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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.

Entropy is a basic parameter that characterizes the state of matter launched by Shannon [

Taking into account the extensive application and its rationality of the axiom system by Szmidt and Kacprzyk [

Definition 1 An fuzzy sets (FS) A in a finite set X is an object with the following form:

A = { 〈 x , μ A ( x ) 〉 | x ∈ X } , μ A ( x ) ∈ [ 0 , 1 ] .

where μ A ( x ) : X → [ 0 , 1 ] , x → μ A ( x ) .

A C = { 〈 x , μ A C ( x ) 〉 | x ∈ X } = { 〈 x , 1 − μ A ( x ) 〉 | x ∈ X } ,

where

Definition 2 An intuitionistic fuzzy sets (IFS) A in a finite set X is an object with the following form:

A = { 〈 x , μ A ( x ) , ν A ( x ) 〉 | x ∈ X }

μ A ( x ) : X → [ 0 , 1 ] , x → μ A ( x ) ; ν A ( x ) : X → [ 0 , 1 ] , x → ν A ( x ) , μ A ( x ) + ν A ( x ) ∈ [ 0 , 1 ] .

μ A ( x ) and ν A ( x ) are the degree of membership and non-membership, respectively.

Definition 3 Let A and B be two IFSs, then we have:

1) A = B if and only if

2) A ⊆ B if

3)

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 [

Fuzzy entropy is defined as follows [

Definition 4 A fuzzy set A in the domain X, for each x ∈ X , E ( A ) = f E ( μ A ) is the entropy of A with the following properties:

(EP1) E ( A ) = 0 ⇔ μ A ( x ) = 0 or μ A ( x ) = 1 .

(EP2) E ( A ) = 1 ⇔ μ A ( x ) = 0.5 .

(EP3) E ( A ) = E ( A C ) .

(EP4) For another fuzzy set B, E ( B ) = f E ( μ B ) denotes the entropy of B, and then we have: If μ B ( x ) ≤ 0.5 and A ⊆ B , E ( A ) ≤ E ( B ) ; If μ B ( x ) ≥ 0.5 and B ⊆ A , E ( A ) ≤ E ( B ) .

where μ A = μ A ( x ) and μ B = μ B ( x ) 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.

Intuitionistic fuzzy entropy is defined as follow [

Definition 5 For IFS A in the domain X, E ( A ) = f E ( μ A , ν A , π A ) is the entropy of A with the following properties for x ∈ X :

(EP1) E ( A ) = 0 ⇔ μ A ( x ) = 0 & ν A ( x ) = 1 or μ A ( x ) = 1 & ν A ( x ) = 0 .

(EP2) μ A ( x ) = ν A ( x ) = 0 ⇒ E ( A ) = 1 .

(EP3) E ( A ) = E ( A C ) .

(EP4) For another IFS B, E ( B ) = f E ( μ B , ν B , π B ) denotes the entropy of B, and we have: If μ B ( x ) ≤ ν B ( x ) and A ⊆ B , E ( A ) ≤ E ( B ) ; If μ B ( x ) ≥ ν B ( x ) and B ⊆ A , E ( A ) ≤ E ( B ) .

Where μ A , ν A , π A are the degree of membership, non-membership and hesitancy 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:

(EP4I)

If μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) or μ A ( x ) ≥ μ B ( x ) ≥ ν B ( x ) ≥ ν A ( x ) , then we have E ( A ) ≤ E ( B ) .

Proof. A ⊆ B ⇔ { μ A ( x ) ≤ μ B ( x ) ν B ( x ) ≤ ν A ( x ) , and μ B ( x ) ≤ ν B ( x ) , then μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) .

And if μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) , then we have:

μ B ( x ) ≤ ν B ( x ) & μ A ( x ) ≤ μ B ( x ) & ν B ( x ) ≤ ν A ( x ) ,

and then we get μ B ( x ) ≤ ν B ( x ) and A ⊆ B .

Similarly, B ⊆ A ⇔ { μ A ( x ) ≥ μ B ( x ) ν B ( x ) ≥ ν A ( x ) , and μ B ( x ) ≥ ν B ( x ) , then

μ A ( x ) ≥ μ B ( x ) ≥ ν B ( x ) ≥ ν A ( x ) .

And if μ A ( x ) ≥ μ B ( x ) ≥ ν B ( x ) ≥ ν A ( x ) , then we have: μ B ( x ) ≥ ν B ( x ) & μ A ( x ) ≥ μ B ( x ) & ν B ( x ) ≥ ν A ( x ) , and then we have μ B ( x ) ≥ ν B ( x ) and B ⊆ A .

EP4 is equivalent to EP4I, thus we obtain E ( A ) ≤ E ( B ) .

According to Definition 5, intuitionistic fuzzy knowledge measure can be defined as follows [

Definition 6 For IFS A, K ( A ) = f K ( μ A , ν A , π A ) is an intuitionistic fuzzy knowledge measure of A if K ( A ) have the following properties:

(KP1) μ A ( x ) = ν A ( x ) = 0 ⇒ K ( A ) = 0 .

(KP2) K ( A ) = 1 ⇔ μ A ( x ) = 0 & ν A ( x ) = 1 or μ A ( x ) = 1 & ν A ( x ) = 0 .

(KP3) K ( A ) = K ( A C ) .

(KP4) If B is also an IFS, and K ( B ) = f K ( μ B , ν B , π B ) is the intuitionistic fuzzy knowledge measure of B, then we have: If μ B ( x ) ≤ ν B ( x ) and A ⊆ B , K ( A ) ≥ K ( B ) ; If μ B ( x ) ≥ ν B ( x ) and B ⊆ A , K ( A ) ≥ K ( B ) .

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:

(KP4I)

If μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) or μ A ( x ) ≥ μ B ( x ) ≥ ν B ( x ) ≥ ν A ( x ) , then we have K ( A ) ≥ K ( B ) .

Obviously, KP4Ⅰmeans that for | μ A ( x ) − ν A ( x ) | ≥ | μ B ( x ) − ν B ( x ) | , we infer K ( A ) ≥ K ( B ) . Hence, knowledge measure K ( A ) can be considered to be a positive relation to | μ A ( x ) − ν A ( x ) | .

From the concept of entropy and knowledge measure above, we can define the knowledge measure of IFS A by:

K ( A ) = 1 − E ( A ) (1)

Some intuitionistic fuzzy knowledge measure formulas can be defined according to some classic intuitionistic fuzzy entropy formulas as follows:

E B B ( A ) = 1 n ∑ i = 1 n π A ( x i ) (2)

E S K B ( A ) = 1 2 n ∑ i = 1 n ( min { μ A ( x i ) , ν A ( x i ) } + π A ( x i ) max { μ A ( x i ) , ν A ( x i ) } + π A ( x i ) + π A ( x i ) ) (3)

E G ( A ) = 1 2 n ∑ i = 1 n ( 1 − | μ A ( x i ) − ν A ( x i ) | ) ( 1 + π A ( x i ) ) (4)

E H C α ( A ) = { ∑ i = 1 n 1 − μ A ( x i ) α − ν A ( x i ) α − π A ( x i ) α ( α − 1 ) n , α ≠ 1 ( α > 0 ) − 1 n ∑ i = 1 n ( μ A ( x i ) log ( μ A ( x i ) ) + ν A ( x i ) log ( ν A ( x i ) ) + π A ( x i ) log ( π A ( x i ) ) ) , α = 1 (5)

E R β ( A ) = 1 n ∑ i = 1 n log ( μ A ( x i ) β + ν A ( x i ) β + π A ( x i ) β ) 1 − β , 0 < β < 1 (6)

K B B ( A ) = 1 − 1 n ∑ i = 1 n π A ( x i ) (7)

K S K B ( A ) = S D G M 1 ( U , V ) = 1 − 1 2 n ∑ i = 1 n ( min { μ A ( x i ) , ν A ( x i ) } + π A ( x i ) max { μ A ( x i ) , ν A ( x i ) } + π A ( x i ) + π A ( x i ) ) (8)

K G ( A ) = 1 − 1 2 n ∑ i = 1 n ( 1 − | μ A ( x i ) − ν A ( x i ) | ) ( 1 + π A ( x i ) ) (9)

K H C α ( A ) = 1 − E H C α ( A ) = { 1 − ∑ i = 1 n 1 − μ A ( x i ) α − ν A ( x i ) α − π A ( x i ) α ( α − 1 ) n , α ≠ 1 ( α > 0 ) 1 + 1 n ∑ i = 1 n ( μ A ( x i ) log ( μ A ( x i ) ) + ν A ( x i ) log ( ν A ( x i ) ) + π A ( x i ) log ( π A ( x i ) ) ) , α = 1 (10)

K R β ( A ) = 1 − E R β ( A ) = 1 − 1 n ∑ i = 1 n log ( μ A ( x i ) β + ν A ( x i ) β + π A ( x i ) β ) 1 − β , 0 < β < 1 (11)

K N ( A ) = 1 n ∑ i = 1 n μ A ( x i ) 2 + ν A ( x i ) 2 + ( 1 − π A ( x i ) ) 2 2 (12)

K S K ( A ) = S D G M 2 ( U , V ) = 1 n ∑ i = 1 n max { | μ A ( x i ) | p , | ν A ( x i ) | p , ( 1 − π A ( x i ) ) p } (13)

In 1996, Bustince and Burillo proposed an entropy formula E B B ( A ) [

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 K G ( A ) meet the property of order KP4I, while K B B ( A ) , K S K B ( A ) , K H C α ( A ) , K R β ( A ) , K N ( A ) , K S K ( A ) and S ( U , V ) don’t meet KP4I.

Proof: According to KP4I, K G ( A ) meet the property of order.

K G = 1 − ( 1 − | μ − ν | ) ( 1 + π ) 2 = { μ 2 − μ − ν 2 + 3 ν 2 , μ ≤ ν , − μ 2 + 3 μ + ν 2 − ν 2 , μ ≥ ν .

For μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) , according to 0 ≤ μ A ( x ) + ν A ( x ) ≤ 1 , 0 ≤ μ B ( x ) + ν B ( x ) ≤ 1 , we have:

μ B ( x ) ≤ ν B ( x ) & μ B ( x ) + ν B ( x ) = 1 − π B ( x ) ≤ 1 ⇒ 2 μ B ( x ) ≤ μ B ( x ) + ν B ( x ) ≤ 1 ⇒ μ B ( x ) ≤ 1 2 .

Therefore, we have 0 ≤ μ A ( x ) ≤ μ B ( x ) ≤ 0.5 .

Similarly, We also have:

When μ A ( x ) ≥ μ B ( x ) ≥ ν B ( x ) ≥ ν A ( x ) , 0 ≤ ν A ( x ) ≤ ν B ( x ) ≤ 0.5 . Thus we get:

K G ( A ) − K G ( B ) = { 1 2 [ ( μ A − μ B ) ( μ A + μ B − 1 ) + ( ν A − ν B ) ( 3 − ν A − ν B ) ] ≥ 0 , μ ≤ ν , 1 2 [ ( μ A − μ B ) ( 3 − μ A − μ B ) + ( ν A − ν B ) ( ν A + ν B − 1 ) ] ≥ 0 , μ ≥ ν .

Thus we obtain: For μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) , K G ( A ) ≥ K G ( B ) .

Similarly, we also have:

For μ A ( x ) ≥ μ B ( x ) ≥ ν B ( x ) ≥ ν A ( x ) , K G ( A ) ≥ K G ( B ) .

Therefore, K G ( A ) meet the property of order.

According to KP4I, K B B ( A ) , K S K B ( A ) , K H C α ( A ) and K R β ( A ) don’t meet KP4I.

For μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) , K B B ( A ) = 1 − π A = μ A + ν A , K B B ( B ) = 1 − π B = μ B + ν B . We cannot have K B B ( A ) ≥ K B B ( B ) .

K H C α ( A ) = { μ A α + ν A α + π A α α − 1 , α ≠ 1 ( α > 0 ) 1 + μ A log μ A + ν A log ν A + π A log π A , α = 1 = { μ A α + ν A α + π A α α − 1 , α ≠ 1 ( α > 0 ) 1 + log ( ( μ A ) μ A ( ν A ) ν A ( π A ) π A ) , α = 1 K H C α ( B ) = { μ B α + ν B α + π B α α − 1 , α ≠ 1 ( α > 0 ) 1 + μ B log μ B + ν B log ν B + π B log π B , α = 1 = { μ B α + ν B α + π B α α − 1 , α ≠ 1 ( α > 0 ) 1 + log ( ( μ B ) μ B ( ν B ) ν B ( π B ) π B ) , α = 1

Obviously, we cannot get K H C α ( A ) ≥ K H C α ( B ) .

K R β ( A ) = 1 − log ( ( μ A ) β + ( ν A ) β + ( π A ) β ) 1 − β , 0 < β < 1 ;

K R β ( B ) = 1 − log ( ( μ B ) β + ( ν B ) β + ( π B ) β ) 1 − β , 0 < β < 1 ;

We cannot obtain K R β ( A ) ≥ K R β ( B ) too.

K S K B ( A ) = 1 − 1 2 ( min { μ A , ν A } + π A max { μ A , ν A } + π A + π A ) = 1 2 ( π A + 1 − ν A 1 − μ A ) , μ A ≤ ν A ,

K S K B ( B ) = 1 − 1 2 ( min { μ B , ν B } + π B max { μ B , ν B } + π B + π B ) = 1 2 ( π B + 1 − ν B 1 − μ B ) , μ B ≤ ν B ,

Obviously, K S K B ( A ) ≥ K S K B ( B ) cannot be determined. Hence, K S K B does not meet the property KP4I.

For μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) , we have

μ A ( x ) p ≤ μ B ( x ) p , ν B ( x ) p ≤ ν A ( x ) p ,

and we cannot have π A ( x ) p ≤ π B ( x ) p & π B ( x ) p ≤ π A ( x ) p . Hence, We cannot have K N ( A ) ≥ K N ( B ) and K S K ( A ) ≥ K S K ( B ) , Hence, K N , K S K and S D G M ( U , V ) do not meet the property KP4I.

According to KP4I, knowledge measure K ( A ) can be considered to be a positive relation to | μ A ( x ) − ν A ( x ) | . In addition, when | μ A ( x ) − ν A ( x ) | 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 | μ A ( x ) − ν A ( x ) | 1 − min ( μ A , ν A ) .

Based on the definition of knowledge measure of IFSs and the analysis above, a model can be achieved:

K p ( A ) = 1 n ∑ i = 1 n ( | μ A ( x i ) − ν A ( x i ) | 1 − min ( μ A ( x i ) , ν A ( x i ) ) ) p , p > 0. (14)

K p ( A ) is proved to meet all four properties of Definition 6.

Proof: For each 0 < p and for each A ∈ IFSs, obviously, 0 ≤ K p ( A ) ≤ 1 .

(KP1) μ A ( x ) = ν A ( x ) = 0 ⇒ K p ( A ) = 0 .

(KP2) K p ( A ) = 1 ⇔ μ A ( x ) = 0 & ν A ( x ) = 1 or μ A ( x ) = 1 & ν A ( x ) = 0 .

(KP3) K p ( A ) = | μ A − ν A | p ( 1 − min ( μ A , ν A ) ) p = | ν A − μ A | p ( 1 − min ( ν A , μ A ) ) p = K p ( A C ) .

(KP4I) For μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) ,

K p ( A ) = ( | μ A − ν A | 1 − min ( μ A , ν A ) ) p = ( ν A − μ A ) p ( 1 − μ A ) p , K p ( B ) = ( ν B − μ B ) p ( 1 − μ B ) p .

∵ μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A (x)

⇒ ( ν A − ν B ) ≥ 0 , ( μ B − μ A ) ≥ 0 ⇒ ( ν A − ν B ) ( 1 − μ B ) ≥ 0 , ( μ B − μ A ) ( 1 − ν B ) ≥ 0 ⇒ ( ν A − ν B ) ( 1 − μ B ) + ( μ B − μ A ) ( 1 − ν B ) ≥ 0 ⇒ ( ν A − ν B ) + ( μ B − μ A ) + μ A ν B − μ B ν A ≥ 0 ⇒ ( ν A − μ A ) − ( ν B − μ B ) + μ A ν B − μ B ν A ≥ 0

⇒ ( ν A − μ A ) ( 1 − μ B ) − ( ν B − μ B ) ( 1 − μ A ) ≥ 0 ⇒ ( ν A − μ A ) ( 1 − μ B ) − ( ν B − μ B ) ( 1 − μ A ) ( 1 − μ A ) ( 1 − μ B ) ≥ 0 ⇒ ν A − μ A 1 − μ A − ν B − μ B 1 − μ B ≥ 0 ⇒ ν A − μ A 1 − μ A ≥ ν B − μ B 1 − μ B ⇒ ( ν A − μ A ) p ( 1 − μ A ) p ≥ ( ν B − μ B ) p ( 1 − μ B ) p ⇒ K p ( A ) ≥ K p (B)

Similarly, we also have:

For μ A ( x ) ≤ μ B ( x ) ≤ ν B ( x ) ≤ ν A ( x ) , K p ( A ) ≥ K p ( B ) .

Hence, for each p > 0 , K p ( A ) is a knowledge measure of IFSs A.

According to Lemma 2 and the analysis above, K G ( A ) and K p ( A ) meet the property of order, while K B B ( A ) , K S K B ( A ) , K H C α ( A ) , K R β ( A ) , K N ( A ) , K S K ( A ) and S ( U , V ) don’t meet KP4I. Hence, we compare K p ( A ) with K G ( A ) as follows.

K p ( A ) = ( | μ A − ν A | 1 − min ( μ A , ν A ) ) p (15)

K p ( A ) is affected by the difference between membership and non-membership degree with the positive correlation, which is the same as K G ( A ) . Meanwhile, when | μ A − ν A | , the difference between membership and non-membership degree, is a constant, K p ( A ) is also affected with the positive correlation by the minimum of membership and non-membership degree, while K G ( A ) the negative correlation.

K G ( A ) = 1 − ( 1 − | μ A − ν A | ) ( 1 + π A ) 2 = 1 − ( 1 − | μ A − ν A | ) ( 2 − μ A − ν A ) 2 .

From practical significance, K p ( A ) will be more reasonable than K G ( A ) . If | μ − ν | = d is a constant, then the greater the minimum value of membership and non-membership, the greater the amount of knowledge. For example, if μ A − ν A = μ B − ν B = d ≥ 0 and min ( μ A , ν A ) = ν A ≥ min ( μ B , ν B ) = ν B ≥ 0 , then we have μ A ≥ μ B , ν A ≥ ν B , μ A ≥ ν A , μ B ≥ ν B , π A ≤ π B , and hence we know that the known extent of A is more than that of B under the same difference between membership and non-membership degree. Thus, it means that K ( A ) ≥ K ( B ) , which is the same as K p ( A ) and different from K G ( A ) .

Let K 1 ( A ) = 1 n ∑ i = 1 n | μ A ( x i ) − ν A ( x i ) | 1 − min ( μ A ( x i ) , ν A ( x i ) ) .

Obviously, K p ( A ) is a power function of K 1 ( A ) , and we get K p ( A ) = ( K 1 ( A ) ) p . According to the nature of the power function, we obtain:

1) When p = 1 , K p ( A ) = K 1 ( A ) is linear function of | μ A − ν A | 1 − min ( μ A , ν A ) .

2) When 0 < p < 1 , K p ( A ) is a convex function on the defined domain K 1 ( A ) ∈ [ 0 , 1 ] , which means that if K 1 ( A ) is close to 0, the amount of information K p ( A ) decreases rapidly; when K 1 ( A ) approaches 1, the amount of information K p ( A ) increases slowly.

3) Contrary to 2), when p > 1 , K p ( A ) is a concave function on the defined domain K 1 ( A ) ∈ [ 0 , 1 ] , which means that if K 1 ( A ) is close to 0, the amount of information K p ( A ) decreases slowly; when K 1 ( A ) approaches 1, the amount of information K p ( A ) increases sharply.

According to the analysis above, in practical applications, people can find suitable information measurement models based on parameter adjustments.

Example 1. Consider five IFSs,

D 1 = { 〈 x , 0 , 0.5 〉 } , D 2 = { 〈 x , 0.1 , 0.5 〉 } , D 3 = { 〈 x , 0.2 , 0.5 〉 } , D 4 = { 〈 x , 0.3 , 0.5 〉 } , D 5 = { 〈 x , 0.4 , 0.5 〉 } .

It is clear that

μ D 1 ( x ) = 0 < μ D 2 ( x ) < μ D 3 ( x ) < μ D 4 ( x ) < μ D 5 ( x ) = ν D 5 ( x ) = ν D 4 ( x ) = ν D 3 ( x ) = ν D 2 ( x ) = ν D 1 ( x ) = 0.5 ⇒ K D 1 ( x ) > K D 2 ( x ) > K D 3 ( x ) > K D 4 ( x ) > K D 5 ( x ) .

From Equations (7)-(14), we obtain

Knowledge measure | Accuracy | Right or Wrong | |||||
---|---|---|---|---|---|---|---|

D_{1} | D_{2} | D_{3} | D_{4} | D_{5} | |||

K_{G} | 0.25 | 0.16 | 0.09 | 0.04 | 0.01 | 100% | Right |

K_{SKB} | 0.5 | 0.522 | 0.538 | 0.543 | 0.533 | 40% | Wrong |

K_{BB} | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 0% | Wrong |

K H Y 1 | 0.699 | 0.590 | 0.553 | 0.553 | 0.590 | 60% | Wrong |

K r 0.5 | 0.699 | 0.562 | 0.538 | 0.538 | 0.562 | 60% | Wrong |

K_{N} | 0.5 | 0.557 | 0.624 | 0.7 | 0.781 | 100% | Right |

K S K 2 | 0.25 | 0.36 | 0.49 | 0.64 | 0.81 | 100% | Right |

K_{0.5} | 0.707 | 0.667 | 0.612 | 0.535 | 0.408 | 100% | Right |

K_{1} | 0.5 | 0.444 | 0.375 | 0.286 | 0.167 | 100% | Right |

K_{2} | 0.25 | 0.198 | 0.141 | 0.082 | 0.028 | 100% | Right |

K_{3} | 0.125 | 0.088 | 0.053 | 0.023 | 0.005 | 100% | Right |

Note. Each bold data means the wrong prediction result and the corresponding method.

can be defined as follows:

Accuracy = Number ( Entropies with Right Order in D i ) Number ( D i ) (16)

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}, K H Y 1 , K r 0.5 , K_{N}, K_{SK} and S(U, V), do not meet the property KP4I. According to _{G} and K_{p} will be better than the others. Hence, we conclude that K_{G} and K_{p} are better than the others.

A type of classic intuitionistic fuzzy sets A m are used to compare and analyze the difference of results among the proposed K p ( A ) and all those traditional knowledge measure formulas [

Example 2. Let A = { 〈 x , μ A ( x ) , ν A ( x ) 〉 | x ∈ X } be an IFS in X. For any positive real number m, De et al. define the IFS A m as follows [

A m = { 〈 x , ( μ A ( x ) ) m , 1 − ( 1 − ν A ( x ) ) m 〉 | x ∈ X } .

Obviously, we have 0 ≤ s ≤ t , 0 ≤ ( μ A ( x ) ) t ≤ ( μ A ( x ) ) s ≤ 1 , 0 ≤ 1 − ( 1 − ν A ( x ) ) s ≤ 1 − ( 1 − ν A ( x ) ) t ≤ 1 , and a series of IFSs for contrast experiments can be constructed.

Using the operation above, they defined the concentration and dilation of A as follows:

Concentration: C O N ( A ) = A 2 = { 〈 x , ( μ A ( x ) ) 2 , 1 − ( 1 − ν A ( x ) ) 2 〉 | x ∈ X } .

Dilation: D I L ( A ) = A 1 / 2 = { 〈 x , ( μ A ( x ) ) 1 / 2 , 1 − ( 1 − ν A ( x ) ) 1 / 2 〉 | x ∈ X } .

Like fuzzy sets, C O N ( A ) and D I L ( A ) can be treated as “Very (A)” and “More or less (A)”, respectively.

In the next, we consider an IFS A in X = {6, 7, 8, 9, 10} defined in reference [

Taking into account the characteristics of the value of language variables, De et al. define IFS A 0.5 , A , A 2 , A 3 , A 4 in X to be “More or Less Large”, “Large”, “Very Large”, “Quite Very Large”, “Very Very Large”. In the same way, B 0.5 , B , B 2 , B 3 , B 4 and C 0.5 , C , C 2 , C 3 , C 4 can be defined [

A = { 〈 6 , 0.1 , 0.8 〉 , 〈 7 , 0.3 , 0.5 〉 , 〈 8 , 0.6 , 0.2 〉 , 〈 9 , 0.9 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

A 0.5 = { 〈 6 , 0.316 , 0.553 〉 , 〈 7 , 0.548 , 0.293 〉 , 〈 8 , 0.775 , 0.106 〉 , 〈 9 , 0.949 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

A 2 = { 〈 6 , 0.01 , 0.96 〉 , 〈 7 , 0.09 , 0.75 〉 , 〈 8 , 0.36 , 0.36 〉 , 〈 9 , 0.81 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

A 3 = { 〈 6 , 0.001 , 0.992 〉 , 〈 7 , 0.027 , 0.875 〉 , 〈 8 , 0.216 , 0.488 〉 , 〈 9 , 0.729 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

A 4 = { 〈 6 , 0.000 , 0.998 〉 , 〈 7 , 0.008 , 0.938 〉 , 〈 8 , 0.13 , 0.59 〉 , 〈 9 , 0.656 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } .

B = { 〈 6 , 0.1 , 0.8 〉 , 〈 7 , 0.3 , 0.5 〉 , 〈 8 , 0.5 , 0.4 〉 , 〈 9 , 0.9 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

B 0.5 = { 〈 6 , 0.316 , 0.553 〉 , 〈 7 , 0.548 , 0.293 〉 , 〈 8 , 0.707 , 0.225 〉 , 〈 9 , 0.949 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

B 2 = { 〈 6 , 0.01 , 0.96 〉 , 〈 7 , 0.09 , 0.75 〉 , 〈 8 , 0.25 , 0.64 〉 , 〈 9 , 0.81 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

B 3 = { 〈 6 , 0.001 , 0.992 〉 , 〈 7 , 0.027 , 0.875 〉 , 〈 8 , 0.125 , 0.784 〉 , 〈 9 , 0.729 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

B 4 = { 〈 6 , 0.000 , 0.998 〉 , 〈 7 , 0.008 , 0.938 〉 , 〈 8 , 0.062 , 0.870 〉 , 〈 9 , 0.656 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } .

C = { 〈 6 , 0.1 , 0.8 〉 , 〈 7 , 0.3 , 0.5 〉 , 〈 8 , 0.5 , 0.5 〉 , 〈 9 , 0.9 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

C 0.5 = { 〈 6 , 0.316 , 0.553 〉 , 〈 7 , 0.548 , 0.293 〉 , 〈 8 , 0.707 , 0.293 〉 , 〈 9 , 0.949 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

C 2 = { 〈 6 , 0.01 , 0.96 〉 , 〈 7 , 0.09 , 0.75 〉 , 〈 8 , 0.25 , 0.75 〉 , 〈 9 , 0.81 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

C 3 = { 〈 6 , 0.001 , 0.992 〉 , 〈 7 , 0.027 , 0.875 〉 , 〈 8 , 0.125 , 0.875 〉 , 〈 9 , 0.729 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } ,

C 4 = { 〈 6 , 0.000 , 0.998 〉 , 〈 7 , 0.008 , 0.938 〉 , 〈 8 , 0.062 , 0.938 〉 , 〈 9 , 0.656 , 0.0 〉 , 〈 10 , 1.0 , 0.0 〉 } .

From the data above, for each x ∈ X = { 6 , 7 , 8 , 9 , 10 } , we obtain:

μ A 0.5 ( x ) ≥ μ A ( x ) ≥ μ A 2 ( x ) ≥ μ A 3 ( x ) ≥ μ A 4 ( x ) , ν A 0.5 ( x ) ≤ ν A ( x ) ≤ ν A 2 ( x ) ≤ ν A 3 ( x ) ≤ ν A 4 ( x ) ;

μ B 0.5 ( x ) ≥ μ B ( x ) ≥ μ B 2 ( x ) ≥ μ B 3 ( x ) ≥ μ B 4 ( x ) , ν B 0.5 ( x ) ≤ ν B ( x ) ≤ ν B 2 ( x ) ≤ ν B 3 ( x ) ≤ ν B 4 ( x ) ;

μ C 0.5 ( x ) ≥ μ C ( x ) ≥ μ C 2 ( x ) ≥ μ C 3 ( x ) ≥ μ C 4 ( x ) , ν C 0.5 ( x ) ≤ ν C ( x ) ≤ ν C 2 ( x ) ≤ ν C 3 ( x ) ≤ ν C 4 ( x ) .

According to the definition of knowledge measure of IFSs, obviously we get:

Knowledge ( A 0.5 ) < Knowledge ( A ) < Knowledge ( A 2 ) < Knowledge ( A 3 ) < Knowledge ( A 4 ) ;

Knowledge ( B 0.5 ) < Knowledge ( B ) < Knowledge ( B 2 ) < Knowledge ( B 3 ) < Knowledge ( B 4 ) ;

Knowledge ( C 0.5 ) < Knowledge ( C ) < Knowledge ( C 2 ) < Knowledge ( C 3 ) < Knowledge ( C 4 ) .

The results are shown in the following Tables 2-4.

Where the evaluation index Accuracy is defined as follows:

Accuracy = Number ( Entropies with Right Order in A i + B i + C i ) Number ( A i + B i + C i ) (17)

Knowledge measure | IFSs for comparison experiments | Accuracy | Number of wrong | ||||
---|---|---|---|---|---|---|---|

A_{1} | A_{2} | A_{3} | A_{4} | ||||

K_{G} | 0.785 | 0.788 | 0.805 | 0.854 | 1 | 93.3% | 0 |

K_{SKB} | 0.794 | 0.786 | 0.783 | 0.827 | 1 | 73.3% | 2 |

K_{BB} | 0.908 | 0.880 | 0.868 | 0.866 | 1 | 46.7% | 5 |

K H Y 1 | 0.754 | 0.744 | 0.783 | 0.816 | 1 | 86.7% | 1 |

K r 0.5 | 0.713 | 0.704 | 0.737 | 0.773 | 1 | 86.7% | 1 |

K_{N} | 0.856 | 0.835 | 0.839 | 0.847 | 1 | 80% | 1 |

K S K 2 | 0.827 | 0.780 | 0.764 | 0.765 | 1 | 60% | 2 |

K_{0.5} | 0.805 | 0.814 | 0.746 | 0.874 | 1 | 80% | 1 |

K_{1} | 0.681 | 0.693 | 0.699 | 0.788 | 1 | 86.7% | 0 |

K_{2} | 0.542 | 0.549 | 0.621 | 0.679 | 1 | 93.3% | 0 |

K_{3} | 0.472 | 0.470 | 0.559 | 0.613 | 0 | 93.3% | 1 |

Note. Each bold data means the wrong prediction result and the corresponding method.

Knowledge measure | IFSs for comparison experiments | Accuracy | Number of wrong | ||||
---|---|---|---|---|---|---|---|

B_{1} | B_{2} | B_{3} | B_{4} | B_{5} | |||

K_{G} | 0.767 | 0.761 | 0.865 | 0.911 | 0.926 | 93.3% | 1 |

K_{SKB} | 0.787 | 0.763 | 0.852 | 0.888 | 0.899 | 73.3% | 1 |

K_{BB} | 0.918 | 0.900 | 0.902 | 0.9066 | 0.90662 | 46.7% | 1 |

K H Y 1 | 0.748 | 0.745 | 0.802 | 0.848 | 0.879 | 86.7% | 1 |

K r 0.5 | 0.710 | 0.706 | 0.748 | 0.791 | 0.826 | 86.7% | 1 |

K_{N} | 0.858 | 0.847 | 0.874 | 0.893 | 0.900 | 80% | 1 |

K S K 2 | 0.846 | 0.814 | 0.819 | 0.831 | 0.838 | 60% | 2 |

K_{0.5} | 0.790 | 0.755 | 0.890 | 0.930 | 0.941 | 80% | 1 |

K_{1} | 0.655 | 0.626 | 0.803 | 0.869 | 0.891 | 86.7% | 1 |

K_{2} | 0.507 | 0.505 | 0.675 | 0.768 | 0.810 | 93.3% | 1 |

K_{3} | 0.437 | 0.445 | 0.587 | 0.691 | 0.748 | 93.3% | 0 |

Note. Each bold data means the wrong prediction result and the corresponding method.

Knowledge measure | IFSs for comparison experiments | Accuracy | Number of wrong | ||||
---|---|---|---|---|---|---|---|

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | |||

K_{G} | 0.481 | 0.488 | 0.672 | 0.683 | 0.722 | 93.3% | 0 |

K_{SKB} | 0.790 | 0.756 | 0.878 | 0.907 | 0.913 | 73.3% | 1 |

K_{BB} | 0.932 | 0.920 | 0.924 | 0.925 | 0.920 | 46.7% | 2 |

0.761 | 0.767 | 0.829 | 0.874 | 0.900 | 86.7% | 0 | |

0.499 | 0.489 | 0.312 | 0.218 | 0.164 | 86.7% | 0 | |

K_{N} | 0.868 | 0.864 | 0.895 | 0.911 | 0.913 | 80% | 1 |

0.872 | 0.852 | 0.861 | 0.866 | 0.864 | 60% | 2 | |

K_{0.5} | 0.786 | 0.673 | 0.910 | 0.942 | 0.949 | 80% | 1 |

K_{1} | 0.648 | 0.593 | 0.832 | 0.890 | 0.905 | 86.7% | 1 |

K_{2} | 0.4985 | 0.4993 | 0.709 | 0.802 | 0.835 | 93.3% | 0 |

K_{3} | 0.429 | 0.445 | 0.619 | 0.731 | 0.783 | 93.3% | 0 |

Note. Each bold data means the wrong prediction result and the corresponding method.

Based on the theoretical derivation, K G ( A ) and K p ( A ) satisfy the property of order KP4I, while K B B ( A ) , K S K B ( A ) , K H C α ( A ) , K H C 1 ( A ) , K R β ( A ) , K N ( A ) , K S K ( A ) and S ( U , V ) do not Satisfy this property. From the comparative analysis of the results in Tables 2-4, we found that the overall order accuracy of K p ( A ) is 89%, and that of K 2 ( A ) & K 3 ( A ) & K G ( A ) is 93.3%, owning the highest accuracy among all methods. Moreover, From Example 1 - 2, the order of all the results from K 2 ( A ) and K G ( A ) is exactly the same. And for the order of all the results from K 3 ( A ) and K G ( A ) , there is only slight differences in Example 2 between them. Hence the overall performance of K p ( A ) is acceptable.

In conclusion, the above theoretical and experimental results show that the proposed parametric algorithm is simple and feasible, and it is an effective tool for knowledge measure. In future research, we will apply the constructed knowledge measure model to calculate the information volume of uncertain variables, study the information-based sorting and decision operators, and apply them to the research of management science decision making like references [

On the basis of Szmidt & Kacprzyk’s axiom system, a simple model of knowledge measure with parameters is presented. And we illustrate the validity of the measure tool from the theoretical and empirical evidence. At the same time, this paper also applies the proposed knowledge measure, along with some classical knowledge measure formulas of IFSs, from the theoretical and practical comparison, to verify a conclusion: In most knowledge measures of IFSs, the accuracy of those formulas satisfying the order property will be higher than that of those not satisfying.

This paper is funded by the National statistical research key projects (No. 2016LZ18), Natural Science Projects (No. 2016A030310105, 2018A030313470) & Soft Science Project (No. 2015A070704051, 2016A030313688) & Quality engineering and teaching reform project (No. 125-XCQ16268) of Guangdong Province, Philosophy and Social Science Project of Guangzhou (No. 2017GZYB45), Team Project Guangdong University of Foreign Studies (No. TD1605), National Undergraduate Training Program for Innovation and Entrepreneurship of China (No. 201511846058, 201711846004).

The authors declare no conflicts of interest regarding the publication of this paper.

Zhang, Z.H., Yuan, S.G., Zhang, J., Ma, C., Xu, J.H. and Lin, X.L. (2018) A Knowledge Measure with Parameter of Intuitionistic Fuzzy Sets. Applied Mathematics, 9, 874-889. https://doi.org/10.4236/am.2018.97060