A Knowledge Measure with Parameter of Intuitionistic Fuzzy Sets

A new knowledge measure with parameter of intuitionistic fuzzy sets (IFSs) is presented based on the membership degree and the non-membership degree of IFSs, which complies with the extended form of Szmidt-Kacprzyk axioms for intuitionistic fuzzy entropy. And a sufficient and necessary condition of order property in the Szmidt-Kacprzyk axioms is discussed. Additionally, some numerical examples are given to illustrate the applications of the proposed knowledge measure and some conventional entropies and knowledge measures of IFSs. The experimental results show that the results of the parametric model proposed in this paper are more accurate than those of most of the classic models.


Introduction
Entropy is a basic parameter that characterizes the state of matter launched by Shannon [1], which is used to characterize the degree of disorder in the system, the uncertainty of the system structure and movement, and the degree of irregularity.In 1965, Zadeh launched fuzzy sets (FS) [2], and Atanassov proposed intuitionistic fuzzy sets (IFS) in 1986 by introducing the degree of hesitation, which means that the research of intuitionistic fuzzy sets is more complex than that of fuzzy sets [3].Fuzzy entropy is one of the most important methods to measure the degree of disorder in fuzzy sets, and knowledge measure is one of the most important bases for measuring the degree of order between fuzzy sets [4] [5] [6].In 1972, De Luca and Termini put forward an axiom system of fuzzy entropy in terms of Shannon's entropy function [4] [6]; Yager proposed some fuzzy entropy formulas according to fuzzy distance measure [5].Since then, many scholars began to use various methods to study fuzzy entropy and intuitionistic fuzzy entropy [6]- [11].Because intuitionistic fuzzy sets are an extension of ordinary fuzzy sets, many scholars focus on entropy and knowledge measure of intuitionistic fuzzy sets [7]- [16].In the past 20 years, based on the study of fuzzy sets, many scholars have proposed a variety of methods to calculate intuitionistic fuzzy entropy and knowledge measure [7]- [16].Based on the axioms of intuitionistic fuzzy entropy [10] [11], Szmidt and Kacprzyk presented a standard judgment of intuitionistic fuzzy knowledge measure with a relatively wide application [12]: non-negative boundedness, symmetry and order.Some researchers also studied Szmidt and Kacprzyk's axiom system and introduced some classic knowledge measure formulas [13] [14] [15].According to the order property, Guo put forward a new knowledge measure with order [13], Nguyen presented a model from a classic distance measure [14], and Das et al. proposed a new model based on a series of similarity measures [15].However, most of the existing research only focused on knowledge measure based on membership degree and non-membership, lack of the research of the known extent for information amount.In order to make full use of intuitionistic fuzzy information to construct information measurement tools, this paper studies a fractional knowledge measure.
Taking into account the extensive application and its rationality of the axiom system by Szmidt and Kacprzyk [12], we first put forward a simple necessary & sufficient condition of order property in Section 2. And then, we comprehensively analyze the differences among some classic models of intuitionistic fuzzy knowledge measure.Hence, in Section 3, we bring about a new construction method consisting of the decision-making advantages and the known extent and theoretically prove that this knowledge measure satisfies all the conditions of Szmidt & Kacprzyk's axiom system.It is proved theoretically that the operators with the order condition in Szmidt & Kacprzyk axiom system will be better than those without the order condition.In Section 4, combined with the research results of De et al. [17], an experimental case construction and empirical test scheme are put forward.Experimental results show that the performance of the presented model with parameters is better than that of the majority of classical operators, and the operators with the order condition will be more accurate than those without the order condition.

Intuitionistic Fuzzy Sets
Definition 1 An fuzzy sets (FS) A in a finite set X is an object with the following Definition 2 An intuitionistic fuzzy sets (IFS) A in a finite set X is an object with the following form: µ and ( ) ν are the degree of membership and non-membership, respectively. ( π is the degree of hesitancy.
Definition 3 Let A and B be two IFSs, then we have:

Entropy and Knowledge Measure of IFS
Claudius's entropy is one of the important parameters in physics that characterize the state of matter.It is a measure of the degree of chaos in the physical sense and describes the disorder degree of matter in an isolated system.In 1948 Shannon first launched entropy into information theory in the "Mathematical Principles of Communication", which characterize the degree of disorder, and uncertainty and irregularity of system structure and motion [1].After the creation of fuzzy sets, many scholars proposed a series of fuzzy entropies and its formulas, which are used to express fuzzy uncertainty.Since Atanassov proposed intuitionistic fuzzy sets [3], many scholars presented many intuitionistic fuzzy entropy formulas and knowledge measures [7]- [16].Next some classic entropy formulas and their knowledge measures will be introduced.
Fuzzy entropy is defined as follows [4] [5] [6]: is the entropy of A with the following properties: (EP1) ( ) ( ) (EP2) ( ) ( ) = are the degree of membership of fuzzy sets A and B, respectively.EP1 and EP2 denote the property of non-negative boundedness, EP3 is the property of symmetry, and EP4 is the property of order.

Where
, , µ ν π are the degree of membership, non-membership and hesi- tancy of IFS, respectively.EP1 and EP2 are the property of non-negative boundedness, EP3 is the property of symmetry, and EP4 is the property of order.
In terms of EP4, we obtain the following necessary and sufficient conditions: , then we have: and then we get Similarly,

( ) ( ) ( ) ( )
, and , then we have: , and then we have EP4 is equivalent to EP4I, thus we obtain ( ) ( ) According to Definition 5, intuitionistic fuzzy knowledge measure can be defined as follows [12] [13]: is the intuitionistic fuzzy knowledge measure of B, then we have: If KP1 and KP2 are the property of non-negative boundedness, KP3 is the property of symmetry, and KP4 is the property of order.
In terms of EP4, we obtain the following necessary and sufficient conditions KP4I: we have ( ) ( ) Obviously, KP4Ⅰmeans that for From the concept of entropy and knowledge measure above, we can define the knowledge measure of IFS A by: ( ) ( ) Some intuitionistic fuzzy knowledge measure formulas can be defined according to some classic intuitionistic fuzzy entropy formulas as follows: ( ) )) )) In 1996, Bustince and Burillo proposed an entropy formula It is easy to prove that the classic knowledge measure formulas above meet the property of non-negative boundedness and symmetry.For the property of order, we have the following Lemma 2.

Lemma 2 ( )
G K A meet the property of order KP4I, while and ( )  )( ) Similarly, We also have: Thus we obtain: For Similarly, we also have: Therefore,

( )
G K A meet the property of order.According to KP4I, , 0 1; 1 and we cannot have , DGM S U V do not meet the property KP4I.

Intuitionistic Fuzzy Knowledge Measure Model with Parameter
According to KP4I, knowledge measure ( ) K A can be considered to be a positive relation to ( ) ( ) . In addition, when ( ) ( ) − is a constant, due to the same difference between membership and non-membership, the greater the minimum value of the degree of membership and non-membership is, the greater the maximum value of the degree of membership and non-membership will be, the higher the degree of known information will be, and hence the larger the knowledge measure value should be under the same difference between membership and non-membership.Thus, the knowledge measure should be positively correlated to Based on the definition of knowledge measure of IFSs and the analysis above, a model can be achieved: ( ) p K A is proved to meet all four properties of Definition 6. Proof: For each 0 p < and for each A ∈ IFSs, obviously, ( ) 1 min , 1 min , ) From Equations ( 7)-( 14), we obtain Table 1.The evaluation index Accuracy Results show that for the knowledge measures with the order property, such as K G and K p , the order of their results is completely correct, while the order of the results for the knowledge measures without the order property, such as K SKB , K BB , 1 HY K , 0.5 r K , K N , K SK and S(U, V), do not meet the property KP4I.According to Table 1, K G and K p will be better than the others.Hence, we conclude that K G and K p are better than the others.
According to the definition of knowledge measure of IFSs, obviously we get: The results are shown in the following Tables 2-4.
Where the evaluation index Accuracy is defined as follows: ( ) ( ) Number Entropies with Right Order in Accuracy Number

)
For another fuzzy setB, 2014, Szmidt and Kacprzyk introduced an improved knowledge measure formula ( ) SKB K A , which is derived from the entropy ( ) SKB EA[12].In terms of the Szmidt and Kacprzyk's axiom system[10] [11][12], in 2016 Guo put forto distance measure proposed by Szmidt and Kacprzyk[18], and Szmidt and Kacprzyk presented a knowledge measure the property of order.
Note.Each bold data means the wrong prediction result and the corresponding method.Z. H. Zhang et al.DOI: 10.4236/am.2018.

Table 2 .
Comparison of experimental results from A k .Note.Each bold data means the wrong prediction result and the corresponding method.

Table 3 .
Comparison of experimental results from B k .
Note.Each bold data means the wrong prediction result and the corresponding method.Z. H. Zhang et al.DOI: 10.4236/am.2018.97060887 Applied Mathematics

Table 4 .
Comparison of experimental results from C k .