A Theoretical Development of Distance Measure for Intuitionistic Fuzzy Numbers

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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

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: 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][31][32][33] suggested a generalization of classical fuzzy set, called IFS.An IFS A in X is given by a set of ordered triples: where μ A , ν A : X → 0, 1 are functions such that 0 ≤ μ A x ν A x ≤ 1 for all x ∈ 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: 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 0 for all x ∈ X.

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.

Definition 2.1 Intuitionistic fuzzy numbers . An intuitionistic fuzzy subset
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 for all x ∈ R the relation 0 ≤ μ A x ν A x ≤ 1 holds.

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iii The membership and nonmembership function of A is of the following form: where f 1 x and h 1 x are strictly increasing and decreasing functions in m − α, m and m, m β , respectively:

2.5
Here m is the mean value of A. Membership function is of the form as follows:

2.6
International Journal of Mathematics and Mathematical Sciences 5 Nonmembership function is of the form:

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:

2.9
Now for a TIFN, we can prove the following result.Proof.The membership and nonmembership functions of A TIFN is given above in 2.8 and 2.9 , respectively.

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The LR-representation of intuitionistic fuzzy number A From 2.8 and 2.9 , for m − α ≤ x ≤ m, we can write

2.10
Since x ≤ m, therefore the following can be written:

2.11
Similarly for m ≤ x ≤ m β, we can write

2.12
Since x ≥ m, therefore the following can be written:

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 The ε cut representation of IFN A IFN generates the following pair of intervals and is denoted by where the interval A L μ ε , A R μ ε is defined in the following way:

2.16
Here μ ε is defined by And in a similar manner the interval A L 1−ν ε , A R 1−ν ε can be defined as follows: where ν ε is defined by

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.

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Atanassov 19 suggested the distance measures as follows.

The normalized Hamming distance d A A, B is
The normalized Euclidean distance ρ A A, B is

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 The normalized Euclidean distance ρ SK A, B is

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 International Journal of Mathematics and Mathematical Sciences 9 The normalized Euclidean distance ρ G A, B is

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 {x 1 } and using the notation in 2.2 IFSs A, B, and C are of the following form: and C { x 1 , 0, 0 }.If we use the ten-person-voting model to interpret, it would be noted that A { x 1 , 1, 0 } represents ten personwho all vote for a person; B { x 1 , 0, 1 } represents ten persons who all vote against him; whereas C { x 1 , 0, 0 } represents ten personswho all 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:

2.25
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 {x 1 } and A, B, and C are of the following form: and C { x 1 , 0.5, 0.5 }.If we use the ten-person-voting model to interpret, it would benoted that A { x 1 , 1, 0 } represents ten personswho all vote for a person; B { x 1 , 0, 0 } represents ten person all hesitate; whereas C { x 1 , 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

2.26
In 2006, based on L p metric Hung and Yang 24 defined the following distance measure: 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.

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

2.28
Then Grzegorzewski 28 proposed the distance measures as follows.
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2.29
And for p ∞

2.31
And for p ∞

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.

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 .

Construction of the Distance Measure for IFNs
Let us consider two IFNs A IFN and B IFN as follows: Therefore, ε cut representation of the IFNs A IFN and B IFN is denoted by And denote the ε cut of d IFN in the following way: To calculate the value of d, θ 1 , σ 1 , 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

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In order to consider both the notations together, an indicator variable η is introduced such that

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As L 1−ν ε and R 1−ν ε take the following form:

3.11
From 3.7 and 3.10 , we can find Therefore finally the distance measure between A IFN and B IFN is obtained as

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:
Proof.Here we have considered two TIFNs A and B. Therefore the ε cut representation of A TIFN and B TIFN is as follows:

3.14
Now with the help of 2.8 and 2.9 , respectively, we can write

3.16
And in a similar manner

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 m 1 ≥ m 2 , 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:

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:

3.20
Now from 3.7 and 3.10 we have the following distances for A TIFN and B TIFN :

3.22
As given by Section 3.1, using 3.21 we can obtain In a similar way, θ 2 can be evaluated as International Journal of Mathematics and Mathematical Sciences Now as α 1 > α 1 and 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

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 A 1 IFN and A 2 IFN the following holds: iii For three IFNs A 1 IFN , A 2 IFN , and A 3 IFN , the distance measure satisfies the triangle inequality: 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 A 1 IFN , A 2 IFN , and A 3 IFN be three IFNs.The ε cut representation of IFNs A 1 IFN , A 2 IFN and A 3 IFN is expressed as

3.28
In order to prove the triangle inequality of the distance measure for the above three IFNs A 1 IFN , A 2 IFN and A 3 IFN , we show below that the triangle inequality for the distance measure between the membership functions of A

Situation (I)
When mean of A 1  IFN is less than mean of A 2 IFN which is less than mean of From 3.7 , we have the following distances for

3.30
Therefore we have to prove that

3.31
From the above three options I.a , I.b and I.c , the following eight combinations are possible: Now, from the above eight combinations, vii and viii are not possible.
As from vii , the following can be seen that.

3.35
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3.37
The proof is similar for the following two cases: 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.

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 {x 1 } is as follows: A { x 1 , 0.6, 0.3 } and B { x 1 , 0.7, 0.2 }.
Therefore A can be expressed as a conjunction of two fuzzy numbers A x 1 ; 0, 0 with membership degree 0. 6 and A − x 1 ; 0, 0 with membership degree 0.7.And similarly B can be expressed as a conjunction of B x 1 ; 0, 0 with membership degree 0.7 and B − x 1 ; 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 {x 1 }.

4.5
And B TIFN is defined as in Set 1.Now we have calculated the distance measure between A 1 TIFN , B TIFN and A 2 TIFN , B TIFN by applying 2.29 , 2.31 , and 3.12 and the results are compared in Table 1.
From Table 1

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.

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; θ 1 , σ 1 ; θ 2 , σ 2 , where d is the mean value of the distance measure d IFN , θ 1 , σ 1 and θ 2 , σ 2 are the left spread and right spread of the membership function and nonmembership function of the distance measure d IFN , respectively.
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 International Journal of Mathematics and Mathematical Sciences 19 and Grzegorzewski 21 .After that they proposed the following group of distance measures for IFSs:

Table 1 :
Here a comparison of the proposed distance measure with the existing distance measure through the above set of 2 examples is given.TIFN 4; 2, 2; 3, 3 , where the membership and nonmembership functions of A 2TIFN are given as follows: 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 A 2 TIFN , B TIFN < d A 1 TIFN , B TIFN .Also we utilize 3.12 to find the distance value of A 1 TIFN , B TIFN and A 2 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 A 2 TIFN , TIFN < d A 1 TIFN , B TIFN .This result is matching with the result obtained from 2.29 .
a b B