Some notes on characters of intuitionistic fuzzy sets

1 Introduction

Atanassov [1] introduced intuitionistic fuzzy sets (IFS) as a generalization of fuzzy sets (FS) [2]. Over the last two decades, the intuitionistic fuzzy set theory has been applied to many different fields, such as logic programming, topology, medical diagnosis, pattern recognition and decision making [3–8]. Sometimes, the level of uncertainty should be reduced as much as possible. In such cases, point operators may be valid for this purpose. Moreover, some investigators argue that one effective way to study intuitionistic fuzzy sets is to first transform them into fuzzy sets and then discuss their corresponding attributes. Atanassov’s operator D _α [9] is the first conversion formula which can be utilized to obtain fuzzy sets from each intuitionistic fuzzy set. Another operator D _{γ x} , called Atanassov’s point operator, was proposed by Burillo and Bustince [10], where an inverse conversion formula deals with the building of intuitionistic fuzzy sets from fuzzy sets. Formula f _α , defined by Burillo and Bustince [11], is similar to D _α defined by Atanassov. Liu and Wang [7] discussed the concept of intuitionistic fuzzy point operators and defined their operator F α x β x n.

In many theoretical and practical issues we are faced with the question, that there are two intuitionistic fuzzy sets in the same universe which need to be calculated a difference between them represented by a distance. Szmidt and Kacpreyk [12] proposed some new definitions of distances between intuitionistic fuzzy sets based on a geometrical representation of an intuitionistic fuzzy sets. It is shown that the definitions proposed are consistent with their counterparts traditionally used for fuzzy sets.

Fuzziness, a feature of imperfect information, results from the lack of crisp distinction between the elements belonging and not belonging to a set (i.e. the boundaries of the set under consideration are not sharply defined). Many investigations have been devoted to the fuzzy entropy of intuitionistic fuzzy sets. Different definitions of fuzzy entropy have been proposed by some researchers [13, 18]. For example, Szmidt and Kacpreyk [13] proposed a measure of fuzziness for intuitionistic fuzzy sets. The measure of entropy can be regarded as a result of a geometric interpretation of intuitionistic fuzzy sets and it basically uses a ratio of distances between them [12].

In this paper, after recalling the definition of intuitionistic fuzzy sets, we present interpretations of intuitionistic fuzzy sets for purposes of transformation formula. And then, some constraint conditions for satisfying the requirements of conversion from intuitionistic fuzzy sets into fuzzy sets are introduced. Simultaneously, a practical and reasonable conversion operator, which can transform an intuitionistic fuzzy set into only one fuzzy set and show how to determine the value of parameter γ _x in Atanassov’s point operator, is proposed based on the principle of information diffusion [16]. And a new formula of fuzzy entropy of intuitionistic fuzzy sets, defined in this study between intuitionistic fuzzy sets, is derived which is based on the conversion of intuitionistic fuzzy sets into fuzzy sets and a distance measure. Surprisingly, we find that the formula proposed is equivalent to the one defined by Szmidt and Kacpreyk [13].

Throughout this paper, X is a universal set; IFSs (X) is a class of all IFSs of X; FSs (X) is a class of all fuzzy sets of X; A ˜ expresses an ordinary fuzzy set and also is just one that is transformed from an IFS, of which membership function is expressed byf A ˜ ( x ).

2 Intuitionistic fuzzy sets

3 Transforming of IFSs into FSs

When we obtain an IFS, we clearly know the degree of membership x to A and the degree of non-membership x to A. The presence of the hesitancy degree π _A  (x) suggests that we have no information on the state of x being or not being in A. Hence, μ _A  (x) and ν _A  (x) are pieces of incomplete information. Incomplete information can be treated with the principle of information diffusion proposed by Huang [16], where the main idea is to fill the gap by reasonable fuzzy sets. Information diffusion changes a traditional sample point data into a fuzzy set, making the information carried by the sample point to proliferate around it. The basic function of the principle is to expand a crisp observation so as to fill the gap caused by the lack of data. The principle asserts that there must be reasonable information diffusion functions to improve no diffusion estimator if and only if X is incomplete, and the information diffusion function should become obvious when we understand the fuzziness of the incomplete data. The completion of information diffusion is based on the information diffusion function.

It is clear that the hesitancy degree π _A  (x) is allotted to both the degree of membership and the degree of non-membership in a definite ratio. When IFSs are considered in place of fuzzy sets, the hesitancy degree will drift or be inclined to the degree of membership or degree of non-membership, decided by the effect of their size. Namely, if μ _A  (x) > ν _A  (x), then μ _A  (x) can obtain more ratio from the hesitancy thanν _A  (x) can.

New constraint conditions are proposed to fulfill the requirements for transforming IFSs into fuzzy sets.

Definition 2. Make A be an IFS and let A ˜ represent converted fuzzy set. A function H : A - IFS _s  (X) → FS _s  (X) defines an operator for transforming IFSs into fuzzy sets, if H has the following properties:

(p1)  ∀ x ∈ X , μ A ( x ) = ν A ( x ) ⇒ f A ˜ ( x ) = 1 / 2 .

(p2) ∀x∈ X, π _A  (x) =1 - μ _A  (x) - ν _A  (x) =0 ⇒f A ˜ ( x ) = μ A ( x ).

(p3) f A ˜ ( x ) increases with increasing μ _A  (x), and decreases with increasing ν _A  (x).

(p4) When π _A  (x) ≠0,| f A ˜ ( x ) - μ A ( x ) | > | 1 - ν A ( x ) - f A ˜ ( x ) |, if and only if μ _A  (x) > ν _A  (x).

Remark. For the first constraint condition, the degree of membership equals the degree of non-membership, so the membership function value f A ˜ ( x ) becomes 1/2. Accordingly, the assumption is to average the hesitancy degree to the degree of membership and the degree of non-membership. In the second constraint, the membership function just equals the degree of membership x being in A, because there is no hesitancy degree. It is reasonable that the hesitancy degree is inclined to a greater degree in the third constraint. The fourth constraint tells us when the degree of membership is greater than the degree of non-membership, it will get more hesitancy degree and vice verse.

Let us first analyze some existing conversionformulae.

Atanassov defined the following operators, for A ∈ IFSs (X), α, β ∈ [0, 1] and α + β ≤ 1,D α : IFS s ( X ) → FS s ( X ) D α ( A ) = { < x , μ A ( x ) + α π A ( x ) , 1 - μ A ( x ) - α π A ( x ) > / x ∈ X }(1)

F α , β : IFSs ( X ) → IFSs ( X ) F α , β ( A ) = { < x , μ A ( x ) + α π A ( x ) , ν A ( x ) + β π A ( x ) > / x ∈ X }(2)

Clearly, D _α  (A) is an ordinary fuzzy set which is built from an IFS and F_α, β (A) is still an IFS. Let A ˜ = D α ( A ), the membership function of A ˜ is f A ˜ ( x ) = μ A ( x ) + α π A ( x ).

For each IFS A, the family of all fuzzy sets associated with A by operator D _α will be denoted by {D _α  (A) } _α∈[0,1]. Burillo and Bustince [10] proved that {D _α  (A) } _α∈[0,1] is a totally ordered fuzzy set.

Burillo and Bustince [10] defined an Atanassov’s point operator D _{γ x} for each x ∈ X and γ _x  ∈ [0, 1] for A ∈ IFSs (X) as follows:f A ˜ ( x ) = D γ x ( A ) = μ A ( x ) + γ x π A ( x )(3)

Similarly, Liu and Wang [7] defined another Atanassov’s point operator F_{α x , β x} .

For each x ∈ X, taking α _x ,  β _x  ∈ [0, 1] and α _x  + β _x  ≤ 1, a point operator F_{α x , β x} is defined for A ∈ IFSs (X) as follows:

F α x , β x ( A ) = { < x , μ A ( x ) + α x π A ( x ) , ν A ( x ) + β x π A ( x ) > / x ∈ X }(4)

Generally, F_{α x , β x}  (A) still is an IFS, but if α _x  + β _x  = 1, it is a fuzzy set.

Further, for any positive n, they structured a new operator: F α x , β x n ( A ) = { < x , μ A ( x ) + α x π A ( x ) 1 - ( 1 - α x - β x ) n α x + β x , ν A ( x ) + β x π A ( x ) 1 - ( 1 - α x - β x ) n α x + β x > / x ∈ X }(5)

whereπ F α x , β x n ( A ) = ( 1 - α x - β x ) n π A ( x )(6)

From the above, the following results can be concluded:

Let α _x ,  β _x  ∈ [0, 1] and α _x  + β _x  ≤ 1 for x ∈ X, A ∈ IFSs (X). The following limit is defined:lim n → ∞ F α x , β x n ( A ) = { < x , lim n → ∞ μ F α x , β x n ( A ) ( x ) , lim n → ∞ ν F α x , β x n ( A ) ( x ) > / x ∈ X }(7)

Liu and Wang proved the following theorem:( i ) lim n → ∞ F α x , β x n ( A ) = D α x / ( α x + β x ) ( A )(8) ( ii ) lim n → ∞ π F α x , β x n ( A ) ( x ) = 0(9)

Clearly, formulas (8) and (9) show that the limit state of point operator F α x , β x n ( A ) is an ordinary fuzzy set. It is noted that on integrating formulae (3) and (8), parameter γ _x  = α _x / (α _x  + β _x ) for each x ∈ X.

Now try to derive a new operator for transforming IFSs into fuzzy sets according to the principle of information diffusion, which will improve the characteristics of operators mentioned above, and specially, get the value of γ _x in Burillo and Bustince’s operator D _{γ x} in a simple way. For an IFS, it is assumed that μ _A  (x) and 1 - ν _A  (x) are two sample points (points M and N in Fig. 2) and the hesitancy degree lies in the area MN ¯.

The degree of membership and degree of non-memberships each has a tendency to diffuse to the middle area based upon the principle of information diffusion. In other words, part of the hesitancy degree will convert to μ _A  (x) or ν _A  (x). According to Definition 2, let the sample set W ={ μ _A  (x) , ν _A  (x) } and the universe V = [0, 1]; information diffusion functions F _μ  (y) and F_1-ν (y) about μ _A  (x) and 1 - ν _A  (x) are constructed on [0, 1], respectively. Here, the triangular fuzzy number is chosen as a form of information diffusion function. Making F _μ  (y) and F_1-ν (y) as the medians of triangular fuzzy numbers, respectively, the information diffusion functions about them are as follows:F μ ( y ) = { 1 μ A ( x ) 0 ≤ y < 1 - ν A ( x ) 1 - y 1 - μ A ( x ) μ A ( x ) ≤ y ≤ 1(10) F 1 - ν ( y ) = { y 1 - ν A ( x ) 0 ≤ y < 1 - ν A ( x ) - y - 1 ν A ( x ) 1 - ν A ( x ) ≤ y ≤ 1(11)

Let 1 - y 1 - μ A ( x ) = y 1 - ν A ( x ), we could get the abscissa of point P which is the intersection of segments CT ¯ and DS ¯ as:y = 1 - ν A ( x ) 1 + π A ( x )(12)

In Fig. 2, segment MQ ¯ = 1 - μ A ( x ) 1 + π A ( x ) π A ( x ) and QN ¯ = 1 - ν A ( x ) 1 + π A ( x ) π A ( x ). This method divides the hesitancy degree π _A  (x) into two parts by allotting QN ¯ to the membership degree μ _A  (x) and MQ ¯ to the non-membership degree ν _A  (x). Thus, the membership function of fuzzy set A ˜ is obtained:f A ˜ ( x ) = μ A ( x ) + 1 - ν A ( x ) 1 + π A ( x ) π A ( x )(13)

Then the membership function of its complementary set A ˜ c is written as:f A ˜ c ( x ) = ν A ( x ) + 1 - μ A ( x ) 1 + π A ( x ) π A ( x )(14)

Note that on integrating formulae (3) and (13), parameter γ x = 1 - μ A ( x ) 1 + π A ( x ) for each x ∈ X.

Property 1. Formula (13) is an operator for transforming IFSs into fuzzy sets.

Proof. (p1) ∀x ∈ X, if μ _A  (x) = ν _A  (x), then f A ˜ ( x ) = μ A ( x ) + 1 - μ A ( x ) 1 + 1 - 2 μ A ( x ) ( 1 - 2 μ A ( x ) ) = μ A ( x ) + 1 2 ( 1 - 2 μ A ( x ) ) = 1 2

(p2) ∀x ∈ X, if π _A  (x) =1 - μ _A  (x) - ν _A  (x) =0, then obviously f A ˜ ( x ) = μ A ( x ).

(p3) Partial derivative of f A ˜ with respect to μ _A is: ∂ f A ˜ ∂ μ A = 1 - ( 1 - ν A ) ( 1 + π A ) 2 ≥ 1 - 1 ( 1 + π A ) 2 > 0

Therefore, f A ˜ increases with increasing μ _A .

Partial derivative of f A ˜ to (1 - ν _A ) is: ∂ f A ˜ ∂ ( 1 - ν A ) = 1 - ( 1 + π A ) - ( 1 - ν A ) ( 1 + π A ) 2 = 1 - 1 - μ A ( 1 + π A ) 2 ≥ 1 - 1 ( 1 + π A ) 2 > 0

So f A ˜ decreases with increasing ν _A ;

(p4) We might deduce | f A ˜ - μ A | = 1 - ν A 1 + π A π A, | 1 - ν A - f A ˜ | = | 1 - ν A - μ A - 1 - ν A 1 + π A π A | = 1 - μ A 1 + π A If μ _A  (x) > ν _A  (x), inequality | f A ˜ ( x ) - μ A ( x ) | > | 1 - ν A ( x ) - f A ˜ ( x ) | is right, or If μ _A  (x) < ν _A  (x), inequality | f A ˜ ( x ) - μ A ( x ) | < | 1 - ν A ( x ) - f A ˜ ( x ) | is right too.

We will give an example to illustrate the proposed operator.

Example 1. Suppose X ={ x₁, x₂, x₃ }, an IFS A ={ < x₁, 0 . 0, 0.1 > , < x₂, 0.3, 0.2 > , < x₃, 0.9, 0.0 > } Then the membership degree f A ˜ ( x ) of the converted fuzzy set A ˜ is:

Our operator (13): f A ˜ ( x ) = { ( 0.426 / x 1 , 0.566 / x 2 , 0.991 / x 3 ) }

Let α = 1/2, then Atanassov’s operator(1): f A ˜ ( x ) = { ( 0.45 / x 1 , 0.55 / x 2 , 0.95 / x 3 ) }

Let γ₁ = 1/2, γ₂ = 1/5, γ₃ = 1/4, Burillo and Bustince’s operator (3): f A ˜ ( x ) = { ( 0.45 / x 1 , 0.40 / x 2 , 0.925 / x 3 ) }

Let us concentrate on the ΔCDE in Fig. 3.

According to Szmidt and Kacprzyk [13], an intuitionistic fuzzy set A can be denoted as (μ _A ,  ν _A ,  π _A ), so a non-fuzzy set (a crisp set) corresponds to the point C (the element fully belongs to it when the condition is (μ _A ,  ν _A ,  π _A ) = (1, 0, 0)) and at point D (the element fully does not belongs to it when (μ _A ,  ν _A ,  π _A ) = (0, 1, 0)). Points C and D representing a crisp set separately have the degree of fuzziness equal to 0. Furthermore, point E has maximal fuzziness which equals to 1. An intuitionistic fuzzy set is represented by the triangle CDE and its interior [13].

Now, a new distance formula is derived. Acording to the following conversion formula (13), an intuitionistic fuzzy set A is transformed into a fuzzy set A ˜ whose membership function is f A ˜ ( x ). Consider this intuitionistic fuzzy set A as a new sign using a binary group, i.e.,A ( x ) = { f A ˜ ( x ) , π A ( x ) }(15)

As shown in Fig. 1, Points C and D are two crisp sets, according to formula (15): C ( x ) = { 1 , 0 } D ( x ) = { 0 , 0 }

Assume that A and B are two intuitionistic fuzzy sets, then the Hamming distance between them is defined asd ZS ( A , B ) = | f A ˜ ( x ) - f B ˜ ( x ) | + | π A ( x ) - π B ( x ) |(16)

If A and B are defined with n points, then the Hamming distance between them is equal tod ZS ( A , B ) = 1 n ∑ i = 1 n [ | f A ˜ ( x i ) - f B ˜ ( x i ) | + | π A ( x i ) - π B ( x i ) | ](17)

It is noted that d _ZS  (A,  B) is composed of two parts, | f A ˜ ( x i ) - f B ˜ ( x i ) | and |π _A  (x _i ) - π _B  (x _i ) |, the former one is the difference of fuzziness degreesbetween A and B, and the latter is that of hesitancy degrees between them.

Owing to the existing fuzziness of unknown-degree π _A  (x) and the fuzziness of uncertainty (the membership degree is μ _A  (x) while the non-membership degree is ν _A  (x)), the fuzzy entropy corresponding to intuitionistic fuzzy sets is difficult to determine.

The following fuzzy entropy formula of a fuzzy set was proposed in Kosko [17]:E ( F ) = d ( A , A near ) d ( A , A far )(18)

Based on Kosko [17], the fuzzy entropy of a fuzzy set can be affirmed by the ratio of nearest non-fuzzy set. A _near distance to the furthest non-fuzzy set A _far distance.

For the situation in Fig. 4(a), owing to an intuitionistic fuzzy set A has μ _A  (x) > ν _A  (x), the corresponding fuzzy sets should satisfy f A ˜ ( x ) > f A ˜ c ( x ) and f A ˜ ( x ) > 1 / 2

Hence, the nearest and furthest crisp sets are C and D, respectively. According to (18) and the theory of [10], the fuzzy entropy of intuitionistic fuzzy set A is obtained as:E ( A ) = | f A ˜ ( x ) - 1 | + | π A ( x ) - 0 | | f A ˜ ( x ) - 0 | + | π A ( x ) - 0 | = π A ( x ) + f A ˜ c ( x ) π A ( x ) + f A ˜ ( x )(19)

For the situation in Fig. 4(b), due to intuitionistic fuzzy set A has μ _A  (x) < ν _A  (x), the corresponding fuzzy set should satisfy f A ˜ ( x ) > f A ˜ c ( x ) and f A ˜ ( x ) < 1 / 2

Hence, the nearest and furthest crisp sets are D and C, respectively, according to the same principle, the fuzzy entropy isE ( A ) = | f A ˜ ( x ) - 0 | + | π A ( x ) - 0 | | f A ˜ ( x ) - 1 | + | π A ( x ) - 0 | = π A ( x ) + f A ˜ ( x ) π A ( x ) + f A ˜ c ( x )(20)

Combining with (19) and (20), we getE ( A ) = π A ( x ) + min { f A ˜ ( x ) , f A ˜ c ( x ) } π A ( x ) + max { f A ˜ ( x ) , f A ˜ c ( x ) }(21)

Let A be an intuitionistic fuzzy set in X = {x₁, x₂, ⋯ , x}, then its fuzzy entropy isE ( A ) = 1 n ∑ i = 1 n π A ( x ) + min { f A ˜ ( x ) , f A ˜ c ( x ) } π A ( x ) + max { f A ˜ ( x ) , f A ˜ c ( x ) }(22)

Using (13) and (14), formula (22) changes to

E ( A ) = 1 n ∑ i = 1 n π A + min { μ A , ν A } + π A 1 + π A min { 1 - ν A , 1 - μ A } π A + max { μ A , ν A } + π A 1 + π A max { 1 - ν A , 1 - μ A }(23) Hence, the fuzzy entropy of intuitionistic fuzzy set A can now be written as:

E ZS ( A ) = 1 n ∑ i = 1 n min { 1 - ν A ( x ) , 1 - μ A ( x ) } max { 1 - ν A ( x ) , 1 - μ A ( x ) }(24)

It is easily proved that our fuzzy entropy measure of intuitionistic fuzzy sets equates one proposed by Szmidt and Kacprzyk [13].

Let F be an intuitionistic fuzzy set in X = {x₁,  x₂, ⋯ , x _n }. Burillo and Bustince [18] defined the entropy of F as follows: E BB ( F ) = ∑ i = 1 n π F ( x i )

and Vlachos and Sergiadis [19] defined the entropy of F as follows: E VS ( F ) = - 1 n ln 2 ∑ i = 1 n [ μ A ( x i ) ln μ A ( x i ) + ν A ( x i ) ln ν A ( x i ) - ( 1 - π A ( x i ) ) ln ( 1 - π A ( x i ) ) - π A ( x i ) ln 2 ]

Hung [15] defined the entropy of F as follows: E H ( F ) = 1 - 1 n ∑ i = 1 n | μ F ( x i ) - ν F ( x i ) |

The following examples are used for comparing the proposed entropy measure with E _BB , E _SK and E _H .

Example 2. Let F₁, F₂ and F₃ be one-element intuitinistic fuzzy sets in X = {x} and defined by F 1 = { ( x , 3 4 , 1 6 ) } , F 2 = { ( x , 1 2 , 0 ) } F 3 = { ( x , 1 2 , 1 4 ) }

By (24), we have E ZS ( F 1 ) = 3 10 , E ZS ( F 2 ) = 1 2 , E ZS ( F 3 ) = 2 3

Therefore, E _ZS  (F₁) < E _ZS  (F₂) < E _ZS  (F₃). This result is consistent with Hung [15]. But, we obtained E BB ( F 2 ) ( = 1 2 ) > E BB ( F 3 ) ( = 1 4 ) > E BB ( F 1 ) ( = 1 12 ) E VS ( F 3 ) ( = 0.939 ) > E VS ( F 1 ) ( = 0.710 ) > E VS ( F 2 ) ( = 0.500 )

Example 3. Let F be an intuitionistic fuzzy set in X. For any positive number n, De [20] defined the intuitionistic fuzzy set F ⁿ F n = { ( x , [ μ F ( x ) ] n , 1 - [ 1 - ν F ( x ) ] n ) / x ∈ X }

By taking into account the characterization of linguistic variables, they regarded F as “LARGE” in X. Using the above operation,

F^1/2 may be treated as “More or less LARGE”,

F² may be treated as “Very LARGE”,

F⁴ may be treated as “Very very LARGE”.

In this paper, we consider an intuitionistic fuzzy set F in X = {6, 7, 8, 9, 10} defined by [20] F = { ( 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 ) }

We used these intuitionistic fuzzy sets to compare E _ZS , E _H , E _BB and E _VS , respectively. The comparison results are shown in Table 1.

From the viewpoint of concentration and dilation operations [20], the entropies of these intuitionistic fuzzy sets have the following requirement:E ( F 1 / 2 ) > E ( F ) > E ( F 2 ) > E ( F 4 )(25)

Based on Table 1, the entropy measures E _Z , E _H and E _VS satisfy (25), but E _BB does not.

In this paper, we have mainly analyzed operators transforming IFSs into fuzzy sets. Furthermore, we not only introduce restrictive conditions which any transforming operator must satisfy but also define a new operator for transforming IFSs into fuzzy sets using the principle of information diffusion. And we have provided one new fuzzy entropy formula for intuitionistic fuzzy sets based on a distance, and defined in this study between intuitionistic fuzzy sets. This measure is similar to the consideration for ordinary fuzzy sets. Moreover, we find it can be equivalent to measure proposed by E.Szmidt and J.Kacprzyk [13]. Furthermore, we used some examples to make comparison with [15, 19]. The results show that the proposed entropy formula has a simple format and is convenientfor operation.