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

Authors: Dengying Jiang; Yaxiong Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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.


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## Body

## 1. 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 [14–17]. 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 [13–17]. 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.


## 2. Preliminaries

In this section, we briefly review some basic notions and definitions related to intuitionistic fuzzy sets.Definition 1 (see [2]).
An intuitionistic fuzzy setA in X=x1,x2,···,xn is given by Atanassov as follows: (1)A=x,μAx,υAx∣x∈X,where μA:X→0,1 and υA:X→0,1, with the condition 0≤μAx+υAx≤1, x∈X. The numbers μAx,υAx∈0,1 denote the degree of membership and nonmembership of x to A, respectively. For each intuitionistic fuzzy set inX, the intuitionistic fuzzy index of x in A is denoted by(2)πAx=1-μAx-υAx,which is a hesitancy degree of x to A. It is obvious that(3)μAx∈0,1,υAx∈0,1,πAx∈0,1,∀x∈X.And the complement of intuitionistic fuzzy set A is represented by Ac=x,υAx,μAx∣x∈X.Definition 2 (see [13]).
LetE be a set-to-point mapping E:A→0,1. E is called an entropy measure if it satisfies the following four constraints: (1)
EA=0 iff A is nonfuzzy;(2)
EA=1 iff μAx=υAx for allx;(3)
EA≤EB if A is less fuzzy thanB, i.e., (4)μAx≤μBxandυAx≥υBxforμBx≤υBxorEA≥EB if B is less fuzzy thanA, i.e., (5)μAx≥μBxandυAx≤υBxforμBx≥υBx;(4)
EA≥EAc.


## 3. The Improved Entropy Axiomatic Definition

Mostly the existing study on intuitionistic fuzzy entropy formula is based on Definitions1 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:(6)ESKA=1n∑i=1nmaxCountAi∩AicmaxCountAi∪Aic,where A=A1,A2,⋯,An, Ai=xi,μAxi,υAxi, and(7)Ai∩Aic=min⁡μAxi,μAcxi,max⁡υAxi,υAcxi,Ai∪Aic=max⁡μAxi,μAcxi,min⁡υAxi,υAcxi,And maxCountAi=μAxi+υAxi 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. (8)EZLA=1-1n∑i=1nμAxi-υAxi,(9)EWLA=1n∑i=1nmin⁡μAxi,μAxi+πAximax⁡μAxi,μAxi+πAxi,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:(10)EVS1=1n∑i=1n2μAxiυAxi+πA2xiμA2xi+υA2xi+πA2xi,and(11)EVS2=EfuzzyA+EintuitA,where(12)EfuzzyA=-1nln⁡2∑i=1nμAxiln⁡μAxi+υAxiln⁡υAxi-1-πAxiln⁡1-πAxi,EintuitA=1n∑i=1nπAxi.Based on Trigonometric Function, an intuitionistic fuzzy entropy formula was given by Wei, Gao, and Guo [17] as follows:(13)EWA=1n∑i=1ncos⁡μAxi-υAxi21+πAxiπ.Example 3.
Use the above six entropy formulas and calculate the following intuitionistic fuzzy sets which have the same fuzziness (μAx-υAx):(14)A1=0.4,0.5,A2=0.3,0.4,A3=0.1,0.2.The results are shown in Table 1.Table 1
Comparison of the results calculated using the existing entropy formulas.
E S K E Z L E W L E V S 1 E V S 2 E W H G A 1 0.8333 0.9 0.8333 0.9762 0.9920 0.9898 A 2 0.8571 0.9 0.8571 0.9706 0.9897 0.9927 A 3 0.8889 0.9 0.8889 0.9815 0.9755 0.9957As can be seen from Table1, with the same ambiguity, the ranking results are quite different when different entropy formulas are adopted. The entropy formula EZL, which has the same entropy, i.e., 0.9, does not take into account the effect of intuition πAx. Regarding the entropy formulas EVS1 and EVS2, the entropy is smaller when the intuition is larger (πA2x>πA1x while EA2<EA1), which is obviously not consistent with the subjective understanding. The results of the entropy formulas ESK, EWL, and EWHG 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.Definition 4.
LetE be a set-to-point mapping E:A→0,1, andE is called an entropy measure if it satisfies the following five axioms:(1)
EA=0 iff A is nonfuzzy;(2)
EA=1 iff μAx=υAx for allx;(3)
EA≤EB if A is less fuzzy thanB, i.e., (15)μAx≤μBxandυAx≥υBxforμBx≤υBxorEA≥EB if B is less fuzzy thanA, i.e., (16)μAx≥μBxandυAx≤υBxforμBx≥υBx;(4)
EA≥EAc;(5)
EA≤EB if μAx-υAx=μBx-υBx for πAx≤πBx.


## 4. An Intuitionistic Fuzzy Entropy Formula with Improved Constraints

Definition4 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): (17)EZWHA=1n∑i=1nμAxiυAxi+πA2xiμA2xi+υA2xi-μAxiυAxi+πAxi.Therefore, formula (17) can be modified to be a new intuitionistic fuzzy entropy formula such that the new formula satisfies the constraint condition (5) in Definition 4.Theorem 5.
For any intuitionistic fuzzy setA, let(18)EA=1n∑i=1nμAxiυAxi+πAxiμA2xi+υA2xi-μAxiυAxi+πAxi,then E(A) is an entropy of the intuitionistic fuzzy set A. It can be proved that it satisfies the 5 constraint conditions of the axiomatic definition of entropy.Proof.
Let(19)EAi=μAxiυAxi+πAxiμA2xi+υA2xi-μAxiυAxi+πAxi.To prove Theorem 5, it is only necessary to prove that E(Ai) satisfies 5 constraint conditions in Definition 4.
According to Definition1, we obtain(20)0≤μAxi≤1,0≤υAxi≤1,0≤πAxi=1-μAxi-υAxi≤1.So 0≤E(Ai)≤1.
( 1 ) Let(21)EAi=0,that is(22)μAxiυAxi+πAxiμAxi-υAxi2+μAxiυAxi+πAxi=0,which is equivalent to(23)μAxi=0,υAxi=1orμAxi=1,υAxi=0.This implies that A is nonfuzzy.
( 2 ) Suppose(24)EAi=1,which is equivalent to(25)μAxiυAxi+πAxiμAxi-υAxi2+μAxiυAxi+πAxi=1,if and only if (26)μAxi-υAxi2=0.So it means that (27)μAxi=υAxi.
( 3 ) Let(28)fμ,υ=μυ+1-μ-υμ2+υ2-μυ+1-μ-υ,where μ,υ∈0,1 and 1-μ-υ∈0,1. To prove that EAi satisfies the constraint condition (3) in Definition 4, it only needs to prove that fμ,υ is monotonically increasing relative to μ and monotonically decreasing relative to υ if μ<υ, and fμ,υ is monotonically decreasing relative to μ, and monotonically increasing relative to υ if μ>υ. The proof is as follows.
The partial derivative off with respect to μ is(29)fμμ,υ=μ-υ-υ2-μυ+μ+3υ-2μ2+υ2-μυ+1-μ-υ2.Let(30)gμ,υ=-υ2-μυ+μ+3υ-2,then(31)guu,υ=1-υ≥0,gυu,υ=3-2υ-u≥0.Thus(32)max⁡gμ,υ=g1,1=-1,gμ,υ≤g1,1=-1<0.Therefore we get fμμ,υ>0 if μ<υ, and fμμ,υ<0 if μ>υ.
Similarly, it can be proved thatfυμ,υ>0 if μ>υ, and fυμ,υ<0 if μ<υ.
In summary, we obtain(33)fμμ,υ>0andfυμ,υ<0ifμ<υ,fμμ,υ<0andfυμ,υ>0ifμ>υ.Hence we get the conclusion that EAi satisfies the constraint condition (3) in Definition 4.
( 4 ) For any 1≤i≤n, it holds that(34)EAic=μAxiυAxi+πAxiυAxi2+μAxi2-μAxiυAxi+πAxi=EAi.
( 5 ) Setting μAxi-υAxi=a, a∈0,1. Notice that(35)πAxi=1-μAxi-υAxi,then we can get(36)μAxi=1-πAxi+a2υAxi=1-πAxi-a2orμAxi=1-πAxi-a2υAxi=1-πAxi+a2.Thus, we obtain(37)EAi=1-πAxi2-a2+4πAxi1-πAxi2+3a2+4πAxi.
Let(38)fπAxi=1-πAxi2-a2+4πAxi1-πAxi2+3a2+4πAxi,then it is easy to see that(39)f′πAxi=8a23-πAxi1-πAxi2+3a2+4πAxi2>0,which shows that EAi<EBi if μAxi-υAxi=μBxi-υBxi for πAxi<πBxi.
Theorem5 is proved. And the new entropy formula has the following properties.Property 6.
LetΦ=x,μΦx,υΦx∣x∈X be an intuitionistic fuzzy set satisfying the condition πΦx=1-μΦx-υΦx=bb∈0,1.Then EΦ is monotonically decreasing relative to μΦx-υΦx.Proof.
To prove Property6, it is only necessary to prove that when πCx=πDx∀C,D∈Φ, EC<ED if μCx-υCx>μDx-υDx.
To prove the above, it is only necessary to prove that whenπCxi=πDxi, ECi<EDi if μCxi-υCxi>μDxi-υDxi.
In fact, we have(40)πCxi=πDxi,μCxi-υCxi>μDxi-υDxi,which is equivalent to(41)μCxiυCxi<μDxiυDxi,if and only if(42)0<μCxiυCxi+πCxi<μDxiυDxi+πDxi,and(43)μC2xi+υC2xi-μCxiυCxi+πCxi>μD2xi+υD2xi-μDxiυDxi+πDxi>0.Then(44)ECi=μCxiυCxi+πCxiμC2xi+υC2xi-μCxiυCxi+πCxi<EDi.Hence, the property is proved.Remark 7.
Property6 demonstrates that if the intuitionistic fuzzy sets have the same intuition, the greater the fuzziness is, i.e., the smaller the absolute deviation of membership and nonmembership is, the greater the intuitionistic fuzzy entropy is.Property 8.
LetIFS(X) be a set of intuitionistic fuzzy sets on a nonempty set X. If the set-to-point mapping E:IFSX⟶[0,1] is the intuitionistic fuzzy entropy of the intuitionistic fuzzy set A, then we have EA≥πA.Proof.
To prove thatEA≥πA, it only needs to prove that EAi≥πAxi. By using the reduction to absurdity, its proof can be done as follows.
AssumeEAi<πAxi, then (45)μAxiυAxi+πAxiμA2xi+υA2xi-μAxiυAxi+πAxi<πAxi,if and only if(46)μAxiυAxi<πAxiμAxiμAxi-1-μAxiυAxi+υAxiυAxi-1.So it is easy to get that(47)μAxiυAxi<0,which contradicts with μAxiυAxi≥0. Hence, the property is proved.Remark 9.
Property8 implies that the degree of hesitation is the infimum of intuitionistic fuzzy entropy.Example 10.
Use the formulasEZWHA and EA to calculate the entropy of the following intuitionistic fuzzy sets:(48)A1=0,0.2,A2=0.4,0.5,A3=0.3,0.4,A4=0.1,0.2,A5=0.3,0.3.The results are shown in Table 2.Table 2
Comparison of the results calculated using theEZWHA and EA.
A 1 A 2 A 3 A 4 A 5 E Z W H 0.8729 0.8231 0.6988 0.8358 0.7143 E 0.9524 0.9677 0.9767 0.9863 1.0000According to Table2, the following conclusions can be drawn:① The comparison of intuitionistic fuzzy entropy ofA2, A3, and A4 is based on the same fuzziness, which means that the value of μAx-υAx 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, EA4>EA3>EA2. However, this property is not available in formula (8), (10), (11), and (17).② For the comparison of intuitionistic fuzzy entropy ofA1 and A4, they have different intuitions and ambiguities while they meet the same condition, i.e., μA<υA. Calculated using the formula (18), it can be seen that EA4>EA1 because μA1<μA4 and υA1=υA4, which conforms to the relationship between the intuitionistic fuzzy sets, but it can not be seen in formula (17).③ The intuitionistic fuzzy setA5 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 EA5=1, but it is not seen in EZWH. 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).


## 5. 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.



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