of developing distance measures for IFNs was an open problem. Then Grzegorzewski [28] investigated two families of metrics in space of IFNs as given in the next section.


#### 2.4.2. The Distance Measures between IFNs

Consider thatA={〈x,μA(x),νA(x)〉:x∈R} and B={〈x,μB(x),νB(x)〉:x∈R} are two intuitionistic fuzzy numbers. Now ε cut representation of the IFNs AIFN and BIFN  is denoted by
(2.28)[AIFN]ε={[AμL(ε),AμR(ε)];[A1-νL(ε),A1-νR(ε)]},[BIFN]ε={[BμL(ε),BμR(ε)];[B1-νL(ε),B1-νR(ε)]}∀ε∈[0,1].Then Grzegorzewski [28] proposed the distance measures as follows.For1≤p≤∞(2.29)dGp(A,B)=(14∫01|AμL(ε)-BμL(ε)|pdε+14∫01|AμR(ε)-BμR(ε)|pdε+14∫01|A1-νL(ε)-B1-νL(ε)|pdε+14∫01|A1-νR(ε)-B1-νR(ε)|pdε)1/p.
And for p=∞(2.30)dGp(A,B)=14sup0<ε≤1|AμL(ε)-BμL(ε)|+14sup0<ε≤1|AμR(ε)-BμR(ε)|+14sup0<ε≤1|A1-νL(ε)-B1-νL(ε)|+14sup0<ε≤1|A1-νR(ε)-B1-νR(ε)|.
For 1≤p≤∞(2.31)ρGp(A,B)=max(∫01|AμL(ε)-BμL(ε)|pdεp,∫01|AμR(ε)-BμR(ε)|pdε,p∫01|A1-νL(ε)-B1-νL(ε)|pdεp,∫01|A1-νR(ε)-B1-νR(ε)|pdεp).
And for p=∞(2.32)ρGp(A,B)=max{sup0<ε≤1|AμL(ε)-BμL(ε)|,sup0<ε≤1|AμR(ε)-BμR(ε)|,sup0<ε≤1|A1-νL(ε)-B1-νL(ε)|,sup0<ε≤1|A1-νR(ε)-B1-νR(ε)|}.
After reviewing the existing measures we realize the need to explore new points of view and the need to develop new distance measures that contain more information if we want them to be more logical. We believe that the distance between two uncertain numbers never generates a crisp value. The uncertainty inherent in the number should be intrinsically connected with their distance value.  With this point of view in the next section we define new distance measure for IFNs based on the interval difference.


## 2.1. Intuitionistic Fuzzy Sets—Basic Definition and Notation

LetX denote a universe of discourse. Then a fuzzy set A/ in X is defined as a set of ordered pairs:
(2.1)A/={〈x,μA/(x)〉:x∈X},
where μA/:X→[0,1] and μA/(x) is the grade of belongingness of x into A/ [1]. Thus automatically the grade of nonbelongingness of x into A/ is equal to 1-μA/(x). However, while expressing the degree of membership of any given element in a fuzzy set, the corresponding degree of nonmembership is not always expressed as a compliment to 1. The fact is that in real life, the linguistic negation does not always identify with logical negation [21]. Therefore Atanassov [19, 30–33] suggested a generalization of classical fuzzy set, called IFS.An IFSA in Xis given by a set of ordered triples:
(2.2)A={〈x,μA(x),νA(x)〉:x∈X},
where μA,νA:X→[0,1] are functions such that 0≤μA(x)+νA(x)≤1forallx∈X. For eachx the numbers μA(x) and νA(x) represent the degree of membership and degree of nonmembership of the element x∈X to A⊂X, respectively.It is easily seen thatAIFS={〈x,μA(x),1-μA(x)〉:x∈X} is equivalent to (2.1); that is, each fuzzy set is a particular case of the IFS. We will denote a family of fuzzy sets in X by FS(X), while IFS(X) stands for the family of all IFSs in X.For each elementx∈X we can compute, so called, the intuitionistic fuzzy index of x in A defined as follows:
(2.3)πA(x)=1-μA(x)-νA(x).
The value of πA(x) is called the degree of indeterminacy (or hesitation) of the element x∈X to the IFS A. It is seen immediately that πA(x)∈[0,1]. If A∈FS(X), then πA(x)=0forallx∈X.


## 2.2. Intuitionistic Fuzzy Numbers

Different research works [25–29] were done over Intuitionistic Fuzzy Numbers (IFNs). Taking care of those research works in this section the notion of IFNs is studied. IFN is the generalization of fuzzy number and so it can be represented in the following manner.Figure 1
TheLR- representation of intuitionistic fuzzy number ADefinition 2.1 (Intuitionistic fuzzy numbers).
An intuitionistic fuzzy subsetA={〈x,μA(x),νA(x)〉:x∈X} of the real  line R is called an IFN if the following holds. (i)
There existm∈R,μA(m)=1, and νA(m)=0, (m is called the mean value of A).(ii)
μA is a continuous mapping from R to the closed interval [0,1] and forallx∈R the relation 0≤μA(x)+νA(x)≤1 holds.(iii)
The membership and nonmembership function ofAis of the following form:
(2.4)μA(x)={0for-∞<x≤m-α,f1(x)forx∈[m-α,m],1forx=m,h1(x)forx∈[m,m+β],0form+β≤x<∞,
where f1(x) and h1(x) are strictly increasing and decreasing functions in [m-α,m] and [m,m+β], respectively:
(2.5)νA(x)={1for-∞≤x≤m-α′,f2(x)forx∈[m-α′,m];0≤f1(x)+f2(x)≤1,0forx=mh2(x)forx∈[m,m+β′];0≤h1(x)+h2(x)≤1,1form+β′≤x≤∞.
Herem is the mean value of A.α and β are called left and right spreads of membership function μA(x), respectively. α′ and β′ represented left and right spreads of nonmembership function νA(x), respectively. Symbolically the intuitionistic fuzzy number is represented as AIFN=(m;α,β;α′,β′).
It is to be noted here that the IFNA={〈x,μA(x),νA(x)〉:x∈R}, that is, AIFN=(m;α,β;α′,β′) is a conjunction of two fuzzy numbers: A+=(m;α,β) with a membership function μA+(x)=μA(x) and A-=(m;α′,β′) with a membership function μA-(x)=1-νA(x).Definition 2.2 (LR-type Intuitionistic fuzzy number).
An IFNAIFN is LR-type IFN such that for membership and nonmembership functions 0≤μA(x)+νA(x)≤1 holds and may be defined as follows.
Membership function is of the form as follows:(2.6)μA(x)={0for-∞<x≤m-α,L(m-xα)form-α≤x≤m,α>0,1forx=m,R(x-mβ)form≤x≤m+β,β>0,0form+β≤x<∞.
Nonmembership function is of the form:(2.7)νA(x)={1for-∞≤x≤m-α′,1-L(m-xα/)form-α′≤x≤m,α′>0,0forx=m,1-R(x-mβ/)form≤x≤m+β′,β′>0,1form+β′≤x≤∞.
ProvidedL(1)=R(1)=0, L is for left, and R is for right reference, m is the mean value of A.αand β are called left and right spreads of membership functions, respectively. α′ and β′ represented left and right spreads of nonmembership functions, respectively. Symbolically, we write AIFN=(m;α,β;α′,β′)LR. Here for L(x) and R(x) different functions may be chosen. For example, L(x)=R(x)=max(0,1-|x|p),p≥0, and so forth (Figure 1).Definition 2.3 (Triangle Intuitionistic fuzzy number).
An IFNAIFN=(m;α,β;α′,β′) may be defined as a triangle intuitionistic fuzzy number (TIFN) if and only if its membership and nonmembership functions take the following form:
(2.8)μA(x)={0for-∞<x≤m-α,1-m-xαform-α≤x≤m,α>0,1forx=m,1-x-mβform≤x≤m+β,β>0,0form+β≤x<∞,(2.9)νA(x)={1for-∞≤x≤m-α′,m-xα′form-α′≤x≤m,α′>0,0forx=m,x-mβ′form≤x≤m+β′,β′>0,1form+β′≤x≤∞.Now for a TIFN, we can prove the following result.Proposition 2.4.
Let one consider a TIFN of the formATIFN=(m;α,β;α′,β′); then  α′>α and β′>β.Proof.
The membership and nonmembership functions ofATIFN  is given above in (2.8) and (2.9), respectively.
From (2.8) and (2.9), for m-α≤x≤m, we can write
(2.10)vA(x)+1-m-xα≤1;(sinceμA(x)+νA(x)≤1)⇔νA(x)≤m-xα;⇔m-xα'≤m-xα;⇔(m-x)(1α′-1α)≤0.
Since x≤m, therefore the following can be written:
(2.11)(1α′-1α)≤0⇔(α-α′)≤0⇔α≤α′.
Similarly form≤x≤m+β, we can write
(2.12)νA(x)+1-x-mβ≤1;(sinceμA(x)+νA(x)≤1)⇔νA(x)≤x-mβ;⇔x-mβ′≤x-mβ;⇔(x-m)(1β′-1β)≤0.
Since x≥m, therefore the following can be written:
(2.13)(1β′-1β)≤0⇔(β-β′)≤0⇔β≤β′.
Therefore symbolically a TIFN is represented as ATIFN=(m;α,β;α′,β′:α′>α,β′>β).Definition 2.5 (Positive Intuitionistic fuzzy number).
(i) An IFN denoted asAIFN=(m;α,β;α/,β/) is said to be positive if both m-α≥0 and m-α′≥0.
(ii) A TIFN denoted asATIFN=(m;α,β;α′,β′:α′>α,β′>β) is called positive if  m-α′≥0 (follows from Proposition 2.4).


## 2.3.ε-Cut Representation of IFN

Let us consider an IFNAIFN=(m;α,β;α′,β′) defined on the real line R as described before. The ε cut of IFN AIFN is defined by
(2.14)Aε={〈x,μA(x),νA(x)〉∣μA(x)≥ε&νA(x)≤1-ε}∀ε∈[0,1].Theε cut representation of IFN AIFN generates the following pair of intervals and is denoted by
(2.15)[AIFN]ε={[AμL(ε),AμR(ε)];[A1-νL(ε),A1-νR(ε)]},where the interval[AμL(ε),AμR(ε)] is defined in the following way:
(2.16)AμR(ε)={sup{x∣x∈με}ifε>0,sup{x∣x∈[m-α,m+β]}ifε=0,AμL(ε)={inf{x∣x∈με}ifε>0,inf{x∣x∈[m-α,m+β]}ifε=0.
Here με is defined by με={x∣μA(x)≥ε}.And in a similar manner the interval[A1-νL(ε),A1-νR(ε)] can be defined as follows:
(2.17)A1-νL(ε)={inf{x∣x∈vε}ifε>0,inf{x∣x∈[m-α′,m+β′]}ifε=0,A1-νR(ε)={sup{x∣x∈νε}if1-ε>0,sup{x∣x∈[m-α',m+β']}if1-ε=0,whereνε is defined by νε={x∣νA(x)≤1-ε}={x∣1-νA(x)≥ε}.


## 2.4. Overview of the Existing Distance Measures

### 2.4.1. Notes on the Distance Measures for IFSs

Let us consider twoA,B∈IFS(X) with membership functions μA(x), μB(x) and nonmembership functions νA(x), νB(x), respectively.Atanassov [19] suggested the distance measures as follows.The normalized Hamming distancedA(A,B) is
(2.18)dA(A,B)=12n∑i=1n[|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|].The normalized Euclidean distanceρA(A,B) is
(2.19)ρA(A,B)=12n∑i=1n[(μA(xi)-μB(xi))2+(νA(xi)-νB(xi))2].Then in 2000, it was shown by Szmidt and Kacprzyk [20] that on computing distance for IFSs, all the three parameters, the degree of membership μ, the degree of non membership ν and the hesitation π describing IFSs, should be taken into account. And therefore they modified the concept of distances proposed by Atanassov [19]. The definition of distances presented by Szmidt and Kacprzyk [20] is given as following:The normalized Hamming distancedSK(A,B) is(2.20)dSK(A,B)=12n∑i=1n[|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|+|πA(xi)-πB(xi)|].The normalized Euclidean distanceρSK(A,B) is(2.21)ρSK(A,B)=12n∑i=1n[(μA(xi)-μB(xi))2+(νA(xi)-νB(xi))2]+(πA(xi)-πB(xi))2.Developing the above distance measures, Szmidt and Kacprzyk [20] claim that as their distance measures for IFSs are calculated incorporating all the three parameters describing IFSs, it reflects distances in three-dimensional spaces. On the other hand, the distance measures proposed by Atanassov [19] are the orthogonal projections of the real distances. And in this respect in their opinion their distance measures for IFSs are better than that of Atanassov.But Grzegorzewski [21] was not convinced with the point of view of Szmidt and Kacprzyk [20]. And based on Hausdorff metric, Grzegorzewski [21] proposed another group of distance measures for IFSs as follows.Normalized Hamming distancedG(A,B) is(2.22)dG(A,B)=1n∑i=1nmax{|μA(xi)-μB(xi)|,|νA(xi)-νB(xi)|}.The normalized Euclidean distanceρG(A,B) is(2.23)ρG(A,B)=1n∑i=1nmax{(μA(xi)-μB(xi))2,(νA(xi)-νB(xi))2}.
Obviously, the above distance measures proposed by Grzegorzewski [21] are easy for application. But in reality it may not fit so well. For example, let us consider three IFSs A,B,C∈IFS(X) where X={x1} and using the notation in (2.2) IFSs A, B, and C are of the following form: A={〈x1,1,0〉}, B={〈x1,0,1〉}, and C={〈x1,0,0〉}. If we use the ten-person-voting model to interpret, it would be noted that A={〈x1,1,0〉} represents ten personwhoall vote for a person; B={〈x1,0,1〉} represents ten persons whoall vote against him; whereas C={〈x1,0,0〉} represents ten personswhoall hesitate. So it is quite reasonable for us to think that the difference between A and C is lesser than the difference between A and B. But for the distances defined above, the difference between A and C is just equal to the difference between A and B, which is not so reasonable for us. This is the shortcomings of the distance measures proposed by Grzegorzewski [21].Again in 2005, Wang and Xin [22] first with help of some examples had shown that the distance measure proposed by Szmidt and Kacprzyk [20] is not reasonable for some cases and then developed the following distance measures:
(2.24)dWX(A,B)=1n∑i=1n[|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|4+max(|μA(xi)-μB(xi)|,|νA(xi)-νB(xi)|)2],(2.25)ρWXp(A,B)=1np∑i=1n[|μA(xi)-μB(xi)|2+|νA(xi)-νB(xi)|2]pp.
Though the above distance measures satisfy the properties of a distance measures, but in practice it is realized that the second one is not suitable for some cases. For example, consider three IFSs A,B,C∈IFS(X) where X={x1} and A,B, and C are of the following form: A={〈x1,1,0〉}, B={〈x1,0,0〉}, and C={〈x1,0.5,0.5〉}. If we use the ten-person-voting model to interpret, it would benotedthat A={〈x1,1,0〉} represents ten personswhoall vote for a person; B={〈x1,0,0〉} represents ten person all hesitate; whereas C={〈x1,0.5,0.5〉} represents half of ten person all vote for a person and the rest vote against him; no one is in hesitation. So it is quite reasonable for us to think that the difference between A and C is lesser than the difference between Aand B. But for the second distance defined above, the difference between A and C is just equal to the difference between A and B, which is not so reasonable for us. This is the shortcomings of the distance measure (2.25) proposed by Wang and Xin [22].Now in 2005, Huang et al. [23] suggested several distance measures for IFSs. At first they developed a group of distance measures to unify the distances proposed by Atanassov [19] and Grzegorzewski [21]. After that they proposed the following group of distance measures for IFSs:(2.26)dLW(A,B)=1n∑i=1n2(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|)2+(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|),ρLW(A,B)=2∑i=1n(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|)2n+∑i=1n(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|),ρLW′(A,B)=2∑i=1n(|μA(xi)-μB(xi)|p+|νA(xi)-νB(xi)|p)1/p(2n)1/p+∑i=1n(|μA(xi)-μB(xi)|p+|νA(xi)-νB(xi)|p)1/p.
In 2006, based on Lp metric Hung and Yang [24] defined the following distance measure:
(2.27)dHYp(A,B)=1n∑i=1n(|μA(xi)-μB(xi)|p+|νA(xi)-νB(xi)|p)1/p.
Now after analyzing the above four distance measures we can say that these measures only reflect the difference between μA(x) and νA(x) and their influence to measure the distances; they do not reflect the influence of degree of indeterminacy or hesitation.So after a short review of the existing measures between IFSs, in our opinion, all the measures have some advantages as well as some disadvantages. Therefore we cannot say that one particular distance measure is the best and should replace others. In our opinion all existing distance measures are valuable. From application point of view it can be said that depending on the characteristics of the data and the specific requirements of the problem, we need to decide what measure should be used.However, after reviewing the existing measures it is seen that the distance measures mentioned above calculate distance measures for IFSs of finite universe of discourse. Therefore the problem of developing distance measures for IFNs was an open problem. Then Grzegorzewski [28] investigated two families of metrics in space of IFNs as given in the next section.


### 2.4.2. The Distance Measures between IFNs

Consider thatA={〈x,μA(x),νA(x)〉:x∈R} and B={〈x,μB(x),νB(x)〉:x∈R} are two intuitionistic fuzzy numbers. Now ε cut representation of the IFNs AIFN and BIFN  is denoted by
(2.28)[AIFN]ε={[AμL(ε),AμR(ε)];[A1-νL(ε),A1-νR(ε)]},[BIFN]ε={[BμL(ε),BμR(ε)];[B1-νL(ε),B1-νR(ε)]}∀ε∈[0,1].Then Grzegorzewski [28] proposed the distance measures as follows.For1≤p≤∞(2.29)dGp(A,B)=(14∫01|AμL(ε)-BμL(ε)|pdε+14∫01|AμR(ε)-BμR(ε)|pdε+14∫01|A1-νL(ε)-B1-νL(ε)|pdε+14∫01|A1-νR(ε)-B1-νR(ε)|pdε)1/p.
And for p=∞(2.30)dGp(A,B)=14sup0<ε≤1|AμL(ε)-BμL(ε)|+14sup0<ε≤1|AμR(ε)-BμR(ε)|+14sup0<ε≤1|A1-νL(ε)-B1-νL(ε)|+14sup0<ε≤1|A1-νR(ε)-B1-νR(ε)|.
For 1≤p≤∞(2.31)ρGp(A,B)=max(∫01|AμL(ε)-BμL(ε)|pdεp,∫01|AμR(ε)-BμR(ε)|pdε,p∫01|A1-νL(ε)-B1-νL(ε)|pdεp,∫01|A1-νR(ε)-B1-νR(ε)|pdεp).
And for p=∞(2.32)ρGp(A,B)=max{sup0<ε≤1|AμL(ε)-BμL(ε)|,sup0<ε≤1|AμR(ε)-BμR(ε)|,sup0<ε≤1|A1-νL(ε)-B1-νL(ε)|,sup0<ε≤1|A1-νR(ε)-B1-νR(ε)|}.
After reviewing the existing measures we realize the need to explore new points of view and the need to develop new distance measures that contain more information if we want them to be more logical. We believe that the distance between two uncertain numbers never generates a crisp value. The uncertainty inherent in the number should be intrinsically connected with their distance value.  With this point of view in the next section we define new distance measure for IFNs based on the interval difference.


## 2.4.1. Notes on the Distance Measures for IFSs

Let us consider twoA,B∈IFS(X) with membership functions μA(x), μB(x) and nonmembership functions νA(x), νB(x), respectively.Atanassov [19] suggested the distance measures as follows.The normalized Hamming distancedA(A,B) is
(2.18)dA(A,B)=12n∑i=1n[|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|].The normalized Euclidean distanceρA(A,B) is
(2.19)ρA(A,B)=12n∑i=1n[(μA(xi)-μB(xi))2+(νA(xi)-νB(xi))2].Then in 2000, it was shown by Szmidt and Kacprzyk [20] that on computing distance for IFSs, all the three parameters, the degree of membership μ, the degree of non membership ν and the hesitation π describing IFSs, should be taken into account. And therefore they modified the concept of distances proposed by Atanassov [19]. The definition of distances presented by Szmidt and Kacprzyk [20] is given as following:The normalized Hamming distancedSK(A,B) is(2.20)dSK(A,B)=12n∑i=1n[|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|+|πA(xi)-πB(xi)|].The normalized Euclidean distanceρSK(A,B) is(2.21)ρSK(A,B)=12n∑i=1n[(μA(xi)-μB(xi))2+(νA(xi)-νB(xi))2]+(πA(xi)-πB(xi))2.Developing the above distance measures, Szmidt and Kacprzyk [20] claim that as their distance measures for IFSs are calculated incorporating all the three parameters describing IFSs, it reflects distances in three-dimensional spaces. On the other hand, the distance measures proposed by Atanassov [19] are the orthogonal projections of the real distances. And in this respect in their opinion their distance measures for IFSs are better than that of Atanassov.But Grzegorzewski [21] was not convinced with the point of view of Szmidt and Kacprzyk [20]. And based on Hausdorff metric, Grzegorzewski [21] proposed another group of distance measures for IFSs as follows.Normalized Hamming distancedG(A,B) is(2.22)dG(A,B)=1n∑i=1nmax{|μA(xi)-μB(xi)|,|νA(xi)-νB(xi)|}.The normalized Euclidean distanceρG(A,B) is(2.23)ρG(A,B)=1n∑i=1nmax{(μA(xi)-μB(xi))2,(νA(xi)-νB(xi))2}.
Obviously, the above distance measures proposed by Grzegorzewski [21] are easy for application. But in reality it may not fit so well. For example, let us consider three IFSs A,B,C∈IFS(X) where X={x1} and using the notation in (2.2) IFSs A, B, and C are of the following form: A={〈x1,1,0〉}, B={〈x1,0,1〉}, and C={〈x1,0,0〉}. If we use the ten-person-voting model to interpret, it would be noted that A={〈x1,1,0〉} represents ten personwhoall vote for a person; B={〈x1,0,1〉} represents ten persons whoall vote against him; whereas C={〈x1,0,0〉} represents ten personswhoall hesitate. So it is quite reasonable for us to think that the difference between A and C is lesser than the difference between A and B. But for the distances defined above, the difference between A and C is just equal to the difference between A and B, which is not so reasonable for us. This is the shortcomings of the distance measures proposed by Grzegorzewski [21].Again in 2005, Wang and Xin [22] first with help of some examples had shown that the distance measure proposed by Szmidt and Kacprzyk [20] is not reasonable for some cases and then developed the following distance measures:
(2.24)dWX(A,B)=1n∑i=1n[|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|4+max(|μA(xi)-μB(xi)|,|νA(xi)-νB(xi)|)2],(2.25)ρWXp(A,B)=1np∑i=1n[|μA(xi)-μB(xi)|2+|νA(xi)-νB(xi)|2]pp.
Though the above distance measures satisfy the properties of a distance measures, but in practice it is realized that the second one is not suitable for some cases. For example, consider three IFSs A,B,C∈IFS(X) where X={x1} and A,B, and C are of the following form: A={〈x1,1,0〉}, B={〈x1,0,0〉}, and C={〈x1,0.5,0.5〉}. If we use the ten-person-voting model to interpret, it would benotedthat A={〈x1,1,0〉} represents ten personswhoall vote for a person; B={〈x1,0,0〉} represents ten person all hesitate; whereas C={〈x1,0.5,0.5〉} represents half of ten person all vote for a person and the rest vote against him; no one is in hesitation. So it is quite reasonable for us to think that the difference between A and C is lesser than the difference between Aand B. But for the second distance defined above, the difference between A and C is just equal to the difference between A and B, which is not so reasonable for us. This is the shortcomings of the distance measure (2.25) proposed by Wang and Xin [22].Now in 2005, Huang et al. [23] suggested several distance measures for IFSs. At first they developed a group of distance measures to unify the distances proposed by Atanassov [19] and Grzegorzewski [21]. After that they proposed the following group of distance measures for IFSs:(2.26)dLW(A,B)=1n∑i=1n2(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|)2+(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|),ρLW(A,B)=2∑i=1n(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|)2n+∑i=1n(|μA(xi)-μB(xi)|+|νA(xi)-νB(xi)|),ρLW′(A,B)=2∑i=1n(|μA(xi)-μB(xi)|p+|νA(xi)-νB(xi)|p)1/p(2n)1/p+∑i=1n(|μA(xi)-μB(xi)|p+|νA(xi)-νB(xi)|p)1/p.
In 2006, based on Lp metric Hung and Yang [24] defined the following distance measure:
(2.27)dHYp(A,B)=1n∑i=1n(|μA(xi)-μB(xi)|p+|νA(xi)-νB(xi)|p)1/p.
Now after analyzing the above four distance measures we can say that these measures only reflect the difference between μA(x) and νA(x) and their influence to measure the distances; they do not reflect the influence of degree of indeterminacy or hesitation.So after a short review of the existing measures between IFSs, in our opinion, all the measures have some advantages as well as some disadvantages. Therefore we cannot say that one particular distance measure is the best and should replace others. In our opinion all existing distance measures are valuable. From application point of view it can be said that depending on the characteristics of the data and the specific requirements of the problem, we need to decide what measure should be used.However, after reviewing the existing measures it is seen that the distance measures mentioned above calculate distance measures for IFSs of finite universe of discourse. Therefore the problem of developing distance measures for IFNs was an open problem. Then Grzegorzewski [28] investigated two families of metrics in space of IFNs as given in the next section.


## 2.4.2. The Distance Measures between IFNs

Consider thatA={〈x,μA(x),νA(x)〉:x∈R} and B={〈x,μB(x),νB(x)〉:x∈R} are two intuitionistic fuzzy numbers. Now ε cut representation of the IFNs AIFN and BIFN  is denoted by
(2.28)[AIFN]ε={[AμL(ε),AμR(ε)];[A1-νL(ε),A1-νR(ε)]},[BIFN]ε={[BμL(ε),BμR(ε)];[B1-νL(ε),B1-νR(ε)]}∀ε∈[0,1].Then Grzegorzewski [28] proposed the distance measures as follows.For1≤p≤∞(2.29)dGp(A,B)=(14∫01|AμL(ε)-BμL(ε)|pdε+14∫01|AμR(ε)-BμR(ε)|pdε+14∫01|A1-νL(ε)-B1-νL(ε)|pdε+14∫01|A1-νR(ε)-B1-νR(ε)|pdε)1/p.
And for p=∞(2.30)dGp(A,B)=14sup0<ε≤1|AμL(ε)-BμL(ε)|+14sup0<ε≤1|AμR(ε)-BμR(ε)|+14sup0<ε≤1|A1-νL(ε)-B1-νL(ε)|+14sup0<ε≤1|A1-νR(ε)-B1-νR(ε)|.
For 1≤p≤∞(2.31)ρGp(A,B)=max(∫01|AμL(ε)-BμL(ε)|pdεp,∫01|AμR(ε)-BμR(ε)|pdε,p∫01|A1-νL(ε)-B1-νL(ε)|pdεp,∫01|A1-νR(ε)-B1-νR(ε)|pdεp).
And for p=∞(2.32)ρGp(A,B)=max{sup0<ε≤1|AμL(ε)-BμL(ε)|,sup0<ε≤1|AμR(ε)-BμR(ε)|,sup0<ε≤1|A1-νL(ε)-B1-νL(ε)|,sup0<ε≤1|A1-νR(ε)-B1-νR(ε)|}.
After reviewing the existing measures we realize the need to explore new points of view and the need to develop new distance measures that contain more information if we want them to be more logical. We believe that the distance between two uncertain numbers never generates a crisp value. The uncertainty inherent in the number should be intrinsically connected with their distance value.  With this point of view in the next section we define new distance measure for IFNs based on the interval difference.


## 3. New Distance Measure for IFNs

Human intuition says that the distances between two uncertain numbers should also be an uncertain number. In view of this the distance measure for IFNs is defined here. The proposed distance measure is an extension of the fuzzy distance measure [34] in which the degree of rejection (that is degree of nonmembership) is considered with degrees of satisfaction (degree of membership). It is also seen that when there is no degree of hesitation; that is, when intuitionistic fuzzy number become fuzzy number, this new distance measure converts to the fuzzy distance measure for normalized fuzzy number [34].


### 3.1. Construction of the Distance Measure for IFNs

Let us consider two IFNsAIFN and BIFN as follows:(3.1)AIFN=(m1;α1,β1;α1′,β1′),BIFN=(m2;α2,β2;α2′,β2′).
Therefore, ε cut representation of the IFNs AIFN and BIFN is denoted by
(3.2)[AIFN]ε={[AμL(ε),AμR(ε)];[A1-νL(ε),A1-νR(ε)]},[BIFN]ε={[BμL(ε),BμR(ε)];[B1-νL(ε),B1-νR(ε)]}∀ε∈[0,1].From mathematical point of view we can say that sinceAIFN and BIFN are IFNs, therefore their distance measure should also have membership and nonmembership part.Let us denote the distance measure betweenAIFN and BIFN as dIFN=(d;θ1,σ1;θ2,σ2), where d is the mean value of the distance measure dIFN, θ1,σ1 and θ2,σ2 are the left spread and right spread of the membership function and nonmembership function of the distance measure dIFN, respectively.And denote theε cut of dIFN in the following way:
(3.3)[dIFN]ε={[dμL(ε),dμR(ε)];[d1-νL(ε),d1-νR(ε)]}∀ε∈[0,1].To calculate the value ofd,θ1,σ1, we have to formulate the membership function of the distance between AIFN and BIFN.Clearlyε (for 0≤ε≤1) cut representation of the membership function of AIFN and BIFN is [AμL(ε),AμR(ε)] and [BμL(ε),BμR(ε)], respectively.Now, the distance between[AμL(ε),AμR(ε)] and [BμL(ε),BμR(ε)] for all ε∈[0,1] is one of the following:(3.4)(a)[AμL(ε),AμR(ε)]-[BμL(ε),BμR(ε)]ifm1≥m2,or(b)[BμL(ε),BμR(ε)]-[AμL(ε),AμR(ε)]ifm1<m2.In order to consider both the notations together, an indicator variableη is introduced such that
(3.5)η([AμL(ε),AμR(ε)]-[BμL(ε),BμR(ε)])+(1-η)([BμL(ε),BμR(ε)]-[AμL(ε),AμR(ε)])=[Lμ(ε),Rμ(ε)],forη={1ifm1≥m2,0ifm1≥m2,
where
(3.6)Lμ(ε)=η(AμL(ε)-BμL(ε)+AμR(ε)-BμR(ε))+(BμL(ε)-AμR(ε)),Rμ(ε)=η(AμL(ε)-BμL(ε)+AμR(ε)-BμR(ε))+(BμR(ε)-AμL(ε)).
Thus
(3.7)[dμL(ε),dμR(ε)]={[L(ε),Rμ(ε)];forLμ(ε)≥0,[0,|Lμ(ε)|∨Rμ(ε)];forLμ(ε)≤0≤Rμ(ε)∀ε∈[0,1].
Now using (3.7) θ1,σ1 are defined as follows:
(3.8)θ1=dμL(1)-max{∫01dμL(ε)dε,0},σ1=∫01dμR(ε)dε-dμR(1).Similarlyθ2,σ2 are of the following form:
(3.9)θ2=d1-νL(1)-max{∫01d1-νL(ε)dε,0},σ2=∫01d1-νR(ε)dε-d1-νR(1),
where
(3.10)[d1-νL(ε),d1-νR(ε)]={[L1-ν(ε),R1-ν(ε)];forL1-ν(ε)≥0,[0,|L1-ν(ε)|∨R1-ν(ε)];forL1-ν(ε)≤0≤R1-ν(ε).
As L1-ν(ε)andR1-ν(ε) take the following form:
(3.11)L1-ν(ε)=η(A1-νL(ε)-B1-νL(ε)+A1-νR(ε)-B1-νR(ε))+(B1-νL(ε)-A1-νR(ε)),R1-ν(ε)=η(A1-νL(ε)-B1-νL(ε)+A1-νR(ε)-B1-νR(ε))+(B1-νR(ε)-A1-νL(ε)).
From (3.7) and (3.10), we can find d=dμL(1)=dμR(1)=d1-νL(1)=d1-νR(1). Therefore finally the distance measure between AIFN and BIFN is obtained as
(3.12)dIFN=(d;θ1,σ1;θ2,σ2).


### 3.2. Distance Measure for TIFNs

IfAIFN and BIFN are two TIFNs, then their distance measure with the help of the above approach (Section 3.1) should be a TIFN. It can be proved by the following proposition:Proposition 3.1.
Let one consider two TIFNs as follows:(3.13)ATIFN=(m1;α1,β1;α1/,β1/:α1/>α1,β1/>β1),BTIFN=(m2;α2,β2;α2/,β2/:α2/>α2,β2/>β2).
Then their distance measure dIFN=(d;θ1,σ1;θ2,σ2) is a TIFN.Proof.
Here we have considered two TIFNsA and B. Therefore the ε cut representation of  ATIFN and BTIFN is as follows:
(3.14)[ATIFN]ε={[AμL(ε),AμR(ε)];[A1-νL(ε),A1-νR(ε)]},[BTIFN]ε={[BμL(ε),BμR(ε)];[B1-νL(ε),B1-νR(ε)]}∀ε∈[0,1].
Now with the help of (2.8) and (2.9), respectively, we can write
(3.15)[AμL(ε),AμR(ε)]=[m1-α1(1-ε),m1+β1(1-ε)],(3.16)[A1-νL(ε),A1-νR(ε)]=[m1-α1′(1-ε),m1+β1′(1-ε)]∀ε∈[0,1].
And in a similar manner
(3.17)[BμL(ε),BμR(ε)]=[m2-α2(1-ε),m2+β2(1-ε)],(3.18)[B1-νL(ε),B1-νR(ε)]=[m2-α2′(1-ε),m2+β2′(1-ε)]∀ε∈[0,1].
Here two possibilities can arise depending on the position of the mean values ofATIFN and BTIFN, which are given as follows.Case 1.
Forη=1, that is, when m1≥m2, we proceed in the following way.
In (3.6), putting the value from (3.15) and (3.17), we can express Lμ(ε) and Rμ(ε) as follows:
(3.19)Lμ(ε)=[(m1-m2)-α1(1-ε)-β2(1-ε)],Rμ(ε)=[(m1-m2)+α2(1-ε)+β1(1-ε)].
Similarly from (3.11), with the help of (3.16) and (3.18), L1-ν(ε),R1-ν(ε) can be written in the following form:
(3.20)L1-ν(ε)=[(m1-m2)-α1′(1-ε)-β2′(1-ε)],R1-ν(ε)=[(m1-m2)+α2′(1-ε)+β1′(1-ε)].
Now from (3.7) and (3.10) we have the following distances for ATIFN and BTIFN:(3.21)dμL(ε)=[(m1-m2)-α1(1-ε)-β2(1-ε)]or0,dμR(ε)=[(m1-m2)+α2(1-ε)+β1(1-ε)],(3.22)d1-νL(ε)=[(m1-m2)-α1′(1-ε)-β2′(1-ε)]or0,d1-νR(ε)=[(m1-m2)+α2′(1-ε)+β1′(1-ε)].
As given by Section 3.1, using (3.21) we can obtain
(3.23)θ1=dμL(1)-max{∫01dμL(ε)dε,0}=(m1-m2)-max{[(m1-m2)-(α1+β2)/2],0}.In a similar way, θ2 can be evaluated as
(3.24)θ2=(m1-m2)-max{[(m1-m2)-(α1/+β2/)/2],0}.Now as α1′>α1andβ2′>β2(3.25)⇒(α1′+β2′)/2>(α1+β2)/2⇒max{[(m1-m2)-(α1/+β2/)/2],0}≤max{[(m1-m2)-(α1+β2)/2],0}⇒θ2>θ1Similarly for σ1=(α2+β1)/2 and σ2=(α2′+β1′)/2, it is clear that σ2>σ1 (as α1′>α1 and β2′>β2).              Now from Proposition 2.4, we can conclude that as θ2>θ1 and σ2>σ1, dIFN is a TIFN.Case 2.
Forη=0, the proof can be done in a similar manner as for Case 1. Hence the proof is completed.Therefore it is now proved that the distance measure betweenATIFN and BTIFN is a TIFN denoted by dTIFN=(d;θ1,σ1;θ2,σ2:θ2>θ1,σ2>σ1).


### 3.3. Metric Properties

The new distance measure satisfies the following properties of a distance metric.(i)
The distance measure proposed in the Section3.1 is a nonnegative intuitionistic fuzzy number.(ii)
For any two intuitionistic fuzzy numbersAIFN1and AIFN2 the following holds:
(3.26)d(AIFN1,AIFN2)=d(AIFN2,AIFN1).(iii)
For three IFNsAIFN1,AIFN2, and AIFN3, the distance measure satisfies the triangle inequality:
(3.27)d(AIFN1,AIFN2)+d(AIFN2,AIFN3)≥d(AIFN1,AIFN3).Proof.
Proof of property (i) follows from (3.12) and property (ii) can be proved with the help of (3.5).
Proof of property (iii) is given here.
LetAIFN1, AIFN2, and AIFN3 be three IFNs. The ε cut representation of IFNs AIFN1, AIFN2 and AIFN3 is expressed as
(3.28)[AIFNi]ε={[AμiL(ε),AμiR(ε)];[A1-νiL(ε),A1-νiR(ε)]}fori=1,2,3forε∈[0,1].
In order to prove the triangle inequality of the distance measure for the above three IFNsAIFN1, AIFN2 and AIFN3, we show below that the triangle inequality for the distance measure between the membership functions of AIFN1, AIFN2 and AIFN3 and nonmembership function of AIFN1, AIFN2, and AIFN3 should hold separately.
Hence, for membership function of the distance measure, the triangle inequality is established in the following way.
From (3.28) consider the interval number [AμiL(ε),AμiR(ε)] that is, the ε cut of membership function of AIFN1, AIFN2, and AIFN3.
Depending on the relative positions of the means ofAIFN1, AIFN2 and AIFN3, three situations arise.
Situation (I)
When mean ofAIFN1 is less than mean of AIFN2 which is less than mean of AIFN3(3.29)Aμ1L(1)+Aμ1R(1)2≤Aμ2L(1)+Aμ2R(1)2≤Aμ3L(1)+Aμ3R(1)2.
From (3.7), we have the following distances for
(3.30)(I.a)AIFN1,AIFN2:{dμL(ε)=Aμ2L(ε)-Aμ1R(ε),dμR(ε)=Aμ2R(ε)-Aμ1L(ε),(I.b)AIFN2,AIFN3:{d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε)or0,d̅μR(ε)=Aμ3R(ε)-Aμ2L(ε),(I.c)AIFN1,AIFN3:{d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε)or0,d̂μR(ε)=Aμ3R(ε)-Aμ1L(ε).
Therefore we have to prove that(3.31)dR+dL+d̅R+d̅L≥d̂R+d̂L.
From the above three options (I.a), (I.b) and (I.c), the following eight combinations are possible:(i)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε),d̅μL(ε)=0,d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(ii)
dμL(ε)=0,d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε),d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(iii)
dμL(ε)=0,d̅μL(ε)=0,d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(iv)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε),d̅μL(ε)=0,d̂μL(ε)=0,(v)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε),d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε),d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(vi)
dμL(ε)=0,d̅μL(ε)=0,d̂μL(ε)=0,(vii)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε),d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε),d̂μL(ε)=0,(viii)
dμL(ε)=0,d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε),d̂μL(ε)=0.
Now, from the above eight combinations, (vii) and (viii) are not possible.
As from (vii), the following can be seen that.(i)
The membership functions ofAIFN1 and AIFN2 are disjoint.(ii)
The membership functions ofAIFN2 and AIFN3 are disjoint.(iii)
The membership functions ofAIFN1 and AIFN3 intersect.
It is very clear that the above situation cannot be happened.
From (viii), we observe the following.(i)
The membership functions ofAIFN1 and AIFN2 intersect.(ii)
The memberships functions ofAIFN2 and AIFN3 are disjoint.(iii)
The memberships functions ofAIFN1 and AIFN3 intersect.
The situation (viii) is also not possible.
Therefore, for the rest six different cases, the proof of inequality (3.31) is given as follows:(i)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε), d̅μL(ε)=0, d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(3.32)dR+dL+d̅R+d̅L=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε+∫01[Aμ2R(ε)-Aμ2L(ε)]dε=∫01[Aμ2R(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε+∫01[Aμ2L(ε)-Aμ2L(ε)]dε≥∫01[Aμ3L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].(ii)
dμL(ε)=0, d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε), d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε)(3.33)dR+dL+d̅R+d̅L=∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3L(ε)-Aμ2R(ε)]dε+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ3L(ε)-Aμ2L(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε+∫01[Aμ2R(ε)-Aμ2R(ε)]dε≥∫01[Aμ3L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].(iii)
dμL(ε)=0, d̅μL(ε)=0, d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(3.34)dR+dL+d̅R+d̅L=∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ2R(ε)-Aμ2L(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥∫01[Aμ3L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].(iv)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε), d̅μL(ε)=0, d̂μL(ε)=0,(3.35)dR+dL+d̅R+d̅L=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε+∫01[Aμ2R(ε)-Aμ2L(ε)]dε≥∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].(v)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε), d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε), d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(3.36)dR+dL+d̅R+d̅L=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3L(ε)-Aμ2R(ε)]+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ3L(ε)-Aμ2R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥∫01[Aμ3L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].(vi)
dμL(ε)=0, d̅μL(ε)=0, d̂μL(ε)=0,(3.37)dR+dL+d̅R+d̅L=∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ3R(ε)-Aμ1L(ε)]dε+∫01[Aμ2R(ε)-Aμ2L(ε)]dε≥∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].
The proof is similar for the following two cases:(I)
([Aμ1L(1)+Aμ1R(1)]/2)≤([Aμ3L(1)+Aμ3R(1)]/2)≤([Aμ2L(1)+Aμ2R(1)]/2),(II)
([Aμ2L(1)+Aμ2R(1)]/2)≤([Aμ1L(1)+Aμ1R(1)]/2)≤([Aμ3L(1)+Aμ3R(1)]/2).
With the help of (3.28), considering the interval number[A1-νiL(ε),A1-νiR(ε)] for all ε∈[0,1], we can prove the triangle inequality for nonmembership function of the distance measure, in same way as for membership function.


## 3.1. Construction of the Distance Measure for IFNs

Let us consider two IFNsAIFN and BIFN as follows:(3.1)AIFN=(m1;α1,β1;α1′,β1′),BIFN=(m2;α2,β2;α2′,β2′).
Therefore, ε cut representation of the IFNs AIFN and BIFN is denoted by
(3.2)[AIFN]ε={[AμL(ε),AμR(ε)];[A1-νL(ε),A1-νR(ε)]},[BIFN]ε={[BμL(ε),BμR(ε)];[B1-νL(ε),B1-νR(ε)]}∀ε∈[0,1].From mathematical point of view we can say that sinceAIFN and BIFN are IFNs, therefore their distance measure should also have membership and nonmembership part.Let us denote the distance measure betweenAIFN and BIFN as dIFN=(d;θ1,σ1;θ2,σ2), where d is the mean value of the distance measure dIFN, θ1,σ1 and θ2,σ2 are the left spread and right spread of the membership function and nonmembership function of the distance measure dIFN, respectively.And denote theε cut of dIFN in the following way:
(3.3)[dIFN]ε={[dμL(ε),dμR(ε)];[d1-νL(ε),d1-νR(ε)]}∀ε∈[0,1].To calculate the value ofd,θ1,σ1, we have to formulate the membership function of the distance between AIFN and BIFN.Clearlyε (for 0≤ε≤1) cut representation of the membership function of AIFN and BIFN is [AμL(ε),AμR(ε)] and [BμL(ε),BμR(ε)], respectively.Now, the distance between[AμL(ε),AμR(ε)] and [BμL(ε),BμR(ε)] for all ε∈[0,1] is one of the following:(3.4)(a)[AμL(ε),AμR(ε)]-[BμL(ε),BμR(ε)]ifm1≥m2,or(b)[BμL(ε),BμR(ε)]-[AμL(ε),AμR(ε)]ifm1<m2.In order to consider both the notations together, an indicator variableη is introduced such that
(3.5)η([AμL(ε),AμR(ε)]-[BμL(ε),BμR(ε)])+(1-η)([BμL(ε),BμR(ε)]-[AμL(ε),AμR(ε)])=[Lμ(ε),Rμ(ε)],forη={1ifm1≥m2,0ifm1≥m2,
where
(3.6)Lμ(ε)=η(AμL(ε)-BμL(ε)+AμR(ε)-BμR(ε))+(BμL(ε)-AμR(ε)),Rμ(ε)=η(AμL(ε)-BμL(ε)+AμR(ε)-BμR(ε))+(BμR(ε)-AμL(ε)).
Thus
(3.7)[dμL(ε),dμR(ε)]={[L(ε),Rμ(ε)];forLμ(ε)≥0,[0,|Lμ(ε)|∨Rμ(ε)];forLμ(ε)≤0≤Rμ(ε)∀ε∈[0,1].
Now using (3.7) θ1,σ1 are defined as follows:
(3.8)θ1=dμL(1)-max{∫01dμL(ε)dε,0},σ1=∫01dμR(ε)dε-dμR(1).Similarlyθ2,σ2 are of the following form:
(3.9)θ2=d1-νL(1)-max{∫01d1-νL(ε)dε,0},σ2=∫01d1-νR(ε)dε-d1-νR(1),
where
(3.10)[d1-νL(ε),d1-νR(ε)]={[L1-ν(ε),R1-ν(ε)];forL1-ν(ε)≥0,[0,|L1-ν(ε)|∨R1-ν(ε)];forL1-ν(ε)≤0≤R1-ν(ε).
As L1-ν(ε)andR1-ν(ε) take the following form:
(3.11)L1-ν(ε)=η(A1-νL(ε)-B1-νL(ε)+A1-νR(ε)-B1-νR(ε))+(B1-νL(ε)-A1-νR(ε)),R1-ν(ε)=η(A1-νL(ε)-B1-νL(ε)+A1-νR(ε)-B1-νR(ε))+(B1-νR(ε)-A1-νL(ε)).
From (3.7) and (3.10), we can find d=dμL(1)=dμR(1)=d1-νL(1)=d1-νR(1). Therefore finally the distance measure between AIFN and BIFN is obtained as
(3.12)dIFN=(d;θ1,σ1;θ2,σ2).


## 3.2. Distance Measure for TIFNs

IfAIFN and BIFN are two TIFNs, then their distance measure with the help of the above approach (Section 3.1) should be a TIFN. It can be proved by the following proposition:Proposition 3.1.
Let one consider two TIFNs as follows:(3.13)ATIFN=(m1;α1,β1;α1/,β1/:α1/>α1,β1/>β1),BTIFN=(m2;α2,β2;α2/,β2/:α2/>α2,β2/>β2).
Then their distance measure dIFN=(d;θ1,σ1;θ2,σ2) is a TIFN.Proof.
Here we have considered two TIFNsA and B. Therefore the ε cut representation of  ATIFN and BTIFN is as follows:
(3.14)[ATIFN]ε={[AμL(ε),AμR(ε)];[A1-νL(ε),A1-νR(ε)]},[BTIFN]ε={[BμL(ε),BμR(ε)];[B1-νL(ε),B1-νR(ε)]}∀ε∈[0,1].
Now with the help of (2.8) and (2.9), respectively, we can write
(3.15)[AμL(ε),AμR(ε)]=[m1-α1(1-ε),m1+β1(1-ε)],(3.16)[A1-νL(ε),A1-νR(ε)]=[m1-α1′(1-ε),m1+β1′(1-ε)]∀ε∈[0,1].
And in a similar manner
(3.17)[BμL(ε),BμR(ε)]=[m2-α2(1-ε),m2+β2(1-ε)],(3.18)[B1-νL(ε),B1-νR(ε)]=[m2-α2′(1-ε),m2+β2′(1-ε)]∀ε∈[0,1].
Here two possibilities can arise depending on the position of the mean values ofATIFN and BTIFN, which are given as follows.Case 1.
Forη=1, that is, when m1≥m2, we proceed in the following way.
In (3.6), putting the value from (3.15) and (3.17), we can express Lμ(ε) and Rμ(ε) as follows:
(3.19)Lμ(ε)=[(m1-m2)-α1(1-ε)-β2(1-ε)],Rμ(ε)=[(m1-m2)+α2(1-ε)+β1(1-ε)].
Similarly from (3.11), with the help of (3.16) and (3.18), L1-ν(ε),R1-ν(ε) can be written in the following form:
(3.20)L1-ν(ε)=[(m1-m2)-α1′(1-ε)-β2′(1-ε)],R1-ν(ε)=[(m1-m2)+α2′(1-ε)+β1′(1-ε)].
Now from (3.7) and (3.10) we have the following distances for ATIFN and BTIFN:(3.21)dμL(ε)=[(m1-m2)-α1(1-ε)-β2(1-ε)]or0,dμR(ε)=[(m1-m2)+α2(1-ε)+β1(1-ε)],(3.22)d1-νL(ε)=[(m1-m2)-α1′(1-ε)-β2′(1-ε)]or0,d1-νR(ε)=[(m1-m2)+α2′(1-ε)+β1′(1-ε)].
As given by Section 3.1, using (3.21) we can obtain
(3.23)θ1=dμL(1)-max{∫01dμL(ε)dε,0}=(m1-m2)-max{[(m1-m2)-(α1+β2)/2],0}.In a similar way, θ2 can be evaluated as
(3.24)θ2=(m1-m2)-max{[(m1-m2)-(α1/+β2/)/2],0}.Now as α1′>α1andβ2′>β2(3.25)⇒(α1′+β2′)/2>(α1+β2)/2⇒max{[(m1-m2)-(α1/+β2/)/2],0}≤max{[(m1-m2)-(α1+β2)/2],0}⇒θ2>θ1Similarly for σ1=(α2+β1)/2 and σ2=(α2′+β1′)/2, it is clear that σ2>σ1 (as α1′>α1 and β2′>β2).              Now from Proposition 2.4, we can conclude that as θ2>θ1 and σ2>σ1, dIFN is a TIFN.Case 2.
Forη=0, the proof can be done in a similar manner as for Case 1. Hence the proof is completed.Therefore it is now proved that the distance measure betweenATIFN and BTIFN is a TIFN denoted by dTIFN=(d;θ1,σ1;θ2,σ2:θ2>θ1,σ2>σ1).


## 3.3. Metric Properties

The new distance measure satisfies the following properties of a distance metric.(i)
The distance measure proposed in the Section3.1 is a nonnegative intuitionistic fuzzy number.(ii)
For any two intuitionistic fuzzy numbersAIFN1and AIFN2 the following holds:
(3.26)d(AIFN1,AIFN2)=d(AIFN2,AIFN1).(iii)
For three IFNsAIFN1,AIFN2, and AIFN3, the distance measure satisfies the triangle inequality:
(3.27)d(AIFN1,AIFN2)+d(AIFN2,AIFN3)≥d(AIFN1,AIFN3).Proof.
Proof of property (i) follows from (3.12) and property (ii) can be proved with the help of (3.5).
Proof of property (iii) is given here.
LetAIFN1, AIFN2, and AIFN3 be three IFNs. The ε cut representation of IFNs AIFN1, AIFN2 and AIFN3 is expressed as
(3.28)[AIFNi]ε={[AμiL(ε),AμiR(ε)];[A1-νiL(ε),A1-νiR(ε)]}fori=1,2,3forε∈[0,1].
In order to prove the triangle inequality of the distance measure for the above three IFNsAIFN1, AIFN2 and AIFN3, we show below that the triangle inequality for the distance measure between the membership functions of AIFN1, AIFN2 and AIFN3 and nonmembership function of AIFN1, AIFN2, and AIFN3 should hold separately.
Hence, for membership function of the distance measure, the triangle inequality is established in the following way.
From (3.28) consider the interval number [AμiL(ε),AμiR(ε)] that is, the ε cut of membership function of AIFN1, AIFN2, and AIFN3.
Depending on the relative positions of the means ofAIFN1, AIFN2 and AIFN3, three situations arise.
Situation (I)
When mean ofAIFN1 is less than mean of AIFN2 which is less than mean of AIFN3(3.29)Aμ1L(1)+Aμ1R(1)2≤Aμ2L(1)+Aμ2R(1)2≤Aμ3L(1)+Aμ3R(1)2.
From (3.7), we have the following distances for
(3.30)(I.a)AIFN1,AIFN2:{dμL(ε)=Aμ2L(ε)-Aμ1R(ε),dμR(ε)=Aμ2R(ε)-Aμ1L(ε),(I.b)AIFN2,AIFN3:{d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε)or0,d̅μR(ε)=Aμ3R(ε)-Aμ2L(ε),(I.c)AIFN1,AIFN3:{d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε)or0,d̂μR(ε)=Aμ3R(ε)-Aμ1L(ε).
Therefore we have to prove that(3.31)dR+dL+d̅R+d̅L≥d̂R+d̂L.
From the above three options (I.a), (I.b) and (I.c), the following eight combinations are possible:(i)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε),d̅μL(ε)=0,d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(ii)
dμL(ε)=0,d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε),d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(iii)
dμL(ε)=0,d̅μL(ε)=0,d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(iv)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε),d̅μL(ε)=0,d̂μL(ε)=0,(v)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε),d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε),d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(vi)
dμL(ε)=0,d̅μL(ε)=0,d̂μL(ε)=0,(vii)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε),d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε),d̂μL(ε)=0,(viii)
dμL(ε)=0,d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε),d̂μL(ε)=0.
Now, from the above eight combinations, (vii) and (viii) are not possible.
As from (vii), the following can be seen that.(i)
The membership functions ofAIFN1 and AIFN2 are disjoint.(ii)
The membership functions ofAIFN2 and AIFN3 are disjoint.(iii)
The membership functions ofAIFN1 and AIFN3 intersect.
It is very clear that the above situation cannot be happened.
From (viii), we observe the following.(i)
The membership functions ofAIFN1 and AIFN2 intersect.(ii)
The memberships functions ofAIFN2 and AIFN3 are disjoint.(iii)
The memberships functions ofAIFN1 and AIFN3 intersect.
The situation (viii) is also not possible.
Therefore, for the rest six different cases, the proof of inequality (3.31) is given as follows:(i)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε), d̅μL(ε)=0, d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(3.32)dR+dL+d̅R+d̅L=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε+∫01[Aμ2R(ε)-Aμ2L(ε)]dε=∫01[Aμ2R(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε+∫01[Aμ2L(ε)-Aμ2L(ε)]dε≥∫01[Aμ3L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].(ii)
dμL(ε)=0, d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε), d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε)(3.33)dR+dL+d̅R+d̅L=∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3L(ε)-Aμ2R(ε)]dε+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ3L(ε)-Aμ2L(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε+∫01[Aμ2R(ε)-Aμ2R(ε)]dε≥∫01[Aμ3L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].(iii)
dμL(ε)=0, d̅μL(ε)=0, d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(3.34)dR+dL+d̅R+d̅L=∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ2R(ε)-Aμ2L(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥∫01[Aμ3L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].(iv)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε), d̅μL(ε)=0, d̂μL(ε)=0,(3.35)dR+dL+d̅R+d̅L=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε+∫01[Aμ2R(ε)-Aμ2L(ε)]dε≥∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].(v)
dμL(ε)=Aμ2L(ε)-Aμ1R(ε), d̅μL(ε)=Aμ3L(ε)-Aμ2R(ε), d̂μL(ε)=Aμ3L(ε)-Aμ1R(ε),(3.36)dR+dL+d̅R+d̅L=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3L(ε)-Aμ2R(ε)]+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ2L(ε)-Aμ1R(ε)]dε+∫01[Aμ3L(ε)-Aμ2R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥∫01[Aμ3L(ε)-Aμ1R(ε)]dε+∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].(vi)
dμL(ε)=0, d̅μL(ε)=0, d̂μL(ε)=0,(3.37)dR+dL+d̅R+d̅L=∫01[Aμ2R(ε)-Aμ1L(ε)]dε+∫01[Aμ3R(ε)-Aμ2L(ε)]dε=∫01[Aμ3R(ε)-Aμ1L(ε)]dε+∫01[Aμ2R(ε)-Aμ2L(ε)]dε≥∫01[Aμ3R(ε)-Aμ1L(ε)]dε≥d̂L+d̂R[Proved].
The proof is similar for the following two cases:(I)
([Aμ1L(1)+Aμ1R(1)]/2)≤([Aμ3L(1)+Aμ3R(1)]/2)≤([Aμ2L(1)+Aμ2R(1)]/2),(II)
([Aμ2L(1)+Aμ2R(1)]/2)≤([Aμ1L(1)+Aμ1R(1)]/2)≤([Aμ3L(1)+Aμ3R(1)]/2).
With the help of (3.28), considering the interval number[A1-νiL(ε),A1-νiR(ε)] for all ε∈[0,1], we can prove the triangle inequality for nonmembership function of the distance measure, in same way as for membership function.


## 4. Numerical Illustration

Here we have considered the following numerical examples to illustrate the proposed measure.Example 4.1.
Let us consider two IFSsA and B defined over the universe of discourse X where X={x1} is as follows: A={〈x1,0.6,0.3〉} and B={〈x1,0.7,0.2〉}.
ThereforeA can be expressed as a conjunction of two fuzzy numbers A+=(x1;0,0) with membership degree 0.6 and A-=(x1;0,0) with membership degree 0.7. And similarly B can be expressed as a conjunction of B+=(x1;0,0) with membership degree 0.7 and B-=(x1;0,0) with membership degree 0.8.
Applying the proposed distance method we will find the required distance measure asd={〈0,0.6,0.3〉}. In this way the proposed distance measure covers the case for X={x1}.Example 4.2.
Let us consider two IFNs sayAIFN=(5;1,2;2,3) and BIFN=(9;1,1;2,1) characterized by their membership functions and nonmembership functions as follows:
(4.1)μA(x)={0for-∞<x≤4,1-(5-x)3for4≤x≤5,1forx=5,1-((x-5)/2)3for5≤x≤7,0for7≤x<∞,νA(x)={1for-∞≤x≤3,((5-x)/2)3for3≤x≤5,0forx=5,((x-5)/2)3for5≤x≤7,1for7≤x≤∞,μB(x)={0for-∞<x≤8,1-9-xfor8≤x≤9,1forx=9,1-x-9for9≤x≤10,0for10≤x<∞,νB(x)={1for-∞≤x≤7,(9-x)/2for7≤x≤9,0forx=9,(x-9)for9≤x≤10,1for10≤x≤∞.
The ε cut representation of AIFN and BIFN is obtained as follows:
(4.2)[AIFN]ε={[5-(1-ε)1/3,5+2(1-ε)1/3];[5-2(1-ε)1/3,5+2(1-ε)1/3]},[BIFN]ε={[9-(1-ε)2,9+(1-ε)2];[9-2(1-ε)2,9+2(1-ε)2]}∀ε∈[0,1].
Theε cut of the distance measure dIFN using (3.7) and (3.10) is obtained as
(4.3)[dIFN]ε={[4-(1-ε)2-2(1-ε)1/3,4+(1-ε)2+(1-ε)1/3];[4-2(1-ε)2-2(1-ε)1/3,4+(1-ε)2+2(1-ε)1/3]}∀ε∈[0,1].
Finally the distance measure ofAIFN and BIFN, computed by the proposed method is given by dIFN=(4;1.84,1.08;2.17,1.83).Example 4.3.
In the following, 2 sets of TIFNs are given to compare the proposed distance method with the existing distance measures presented by Grzegorzewski [28]. A comparison between the results of the proposed distance measure and the results of the existing method is shown in Table 1.Table 1
Here a comparison of the proposed distance measure with the existing distance measure through the above set of 2 examples is given.
Distance measuresGrzegorzewski’s method [28]The Proposed Methodfor IFNsEquation (2.29); p=1Equation (2.31); 1≤p<∞Equation (3.12)Set 1:d(ATIFN1,BTIFN)7/81dTIFN=(1;1,2;1,3)Set 2:d(ATIFN2,BTIFN)5/81dTIFN′=(1;1,1.5;1,3)Set1.
ConsiderATIFN1=(3;2,2;3,4) and BTIFN=(4;2,2;3,3). The membership and nonmembership functions of ATIFN1 and BTIFN are, respectively, of the following form:
(4.4)μA1(x)={12(x-1);1≤x≤3,12(5-x);3≤x≤5,νA1(x)={13(3-x);0≤x≤3,14(x-3);3≤x≤7,μB(x)={12(x-2);2≤x≤4,12(6-x);4≤x≤6,νB(x)={13(4-x);2≤x≤4,13(x-4);4≤x≤7.Set2.
ConsiderATIFN2 = (3;1,3;3,4) and BTIFN = (4;2,2;3,3), where the membership and nonmembership functions of A2TIFN are given as follows:
(4.5)μA2(x)={(x-2);1≤x≤3,13(6-x);3≤x≤6,νA2(x)={13(3-x);0≤x≤3,14(x-3);3≤x≤7.
AndBTIFN is defined as in Set 1.
Now we have calculated the distance measure betweenATIFN1,BTIFN and ATIFN2, BTIFN by applying (2.29), (2.31), and (3.12) and the results are compared in Table 1.From Table1 we can see that Set 1 and Set 2 are different sets of TIFNs, but Equation (2.31) gives the same distance value. Therefore from Table 1, it can be said that Equation (2.31) is not so well fit in real life applications.Applying (2.29) and choosing p=1, we can say from Table 1 that d(ATIFN2,BTIFN)<d(ATIFN1,BTIFN).Also we utilize (3.12) to find the distance value of ATIFN1, BTIFN and ATIFN2, BTIFN respectively. Then we analyze the results by applying the defuzzification procedure proposed by Chang et al. [35] to the both distance measures, respectively, and we obtain d(ATIFN2,BTIFN)<d(ATIFN1,BTIFN). This result is matching with the result obtained from (2.29).


## 5. Conclusion

IFSs theory provides a flexible framework to cope with imperfect and/or imprecise information often present in real world application. The concept of IFS can be viewed as an alternative approach to define a fuzzy set in the case when available information is not sufficient to define a conventional fuzzy set. In this paper a new distance measure for computing distance for IFNs is introduced. We believe that the distances between two uncertain numbers should be an uncertain number. If the uncertainty is inherent within the numbers, this uncertainty should be intrinsically connected with their distance value. With this view point here a new method to measure the distance for IFNs is presented. What is important, the proposed distance method gives new viewpoints for the study of similarity of IFNs. This will be a topic of our future research work.



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*Source: 949143-2010-03-31.xml*