A New Entropy and Its Properties Based on the Improved Axiomatic Definition of Intuitionistic Fuzzy Entropy

Regarding the problem of the existing intuitionistic fuzzy entropy formulas in ordering the partial entropy, the constraint condition that is consistent with the intuitionistic facts is proposed in this paper, the axiomatic definition of entropy which fully reflects the intuition and fuzziness of intuitionistic fuzzy sets is given, and the improved intuitionistic fuzzy entropy formula is constructed according to the entropy axiomatic definition and its properties are studied. Finally, we compare the improved formula with the existing intuitionistic fuzzy entropy formulas, and the result turns out that the improved formula can solve the problem in the entropy ordering theoretically and practically.


Introduction
Atanassov extended the fuzzy set theory given by Zadeh [1] to the intuitionistic fuzzy set theory [2].Vague sets [3] and interval-valued fuzzy sets [4] are the other two generalizations of fuzzy sets which were proved to be equivalent to the intuitionistic fuzzy sets theoretically by Bustince and Burillo [5], Atanassov and Gargov [6], Cornelis, Atanassov, and Kerre [7], and Deschrijver and Kerre [8], respectively.At present, the intuitionistic fuzzy set theory is widely used in many fields, such as decision making [9], image [10], and medicine [11].
It is important to investigate the fuzzy entropy, which is used to describe the degree of uncertainty of fuzzy sets and is determined by the absolute deviation of membership and nonmembership degree of fuzzy sets.As the expansion of fuzzy sets, the degree of uncertainty of intuitionistic fuzzy sets includes not only the fuzziness of known information but also the intuition of unknown information.The ambiguity of known information is determined by the absolute deviation of membership degree and nonmembership degree, and the intuition of unknown information is determined by the degree of hesitation.The degree of uncertainty of intuitionistic fuzzy sets is described by intuitionistic fuzzy entropy which is first defined by Burillo and Bustince [12], while this axiom only describes the intuition of intuitionistic fuzzy sets but ignores its ambiguity.Based on the study of Burillo and Bustince, Szmidt and Kacprzyk proposed the axiomatic definition of intuitionistic fuzzy entropy which can reflect the intuition and fuzziness and constructed an entropy formula by using the geometric meaning of intuitionistic fuzzy sets [13], which provides different points of views for many scholars to build new intuitionistic fuzzy entropy formulas [14][15][16][17].However, it is worth noting that the entropy formulas satisfying the entropy axiomatic definition of Szmidt and Kacprzyk demonstrate significant diversity in the sorting results under the same conditions of fuzziness in [13][14][15][16][17].The reason lies in the fact that the axiomatic definition of entropy does not fully reflect the intuition of entropy under the same ambiguity.Therefore, the authors construct a new intuitionistic fuzzy entropy formula which fully reflects the fuzziness and intuition based on the pioneering study of Szmidt and Kacprzyk [13] and prove that the formula has two better properties in the following sections.

Preliminaries
In this section, we briefly review some basic notions and definitions related to intuitionistic fuzzy sets.

The Improved Entropy Axiomatic Definition
Mostly the existing study on intuitionistic fuzzy entropy formula is based on Definitions 1 and 2, among which Szmidt and Kacprzyk combined using the geometric meaning of intuitionistic fuzzy sets to give an intuitionistic fuzzy entropy formula as follows: where  = { 1 , And max(  ) =   (  ) +   (  ) which is called the maximum potential.The following two intuitionistic fuzzy entropy formulas were given by Zeng and Li [15] and Wang and Lei [16], respectively.
It was proved that formula (6) was equivalent to (9) by Wei, Wang et al. [18].Vlachos and Sergiadis [14] gave two intuitionistic fuzzy entropy formulas as follows: and where Based on Trigonometric Function, an intuitionistic fuzzy entropy formula was given by Wei, Gao, and Guo [17] as follows: Example 3. Use the above six entropy formulas and calculate the following intuitionistic fuzzy sets which have the same fuzziness (|  () −   ()|): The results are shown in Table 1.As can be seen from Table 1, with the same ambiguity, the ranking results are quite different when different entropy formulas are adopted.The entropy formula   , which has the same entropy, i.e., 0.9, does not take into account the effect of intuition (  ()).Regarding the entropy formulas  1 and  2 , the entropy is smaller when the intuition is larger (  2 () >   1 () while ( 2 ) < ( 1 )), which is obviously not consistent with the subjective understanding.The results of the entropy formulas   ,   , and   are in line with intuitive facts.Thus, it can be seen that although the above six formulas all satisfy the four axioms of Szmidt and Kacprzyk, they show different sorting results.The authors find out that the reason is due to the fact that the axiomatic definition of entropy does not fully reflect the intuition of entropy when the fuzziness is the same.So on the basis of Definition 2, the constraint condition (5) is strengthened to be as follows: when the fuzziness of the intuitionistic fuzzy sets is the same, the entropy increases monotonically with the intuition.Therefore, the improved axiomatic definition of entropy proposed in this paper is defined as follows.

An Intuitionistic Fuzzy Entropy Formula with Improved Constraints
Definition 4 fully reflects the ambiguity of the known information and the intuition of the unknown information of the intuitionistic fuzzy sets, where the fuzziness is reflected by the constraint condition (3), and the intuition is embodied by the constraint condition (5).However, the following intuitionistic fuzzy entropy formula given by Zhao, Wang, and Hao [19] produces counterintuitive results in sorting the partial intuitionistic fuzzy entropy (see Example 10 for details): Therefore, formula (17) To prove Theorem 5, it is only necessary to prove that (  ) satisfies 5 constraint conditions in Definition 4.
According to Definition 1, we obtain This implies that  is nonfuzzy.
(2) Suppose which is equivalent to if and only if So it means that where ,  ∈ [0, 1] and 1 −  −  ∈ [0, 1].To prove that (  ) satisfies the constraint condition (3) in Definition 4, it only needs to prove that (, ) is monotonically increasing relative to  and monotonically decreasing relative to  if  < , and (, ) is monotonically decreasing relative to , and monotonically increasing relative to  if  > .The proof is as follows.
The partial derivative of  with respect to  is then Therefore we get   (, ) > 0 if  < , and   (, ) < 0 if  > .
In =  (  ) . ( then we can get Thus, we obtain Let then it is easy to see that which shows that ( Theorem 5 is proved.And the new entropy formula has the following properties.Proof.To prove Property 6, it is only necessary to prove that when To prove the above, it is only necessary to prove that when In So it is easy to get that which contradicts with   (  )  (  ) ≥ 0. Hence, the property is proved.
The results are shown in Table 2.
According to Table 2, the following conclusions can be drawn: A The comparison of intuitionistic fuzzy entropy of  2 ,  3 , and  4 is based on the same fuzziness, which means that the value of |  () −   ()| of the three sets is equal.And the results calculated by formula (18) show that the greater the intuition of the unknown information is, the greater the intuitionistic fuzzy entropy is; that is, ( 4 ) > ( 3 ) > ( 2 ).However, this property is not available in formula ( 8), (10), (11), and (17).
B For the comparison of intuitionistic fuzzy entropy of  1 and  4 , they have different intuitions and ambiguities while they meet the same condition, i.e.,   <   .Calculated using the formula (18), it can be seen that ( 4 ) > ( 1 ) because   1 <   4 and   1 =   4 , which conforms to the relationship between the intuitionistic fuzzy sets, but it can not be seen in formula (17).
C The intuitionistic fuzzy set  5 has the same degree of membership and nonmembership, which implies that there is no enough information to support or oppose a proposition.In this way, the entropy reaches a maximum of 1, that is ( 5 ) = 1, but it is not seen in   .In summary, the results calculated by the entropy formula proposed in this paper are consistent with intuitionistic facts, which shows that it is better than the entropy formula (17).

Conclusion
The entropy of the intuitionistic fuzzy set is used to describe the degree of uncertainty of the intuitionistic fuzzy set, including the fuzziness of known information and the intuition of unknown information.Firstly this paper analyzes the existing intuitionistic fuzzy entropy formulas comprehensively and explores the reason why the sorting of some entropies of intuitionistic fuzzy sets is not consistent with intuitionistic facts.Secondly, the constraint conditions are given in accordance with the intuitionistic facts based on the analysis of the differences in entropy formulas, and then a new intuitionistic fuzzy entropy formula is constructed and its properties are analyzed.Finally, the comparison analysis shows that the method and the formula proposed in this paper can better solve the problem of entropy ordering.

Table 1 :
Comparison of the results calculated using the existing entropy formulas.
can be modified to be a new intuitionistic fuzzy entropy formula such that the new formula satisfies the constraint condition (5) in Definition 4.

Table 2 :
Comparison of the results calculated using the   () and ().