A Theoretical Development of Distance Measure for Intuitionistic Fuzzy Numbers

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.

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. Section 2 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

Let X denote a universe of discourse. Then a fuzzy set 𝐴^/ in X is defined as a set of ordered pairs: 𝐴^/={⟨𝑥,𝜇_𝐴/(𝑥)⟩∶𝑥∈X}, (2.1) where 𝜇_𝐴/∶X→[0,1] and 𝜇_𝐴/(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−𝜇_𝐴/(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 IFS A in X is given by a set of ordered triples: 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)⟩∶𝑥∈𝑋}, (2.2) where 𝜇_𝐴,𝜈_𝐴∶𝑋→[0,1] are functions such that 0≤𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1forall𝑥∈𝑋. For each x the numbers 𝜇_𝐴(𝑥) and 𝜈_𝐴(𝑥) represent the degree of membership and degree of nonmembership of the element 𝑥∈𝑋 to 𝐴⊂𝑋, respectively.

It is easily seen that 𝐴_IFS={⟨𝑥,𝜇_𝐴(𝑥),1−𝜇_𝐴(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 element 𝑥∈𝑋 we can compute, so called, the intuitionistic fuzzy index of 𝑥 in 𝐴 defined as follows: 𝜋_𝐴(𝑥)=1−𝜇_𝐴(𝑥)−𝜈_𝐴(𝑥). (2.3) The value of 𝜋_𝐴(𝑥) is called the degree of indeterminacy (or hesitation) of the element 𝑥∈𝑋 to the IFS A. It is seen immediately that 𝜋_𝐴(𝑥)∈[0,1]. If 𝐴∈FS(𝑋), then 𝜋_𝐴(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 

The LR- representation of intuitionistic fuzzy number A

Definition 2.1 (Intuitionistic fuzzy numbers). An intuitionistic fuzzy subset 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)⟩∶𝑥∈𝑋} of the real line R is called an IFN if the following holds. (i)There exist𝑚∈𝑅,𝜇_𝐴(𝑚)=1, and 𝜈_𝐴(𝑚)=0, (m is called the mean value of A).(ii)𝜇_𝐴 is a continuous mapping from R to the closed interval [0,1] and forall𝑥∈𝑅 the relation 0≤𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1 holds. (iii)The membership and nonmembership function of A is of the following form: 𝜇_𝐴𝑓(𝑥)=0for−∞<𝑥≤𝑚−𝛼,₁ℎ(𝑥)for𝑥∈[𝑚−𝛼,𝑚],1for𝑥=𝑚,₁(𝑥)for𝑥∈[𝑚,𝑚+𝛽],0for𝑚+𝛽≤𝑥<∞, (2.4) where 𝑓₁(𝑥) and ℎ₁(𝑥) are strictly increasing and decreasing functions in [𝑚−𝛼,𝑚] and [𝑚,𝑚+𝛽], respectively: 𝜈_𝐴(𝑥)=1for−∞≤𝑥≤𝑚−𝛼^′,𝑓₂(𝑥)for𝑥∈[𝑚−𝛼^′,𝑚];0≤𝑓₁(𝑥)+𝑓₂ℎ(𝑥)≤1,0for𝑥=𝑚₂(𝑥)for𝑥∈[𝑚,𝑚+𝛽^′];0≤ℎ₁(𝑥)+ℎ₂(𝑥)≤1,1for𝑚+𝛽^′≤𝑥≤∞. (2.5) Here m is the mean value of A. 𝛼 and 𝛽 are called left and right spreads of membership function 𝜇_𝐴(𝑥), respectively. 𝛼^′ and 𝛽^′ represented left and right spreads of nonmembership function 𝜈_𝐴(𝑥), respectively. Symbolically the intuitionistic fuzzy number is represented as 𝐴_IFN=(𝑚;𝛼,𝛽;𝛼^′,𝛽^′).
It is to be noted here that the IFN 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)⟩∶𝑥∈𝑅}, that is, 𝐴_IFN=(𝑚;𝛼,𝛽;𝛼^′,𝛽^′) is a conjunction of two fuzzy numbers: 𝐴⁺=(𝑚;𝛼,𝛽) with a membership function 𝜇_𝐴+(𝑥)=𝜇_𝐴(𝑥) and 𝐴⁻=(𝑚;𝛼^′,𝛽^′) with a membership function 𝜇_𝐴−(𝑥)=1−𝜈_𝐴(𝑥).

Definition 2.2 (𝐿𝑅-type Intuitionistic fuzzy number). An IFN 𝐴_IFN is 𝐿𝑅-type IFN such that for membership and nonmembership functions 0≤𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1 holds and may be defined as follows.
Membership function is of the form as follows: 𝜇_𝐴(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<∞. (2.6)
Nonmembership function is of the form: 𝜈_𝐴(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≤∞. (2.7)
Provided 𝐿(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 𝐴_IFN=(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 IFN 𝐴_IFN=(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: 𝜇_𝐴(𝑥)=0for−∞<𝑥≤𝑚−𝛼,1−𝑚−𝑥𝛼for𝑚−𝛼≤𝑥≤𝑚,𝛼>0,1for𝑥=𝑚,1−𝑥−𝑚𝛽𝜈for𝑚≤𝑥≤𝑚+𝛽,𝛽>0,0for𝑚+𝛽≤𝑥<∞,_𝐴(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≤∞. (2.8)(2.9)

Now for a TIFN, we can prove the following result.

Proposition 2.4. Let one consider a TIFN of the form 𝐴_TIFN=(𝑚;𝛼,𝛽;𝛼^′,𝛽^′); then 𝛼^′>𝛼 and 𝛽^′>𝛽.

Proof. The membership and nonmembership functions of 𝐴_TIFN is given above in (2.8) and (2.9), respectively.
From (2.8) and (2.9), for 𝑚−𝛼≤𝑥≤𝑚, we can write 𝑣_𝐴(𝑥)+1−𝑚−𝑥𝛼≤1;(since𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1)⇔𝜈_𝐴(𝑥)≤𝑚−𝑥𝛼;⇔𝑚−𝑥𝛼^′≤𝑚−𝑥𝛼;1⇔(𝑚−𝑥)(𝛼^′−1𝛼)≤0. (2.10) Since 𝑥≤𝑚, therefore the following can be written: (1𝛼^′−1𝛼)≤0⇔(𝛼−𝛼^′)≤0⇔𝛼≤𝛼^′. (2.11)
Similarly for 𝑚≤𝑥≤𝑚+𝛽, we can write 𝜈_𝐴(𝑥)+1−𝑥−𝑚𝛽≤1;(since𝜇_𝐴(𝑥)+𝜈_𝐴(𝑥)≤1)⇔𝜈_𝐴(𝑥)≤𝑥−𝑚𝛽;⇔𝑥−𝑚𝛽^′≤𝑥−𝑚𝛽;1⇔(𝑥−𝑚)(𝛽^′−1𝛽)≤0. (2.12) Since 𝑥≥𝑚, therefore the following can be written: (1𝛽^′−1𝛽)≤0⇔(𝛽−𝛽^′)≤0⇔𝛽≤𝛽^′. (2.13) Therefore symbolically a TIFN is represented as 𝐴_TIFN=(𝑚;𝛼,𝛽;𝛼^′,𝛽^′∶𝛼^′>𝛼,𝛽^′>𝛽).

Definition 2.5 (Positive Intuitionistic fuzzy number). (i) An IFN denoted as 𝐴_IFN=(𝑚;𝛼,𝛽;𝛼^/,𝛽^/) is said to be positive if both 𝑚−𝛼≥0 and 𝑚−𝛼^′≥0.
(ii) A TIFN denoted as 𝐴_TIFN=(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 IFN 𝐴_IFN=(𝑚;𝛼,𝛽;𝛼^′,𝛽^′) defined on the real line R as described before. The 𝜀 cut of IFN 𝐴_IFN is defined by 𝐴_𝜀={⟨𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)⟩∣𝜇_𝐴(𝑥)≥𝜀&𝜈_𝐴(𝑥)≤1−𝜀}∀𝜀∈[0,1]. (2.14)

The 𝜀 cut representation of IFN 𝐴_IFN generates the following pair of intervals and is denoted by [𝐴_IFN]_𝜀={[𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)];[𝐴^𝐿_1−𝜈(𝜀),𝐴^𝑅_1−𝜈(𝜀)]}, (2.15)

where the interval [𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)] is defined in the following way: 𝐴^𝑅_𝜇(𝜀)=sup{𝑥∣𝑥∈𝜇_𝜀𝐴}if𝜀>0,sup{𝑥∣𝑥∈[𝑚−𝛼,𝑚+𝛽]}if𝜀=0,_𝐿𝜇(𝜀)=inf{𝑥∣𝑥∈𝜇_𝜀}if𝜀>0,inf{𝑥∣𝑥∈[𝑚−𝛼,𝑚+𝛽]}if𝜀=0. (2.16) Here 𝜇_𝜀 is defined by 𝜇_𝜀={𝑥∣𝜇_𝐴(𝑥)≥𝜀}.

And in a similar manner the interval[𝐴^𝐿_1−𝜈(𝜀),𝐴^𝑅_1−𝜈(𝜀)] can be defined as follows: 𝐴^𝐿_1−𝜈(𝜀)=inf{𝑥∣𝑥∈𝑣_𝜀}if𝜀>0,inf{𝑥∣𝑥∈[𝑚−𝛼^′,𝑚+𝛽^′𝐴]}if𝜀=0,_𝑅1−𝜈(𝜀)=sup{𝑥∣𝑥∈𝜈_𝜀}if1−𝜀>0,sup{𝑥∣𝑥∈[𝑚−𝛼^′,𝑚+𝛽^′]}if1−𝜀=0, (2.17)

where 𝜈_𝜀 is defined by 𝜈_𝜀={𝑥∣𝜈_𝐴(𝑥)≤1−𝜀}={𝑥∣1−𝜈_𝐴(𝑥)≥𝜀}.

2.4. Overview of the Existing Distance Measures

2.4.1. Notes on the Distance Measures for IFSs

Let us consider two 𝐴,𝐵∈IFS(𝑋) with membership functions 𝜇_𝐴(𝑥), 𝜇_𝐵(𝑥) and nonmembership functions 𝜈_𝐴(𝑥), 𝜈_𝐵(𝑥), respectively.

Atanassov [19] suggested the distance measures as follows.

The normalized Hamming distance 𝑑_𝐴(𝐴,𝐵) is 𝑑_𝐴1(𝐴,𝐵)=2𝑛^𝑛_𝑖=1[|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|]. (2.18)

The normalized Euclidean distance 𝜌_𝐴(𝐴,𝐵) is 𝜌_𝐴(𝐴,𝐵)=12𝑛^𝑛_𝑖=1[(𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖))²+(𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖))²]. (2.19)

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 distance 𝑑_SK(𝐴,𝐵) is

𝑑_SK1(𝐴,𝐵)=2𝑛^𝑛_𝑖=1[|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|+|𝜋_𝐴(𝑥_𝑖)−𝜋_𝐵(𝑥_𝑖)|]. (2.20)

The normalized Euclidean distance 𝜌_SK(𝐴,𝐵) is

𝜌_SK(𝐴,𝐵)=12𝑛^𝑛_𝑖=1[(𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖))²+(𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖))²]+(𝜋_𝐴(𝑥_𝑖)−𝜋_𝐵(𝑥_𝑖))². (2.21)

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 distance 𝑑_𝐺(𝐴,𝐵) is

𝑑_G1(𝐴,𝐵)=𝑛^𝑛_𝑖=1max{|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|,|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|}. (2.22)

The normalized Euclidean distance 𝜌_𝐺(𝐴,𝐵) is

𝜌_𝐺(𝐴,𝐵)=1𝑛^𝑛_𝑖=1max{(𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖))²,(𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖))²}. (2.23) 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: 𝑑_WX1(𝐴,𝐵)=𝑛^𝑛_𝑖=1[|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|4+max(|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|,|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|)2],𝜌^𝑝_WX1(𝐴,𝐵)=_𝑝√𝑛^𝑝𝑛_𝑖=1[|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|2+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|2]^𝑝. (2.24)(2.25) 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 A and 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:

𝑑_LW1(𝐴,𝐵)=𝑛∑^𝑛_𝑖=12(|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|)2+(|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖,𝜌)|)_LW2∑(𝐴,𝐵)=^𝑛𝑖=1(|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|)2𝑛+∑^𝑛_𝑖=1(|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖,𝜌)|)_′LW(𝐴,𝐵)=2∑^𝑛_𝑖=1(|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|^𝑝+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|^𝑝)^1/𝑝(2𝑛)^1/𝑝+∑^𝑛_𝑖=1(|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|^𝑝+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|^𝑝)^1/𝑝. (2.26) In 2006, based on 𝐿_𝑝 metric Hung and Yang [24] defined the following distance measure: 𝑑^𝑝_HY1(𝐴,𝐵)=𝑛^𝑛_𝑖=1(|𝜇_𝐴(𝑥_𝑖)−𝜇_𝐵(𝑥_𝑖)|^𝑝+|𝜈_𝐴(𝑥_𝑖)−𝜈_𝐵(𝑥_𝑖)|^𝑝)^1/𝑝. (2.27) Now after analyzing the above four distance measures we can say that these measures only reflect the difference between 𝜇_𝐴(𝑥) and 𝜈_𝐴(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 that 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝜈_𝐴(𝑥)⟩∶𝑥∈𝑅} and 𝐵={⟨𝑥,𝜇_𝐵(𝑥),𝜈_𝐵(𝑥)⟩∶𝑥∈𝑅} are two intuitionistic fuzzy numbers. Now 𝜀 cut representation of the IFNs 𝐴_IFN and 𝐵_IFN is denoted by [𝐴_IFN]_𝜀={[𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)];[𝐴^𝐿_1−𝜈(𝜀),𝐴^𝑅_1−𝜈(𝜀)]},[𝐵_IFN]_𝜀={[𝐵^𝐿_𝜇(𝜀),𝐵^𝑅_𝜇(𝜀)];[𝐵^𝐿_1−𝜈(𝜀),𝐵^𝑅_1−𝜈(𝜀)]}∀𝜀∈[0,1]. (2.28)

Then Grzegorzewski [28] proposed the distance measures as follows.

For 1≤𝑝≤∞

𝑑^𝑝_𝐺1(𝐴,𝐵)=(4¹₀|𝐴^𝐿_𝜇(𝜀)−𝐵^𝐿_𝜇(𝜀)|^𝑝1𝑑𝜀+4¹₀|𝐴^𝑅_𝜇(𝜀)−𝐵^𝑅_𝜇(𝜀)|^𝑝+1𝑑𝜀4¹₀|𝐴^𝐿_1−𝜈(𝜀)−𝐵^𝐿_1−𝜈(𝜀)|^𝑝1𝑑𝜀+4¹₀|𝐴^𝑅_1−𝜈(𝜀)−𝐵^𝑅_1−𝜈(𝜀)|^𝑝𝑑𝜀)^1/𝑝. (2.29) And for 𝑝=∞

𝑑^𝑝_𝐺1(𝐴,𝐵)=4sup_0<𝜀≤1|𝐴^𝐿_𝜇(𝜀)−𝐵^𝐿_𝜇1(𝜀)|+4sup_0<𝜀≤1|𝐴^𝑅_𝜇(𝜀)−𝐵^𝑅_𝜇+1(𝜀)|4sup_0<𝜀≤1|𝐴^𝐿_1−𝜈(𝜀)−𝐵^𝐿_1−𝜈1(𝜀)|+4sup_0<𝜀≤1|𝐴^𝑅_1−𝜈(𝜀)−𝐵^𝑅_1−𝜈(𝜀)|. (2.30) For 1≤𝑝≤∞𝜌^𝑝_𝐺(𝐴,𝐵)=max(^𝑝1₀|𝐴^𝐿_𝜇(𝜀)−𝐵^𝐿_𝜇(𝜀)|^𝑝𝑑𝜀,^𝑝1₀|𝐴^𝑅_𝜇(𝜀)−𝐵^𝑅_𝜇(𝜀)|^𝑝𝑑𝜀,_𝑝10|𝐴^𝐿_1−𝜈(𝜀)−𝐵^𝐿_1−𝜈(𝜀)|^𝑝𝑑𝜀,^𝑝1₀|𝐴^𝑅_1−𝜈(𝜀)−𝐵^𝑅_1−𝜈(𝜀)|^𝑝𝑑𝜀). (2.31) And for 𝑝=∞

𝜌^𝑝_𝐺(𝐴,𝐵)=max{sup_0<𝜀≤1|𝐴^𝐿_𝜇(𝜀)−𝐵^𝐿_𝜇(𝜀)|,sup_0<𝜀≤1|𝐴^𝑅_𝜇(𝜀)−𝐵^𝑅_𝜇(𝜀)|,sup_0<𝜀≤1|𝐴^𝐿_1−𝜈(𝜀)−𝐵^𝐿_1−𝜈(𝜀)|,sup_0<𝜀≤1|𝐴^𝑅_1−𝜈(𝜀)−𝐵^𝑅_1−𝜈(𝜀)|}. (2.32) 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 IFNs 𝐴_IFN and 𝐵_IFN as follows:

𝐴_IFN=(𝑚₁;𝛼₁,𝛽₁;𝛼₁^′,𝛽₁^′),𝐵_IFN=(𝑚₂;𝛼₂,𝛽₂;𝛼₂^′,𝛽₂^′). (3.1) Therefore, 𝜀 cut representation of the IFNs 𝐴_IFN and 𝐵_IFN is denoted by [𝐴_IFN]_𝜀={[𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)];[𝐴^𝐿_1−𝜈(𝜀),𝐴^𝑅_1−𝜈(𝜀)]},[𝐵_IFN]_𝜀={[𝐵^𝐿_𝜇(𝜀),𝐵^𝑅_𝜇(𝜀)];[𝐵^𝐿_1−𝜈(𝜀),𝐵^𝑅_1−𝜈(𝜀)]}∀𝜀∈[0,1]. (3.2)

From mathematical point of view we can say that since 𝐴_IFN and 𝐵_IFN are IFNs, therefore their distance measure should also have membership and nonmembership part.

Let us denote the distance measure between 𝐴_IFN and 𝐵_IFN as 𝑑_IFN=(𝑑;𝜃₁,𝜎₁;𝜃₂,𝜎₂), where d is the mean value of the distance measure 𝑑_IFN, 𝜃₁,𝜎₁ and 𝜃₂,𝜎₂ are the left spread and right spread of the membership function and nonmembership function of the distance measure 𝑑_IFN, respectively.

And denote the 𝜀 cut of 𝑑_IFN in the following way: [𝑑_IFN]_𝜀={[𝑑^𝐿_𝜇(𝜀),𝑑^𝑅_𝜇(𝜀)];[𝑑^𝐿_1−𝜈(𝜀),𝑑^𝑅_1−𝜈(𝜀)]}∀𝜀∈[0,1]. (3.3)

To calculate the value of 𝑑,𝜃₁,𝜎₁, we have to formulate the membership function of the distance between 𝐴_IFN and 𝐵_IFN.

Clearly 𝜀 (for 0≤𝜀≤1) cut representation of the membership function of 𝐴_IFN and 𝐵_IFN is [𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)] and [𝐵^𝐿_𝜇(𝜀),𝐵^𝑅_𝜇(𝜀)], respectively.

Now, the distance between [𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)] and [𝐵^𝐿_𝜇(𝜀),𝐵^𝑅_𝜇(𝜀)] for all 𝜀∈[0,1] is one of the following:

(𝑎)[𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)]−[𝐵^𝐿_𝜇(𝜀),𝐵^𝑅_𝜇(𝜀)]if𝑚₁≥𝑚₂,or(𝑏)[𝐵^𝐿_𝜇(𝜀),𝐵^𝑅_𝜇(𝜀)]−[𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)]if𝑚₁<𝑚₂. (3.4)

In order to consider both the notations together, an indicator variable 𝜂 is introduced such that 𝜂([𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)]−[𝐵^𝐿_𝜇(𝜀),𝐵^𝑅_𝜇(𝜀)])+(1−𝜂)([𝐵^𝐿_𝜇(𝜀),𝐵^𝑅_𝜇(𝜀)]−[𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)])=[𝐿_𝜇(𝜀),𝑅_𝜇(𝜀)],for𝜂=1if𝑚₁≥𝑚₂,0if𝑚₁≥𝑚₂, (3.5) where 𝐿_𝜇(𝜀)=𝜂(𝐴^𝐿_𝜇(𝜀)−𝐵^𝐿_𝜇(𝜀)+𝐴^𝑅_𝜇(𝜀)−𝐵^𝑅_𝜇(𝜀))+(𝐵^𝐿_𝜇(𝜀)−𝐴^𝑅_𝜇𝑅(𝜀)),_𝜇(𝜀)=𝜂(𝐴^𝐿_𝜇(𝜀)−𝐵^𝐿_𝜇(𝜀)+𝐴^𝑅_𝜇(𝜀)−𝐵^𝑅_𝜇(𝜀))+(𝐵^𝑅_𝜇(𝜀)−𝐴^𝐿_𝜇(𝜀)). (3.6) Thus [𝑑^𝐿_𝜇(𝜀),𝑑^𝑅_𝜇(𝜀)]=[𝐿(𝜀),𝑅_𝜇(𝜀)];for𝐿_𝜇(𝜀)≥0,[0,|𝐿_𝜇(𝜀)|∨𝑅_𝜇(𝜀)];for𝐿_𝜇(𝜀)≤0≤𝑅_𝜇(𝜀)∀𝜀∈[0,1]. (3.7) Now using (3.7) 𝜃₁,𝜎₁ are defined as follows: 𝜃₁=𝑑^𝐿_𝜇(1)−max{¹₀𝑑^𝐿_𝜇(𝜀)𝑑𝜀,0},𝜎₁=¹₀𝑑^𝑅_𝜇(𝜀)𝑑𝜀−𝑑^𝑅_𝜇(1). (3.8)

Similarly 𝜃₂,𝜎₂ are of the following form: 𝜃₂=𝑑^𝐿_1−𝜈(1)−max{¹₀𝑑^𝐿_1−𝜈(𝜀)𝑑𝜀,0},𝜎₂=¹₀𝑑^𝑅_1−𝜈(𝜀)𝑑𝜀−𝑑^𝑅_1−𝜈(1), (3.9) where [𝑑^𝐿_1−𝜈(𝜀),𝑑^𝑅_1−𝜈(𝜀)]=[𝐿_1−𝜈(𝜀),𝑅_1−𝜈(𝜀)];for𝐿_1−𝜈(𝜀)≥0,[0,|𝐿_1−𝜈(𝜀)|∨𝑅_1−𝜈(𝜀)];for𝐿_1−𝜈(𝜀)≤0≤𝑅_1−𝜈(𝜀). (3.10) As 𝐿_1−𝜈(𝜀)and𝑅_1−𝜈(𝜀) take the following form: 𝐿_1−𝜈(𝜀)=𝜂(𝐴^𝐿_1−𝜈(𝜀)−𝐵^𝐿_1−𝜈(𝜀)+𝐴^𝑅_1−𝜈(𝜀)−𝐵^𝑅_1−𝜈(𝜀))+(𝐵^𝐿_1−𝜈(𝜀)−𝐴^𝑅_1−𝜈𝑅(𝜀)),_1−𝜈(𝜀)=𝜂(𝐴^𝐿_1−𝜈(𝜀)−𝐵^𝐿_1−𝜈(𝜀)+𝐴^𝑅_1−𝜈(𝜀)−𝐵^𝑅_1−𝜈(𝜀))+(𝐵^𝑅_1−𝜈(𝜀)−𝐴^𝐿_1−𝜈(𝜀)). (3.11) From (3.7) and (3.10), we can find 𝑑=𝑑^𝐿_𝜇(1)=𝑑^𝑅_𝜇(1)=𝑑^𝐿_1−𝜈(1)=𝑑^𝑅_1−𝜈(1). Therefore finally the distance measure between 𝐴_IFN and 𝐵_IFN is obtained as 𝑑_IFN=(𝑑;𝜃₁,𝜎₁;𝜃₂,𝜎₂). (3.12)

3.2. Distance Measure for TIFNs

If 𝐴_IFN and 𝐵_IFN 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: 𝐴_TIFN=(𝑚₁;𝛼₁,𝛽₁;𝛼₁^/,𝛽₁^/∶𝛼₁^/>𝛼₁,𝛽₁^/>𝛽₁𝐵),_TIFN=(𝑚₂;𝛼₂,𝛽₂;𝛼₂^/,𝛽₂^/∶𝛼₂^/>𝛼₂,𝛽₂^/>𝛽₂). (3.13) Then their distance measure 𝑑_IFN=(𝑑;𝜃₁,𝜎₁;𝜃₂,𝜎₂) is a TIFN.

Proof. Here we have considered two TIFNs A and B. Therefore the 𝜀 cut representation of 𝐴_TIFN and 𝐵_TIFN is as follows: [𝐴_TIFN]_𝜀={[𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)];[𝐴^𝐿_1−𝜈(𝜀),𝐴^𝑅_1−𝜈(𝜀)]},[𝐵_TIFN]_𝜀={[𝐵^𝐿_𝜇(𝜀),𝐵^𝑅_𝜇(𝜀)];[𝐵^𝐿_1−𝜈(𝜀),𝐵^𝑅_1−𝜈(𝜀)]}∀𝜀∈[0,1]. (3.14) Now with the help of (2.8) and (2.9), respectively, we can write [𝐴^𝐿_𝜇(𝜀),𝐴^𝑅_𝜇(𝜀)]=[𝑚₁−𝛼₁(1−𝜀),𝑚₁+𝛽₁(1−𝜀)],[𝐴^𝐿_1−𝜈(𝜀),𝐴^𝑅_1−𝜈(𝜀)]=[𝑚₁−𝛼^′₁(1−𝜀),𝑚₁+𝛽^′₁(1−𝜀)]∀𝜀∈[0,1]. (3.15)(3.16) And in a similar manner =[𝑚₂−𝛼₂(1−𝜀),𝑚₂+𝛽₂(1−𝜀)],[𝐵^𝐿_1−𝜈(𝜀),𝐵^𝑅_1−𝜈(𝜀)]=[𝑚₂−𝛼^′₂(1−𝜀),𝑚₂+𝛽^′₂(1−𝜀)]∀𝜀∈[0,1]. (3.17)(3.18)
Here two possibilities can arise depending on the position of the mean values of 𝐴_TIFN and 𝐵_TIFN, which are given as follows.Case 1. For 𝜂=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 𝐿_𝜇(𝜀) and 𝑅_𝜇(𝜀) as follows: 𝐿_𝜇(𝜀)=[(𝑚₁−𝑚₂)−𝛼₁(1−𝜀)−𝛽₂𝑅(1−𝜀)],_𝜇(𝜀)=[(𝑚₁−𝑚₂)+𝛼₂(1−𝜀)+𝛽₁(1−𝜀)]. (3.19)
Similarly from (3.11), with the help of (3.16) and (3.18), 𝐿_1−𝜈(𝜀),𝑅_1−𝜈(𝜀) can be written in the following form: 𝐿_1−𝜈(𝜀)=[(𝑚₁−𝑚₂)−𝛼^′₁(1−𝜀)−𝛽^′₂𝑅(1−𝜀)],_1−𝜈(𝜀)=[(𝑚₁−𝑚₂)+𝛼^′₂(1−𝜀)+𝛽^′₁(1−𝜀)]. (3.20)
Now from (3.7) and (3.10) we have the following distances for 𝐴_TIFN and 𝐵_TIFN∶𝑑^𝐿_𝜇(𝜀)=[(𝑚₁−𝑚₂)−𝛼₁(1−𝜀)−𝛽₂𝑑(1−𝜀)]or0,_𝑅𝜇(𝜀)=[(𝑚₁−𝑚₂)+𝛼₂(1−𝜀)+𝛽₁𝑑(1−𝜀)],_𝐿1−𝜈(𝜀)=[(𝑚₁−𝑚₂)−𝛼^′₁(1−𝜀)−𝛽^′₂𝑑(1−𝜀)]or0,_𝑅1−𝜈(𝜀)=[(𝑚₁−𝑚₂)+𝛼^′₂(1−𝜀)+𝛽^′₁(1−𝜀)]. (3.21)(3.22) As given by Section 3.1, using (3.21) we can obtain 𝜃₁=𝑑^𝐿_𝜇(1)−max{¹₀𝑑^𝐿_𝜇(𝜀)𝑑𝜀,0}=(𝑚₁−𝑚₂)−max{[(𝑚₁−𝑚₂)−(𝛼₁+𝛽₂)/2],0}. (3.23)In a similar way, 𝜃₂ can be evaluated as 𝜃₂=(𝑚₁−𝑚₂)−max{[(𝑚₁−𝑚₂)−(𝛼^/₁+𝛽^/₂)/2],0}. (3.24)Now as 𝛼^′₁>𝛼₁and𝛽^′₂>𝛽₂⇒(𝛼^′₁+𝛽^′₂)/2>(𝛼₁+𝛽₂)/2⇒max{[(𝑚₁−𝑚₂)−(𝛼^/₁+𝛽^/₂)/2],0}≤max{[(𝑚₁−𝑚₂)−(𝛼₁+𝛽₂)/2],0}⇒𝜃₂>𝜃₁ (3.25)Similarly for 𝜎₁=(𝛼₂+𝛽₁)/2 and 𝜎₂=(𝛼^′₂+𝛽^′₁)/2, it is clear that 𝜎₂>𝜎₁ (as 𝛼^′₁>𝛼₁ and 𝛽^′₂>𝛽₂). Now from Proposition 2.4, we can conclude that as 𝜃₂>𝜃₁ and 𝜎₂>𝜎₁, 𝑑_IFN is a TIFN.Case 2. For 𝜂=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 between 𝐴_TIFN and 𝐵_TIFN is a TIFN denoted by 𝑑_TIFN=(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 Section 3.1 is a nonnegative intuitionistic fuzzy number. (ii)For any two intuitionistic fuzzy numbers 𝐴¹_IFNand 𝐴²_IFN the following holds: 𝑑(𝐴¹_IFN,𝐴²_IFN)=𝑑(𝐴²_IFN,𝐴¹_IFN). (3.26)(iii)For three IFNs 𝐴¹_IFN,𝐴²_IFN, and 𝐴³_IFN, the distance measure satisfies the triangle inequality: 𝑑(𝐴¹_IFN,𝐴²_IFN)+𝑑(𝐴²_IFN,𝐴³_IFN)≥𝑑(𝐴¹_IFN,𝐴³_IFN). (3.27)

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.
Let 𝐴¹_IFN, 𝐴²_IFN, and 𝐴³_IFN be three IFNs. The 𝜀 cut representation of IFNs 𝐴¹_IFN, 𝐴²_IFN and 𝐴³_IFN is expressed as [𝐴^𝑖_IFN]_𝜀={[𝐴_𝜇^𝑖𝐿(𝜀),𝐴_𝜇^𝑖𝑅(𝜀)];[𝐴^𝑖𝐿_1−𝜈(𝜀),𝐴^𝑖𝑅_1−𝜈(𝜀)]}for𝑖=1,2,3for𝜀∈[0,1]. (3.28)
In order to prove the triangle inequality of the distance measure for the above three IFNs 𝐴¹_IFN, 𝐴²_IFN and 𝐴³_IFN, we show below that the triangle inequality for the distance measure between the membership functions of 𝐴¹_IFN, 𝐴²_IFN and 𝐴³_IFN and nonmembership function of 𝐴¹_IFN, 𝐴²_IFN, and 𝐴³_IFN 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 [𝐴_𝜇^𝑖𝐿(𝜀),𝐴_𝜇^𝑖𝑅(𝜀)] that is, the 𝜀 cut of membership function of 𝐴¹_IFN, 𝐴²_IFN, and 𝐴³_IFN.
Depending on the relative positions of the means of 𝐴¹_IFN, 𝐴²_IFN and 𝐴³_IFN, three situations arise.
Situation (I)
When mean of 𝐴¹_IFN is less than mean of 𝐴²_IFN which is less than mean of 𝐴³_IFN𝐴_𝜇^1𝐿(1)+𝐴_𝜇^1𝑅(1)2≤𝐴_𝜇^2𝐿(1)+𝐴_𝜇^2𝑅(1)2≤𝐴_𝜇^3𝐿(1)+𝐴_𝜇^3𝑅(1)2. (3.29)
From (3.7), we have the following distances for (I.a)𝐴¹_IFN,𝐴²_IFN∶𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅𝑑(𝜀),_𝑅𝜇(𝜀)=𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀),(I.b)𝐴²_IFN,𝐴³_IFN∶𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝑅(𝜀)or0,𝑑^𝑅_𝜇(𝜀)=𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^2𝐿𝐴(𝜀),(I.c)¹_IFN,𝐴³_IFN∶𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅𝑑(𝜀)or0,_𝑅𝜇(𝜀)=𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀). (3.30)
Therefore we have to prove that 𝑑^𝑅+𝑑^𝐿+𝑑^𝑅+𝑑^𝐿≥𝑑^𝑅+𝑑^𝐿. (3.31)
From the above three options (I.a), (I.b) and (I.c), the following eight combinations are possible:(i)𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),𝑑^𝐿_𝜇(𝜀)=0,𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),(ii)𝑑^𝐿_𝜇(𝜀)=0,𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝑅(𝜀),𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),(iii)𝑑^𝐿_𝜇(𝜀)=0,𝑑^𝐿_𝜇(𝜀)=0,𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),(iv)𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),𝑑^𝐿_𝜇(𝜀)=0,𝑑^𝐿_𝜇(𝜀)=0,(v)𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝑅(𝜀),𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),(vi)𝑑^𝐿_𝜇(𝜀)=0,𝑑^𝐿_𝜇(𝜀)=0,𝑑^𝐿_𝜇(𝜀)=0,(vii)𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝑅(𝜀),𝑑^𝐿_𝜇(𝜀)=0,(viii)𝑑^𝐿_𝜇(𝜀)=0,𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝑅(𝜀),𝑑^𝐿_𝜇(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 of 𝐴¹_IFN and 𝐴²_IFN are disjoint. (ii)The membership functions of 𝐴²_IFN and 𝐴³_IFN are disjoint. (iii)The membership functions of 𝐴¹_IFN and 𝐴³_IFN intersect.
It is very clear that the above situation cannot be happened.
From (viii), we observe the following.(i)The membership functions of 𝐴¹_IFN and 𝐴²_IFN intersect. (ii)The memberships functions of 𝐴²_IFN and 𝐴³_IFN are disjoint. (iii)The memberships functions of 𝐴¹_IFN and 𝐴³_IFN 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)𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀), 𝑑^𝐿_𝜇(𝜀)=0, 𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),𝑑^𝑅+𝑑^𝐿+𝑑^𝑅+𝑑^𝐿=¹₀[𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^2𝐿=(𝜀)]𝑑𝜀₁₀[𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^2𝐿(𝜀)]𝑑𝜀=¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^2𝐿≥(𝜀)]𝑑𝜀₁₀[𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿≥𝑑(𝜀)]𝑑𝜀_𝐿+𝑑^𝑅[Proved]. (3.32)(ii)𝑑^𝐿_𝜇(𝜀)=0, 𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝑅(𝜀), 𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)𝑑^𝑅+𝑑^𝐿+𝑑^𝑅+𝑑^𝐿=¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^2𝐿=(𝜀)]𝑑𝜀₁₀[𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^2𝑅(𝜀)]𝑑𝜀≥¹₀[𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿≥𝑑(𝜀)]𝑑𝜀_𝐿+𝑑^𝑅[Proved]. (3.33)(iii)𝑑^𝐿_𝜇(𝜀)=0, 𝑑^𝐿_𝜇(𝜀)=0, 𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),𝑑^𝑅+𝑑^𝐿+𝑑^𝑅+𝑑^𝐿=¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^2𝐿=(𝜀)]𝑑𝜀₁₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^2𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀≥¹₀[𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿≥𝑑(𝜀)]𝑑𝜀_𝐿+𝑑^𝑅[Proved]. (3.34)(iv)𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀), 𝑑^𝐿_𝜇(𝜀)=0, 𝑑^𝐿_𝜇(𝜀)=0,𝑑^𝑅+𝑑^𝐿+𝑑^𝑅+𝑑^𝐿=¹₀[𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^2𝐿=(𝜀)]𝑑𝜀₁₀[𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^2𝐿(𝜀)]𝑑𝜀≥¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿≥𝑑(𝜀)]𝑑𝜀_𝐿+𝑑^𝑅[Proved]. (3.35)(v)𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀), 𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝑅(𝜀), 𝑑^𝐿_𝜇(𝜀)=𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀),𝑑^𝑅+𝑑^𝐿+𝑑^𝑅+𝑑^𝐿=¹₀[𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝑅+(𝜀)]₁₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^2𝐿(𝜀)]𝑑𝜀=¹₀[𝐴_𝜇^2𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^2𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿≥(𝜀)]𝑑𝜀₁₀[𝐴_𝜇^3𝐿(𝜀)−𝐴_𝜇^1𝑅(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿≥𝑑(𝜀)]𝑑𝜀_𝐿+𝑑^𝑅[Proved]. (3.36)(vi)𝑑^𝐿_𝜇(𝜀)=0, 𝑑^𝐿_𝜇(𝜀)=0, 𝑑^𝐿_𝜇(𝜀)=0,𝑑^𝑅+𝑑^𝐿+𝑑^𝑅+𝑑^𝐿=¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^2𝐿=(𝜀)]𝑑𝜀₁₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿(𝜀)]𝑑𝜀+¹₀[𝐴_𝜇^2𝑅(𝜀)−𝐴_𝜇^2𝐿(𝜀)]𝑑𝜀≥¹₀[𝐴_𝜇^3𝑅(𝜀)−𝐴_𝜇^1𝐿≥𝑑(𝜀)]𝑑𝜀_𝐿+𝑑^𝑅[Proved]. (3.37)
The proof is similar for the following two cases: (I)([𝐴_𝜇^1𝐿(1)+𝐴_𝜇^1𝑅(1)]/2)≤([𝐴_𝜇^3𝐿(1)+𝐴_𝜇^3𝑅(1)]/2)≤([𝐴_𝜇^2𝐿(1)+𝐴_𝜇^2𝑅(1)]/2),(II)([𝐴_𝜇^2𝐿(1)+𝐴_𝜇^2𝑅(1)]/2)≤([𝐴_𝜇^1𝐿(1)+𝐴_𝜇^1𝑅(1)]/2)≤([𝐴_𝜇^3𝐿(1)+𝐴_𝜇^3𝑅(1)]/2).
With the help of (3.28), considering the interval number[𝐴^𝑖𝐿_1−𝜈(𝜀),𝐴^𝑖𝑅_1−𝜈(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 IFSs A 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⟩}.
Therefore A 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 as 𝑑={⟨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 say 𝐴_IFN=(5;1,2;2,3) and 𝐵_IFN=(9;1,1;2,1) characterized by their membership functions and nonmembership functions as follows: 𝜇_𝐴(𝑥)=0for−∞<𝑥≤4,1−(5−𝑥)³for4≤𝑥≤5,1for𝑥=5,1−((𝑥−5)/2)³𝜈for5≤𝑥≤7,0for7≤𝑥<∞,_𝐴(𝑥)=1for−∞≤𝑥≤3,((5−𝑥)/2)³for3≤𝑥≤5,0for𝑥=5,((𝑥−5)/2)³𝜇for5≤𝑥≤7,1for7≤𝑥≤∞,_𝐵√(𝑥)=0for−∞<𝑥≤8,1−√9−𝑥for8≤𝑥≤9,1for𝑥=9,1−𝜈𝑥−9for9≤𝑥≤10,0for10≤𝑥<∞,_𝐵(𝑥)=1for−∞≤𝑥≤7,(9−𝑥)/2for7≤𝑥≤9,0for𝑥=9,(𝑥−9)for9≤𝑥≤10,1for10≤𝑥≤∞. (4.1) The 𝜀 cut representation of 𝐴_IFN and 𝐵_IFN is obtained as follows: [𝐴_IFN]_𝜀={[5−(1−𝜀)^1/3,5+2(1−𝜀)^1/3];[5−2(1−𝜀)^1/3,5+2(1−𝜀)^1/3]},[𝐵_IFN]_𝜀={[9−(1−𝜀)²,9+(1−𝜀)²];[9−2(1−𝜀)²,9+2(1−𝜀)²]}∀𝜀∈[0,1]. (4.2)
The 𝜀 cut of the distance measure 𝑑_IFN using (3.7) and (3.10) is obtained as [𝑑_IFN]_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]. (4.3)
Finally the distance measure of 𝐴_IFN and 𝐵_IFN, computed by the proposed method is given by 𝑑_IFN=(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.

Set 1. Consider 𝐴¹_TIFN=(3;2,2;3,4) and 𝐵_TIFN=(4;2,2;3,3). The membership and nonmembership functions of 𝐴¹_TIFN and 𝐵_TIFN are, respectively, of the following form: 𝜇_𝐴11(𝑥)=2(𝑥1−1);1≤𝑥≤3,𝜈2(5−𝑥);3≤𝑥≤5,_𝐴11(𝑥)=3(31−𝑥);0≤𝑥≤3,𝜇4(𝑥−3);3≤𝑥≤7,_𝐵1(𝑥)=2(𝑥1−2);2≤𝑥≤4,𝜈2(6−𝑥);4≤𝑥≤6,_𝐵1(𝑥)=3(41−𝑥);2≤𝑥≤4,3(𝑥−4);4≤𝑥≤7. (4.4)

Set 2. Consider 𝐴²_TIFN = (3;1,3;3,4) and 𝐵_TIFN = (4;2,2;3,3), where the membership and nonmembership functions of 𝐴_2TIFN are given as follows: 𝜇_𝐴2(𝑥)=1(𝑥−2);1≤𝑥≤3,𝜈3(6−𝑥);3≤𝑥≤6,_𝐴21(𝑥)=3(31−𝑥);0≤𝑥≤3,4(𝑥−3);3≤𝑥≤7. (4.5)
And 𝐵_TIFN is defined as in Set 1.
Now we have calculated the distance measure between 𝐴¹_TIFN,𝐵_TIFN and 𝐴²_TIFN, 𝐵_TIFN by applying (2.29), (2.31), and (3.12) and the results are compared in Table 1.

From Table 1 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(𝐴²_TIFN,𝐵_TIFN)<𝑑(𝐴¹_TIFN,𝐵_TIFN).

Also we utilize (3.12) to find the distance value of 𝐴¹_TIFN, 𝐵_TIFN and 𝐴²_TIFN, 𝐵_TIFN 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(𝐴²_{𝑇𝐼𝐹N},𝐵_TIFN)<𝑑(𝐴¹_TIFN,B_TIFN). 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.

Acknowledgments

The authors would like to thank the anonymous reviewers for their constructive suggestions. The first author gratefully acknowledges the financial support provided by the Council of Scientific and Industrial Research, India (Award no. 9/81(712)/08-EMR-I).