On Intuitionistic Fuzzy Entropy of Order-α

Using the idea of Rènyi’s entropy, intuitionistic fuzzy entropy of order-α is proposed in the setting of intuitionistic fuzzy sets theory. This measure is a generalized version of fuzzy entropy of order-α proposed by Bhandari and Pal and intuitionistic fuzzy entropy defined by Vlachos and Sergiadis. Our study of the four essential and some other properties of the proposed measure clearly establishes the validity of themeasure as intuitionistic fuzzy entropy. Finally, a numerical example is given to show that the proposed entropy measure for intuitionistic fuzzy set is reasonable by comparing it with other existing entropies.


Introduction
In 1965, Zadeh [1] proposed the notion of fuzzy set (FS) to model nonstatistical imprecise or vague phenomena.Since then, the theory of fuzzy set has become a vigorous area of research in different disciplines such as engineering, artificial intelligence, medical science, signal processing, and expert systems.Fuzziness, a feature of uncertainty, results from the lack of sharp distinction of being or not being a member of a set; that is, the boundaries of the set under consideration are not sharply defined.A measure of fuzziness used and cited in the literature is fuzzy entropy, also first mentioned in 1968 by Zadeh [2].In 1972, De Luca and Termini [3] first provided axiomatic structure for the entropy of fuzzy sets and defined an entropy measure of a fuzzy set based on Shannon's entropy function [4].Kaufmann [5] introduced a fuzzy entropy measure by a metric distance between the fuzzy set and that of its nearest crisp set.In addition, Yager [6] defined an entropy measure of a fuzzy set in terms of a lack of distinction between fuzzy set and its complement.In 1989, N. R. Pal and S. K. Pal [7] proposed fuzzy entropy based on exponential function to measure the fuzziness called exponential fuzzy entropy.Further, Bhandari and Pal [8] proposed fuzzy entropy of order- corresponding to Rènyi entropy [9].Recently, Verma and Sharma [10] have introduced a parametric generalized entropy measure for fuzzy sets called "exponential fuzzy entropy of order-." Atanassov [11,12] introduced the concept of intuitionistic fuzzy set (IFS), which is a generalization of the notion of fuzzy set.The distinguishing characteristic of intuitionistic fuzzy set is that it assigns to each element a membership degree, a nonmembership degree, and the hesitation degree.Firstly, Burillo and Bustince [13] defined the entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets.Szmidt and Kacprzyk [14] used a different approach from that of Burillo and Bustince to introduce the entropy measure for IFS based on the geometric interpretation of IFS and the distances between them.Zeng and Li [15] considered the axioms of Szmidt and Kacprzyk using the notion of IVFSs and discussed the relationship between similarity measure and entropy.Hung and Yang [16] gave their axiomatic definition of entropy for IFSs by exploiting the concept of probability.Vlachos and Sergiadis [17] proposed a measure of intuitionistic fuzzy entropy and revealed an intuitive and mathematical connection between the notions of entropy for fuzzy set and intuitionistic fuzzy set.Zhang and Jiang 2 Advances in Fuzzy Systems [18] proposed intuitionistic (vague) fuzzy entropy by means of intersection and union of the membership degree and nonmembership degree of the intuitionistic (vague) fuzzy set.
Having come to rather rigid measures, it is natural to see other possible measures and those having parametric generalization that covers existing measures and leads to many more.The parameters involved give flexibility in applications and their values can be determined from the data itself or experiment.In this paper, a new entropy measure for IFSs is defined.This is called "intuitionistic fuzzy entropy of order-." The paper is organized as follows.In Section 2, some basic definitions related to fuzzy set theory and intuitionistic fuzzy set theory are briefly given.In Section 3, a new entropy measure called "intuitionistic fuzzy entropy of order-" is proposed, and its axiomatic justification is established.Some mathematical properties of the proposed measure are also studied in this section.In Section 4, a numerical example is given to demonstrate the effectiveness of the proposed measure of intuitionistic fuzzy entropy with existing intuitionistic fuzzy entropies [13][14][15][16][17][18] by the comparison.Brief conclusions are presented in Section 5.

Preliminaries
In the following, some needed basic concepts and definitions related to fuzzy sets and intuitionistic fuzzy sets are introduced.
The number  Ã() describes the degree of belongingness of  ∈  in Ã.
De Luca and Termini [3] defined fuzzy entropy for a fuzzy set Ã corresponding to Shannon's entropy [4] as Later, Bhandari and Pal [8] made a survey of information measures on fuzzy sets and proposed some new measures of fuzzy entropy.Corresponding to Rènyi entropy [9], they defined the following measure: Atanassov [11,12], as mentioned earlier, generalized Zadeh's idea of fuzzy sets, by what is called intuitionistic fuzzy sets, defined as follows.
Definition 2 (intuitionistic fuzzy set [11]).An intuitionistic fuzzy set  in a finite universe of discourse  = { 1 ,  2 , . . .,   } is given by where with the condition In this definition numbers   () and ]  (), respectively, denote the degree of membership and degree of nonmembership of  ∈  to the set .
Definition 3 (set operations on intuitionistic fuzzy sets [12]).Let IFS() denote the family of all IFSs in the universe , and let ,  ∈ IFS() given by Then usual set relations and operations are defined as follows: (i)  ⊆  if and only if   () ≤   () and ]  () ≥ ]  () for all  ∈ ; (ii)  =  if and only if  ⊆  and  ⊆ ; Szmidt and Kacprzyk [14] extended the axioms of De Luca and Termini [3] for proposing the entropy measure in the setting of IFSs.
Throughout this paper, we will denote the set of all intuitionistic fuzzy sets in  by IFS() and by FS() the set of all fuzzy sets defined in .
With these background ideas and concepts, we, in the next section, introduce a new measure called "intuitionistic fuzzy entropy of order-" for intuitionistic fuzzy sets, which has a parameter.

Intuitionistic Fuzzy Entropy of Order-𝛼
Definition 5.The intuitionistic fuzzy entropy of order- for intuitionistic fuzzy set  is defined as × (  (  ) + ]  (  )) Theorem 6.The   () measure in (9) of the intuitionistic fuzzy entropy of order- is an entropy measure for IFSs; that is, it satisfies all the axioms given in Definition 4 above.
Let us further suppose that all   ∈  1 are dominated by the condition while for all   ∈  2 satisfy Then from the monotonicity of the function  and (9), we obtain that   () ≤   () when  ⊆ .
Then, from (9), we straightforwardly have Hence,   () is a measure of intuitionistic fuzzy entropy.This proves the theorem.
The values of different entropy measures under  Therefore, the behavior of intuitionistic fuzzy entropy of order-,   (), is good for the viewpoint of structured linguistic variables.

Conclusions
This work introduces a new entropy measure called intuitionistic fuzzy entropy of order- in the setting of intuitionistic fuzzy set theory.Some properties of this measure have been also studied.This measure generalizes Bhandari and Pal [8] fuzzy entropy of order- and Vlachos and Sergiadis [17] logarithmic intuitionistic fuzzy entropy.Introduction of parameter provides new flexibility and wider application of intuitionistic fuzzy entropy to different situations.
() attains the maximum value when the set is most intuitionistic fuzzy set and the minimum value when the set is crisp set; moreover, maximum and minimum values are independent of .Proof.It has already been proved that   () is maximum if and only if  is most intuitionistic fuzzy set, that is,   (  ) = ]  (  ), for all   ∈ , and minimum when  is a crisp set.So, it is enough to prove that the maximum and minimum values are independent of .Let  be the most intuitionistic fuzzy set; that is, Corollary 8.For any  ∈ IFS() and   the complement of intuitionistic fuzzy set ,   () =   (  ) =   ( ∪   ) =   ( ∩   ) .(39) Theorem 9.