Distance-Based Knowledge Measure for Intuitionistic Fuzzy Sets with Its Application in Decision Making

Atanassov [1,2] developed the concept of intuitionistic fuzzy set on the basis of Zadeh’s fuzzy set [3]. Atanassov’s intuitionistic fuzzy sets (AIFSs) relax the condition that the non-membership degree and the membership degree sum to 1. AIFSs are a generalization of fuzzy sets, i.e., a particular case of other types of generalized fuzzy sets [4,5]. Moreover, AIFSs are identical to interval-valued fuzzy sets (IVFSs) from a mathematical perspective [6]. In an AIFS, the hesitation degree is the difference between one and the sum of membership and non-membership grades. The hesitation degree contributes much serviceability to the depiction of uncertain information. Researchers have paid much attention on the intuitionistic fuzzy set theory since its advantage in modeling uncertain information systems [7]. The theory of intuitionistic fuzzy sets has been successfully applied in many fields, including uncertainty reasoning [8] and decision making [9,10]. The connection between AIFSs and other uncertain theories is also attracting increasingly much interest [11,12,13,14,15,16,17,18].

Zadeh [3] first introduced the notion of entropy to fuzzy sets to measure the uncertainty or fuzziness in a fuzzy set. The notion of fuzzy entropy defined for fuzzy sets is partially similar to the concept of Shannon entropy [19], which was initially defined in probability theory. Luca and Termini [20] developed the axiomatic definition of entropy, and then proposed a kind of non-probabilistic fuzzy entropy. Then, Burillo and Bustince [21] first axiomatically defined the measure of intuitionistic entropy, which was merely determined by hesitation degree. Unlike the entropy measures created by Burillo and Bustince [21], the entropy measure for intuitionistic fuzzy sets developed by Szmidt and Kacprzyk [22] was defined based on the ratio of two distance values. Axiomatic definition for intuitionistic fuzzy entropy was also presented by Szmidt and Kacprzyk [22]. Following the work of Szmidt and Kacprzyk [22], many authors [23,24,25,26] have done a great deal of work concentrating on the definition of entropy measures. Some research has also focused on the entropy of AIFSs and their application in the evaluation of attribution weighting vector [9,10]. It has been pointed out by Szmidt et al. [27] that entropy measure cannot capture all uncertainty hidden in an AIFS. Thus, it may be difficult to develop a satisfactory uncertainty measure for AIFSs merely by entropy measure. The difference between entropy and hesitation in measuring the uncertainty of AIFSs has been pointed out by Pal et al. [28]. In [28], it was claimed that the combination of entropy and hesitation may furnish an effective way to measure the total uncertainty hidden in an AIFS.

Generally, knowledge measure is related to the useful information provided by an AIFS. From the perspective of information theory, much information indicates a great amount of knowledge, which is helpful for decision making. Therefore, the notion of knowledge measure can be regarded as the complementary concept of total uncertainty measure, rather than of entropy measure. This means that less total uncertainty always accompanies a greater amount of knowledge. With the purpose of making an evident distinction between types of intuitionistic fuzzy information, Szmidt et al. [27] took both intuitionistic fuzzy entropy and hesitation into consideration to develop a knowledge measure for AIFS, in which the intuitionistic fuzzy entropy was defined by quantifying the ration between the nearer distance and farer distance. This knowledge measure has been used to estimate the weight of each attribute to solve multi-attribute decision making (MADM) problems [29]. Nguyen [30] has developed a novel knowledge measure by measuring the distance from an AIFS to the most uncertain AIFS. It seems that this knowledge measure can well describe fuzziness and intuitionism in AIFSs. However, the use of normalized Euclidean distance may bring another problem, namely that the relation between fuzziness and knowledge cannot be completely reflected. Recently, Guo [29] put forward a new axiomatic definition for the knowledge measure of AIFS. A new and highly robust model was introduced in [31] to quantify the knowledge amount of AIFS. By measuring the difference between an AIFS and its complement, the new model proposed by Guo [31] has been widely used to defined entropy measure for AIFSs [32,33]. Moreover, the combination of the two parts in Guo’s model [31] lacks a clear physical interpretation. Several years ago, Das et al. [34] performed a comprehensive review of axiomatic definitions of information measures of AIFSs and investigated their relationships, in which entropy measure, knowledge measure, distance measure, and similarity measure are all concerned.

The above analysis demonstrates that the topic of knowledge measure for AIFSs is still open for debate, and commanding prodigious attention. Most research on knowledge and uncertainty measures of AIFSs mainly focus on the difference between AIFS and its complement. Only a few knowledge measures are constructed by measuring the distinction between an AIFS and the AIFS with maximum uncertainty or minimum uncertainty. Although Nguyen [30] opened up this new way of studying knowledge measures of AIFSs, further exploration is needed to improve this kind of knowledge measure and realize a desirable knowledge measure for AIFSs. This motivates us to present a new method to measure the knowledge of AIFSs based on a novel intuitionistic fuzzy distance, which is defined based on the transformation from an intuitionistic fuzzy value (IFV) to an interval value. An axiomatic definition of the knowledge measure of AIFSs will also be formulated from a more general point of view. Moreover, we will further explore the proposed knowledge measure’s properties, and we compare it with other measures based on numerical examples to demonstrate its performance. Then we will apply it to the problem of intuitionistic fuzzy multi-attribute group decision making (MAGDM).

The remainder of this study is structured as follows. Several concepts regarding AIFSs are explained in Section 2. In Section 3, a new type of distance measure for AIFSs is developed, followed by the proposal and discussion of the distance-based knowledge measure in Section 4. In Section 5, the proposed distance and knowledge measures are used to develop a new method to solve MAGDM problems in intuitionistic fuzzy condition. Application of the new method for MAGDM is presented in Section 6 to illustrate the performance of the proposed method. Some conclusions of this paper are presented in Section 7.

Here, we briefly recount some background knowledge about AIFSs to for ease of subsequent exposition.

Apparently, we can obtain ${\pi }_B(x)\in [0,1]$, $\forall x\in X$. ${\pi }_B(x)$ is also referred to as the intuitionistic index of $x$ to $B$. Greater ${\pi }_B(x)$ indicates more vagueness. It is apparent that when ${\pi }_B(x)=0$, $\forall x\in X$, the AIFS degenerates into an ordinary fuzzy set.

For two AIFSs $A$ and $B$ defined in $X$, the following relations were defined in [1]: $A\supseteq B$ if and only if ${\mu }_A(x)\ge {\mu }_B(x)$, $v_A(x)\le v_B(x)$ for each $x\in X$. The complement of $B$ is denoted $B^C$ [1], and can be obtained by $B^C=\left\{〈x,v_B(x),{\mu }_B(x)〉\left|x\in X\right.\right\}$.

It has been proved that AIFSs and IVFSs are mathematically identical [4,6]. They can be converted to each other. Thus, For an AIFS $B$ defined in $X$ and $x\in X$, we can use an interval $\left[{\mu }_B(x),1-v_B(x)\right]$ to express the membership and non-membership grades of $x$ with respect to B. We can see this as the interval-valued interpretation of AIFS, in which ${\mu }_B(x)$ and $1-v_B(x)$ represent the lower bound and upper bound of membership degree, respectively. Apparently, $\left[{\mu }_B(x),1-v_B(x)\right]$ is a valid interval, since ${\mu }_B(x)\le 1-v_B(x)$ always holds for ${\mu }_B(x)+v_B(x)\le 1$. The correspondence relation between AIFSs and IVFSs holds only from the mathematical point of view. If we explore their conceptual explanation and practical application, they may differ in the description of uncertainty [9,35].

In what follows, $AIFSs(X)$ is used to denote the set consisted of all AIFSs defined in $X$. Generally, the couple $〈{\mu }_B(x),v_B(x)〉$ is also called an IFV for clarity.

For all IFVs, based on the partial ranking order, we can obtain the smallest IFV as $〈0,1〉$, denoted by 0, and the largest IFV as $〈1,0〉$, denoted by 1.

For a linear order of IFVs, to rank multiple IFVs, Chen and Tan [36] defined the score function of an IFV as $S(a)={\mu }_a-v_a$. Following the concept of score function for IFVs, Hong and Choi [37] developed an accuracy function $H(a)={\mu }_a+v_a$ to depict the accuracy of IFV $a=〈{\mu }_a,v_a〉$. Xu [38] then proposed a ranking-order relation between two IFVs $a$ and $b$, which can be equivalently shown as follows.

For their score functions, if $S(a)$ is greater than $S(b)$, then $a$ is greater than $b$, and vice versa.

If $S(a)$ and $S(b)$ are equal, we consider the following cases: (1) if $H(a)$ is equal to $H(b)$, then $a$ and $b$ are equal; and (2) if $H(a)$ is greater than $H(b)$, then $a$ is greater than $b$; and vice versa.

Based on above order relation, the linear order relation of multiple IFVs can be obtained.

We know that similarity measure and distance measure are important in the research of fuzzy set theory [39]. Similarly, the construction of similarity measure and distance measures for AIFSs plays an important role in AIFSs [23,40,41,42,43,44,45,46,47,48,49,50], and they are helpful for the comparison of intuitionistic fuzzy information [24,25].

Similarity measure and distance measure usually are regarded as a couple of dual concepts. Thus, distance measures can be used to define similarity measures, and vice versa.

In past years, numerous similarity measure and distance measure have been advanced [7,39,45]. However, some may lead to unreasonable results in practical applications [7]. Some new defined distance/similarity measures may have complicated expressions [39,45], which are not suitable for constructing knowledge measure for AIFSs. Thus, it is necessary to define a desirable distance measure to assist us in developing a new knowledge measure. Here, we propose a new distance measure for AIFSs by borrowing a distance measure for interval values. It has been claimed that an AIFS can be represented in the form of interval-valued fuzzy set [5]. Based on such relation, an intuitionistic fuzzy distance measure can be developed based on interval comparison.

Suppose that $A=\left\{〈x,{\mu }_A(x),v_A(x)〉\left|x\in X\right.\right\}$ is an AIFS defined in $X=\{x_1,x_2,\cdots ,x_n\}$, its knowledge measure K should intuitively satisfy some properties. It is rational that the knowledge measure K must be a non-negative function determined by ${\mu }_A(x)$ and $v_A(x)$. The knowledge amount of A should be identical to the knowledge amount of its complement, i.e., K(A) = K(A^C). When the AIFS A is reduced to classical Zadeh’s fuzzy set, a negative correlation should exist between the knowledge measure and fuzziness. It has been in our mind that the fuzziness of Zadeh’s fuzzy set determines its fuzzy entropy, and they are both negatively correlated to $\left|{\mu }_A(x)-v_A(x)\right|$ [22]. So the knowledge measure K(A) should be monotonously increasing with respect to $\left|{\mu }_A(x)-v_A(x)\right|$. Moreover, we note that a crisp set provides the maximum amount of information, so the knowledge amount of a crisp set reaches the maximum value $K_{\max }=1$. Conversely, the case that $\forall x\in X$, ${\mu }_A(x)=v_A(x)=0$ means full ignorance, so the knowledge amount reaches its minimum value $K_{\min }=0$. In addition, in the case of ${\mu }_A(x_i)=v_A(x_i)=a\ne 0$, we have ${\pi }_A(x_i)=1-2a$. Thus, the less a indicates greater the greater hesitant degree ${\pi }_A(x_i)$, which leads to the greater uncertainty degree and smaller knowledge amount.

Considering these intuitive properties, we give the following definition to describe the axiomatic properties of the knowledge measure for AIFSs.

Since both knowledge and entropy measures are always regarded as two complementary concepts, we discuss these properties by comparing them with those of entropy measure. We can see that the third property in [22] defined for intuitionistic fuzzy entropy, denoted as E, is stated as: $E(B)\le E(A)$ if B is less fuzzy than A, i.e., $\forall x\in X$, (1) ${\mu }_B(x)\le {\mu }_A(x)$ and $v_B(x)\ge v_A(x)$ for ${\mu }_A(x)\le v_A(x)$, or (2) ${\mu }_B(x)\ge {\mu }_A(x)$ and $v_B(x)\le v_A(x)$ for ${\mu }_A(x)\ge v_A(x)$.

The first condition indicates that ${\mu }_B(x_i)\le {\mu }_A(x_i)\le v_A(x_i)\le v_B(x_i)$ and $\left|{\mu }_B(x_i)-v_B(x_i)\right|$$\ge \left|{\mu }_A(x_i)-v_A(x_i)\right|$. Similarly, the second condition implies that ${\mu }_B(x_i)\ge {\mu }_A(x_i)$$\ge v_A(x_i)\ge v_B(x_i)$ and $\left|{\mu }_B(x_i)-v_B(x_i)\right|$$\ge \left|{\mu }_A(x_i)-v_A(x_i)\right|$. Therefore, the entropy measure of AIFS decreases with $\left|{\mu }_A(x_i)-v_A(x_i)\right|$, i.e., $E(B)\le E(A)$ if $\left|{\mu }_B(x_i)-v_B(x_i)\right|\ge \left|{\mu }_A(x_i)-v_A(x_i)\right|$, $i=1,2,\cdots ,n$, which is related to the property of KP3. However, this property of intuitionistic fuzzy entropy does not consider the influence of hesitation degree. It may not be sensible to discuss the relationship between fuzziness and intuitionistic fuzzy entropy if the hesitance degree is not fixed. Moreover, since $\left|{\mu }_A(x_i)-v_A(x_i)\right|\ge \left|{\mu }_B(x_i)-v_B(x_i)\right|$ cannot always induce ${\mu }_A(x_i)\le {\mu }_B(x_i)\le v_B(x_i)\le v_A(x_i)$ or ${\mu }_A(x_i)\ge {\mu }_B(x_i)\ge v_B(x_i)\ge v_A(x_i)$, the property $E(A)\le E(B)$ if $\left|{\mu }_A(x_i)-v_A(x_i)\right|\ge \left|{\mu }_B(x_i)-v_B(x_i)\right|$ is more general than the third property listed in [22]. Thus, for the relation between knowledge and fuzziness, our proposed axiomatic property is made more general by relaxing the formal constraint by using $\left|\mu (x)-v(x)\right|$. However, such relaxation does not cause an unreliable measure of the knowledge amount because of the limitation of hesitation degree, which will be illustrated later. This also demonstrates the possibility and reasonability of further exploring the relation between the entropy measure and knowledge measure of AIFSs. We point out that the entropy of an AIFS reaches its peak value when the membership degree and non-membership degree are identical for all elements [22]. This is analogous to the entropy measure of fuzzy sets, which solely concerns the relation between membership degree and non-membership degree. Therefore, the notions of entropy and knowledge measure are not just complementary concepts, but rather they differ from each other not only in the aspect of viewpoint, but also in the point they focus on. The fuzzy entropy merely depicts the difference between an AIFS and a crisp set, which is denoted as fuzziness, while knowledge measure is defined to measure the closeness between AIFS and a crisp set, which takes both fuzziness and hesitancy into account.

Following the axiomatic properties in Definition 7, we can create knowledge measures for AIFSs by a mapping F: D→[0,1], where $D=\{(x,y)\in [0,1]\times [0,1]|x+y\le 1\}$, and F must satisfy the following conditions:

For $(x,y)\in [0,1]\times [0,1]$, we can effortlessly obtain many F functions satisfying the above conditions, such as F(x,y) = (|x−y|+x+y)/2 and F(x,y) = x²+y². Using these functions, we can construct knowledge measures for AIFSs. Given an AIFS $A=\left\{〈x,{\mu }_A(x),v_A(x)〉\left|x\in X\right.\right\}$ defined in $X=\{x_1,x_2,\cdots ,x_n\}$, its knowledge measure K can be expressed by $K={\displaystyle \sum _{i=1}^{n}F({\mu }_A(x_i),v_A(x_i))}/n$. In this way, many knowledge measures can be created for AIFSs, but most may lack of specific physical meaning. This motivates us to construct knowledge measures with both clear physical significance and axiomatic mathematical properties.