How to measure non-specificity of intuitionistic fuzzy sets

1 Introduction

In 1965, Zadeh [1] defined fuzzy sets as a very good tool on handling uncertainty, vagueness, and imprecision. For any fuzzy set A, each element x from the universe X has associated a function μ _A  (x) indicating the degree of membership of x to A. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines, including medical and life sciences, management sciences, social sciences, engineering, statistics, graph theory, artificial intelligence, signal processing, multi-agent systems, pattern recognition, robotics, computer networks, expert systems, decision making and automata theory.

One of the main criticisms to this theory is that the success of the use of fuzzy sets depends on the choice of the membership function made by expert, i.e., an expert has to choose μ _A  (x) for each element of the fuzzy set. But there are applications in which experts do not have precise knowledge about the function that should be taken. To solve this problem, Atanassov [2–5]introduced the concept of intuitionistic fuzzy sets (IFSs) as a generalization of fuzzy sets. Atanassov added in the definition of fuzzy set a new component which determines the degree of non-membership. Fuzzy sets give the degree of membership of an element in a given set (the non-membership of degree equals one minus the degree of membership), while intuitionistic fuzzy sets give both a degree of membership and a degree of non-membership. For an IFS, its membership and non-membership degrees are more or less independent from each other, with the only requirement that the sum of these two degrees is not greater than 1. Intuitionistic fuzzy sets are higher order fuzzy sets. Application of higher order fuzzy sets makes the solution procedure more complex, but if the complexity on computation time, computation volume or memory space are not the matter of concern then a better result could be achieved. Therefore, since its inception, IFS has attracted more and more attentions from researchers [6, 16–23, 25].

Uncertainty measures can supply new viewpoints for analyzing data, which can help us in disclosing the substantive characteristics of data. So the uncertainty measurement issue has become a key topic in the IFS theory, analogous to its role in the fuzzy set theory. There have been several attempts to quantify the uncertainty associated with IFSs. Usually, such uncertainty measure can be regarded as entropy-based uncertainty measure, since entropy is often viewed as a measure of uncertainty. Following Zdeh’s [7] concept on entropy of fuzzy sets, many researchers have proposed definitions of entropy for fuzzy sets [10–13]. These measures try to quantify only one aspect of uncertainty, i.e., fuzziness. But fuzzy sets are associated with another kind of uncertainty, which is related to lack of specificity. To make a distinction between the two, we can say that fuzziness measures gradualarity while specificity is related to granularity. In [14, 15] Yager proposed different measures of specificity for a fuzzy set, which quantifies the degree to which a fuzzy subset focuses on (points to) one and only one element as its member.

An axiomatic foundation for entropy of IFS, on the other hand, was first proposed by Burillo and Bustince in 1996 [16]. Szmidt and Kacprzyk [17] have also proposed another set of axioms. Both of these axiomatic structures are non-probabilistic type. However, it is demonstrated that these two interesting approaches cannot capture all facets of uncertainty that are associated with IFSs. Szmidt et al. [18, 20] also acknowledged that an entropy measure alone is not a satisfactory measure of knowledge that is useful for decision making using an IFS. A measure of specificity associated with an IFS is also proposed by Yager [19]. In [19] Yager proposed different measures of specificity for fuzzy sets, including a class of measures called linear specificity measures. An important member of this class computes specificity as: largest membership value the average of the other membership values. The specificity measure for an IFS in [19] is also defined by the similar expression.

However, the entropy alone may not be a satisfactory uncertainty measure for IFSs. The reason is that an entropy measures fuzziness or the difference between the membership and non-membership degrees, but does not consider any peculiarities of the membership and non-membership degrees. Hence, Vlachos et al. [21] presented the definition of intuitionistic fuzzy entropy based on cross entropy measure of IFSs. Their proposed intuitionistic fuzzy entropy can be decomposed into two terms, namely fuzziness and intuitionism. It can be found that this intuitionistic fuzzy entropy measure gets its maximum value for all IFSs with μ (x) = v (x) , ∀ x ∈ X. Thus, it may not be sensitive enough to the change of IFSs, especially for IFSs with equal membership and non-membership grades. So the aforementioned problem still exists.

Following Vlachos’ work, Mao et al. [23] proposed a novel entropy measure for IFSs based on their modified cross-entropy of IFSs. Their proposed entropy measure is a real-valued continuous function being monotonically increasing with respect to the variable π (x), and decreasing with |μ (x) - v (x) |. That is to say, the entropy value will increase along with the enhanced fuzziness under the same intuitionism. Equivalently, the entropy value will decrease along with the weakened intuitionism under the same fuzziness. The decomposition of the novel entropy measure depicts that uncertain information of IFSs consists of intuitionism and fuzziness. Intuitionism is determined by hesitancy degree π (x) of IFSs, while fuzziness is controlled by the closeness of membership and non-membership, i.e., |μ (x) - v (x) |. This measure essentially uncovers the property of uncertainty measure for IFSs. Moreover, Pal et al. [22] also addressed this problem by defining two kinds of entropy, one of which is related to fuzziness while the other is related to lack of knowledge.

Although the combination of fuzziness and intuitionism can measure the uncertainty of IFSs well, all existing entropy measures for IFSs cannot discriminate uncertainty degrees of crisp sets, which will be discussed in Section 3. For a crisp set, its non-specificity plays an important role in quantifying uncertainty associated to its cardinality. We would have in mind that crisp sets can be regarded as a special IFS with all membership and non-membership degrees as 0 or 1, hesitancy degrees as 0, ∀x ∈ X. So uncertainty measures for IFSs should be capable of quantifying uncertainty degree for crisp sets. In such sense, non-specificity-based uncertainty must be taken into account when defining uncertainty measure for IFSs.

So far, however, there are few uncertainty measures containing non-specificity. In this paper, we address the problem of how to define a non-specificity measure for IFSs. The proposed non-specificity measure is developed from Hartley measure [24]. Meanwhile, we uncover some axiomatic properties of the proposed non-specificity measure. To do so, we organize the rest of this paper as follows. In Section 2, we briefly recall some concepts related to IFSs. In Section 3, we critically review and analyze existing axiomatic entropy measures for IFSs. We put forward a non-specificity measure is in Section 4. In this section we also provide some properties of proposed non-specificity measure along with their proofs. In Section 5, illustrative examples are given to show the performance of the proposed measure. Section 6 gives a conclusion of this paper.

2 Preliminaries

In this section, we briefly recall basic concepts related to IFS, and then list the relations between IFSs. The connection between IFSs and crisp sets is also presented.

Definition 1. [1] Let X = {x ₁,  x ₂, ⋯ ,  x _n } be the universe of discourse, then a fuzzy set A in X is defined as follows: A = { 〈 x , μ A ( x ) 〉 | x ∈ X }(1) where μ _A  (x) : X → [0, 1] is the membership degree.

Definition 2. [2] An intuitionistic fuzzy set A in X defined by Atanassov can be written as: A = { 〈 x , μ A ( x ) , v A ( x ) 〉 | x ∈ X }(2) where μ _A  (x) : X → [0, 1] and v _A  (x) : X → [0, 1] are membership degree and non-membership degree, respectively, satisfying the requirement: 0 ≤ μ A ( x ) + v A ( x ) ≤ 1(3)

1 - μ _A  (x) - v _A  (x) is called the hesitancy degree of the element x ∈ X to the set A, denoted by π _A  (x). π _A  (x) is also called the intuitionistic index of x to A. Greater π _A  (x) indicates more vagueness on x. Obviously, when π _A  (x) =0, ∀x ∈ X, the IFS degenerates into an ordinary fuzzy set. In the sequel, all IFSs in X is denoted as IFSs (X).

Definition 3. [2] For A ∈ IFSs (X) and B ∈ IFSs (X), some relations between them are defined as:

(R1) A ⊆ B iff ∀x ∈ X μ _A  (x) ≤ μ _B  (x) , v _A  (x) ≥ v _B  (x);

(R2) A = B iff ∀x ∈ X μ _A  (x) = μ _B  (x) , v _A  (x) = v _B  (x);

(R3) A ^C  ={ 〈 x, v _A  (x) , μ _A  (x) 〉 |x ∈ X  }, where A ^C is the complement of A.

Definition 4. Let X = {x ₁, x ₂, ⋯ , x _n } be the universe of discourse. A denotes an arbitrary nonempty subset of X, i.e., A ⊆ X and A≠ ∅. Then A can be expressed in the form of IFS as: A ˜ = { 〈 x i , μ A ˜ ( x i ) , v A ˜ ( x i ) 〉 }, where μ A ˜ ( x i ) = 1 for x _i  ∈ A and μ A ˜ ( x i ) = 0 for x _i  ∉ A, i = 1, 2, ⋯ , n; v A ˜ ( x i ) = 1 - μ A ˜ ( x i ), and π A ˜ ( x i ) = 0, ∀x _i  ∈ X.

For example, in the universe X = {x ₁, x ₂, x ₃, x ₄}, the subset A = {x ₁, x ₂}, B = {x ₁, x ₂, x ₃} can, respectively, be transformed to IFSs as: A ˜ = { 〈 x 1 , 1 , 0 〉 , 〈 x 2 , 1 , 0 〉 , 〈 x 3 , 0 , 1 〉 , 〈 x 4 , 0 , 1 〉 } , B ˜ = { 〈 x 1 , 1 , 0 〉 , 〈 x 2 , 1 , 0 〉 , 〈 x 3 , 1 , 0 〉 , 〈 x 4 , 0 , 1 〉 } .

In such sense, IFSs can be considered as a generalized form of fuzzy sets and crisp sets. For clarity, in the sequel, we do not intend to differentiate the denotation of crisp sets, fuzzy sets, and IFSs.

As discussed earlier, most of the existing uncertainty measures are defined based on entropy. Burillo and Bustince [16] first presented the definition of entropy for IFSs as following.

Definition 5. [16] Let IFSs (X) denote the set of all IFSs in X. A mapping E _I  : IFSs (X) → [0, 1] is called a (normalized) entropy if it satisfies the followingproperties:

(E _I 1) E _I  (A) =0 if and only if A is a fuzzy set;

(E _I 2) E _I  (A) =1 if and only if μ _A  (x) = v _A  (x) =0 for every x ∈ X;

(E _I 3) E _I  (A) = E _I  (A ^C );

(E _I 4) E _I  (A) ≥ E _I  (B) if μ _A  (x) ≤ μ _B  (x) and v _A  (x) ≤ v _B  (x) for every x ∈ X.

We can note that this entropy measure is increasing with the hesitancy degree. This definition basically considers the entropy as a measure of how far the IFS is from being a fuzzy set. Therefore, the uncertainty degree of fuzzy sets cannot be captured by this definition.

By describing the difference between fuzzy sets and crisp sets, De Luca and Termini [10] has defined an entropy measure E LT FS for fuzzy sets as E LT FS ( A ) = - 1 n ∑ i = 1 n [ μ A ( x i ) log 2 μ A ( x i ) + ( 1 - μ A ( x i ) ) log 2 ( 1 - μ A ( x i ) ) ](4)

Then Szmidt and Kacprzyk [17] generalize this definition to measure the entropy of IFSs as following.

Definition 6. [17] A mapping E _F  : IFSs (X) → [0, 1]is said to be an entropy if it satisfies the following axioms:

(E _F 1) E _F  (A) =0 if and only if A is a crisp set;

(E _F 2) E _F  (A) =1 if and only if μ _A  (x _i ) = v _A  (x _i ) for every x _i  ∈ X;

(E _F 3) E _F  (A) = E _F  (A ^C );

(E _F 4) E _F  (A) ≤ E _F  (B) if μ _A  (x _i ) ≤ μ _B  (x _i ) and v _A  (x _i ) ≥ v _B  (x _i ) for μ _B  (x _i ) ≤ v _B  (x _i ) ∀x _i  ∈ X or μ _A  (x _i ) ≥ μ _B  (x _i ) and v _A  (x _i ) ≤ v _B  (x _i ) for μ _B  (x _i ) ≥ v _B  (x _i ) ∀x _i  ∈ X.

We can find that this definition only focus on fuzziness. It ignores the entropy changes caused by hesitancy degree while μ _A  (x) = v _A  (x) for all x ∈ X.

These two definitions provide us with two different ways of measuring uncertainty for IFSs. In fact, these two kinds of entropy exist in an IFS simultaneously. On the one hand, an IFS is a fuzzy set and hence is associated with fuzziness. A measure of such fuzziness would be related to the departure of the IFS from its closest crisp set. Such uncertainty may be called the fuzziness associated with an IFS. On the other hand, in an IFS, there exists another kind of uncertainty that is related to lack of knowledge. This aspect of uncertainty is related to how much we do not know about membership and non-membership, i.e., related to hesitancy 1 - μ _A  (x) - v _A  (x). The uncertainty of this kind may be called intuitionism type uncertainty. Here the E _I  (A) in Definition 5 models the intuitionism aspect of uncertainty. E _F  (A) in Definition 6 is defined to model the fuzzy aspect of uncertainty. Thus although, both of these measures are interesting measures of uncertainty, neither of them captures both facets of uncertainty.

To illustrate these arguments, let us consider the following examples.

Example 1. Let X be the universe, A and B be two IFSs in X. For some fixed x ∈ X, suppose μ _A  (x) =0.2, v _A  (x) =0.5; μ _B  (x) =0.6, v _B  (x) =0.3. Thus, π _A  (x) =0.3 whereas π _B  (x) =0.1. E _I  (A) is a model of entropy in terms of intuitionism, so, in principle E _I  (A) ≥ E _I  (B). But clearly the conditions μ _A  (x) ≤ μ _B  (x) and v _A  (x) ≤ v _B  (x) do not hold at least for one x ∈ X.

Example 2. Let A = {< x ₁, 0.5, 0.5 > , < x ₂, 0.3, 0.3 >} and B = {< x ₁, 0.2, 0.2 > , < x ₂, 0.1, 0.1 >} be two IFSs on X = {x ₁,  x ₂}. Then we have that E _F  (A) = E _F  (B) =1. But clearly in terms of lack of knowledge (hesitancy, intuitionism), these two sets differ significantly. So as an intrinsic property of IFSs, the intuitionism-type uncertainty, which is very special, is not reflected in this formulation.

In order to construct a comprehensive uncertainty measure to capture all the uncertain information of IFSs, Mao et al. [23] refined the axiomatic criterions of the intuitionistic fuzzy entropy measure and propose a general model as follows:

Definition 7. The intuitionistic fuzzy entropy measure of an IFS A is a real-valued function E _IF  (A) with π _A and Δ _A  = |μ _A  - v _A |, i.e., E _IF  (A) = g (π _A , Δ _A ) : IFS (X) → [0, 1], satisfying the following axiomatic principles:

(E _IF 1) E _IF  (A) =0 if and only if A is a crisp set;

(E _IF 2) E _IF  (A) =1 if and only if μ _A  (x _i ) = v _A  (x _i ) =0 for every x _i  ∈ X;

(E _IF 3) E _IF  (A) = E _IF  (A ^C );

(E _IF 4) g (π _A , Δ _A ) is a real-valued continuous function being increasing with respect to the first variable π _A , and decreasing with second variable Δ _A  = |μ _A  - v _A |. That is to say, the entropy value will increase along with the enhanced fuzziness under the same intuitionism, and it will decrease along with the weakened intuitionism under the same fuzziness.

Pal et al. [22] have also reformulated the axioms for measures of uncertainty associated with IFSs in a manner such that it can capture both types of uncertainties present in IFSs. They applied the couple (E _I , E _F ) to quantify the uncertainty hidden in IFSs. Definitions of E _I and E _F are in analogy with Definition 5 and Definition 6, respectively. Moreover, we can obtain the definition of E _IF by taking both E _I and E _F into account. So we can conclude that the uncertainty degree of IFSs can be well measured by the combination of fuzziness and intuitionism.

It is obvious that intuitionistic fuzzy entropy measure reduces into a fuzziness measure when an IFS reduces to an ordinary fuzzy set. So an intuitionistic fuzzy entropy satisfying Definition 7 can be also used to quantify fuzziness of fuzzy sets. Since the crisp sets can be regarded as a special type of IFSs, the uncertainty measure for IFSs should be capable of discriminating the uncertainty degree of crisp sets. However, neither fuzziness nor intuitionism nor the combination of them makes sense for crisp sets. The reason is that the fuzziness and intuitionism of a crisp set are both zero, but it contains another kind of uncertainty, namely non-specificity, which is related to its cardinality. As for an IFS, the non-specificity can be regarded as the inverse concept of the degree to which it focuses on one and only one element as its member. Obviously, the non-specificity-based uncertainty is out of definitions of fuzziness and intuitionism.

This can be demonstrated by the next example.

Example 3. Let X = {x ₁,  x ₂,  x ₃} be the universe. Three subsets of X A = {x ₁}, B = {x ₁,  x ₂} and C = {x ₁, x ₂, x ₃} can, respectively, be expressed by IFSs as: A = { < x 1 , 1 , 0 > , < x 2 , 0 , 1 > , < x 3 , 0 , 1 > } , B = { < x 1 , 1 , 0 > , < x 2 , 1 , 0 > , < x 3 , 0 , 1 > } , C = { < x 1 , 1 , 0 > , < x 2 , 1 , 0 > , < x 3 , 1 , 0 > } .

Given Definition 7, we have: E I ( A ) = E I ( B ) = E I ( C ) = 0 , E F ( A ) = E F ( B ) = E F ( C ) = 0 , which mean these three subsets contain equivalent uncertainty information. But, apparently, the uncertainty degree of full set C is largest, while A is a crisp set taking its uncertainty degree as 0.

We can find that for a crisp set, its fuzziness and intuitionism are both taking the value of 0. So the intuitionistic fuzzy entropy measure E _IF cannot discriminate uncertainty degrees of crisp sets with different elements. Hence, another kind of uncertainty, non-specificity, must be considered. By combining non-specificity measure with intuitionistic fuzzy entropy, we can construct a unified uncertainty measure that is applicable for both IFSs and crisp sets. Since there already exist many intuitionistic fuzzy entropy measures for IFSs, and they are beyond the scope of this paper, in the following, we merely focus on the most desirable non-specificity measure of IFSs.

Let X = {x ₁, x ₂, ⋯ , x _n } be the universe of discourse. Since a singleton element crisp set contains only one element, it is implicitly specific. Its non-specificity degree should be zero. Intuitively, the full set X = {x ₁, x ₂, ⋯ , x _n } representing full ignorance implies maximum non-specificity. Since the fuzziness and intuition of crisp sets are both 0, the only measurable kind of uncertainty in the crisp set is the non-specificity, directly referring to its cardinality. Regarding the well-known Hartley measure [24] shown in next definition as the standard measure of non-specificity of crisp sets, we can determine the range of non-specificity as [0, log ₂ n].

Definition 8. [24] Let X = {x ₁, x ₂, ⋯ , x _n } be the universe of discourse, and A be any subset of X. Then, the Hartley measure is defined as: H ( A ) = log 2 ( | A | )(5) where | · | denotes the cardinality.

Yager has investigated the specificity of IFSs based on the relationship between the membership degrees of different elements [19]. Since the non-specificity is the inverse concept of specificity, we can define it in the same way. Motivated by Yager’s work [19], we can provide the following procedure for calculating the non-specificity of IFS A on X = {x ₁, x ₂, ⋯ , x _n }.

Determine the largest membership value,denoted by α = max [μ _A  (x ₁) , μ _A  (x ₂) , …, μ _A  (x _n )]. We can assume this occurs at x ^*, thus α = μ _A  (x ^*);

Considering μ _A  (x ^*) ≤1 - v _A  (x ^*), we have M _A  (x ^*) - α = 0, thus M _A  (x ^*) = min [α, 1 - v _A  (x ^*)] = α. So Equation (6) is equivalent to the followingformula: N ( A ) = log 2 ( n + ∑ x ( M A ( x ) - α ) )(7)

In the following example, we will illustrate thisalgorithm.

Example 4. Assume X = {x ₁, x ₂, x ₃, x ₄, x ₅} be the universe of discourse. Let A be an IFS on X given as: A = { 〈 x 1 , 0.3 , 0.6 〉 , 〈 x 2 , 0.6 , 0 〉 , 〈 x 3 , 0.4 , 0.6 〉 , 〈 x 4 , 0.2 , 0.7 〉 , 〈 x 5 , 0 , 0.5 〉 }

In this case, we have:

α = max [μ (x ₁) , μ (x ₂) , μ (x ₃) , μ (x ₄) , μ (x ₅)] =0.6 and it occurs at x ₂. Then we can get: M A ( x 1 ) = min [ α , 1 - v ( x 1 ) ] = min [ 0.6 , 0.4 ] = 0.4 ; M A ( x 3 ) = min [ α , 1 - v ( x 3 ) ] = min [ 0.6 , 0.4 ] = 0.4 ; M A ( x 4 ) = min [ α , 1 - v ( x 4 ) ] = min [ 0.6 , 0.3 ] = 0.3 ; M A ( x 5 ) = min [ α , 1 - v ( x 5 ) ] = min [ 0.6 , 0.5 ] = 0.5 .

The non-specificity of A can be obtained as: N ( A ) = log 2 ( 5 + ( 0.4 - 0.6 ) + ( 0.4 - 0.6 ) + ( 0.3 - 0.6 ) + ( 0.5 - 0.6 ) ) = 2.07

Let us now look at some properties of the proposed non-specificity measure.

Theorem 1. For an arbitrary IFS A in the universe X = {x ₁, x ₂, ⋯ , x _n }, its non-specificity satisfies N (A) ∈ [0, log ₂ n].

Proof. Since M _A  (x) ≤ α, we have n + ∑_x≠x^* (M _A  (x) - α) ≤ n. Considering 0 ≤ M _A  (x) and α ≤ 1, we get: n + ∑ x ≠ x * ( M A ( x ) - α ) = n + ∑ x ≠ x * M A ( x ) - ( n - 1 ) α ≥ n - ( n - 1 ) α ≥ n - ( n - 1 ) = 1

Hence, 1 ≤ n + ∑_x≠x^* (M _A  (x) - α) ≤ n.

Consequently, we obtain N (A) ∈ [0, log ₂ n]. □

Theorem 2. If A is a crisp subset of X = {x ₁, x ₂, ⋯ , x _n }, its non-specificity measure N (A) coincides with its Hartley Measure H (A).

Proof. Without any loss of generality, we can suppose that A ={ x ₁, x ₂, ⋯ , x _m  } with 1 ≤ m ≤ n. A can be rewritten as: A = { 〈 x 1 , 1 , 0 〉 , … , 〈 x m , 1 , 0 〉 , 〈 x m + 1 , 0 , 1 〉 , ⋯ , 〈 x n , 0 , 1 〉 }

So we have α = 1 for all x _i  ∈ A. Since no generality will lose by taking x ^* = x ₁, we have: M A ( x 2 ) = ⋯ = M A ( x m ) = 1 , M A ( x m + 1 ) = ⋯ = M A ( x n ) = 0 .

It follows that: n + ∑ x ≠ x * ( M A ( x ) - α ) = n + ∑ x ≠ x * M A ( x ) - ( n - 1 ) α = n + ( m - 1 ) - ( n - 1 ) = m .

Hence, N ( A ) = log 2 ( n + ∑ x ≠ x * ( M A ( x ) - α ) ) = log 2 m = log 2 ( | A | ) = H ( A )

This indicates that its non-specificity N (A) coincides with its Hartley Measure H (A). □

Theorem 3. If A is an ordinary fuzzy set in X = {x ₁, x ₂, ⋯ , x _n }, we have: N ( A ) = log 2 ( n + ∑ x ≠ x * ( μ A ( x ) - α ) ) .

Proof. Without loss of generality, we can take α = μ _A  (x ₁), i.e., x ^* = x ₁. Since A is an ordinary fuzzy set, we have μ _A  (x) + v _A  (x) =1 for every x ∈ X. α = μ (x ₁) = max [μ _A  (x ₁) , μ _A  (x ₂) , ⋯ , μ _A  (x _n )] indicates that μ _A  (x ₁) ≥ μ _A  (x) =1 - v _A  (x) ∀x ∈ X. And then, M A ( x ) = min [ 1 - v A ( x ) , α ] = min [ 1 - v A ( x ) , μ A ( x 1 ) ] = 1 - v A ( x ) = μ A ( x ) Finally we get: N ( A ) = log 2 ( n + ∑ x ≠ x * ( M A ( x ) - α ) ) = log 2 ( n + ∑ x ≠ x * ( μ A ( x ) - α ) )

Thus, N (A) = log ₂ (n + ∑_x≠x^* (μ _A  (x) - α)). □

Theorem 4. If there exists at least one x ∈ X such that μ (x) =1, then N (A) = log ₂ (n - ∑_x≠x^* v _A  (x)).

Proof. Given μ (x) =1, we get α = 1 and M (x) = min [α, 1 - v _A  (x)] =1 - v _A  (x).

Consequently, N ( A ) = log 2 ( n + ∑ x ≠ x * ( M A ( x ) - α ) )

= log 2 ( n + ∑ x ≠ x * ( 1 - v A ( x ) - 1 ) ) = log 2 ( n - ∑ x ≠ x * v A ( x ) )

So we get N (A) = log ₂ (n - ∑_x≠x^* v _A  (x)). □

Theorem 5. N (A) = log ₂ n for an IFS A satisfying: α = μ _A  (x ^*) = max [μ _A  (x ₁) , μ _A  (x ₂) , ⋯ , μ _A  (x _n )] an d v (x)  _A  ≤ v _A  (x ^*).

Proof. Since μ _A  (x ^*) + v _A  (x ^*) ≤1, then μ _A  (x ^*) ≤1 - v _A  (x ^*). Based on v _A  (x) ≤ v _A  (x ^*), we can also get 1 - v _A  (x) ≥1 - v _A  (x ^*) ≥ μ _A  (x ^*) = α. So we have M _A  (x) = min [α, 1 - v _A  (x)] = α.

Hence, N ( A ) = log 2 ( n + ∑ x ≠ x * ( M A ( x ) - α ) ) = log 2 ( n + ∑ x ≠ x * ( α - α ) ) = log 2 n

So we have N (A) = log ₂ n. □

Theorem 5 provides a situation where an IFS contains the greatest non-specificity. We can note that the full set X satisfies this condition, so it implies the maximum non-specificity.

Next, we will give the property of function M for a further investigation on the non-specificity measure.

Theorem 6. [19] For each x ≠ x ^*, there exist a value λ _x  ∈ [0, 1] such that M _A  (x) = μ _A  (x) + λ _x π _A  (x).

Proof. By the definition of M _A  (x) we should consider two cases. One case happens when α ≥ 1 - v _A  (x). Then we have:

M _A  (x) =1 - v _A  (x) = μ _A  (x) + π _A  (x), i.e.,

M _A  (x) = μ _A  (x) + λ _x π _A  (x) for λ _x  = 1. Considering another case where α ≤ 1 - v _A  (x), we have μ _A  (x) ≤ M _A  (x) = α ≤ 1 - v _A  (x). It is obvious that there exists a value λ _x  ∈ [0, 1] such that, M A ( x ) = μ A ( x ) + λ x ( 1 - v A ( x ) - μ A ( x ) ) = μ A ( x ) + λ x π A ( x )

So, for each x ≠ x ^*, there exist a value λ _x  ∈ [0, 1] such that M _A  (x) = μ _A  (x) + λ _x π _A  (x). □

In the light of this result, we can view our formulation for the non-specificity of an IFS A as the non-specificity of a particular ordinary fuzzy subset compatible with A.

Let A be an IFS in X = {x ₁, x ₂, ⋯ , x _n }, and A ˜ be an ordinary fuzzy set defined as: μ A ˜ ( x * ) = min [ μ A ( x * ) , 1 - v A ( x ) ] and

v A ˜ ( x ) = 1 - μ A ˜ ( x ), for all x ∈ X.

By Theorem 3 we get the non-specificity of A ˜ as: N ( A ˜ ) = log 2 ( n + ∑ x ≠ x * ( μ A ˜ ( x ) - α ) ) = log 2 ( n + ∑ x ≠ x * ( min [ μ A ( x * ) , 1 - v A ( x ) ] - α ) )

So we have N ( A ) = N ( A ˜ ). Thus the non-specificity of the IFS A is the equal to that of A ˜, a particular ordinary fuzzy subset compatible with the IFS A. Then a natural question arises, why do we use this special compatible ordinary fuzzy set, instead of using any arbitrary ordinary fuzzy set F associated with A satisfying: μ _F  (x) = μ _A  (x) + λ _x π _A  (x) with λ _x  ∈ [0, 1]. We will borrow a theorem in [19] and modify it to show the uniqueness of our choice.

Theorem 7. Let A ={ 〈 x, μ _A  (x) , v _A  (x) 〉 |x ∈ X } be an IFS on X = {x ₁, x ₂, ⋯ , x _n } and α = max [μ _A  (x)] = μ _A  (x ^*). A ˜ denotes a n ordinary fuzzy set associated with A such that: μ A ˜ ( x * ) = min [ μ A ( x * ) , 1 - v A ( x ) ] and v A ˜ ( x ) = 1 - μ A ˜ ( x ), for all x ∈ X. F be an arbitrary ordinary fuzzy set associated with A satisfying: μ _F  (x) = μ _A  (x) + λ _x π _A  (x) with λ _x  ∈ [0, 1]. Then the non-specificity of A ˜ is not smaller than that of F, i.e., N ( F ) ≤ N ( A ˜ ).

Proof. In this case we have: N ( A ˜ ) = log 2 ( n + ∑ x ≠ x * ( μ A ˜ ( x ) - α ) ) = log 2 ( n + ∑ x ≠ x * ( min [ α , 1 - v A ( x ) ] - α ) )

For convenience, we let β = max [μ _F  (x)] = μ _F  (x ^**).

Since μ _F  (x ^*) = μ _A  (x ^*) + λ _{x ^*} π _A  (x ^*) ≥ α, we have β = max [μ _F  (x)] ≥ α.

The non-specificity grades of F and A ˜ can be respectively written as: N ( F ) = log 2 ( n + ∑ x ≠ x * * ( μ F ( x ) - β ) ) = log 2 ( n - ( n - 1 ) β + μ F ( x * ) + ∑ x ≠ x * * , x * μ F ( x ) ) = log 2 ( n + ( μ F ( x * ) - β ) + ∑ x ≠ x * * , x * ( μ F ( x ) - β ) ) N ( A ˜ ) = log 2 ( n + ∑ x ≠ x * ( μ A ˜ ( x ) - α ) ) = log 2 ( n + ∑ x ≠ x * ( min [ α , 1 - v A ( x ) ] - α ) ) = log 2 ( n - ( n - 1 ) α + ∑ x ≠ x * ( min [ α , 1 - v A ( x ) ] ) ) = log 2 ( n - ( n - 1 ) α + min [ α , 1 - v A ( x * * ) ] + ∑ x ≠ x * , x * * ( min [ α , 1 - v A ( x ) ] ) ) = log 2 ( n + ( min [ α , 1 - v A ( x * * ) ] - α ) + ∑ x ≠ x * , x * * ( min [ α , 1 - v A ( x ) ] - α ) )

When x ≠ x ^*, x ^**, two cases should be considered:

(1) α ≤ 1 - v _A  (x), then min [α, 1 - v _A  (x)] - α = 0, thus μ _F  (x) - β ≤ 0 ≤ min [α, 1 - v _A  (x)] - α.

(2) α ≥ 1 - v _A  (x), then

min [α, 1 - v _A  (x)] - α = 1 - v _A  (x) - α. Moreover, considering the following inequalities: μ F ( x ) = μ A ( x ) + λ x π A ( x ) = μ A ( x ) + λ x ( 1 - μ A ( x ) - v A ( x ) ) = ( 1 - λ x ) μ A ( x ) + λ x ( 1 - v A ( x ) ) ≤ 1 - v A ( x )

we get μ _F  (x) - β ≤ 0 ≤1 - v _A  (x) - β ≤ 1 - v _A  (x) - α.

Thus for any x ≠ x ^*, x ^**, we have μ _F  (x) - β ≤ 0 ≤ min [α, 1 - v _A  (x)] - α.

Given 1 - v _A  (x ^**) ≥ μ _A  (x ^**) = β ≥ α, we have min [α, 1 - v _A  (x ^**)] - α = α - α = 0. Comparing with μ _F  (x ^*) - β ≤ 0, we obtain: μ F ( x * ) - β ≤ min [ α , 1 - v A ( x * * ) ] - α .

Considering the following inequalities: μ F ( x * ) - β ≤ min [ α , 1 - v A ( x * * ) ] - α , μ F ( x ) - β ≤ 0 ≤ min [ α , 1 - v A ( x ) ] - α ,

we have the result that N ( F ) ≤ N ( A ˜ ). □

Thus we can see that our definition for the non-specificity of an IFS is equal to the largest non-specificity of any ordinary fuzzy subsets that are consistent with the IFS. That is to say, the non-specificity of an IFS is greater than any ordinary fuzzy sets associated with it, which coincides with our intuitive cognition.

Then another question arousing our interest is finding the associated ordinary fuzzy subset F with the smallest non-specificity. Inspired by the definition of vortex subset given by Yager [19], we can conclude that the ordinary fuzzy set with the smallest non-specificity must be a vortex. We first introduce the concept of a vortex subset.

Definition 9. [19] Let A ={ 〈 x, μ _A  (x) , v _A  (x) 〉 |x ∈ X } be an IFS on X = {x ₁, x ₂, ⋯ , x _n }. F denotes an ordinary fuzzy set associated with A. F is called a vortex subset of A if F satisfies μ _F  (x ^*) =1 - v _A  (x ^*) and μ _F  (x) = μ _A  (x) for all x ≠ x ^*.

From this definition we can note that there exist n vortex subsets associated with A. We can denote F _j as the vortex subset in which x ^* = x _j . For the vortex subset focused on x _j we have:

μ _F  (x _j ) = μ _A  (x _j ) + λ _x π _A  (x _j ) with λ _x  = 1;

μ _F  (x) = μ _A  (x) + λ _x π _A  (x) for x ≠ x _j with λ _x  = 0.

Theorem 8. Assume A ={ 〈 x, μ _A  (x) , v _A  (x) 〉 |x ∈ X } is an IFS on X = {x ₁, x ₂, ⋯ , x _n }. If F is any consistent ordinary fuzzy subset associated with A then there exists some vortex set F _k such that N (F _k ) ≤ N (F).

Proof. Let μ _F  (x _k ) = max [μ _F  (x)], then we have: N (F) = log ₂ (n + ∑_{x≠x k}  (μ _F  (x) - μ _F  (x _k ))).

Let F _k be the vortex subset of A focused on x _k , then we have:

μ _{F k}  (x _k ) =1 - v _A  (x _k ), μ _{F k}  (x _j ) = μ _A  (x _j ) for x _j  ≠ x _k .

μ _F  (x _k ) = μ _A  (x _k ) + λ _{x k} π _A  (x _k ) ≥ μ _F  (x) = μ _A  (x) + λ _x π _A  (x) also indicates that: μ F k ( x k ) = μ A ( x k ) + π A ( x k ) ≥ μ A ( x ) = μ F k ( x ) .

So the non-specificity of F _k is: N ( F k ) = log 2 ( n + ∑ x ≠ x k ( μ F k ( x ) - μ F k ( x k ) ) ) .

For x ≠ x _k , regarding μ _F  (x) = μ _A  (x) + λ _x π _A  (x) with λ _x  ∈ [0, 1] and μ _{F k}  (x) = μ _A  (x) +0 · π _A  (x), we have μ _{F k}  (x) ≤ μ _F  (x).

Moreover, by μ _F  (x _k ) = μ _A  (x _k ) ≤1 - v _A  (x _k ) = μ _{F k}  (x _k ), we can get μ _{F k}  (x) - μ _{F k}  (x _k ) ≤ μ _F  (x) - μ _F  (x _k ). So we can finally get N (F _k ) ≤ N (F). □

This theorem says that for any ordinary fuzzy set F associated with A, we can find a vortex subset F _k such that N (F _k ) ≤ N (F). So the ordinary fuzzy subset with the smallest non-specificity must be a vortex set.

More performance of the non-specificity uncertainty will be illustrated by following examples.

Example 5. Let X = {x ₁, x ₂, x ₃, x ₄, x ₅} be the universe of discourse. Five crisp subsets of X are given as: A 1 = { x 1 } , A 2 = { x 1 , x 2 } , A 3 = { x 1 , x 2 , x 3 } , A 4 = { x 1 , x 2 , x 3 , x 4 } , A 5 = { x 1 , x 2 , x 3 , x 4 , x 5 } .

By Definition 4, these crisp sets can be written in forms of IFSs as: A 1 = { 〈 x 1 , 1 , 0 〉 , 〈 x 2 , 0 , 1 〉 , 〈 x 3 , 0 , 1 〉 , 〈 x 4 , 0 , 1 〉 , 〈 x 5 , 0 , 1 〉 } , A 2 = { 〈 x 1 , 1 , 0 〉 , 〈 x 2 , 1 , 0 〉 , 〈 x 3 , 0 , 1 〉 , 〈 x 4 , 0 , 1 〉 , 〈 x 5 , 0 , 1 〉 } , A 3 = { 〈 x 1 , 1 , 0 〉 , 〈 x 2 , 1 , 0 〉 , 〈 x 3 , 1 , 0 〉 , 〈 x 4 , 0 , 1 〉 , 〈 x 5 , 0 , 1 〉 } , A 4 = { 〈 x 1 , 1 , 0 〉 , 〈 x 2 , 1 , 0 〉 , 〈 x 3 , 1 , 0 〉 , 〈 x 4 , 1 , 0 〉 , 〈 x 5 , 0 , 1 〉 } , A 5 = { 〈 x 1 , 1 , 0 〉 , 〈 x 2 , 1 , 0 〉 , 〈 x 3 , 1 , 0 〉 , 〈 x 4 , 1 , 0 〉 , 〈 x 5 , 1 , 0 〉 } .

Table 1 gives all kinds of uncertainty for these five sets. We can note that all of the fuzziness, intuitionism, and intuitionistic fuzzy entropy remain unchanged for different sets, thus they cannot discriminate the uncertainty hidden in different sets. From A ₁ to A ₅, the non-specificity degree increases. Moreover, the increasing non-specificity coincides with the positive correlation between the cardinality of a crisp sets and its uncertainty degree. So we can conclude that the intuitionistic fuzzy uncertainty measure generalized by combining intuitionistic fuzzy entropy and non-specificity can be applied to describe the uncertainty of crisp sets.

Example 6. Let X = {x ₁, x ₂, x ₃, x ₄, x ₅} be the universe of discourse. Two IFSs in X are given as: A 1 = { 〈 x 1 , 0.3 , 0.6 〉 , 〈 x 2 , 0.6 , 0 〉 , 〈 x 3 , 0.4 , 0.6 〉 , 〈 x 4 , 0.2 , 0.7 〉 , 〈 x 5 , 0 , 0.5 〉 } , A 2 = A 1 C = { 〈 x 1 , 0.6 , 0.3 〉 , 〈 x 2 , 0 , 0.6 〉 , 〈 x 3 , 0.6 , 0.4 〉 , 〈 x 4 , 0.7 , 0.2 〉 , 〈 x 5 , 0.5 , 0 〉 } .

By Equation (6) we can get: N ( A 1 ) = 2.07 , N ( A 2 ) = 2.20 .

Generally, the information conveyed by an IFS is different from that of its complement set. From this example we can note that the non-specificity measure can reflect this difference. The third property inDefinition 5, Definition 6 and Definition 7 shows that both of fuzziness measure and intuitionism measure miss the capability of discriminating an IFS from its complement. Equation (6) shows that in most cases the non-specificity grades of an IFS and its complement are distinct. In this example, for the IFS A ₁, the membership grade of x ₂ is maximum, and it is quite different from other membership grades. For the IFS A ₂, although its greatest membership grade is 0.7, occurring in x ₄, there exist two identical membership grades μ _{A 2}  (x ₁) = μ _{A 2}  (x ₃) =0.6, which is close to the maxim value 0.7. So the non-specificity of A ₂ is greater than that of A ₁, and A ₂ should be more uncertain than A ₁. Thus, adding non-specificity to intuitionistic fuzzy entropy, we can construct an intuitionistic fuzzy uncertainty measure with the ability of discriminating uncertainty degrees of an IFS and its complement.

Example 7. Let X = {x ₁,  x ₂} be the universe of discourse. A typical IFS in X is defined as: A ={ 〈 x ₁, a, b 〉 , 〈 x ₂, b, a 〉 }.

By the equivalent definition of non-specificity measure in Equation (7), we can get: N ( A ) = log 2 ( 2 + min ( 1 - b , max ( a , b ) ) + min ( 1 - a , max ( a , b ) ) - 2 · max ( a , b ) ) .

Since a + b ≤ 1, we have a ≤ 1 - Cb and b ≤ 1 - a.

Then the following cases should be considered:

(1) b ≤ a ≤ 1 - b, then b ≤ 0.5, and we have:

(i) 1 - a ≥ a, i.e., a ≤ 0.5, we can get N (A) = log ₂ (2 + a + a - 2 · a) = 1;

(ii) 1 - a ≤ a, i.e., a ≥ 0.5, then we obtain that N (A) = log ₂ (2 + a + (1 - a) -2 · a) = log ₂ (3 - 2 · a).

(2) a ≤ b ≤ 1 - a, then a ≤ 0.5, and we have:

(iii) 1 - b ≥ b, i.e., b ≤ 0.5, we get N (A) = log ₂ (2 + b + b - 2 · b) = 1.

(iv) 1 - b ≤ b, i.e., b ≥ 0.5, we have N (A) = log ₂ (2 + b + (1 - b) -2 · b) = log ₂ (3 - 2 · b).

Taking all cases into account, we can get: N ( A ) = { log 2 ( 3 - 2 b ) , a < 0.5 , b > 0.5 log 2 ( 3 - 2 a ) , a > 0.5 , b < 0.5 a ≤ 0.5 , b ≤ 0.5 .

For a more expressive manifestation on the non-specificity of A, we present Fig. 1 to show the value of N (A), which is denoted by the color of each point (a, b) on the simplex.

6 Conclusions

We have presented a non-specificity measure associated with intuitionistic fuzzy sets. Firstly, we critically reviewed existing axiomatic definitions of entropy measures and pointed out that they cannot capture all kinds of uncertainty implying in intuitionistic fuzzy sets. Then we argued that an intuitionistic fuzzy set has another kind of uncertainty, namely non-specificity. The non-specificity is determined by the inverse of the degree to which an intuitionistic fuzzy set points to one and only one element as its member. We defined a non-specificity measure based on the difference between the maximum membership grade and other membership grades. Its other properties are presented with their proofs. Performances of the proposed non-specificity uncertainty measure are demonstrated by examples. Moreover, the non-specificity measure can be used to constructing a unified uncertainty measure for IFSs. The unified uncertainty measure can not only quantify uncertainty degree of crisp sets, but also discriminate an IFS from its complement.

However, we must mention that the forms of non-specificity measure are not unique. There may be many other potential measures that can quantify non-specificity. Moreover, if there is only one element in the universe, our proposed non-specificity measure is undefined. When it is applied to determine weights of attributes in the problem of multi-attribute decision-making, the non-specificity measure lose its applicability. So the follow-up research is dedicated to looking for more effective non-specificity measures with applications.