A Theoretical Development of Distance Measure for Intuitionistic Fuzzy Numbers
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.
The theory of fuzzy set introduced by Zadeh  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  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  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  introduced the concept of vague sets, which is another generalization of fuzzy sets. Bustince and Burillo  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 . 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  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  compute crisp distance measures for IFNs. But the well-known fact that needs to be remembered here is that  “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.
2.1. Intuitionistic Fuzzy Sets—Basic Definition and Notation
Let X denote a universe of discourse. Then a fuzzy set in X is defined as a set of ordered pairs: where and is the grade of belongingness of into . Thus automatically the grade of nonbelongingness of into is equal to . 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 . 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: where are functions such that . For each x the numbers and represent the degree of membership and degree of nonmembership of the element to , respectively.
It is easily seen that 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 by , while stands for the family of all IFSs in .
For each element we can compute, so called, the intuitionistic fuzzy index of in defined as follows: The value of is called the degree of indeterminacy (or hesitation) of the element to the IFS A. It is seen immediately that . If , then .
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.
Definition 2.1 (Intuitionistic fuzzy numbers). An intuitionistic fuzzy subset of the real line R is called an IFN if the following holds. (i)There exist and , (m is called the mean value of A).(ii) is a continuous mapping from R to the closed interval and the relation holds. (iii)The membership and nonmembership function of A is of the following form:
where and are strictly increasing and decreasing functions in and respectively:
Here m is the mean value of A. and are called left and right spreads of membership function respectively. and represented left and right spreads of nonmembership function respectively. Symbolically the intuitionistic fuzzy number is represented as .
It is to be noted here that the IFN that is, is a conjunction of two fuzzy numbers: with a membership function and with a membership function .
Definition 2.2 (-type Intuitionistic fuzzy number). An IFN is -type IFN such that for membership and nonmembership functions holds and may be defined as follows.
Membership function is of the form as follows:
Nonmembership function is of the form:
Provided , L is for left, and R is for right reference, m is the mean value of A. and are called left and right spreads of membership functions, respectively. and represented left and right spreads of nonmembership functions, respectively. Symbolically, we write . Here for L(x) and R(x) different functions may be chosen. For example, and so forth (Figure 1).
Definition 2.3 (Triangle Intuitionistic fuzzy number). An IFN may be defined as a triangle intuitionistic fuzzy number (TIFN) if and only if its membership and nonmembership functions take the following form:
Now for a TIFN, we can prove the following result.
Proposition 2.4. Let one consider a TIFN of the form ; then and .
Proof. The membership and nonmembership functions of is given above in (2.8) and (2.9), respectively.
From (2.8) and (2.9), for , we can write Since therefore the following can be written:
Similarly for , we can write Since , therefore the following can be written: Therefore symbolically a TIFN is represented as .
Definition 2.5 (Positive Intuitionistic fuzzy number). (i) An IFN denoted as is said to be positive if both and .
(ii) A TIFN denoted as is called positive if (follows from Proposition 2.4).
2.3. -Cut Representation of IFN
Let us consider an IFN defined on the real line R as described before. The cut of IFN is defined by
The cut representation of IFN generates the following pair of intervals and is denoted by
where the interval is defined in the following way: Here is defined by
And in a similar manner the interval can be defined as follows:
where is defined by .
2.4. Overview of the Existing Distance Measures
2.4.1. Notes on the Distance Measures for IFSs
Let us consider two with membership functions , and nonmembership functions , respectively.
Atanassov  suggested the distance measures as follows.
The normalized Hamming distance is
The normalized Euclidean distance is
Then in 2000, it was shown by Szmidt and Kacprzyk  that on computing distance for IFSs, all the three parameters, the degree of membership , the degree of non membership and the hesitation describing IFSs, should be taken into account. And therefore they modified the concept of distances proposed by Atanassov . The definition of distances presented by Szmidt and Kacprzyk  is given as following:
The normalized Hamming distance is
The normalized Euclidean distance is
Developing the above distance measures, Szmidt and Kacprzyk  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  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  was not convinced with the point of view of Szmidt and Kacprzyk . And based on Hausdorff metric, Grzegorzewski  proposed another group of distance measures for IFSs as follows.
Normalized Hamming distance is
The normalized Euclidean distance is
Obviously, the above distance measures proposed by Grzegorzewski  are easy for application. But in reality it may not fit so well. For example, let us consider three IFSs where and using the notation in (2.2) IFSs , and are of the following form: , and . If we use the ten-person-voting model to interpret, it would be noted that represents ten personwhoall vote for a person; represents ten persons whoall vote against him; whereas 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 .
Again in 2005, Wang and Xin  first with help of some examples had shown that the distance measure proposed by Szmidt and Kacprzyk  is not reasonable for some cases and then developed the following distance measures: 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 where and and are of the following form: , and . If we use the ten-person-voting model to interpret, it would benotedthat represents ten personswhoall vote for a person; represents ten person all hesitate; whereas 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 .
Now in 2005, Huang et al.  suggested several distance measures for IFSs. At first they developed a group of distance measures to unify the distances proposed by Atanassov  and Grzegorzewski . After that they proposed the following group of distance measures for IFSs:
In 2006, based on metric Hung and Yang  defined the following distance measure: Now after analyzing the above four distance measures we can say that these measures only reflect the difference between and 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  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 and are two intuitionistic fuzzy numbers. Now cut representation of the IFNs and is denoted by
Then Grzegorzewski  proposed the distance measures as follows.
For And for
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  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 .
3.1. Construction of the Distance Measure for IFNs
Let us consider two IFNs and as follows:
Therefore, cut representation of the IFNs and is denoted by
From mathematical point of view we can say that since and are IFNs, therefore their distance measure should also have membership and nonmembership part.
Let us denote the distance measure between and as , where d is the mean value of the distance measure , and are the left spread and right spread of the membership function and nonmembership function of the distance measure respectively.
And denote the cut of in the following way:
To calculate the value of , we have to formulate the membership function of the distance between and .
Clearly (for ) cut representation of the membership function of and is and , respectively.
Now, the distance between and for all is one of the following:
In order to consider both the notations together, an indicator variable is introduced such that where Thus Now using (3.7) are defined as follows:
Similarly are of the following form: where As take the following form: From (3.7) and (3.10), we can find . Therefore finally the distance measure between and is obtained as
3.2. Distance Measure for TIFNs
If and 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: Then their distance measure is a TIFN.
Proof. Here we have considered two TIFNs A and B. Therefore the cut representation of and is as follows:
Now with the help of (2.8) and (2.9), respectively, we can write
And in a similar manner
Here two possibilities can arise depending on the position of the mean values of and , which are given as follows.Case 1. For that is, when , we proceed in the following way.
In (3.6), putting the value from (3.15) and (3.17), we can express and as follows:
Similarly from (3.11), with the help of (3.16) and (3.18), can be written in the following form:
Now from (3.7) and (3.10) we have the following distances for and As given by Section 3.1, using (3.21) we can obtain In a similar way, can be evaluated as Now as Similarly for and , it is clear that (as and ). Now from Proposition 2.4, we can conclude that as and , is a TIFN.Case 2. For , 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 and is a TIFN denoted by .
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 and the following holds: (iii)For three IFNs , and , 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 , and be three IFNs. The cut representation of IFNs , and is expressed as
In order to prove the triangle inequality of the distance measure for the above three IFNs , and , we show below that the triangle inequality for the distance measure between the membership functions of , and and nonmembership function of , and 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 that is, the cut of membership function of , and
Depending on the relative positions of the means of , and , three situations arise.
When mean of is less than mean of which is less than mean of
From (3.7), we have the following distances for
Therefore we have to prove that
From the above three options (I.a), (I.b) and (I.c), the following eight combinations are possible:(i)(ii),(iii),(iv)(v)(vi),(vii)(viii)
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 and are disjoint. (ii)The membership functions of and are disjoint. (iii)The membership functions of and intersect.
It is very clear that the above situation cannot be happened.
From (viii), we observe the following.(i)The membership functions of and intersect. (ii)The memberships functions of and are disjoint. (iii)The memberships functions of and 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), , (ii), , (iii), , (iv), , (v), , (vi), , ,
The proof is similar for the following two cases: (I)(II)
With the help of (3.28), considering the interval number for all , 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 is as follows: and .
Therefore A can be expressed as a conjunction of two fuzzy numbers with membership degree 0.6 and with membership degree 0.7. And similarly B can be expressed as a conjunction of with membership degree 0.7 and with membership degree .
Applying the proposed distance method we will find the required distance measure as . In this way the proposed distance measure covers the case for .
Example 4.2. Let us consider two IFNs say and characterized by their membership functions and nonmembership functions as follows: