A New Type of Compositive Information Entropy for IvIFS and Its Applications

We first show the interval-valued intuitionistic fuzzy entropy which reflects intuitionism and fuzziness of interval-valued intuitionistic fuzzy set (IvIFS) based on interval-valued intuitionistic fuzzy cross-entropy. As for intuitionism and fuzziness of IvIFS, we propose interval-valued intuitionistic entropy and interval-valued fuzzy entropy, respectively. Furthermore, we establish the interval-valued span entropy describing the uncertainty of membership degree and nonmembership degree and show some concrete measure formulas. Combining intuitionistic factor, fuzzy factor, and span factor, we ultimately put forward the axiomatic definition of the compositive entropy and give a measure formula of compositive entropy. In addition, the effectiveness of the compositive entropy measure is illuminated by comparison with other entropy measures. Furthermore, the compositive entropy is applied tomultiple attributes’ decision-making by using theweighted correlation coefficient between IvIFSs and pattern recognition by a similarity measure transformed from the compositive entropy.


Introduction
Since Zadeh [1] first introduced fuzzy set (FS) in 1965, many theories of higher order fuzzy set have been proposed.In 1986, Atanassov [2] generalized FS to intuitionistic fuzzy set (IFS) described by the membership degree and nonmembership degree for each element of the universe.In addition, the interval-valued fuzzy set (IvFS) [3] is conceived by Zadeh to specify the interval-valued degree of membership to each element of the universe.The concept of vague set (VS) [4] introduced by Gau and Buehrer is another generalization of fuzzy set, which is identified with IvFS pointed out in [5].Torra [6] proposed the concept of hesitant fuzzy set (HFS) to permit the membership of an element to be a set of several possible values between 0 and 1.In 1989, Atanassov and Gargov [7] combined IFS with IvFS and introduced the notion of interval-valued intuitionistic fuzzy set (IvIFS) whose membership degree and nonmembership degree were intervals rather than real numbers.The IvIFS could also be described by a membership interval, a nonmembership interval, and a hesitancy interval, which made IvIFS more powerful and flexible in dealing with complexity and uncertainty than IvFS and IFS.In recent years, further researches about IvIFS have gained a series of achievements.Atanassov [8] presented some operations of IvIFS and studied their basic properties.Park et al. [9] investigated the correlation coefficients of IvIFS, which considered three parameters feature of IvIFS.Deschrijver and Kerre [10] established the relationships among IvIFS, IFS, and -fuzzy set.In [11], Xu et al. investigated the clustering operations of IvIFS and in [12], Xu and Chen defined a variety of distance measures and similarity measures of IvIFS for decision-making.
In 1968, Zadeh [13] first introduced the entropy of fuzzy event to measure uncertain information by probabilistic methods.In the past decades, fuzzy entropy, as a very important notion for measuring fuzziness degree or uncertain 2 Mathematical Problems in Engineering information in fuzzy set theory, has received great attention.de Luca and Termini [14] presented the axiomatic definition of fuzzy entropy.Kaufmann [15], Yager [16], and Liu [17] defined some fuzzy entropy formulas for fuzzy set by utilizing a distance measure that describes the difference between FSs.In order to measure the uncertain information of IFS, Burillo and Bustince [18,19] defined intuitionistic fuzzy entropy and extended this concept to interval-valued version.Szmidt and Kacprzyk [20] proposed another intuitionistic fuzzy entropy by employing a geometric interpretation of IFS.Hung and Yang [21] and Wang and Lei [22] improved entropy formula and its constructive principles, but they still ignored the effect induced by changes of hesitancy degree when membership degree is equal to nonmembership degree.So Mao et al. [23] established a novel entropy of IFS, which included two factors, the intuitionistic factor and the fuzzy factor.For IvIFS, Zhang et al. [24] and Wei et al. [25] defined the entropy of IvIFS, which generalized the entropy of IFS in [20].Zhang et al. [26] showed the entropy measure by transforming the IvIFS into IFS.Jin et al. [27] proposed the interval-valued intuitionistic fuzzy continuous weighted entropy on the basis of the continuous ordered weighted averaging (COWA) operator.Ye [28] proposed two entropy measures for IvIFSs and established an entropy weighted model to determine the entropy weights with respect to a decision matrix provided as IvIFS.Besides, Qu et al. [29] showed a reasonable entropy formula which considered the span of membership degree and nonmembership degree.
In order to measure the discrimination information for different kinds of fuzzy sets, the cross-entropy has been widely studied.Shang and Jiang [30] defined the fuzzy cross-entropy between two FSs.Vlachos and Sergiadis [31] introduced the concept of intuitionistic fuzzy cross-entropy of IFSs and applied a kind of intuitionistic fuzzy cross-entropy measure to pattern recognition, medical diagnosis, and image segmentation.Based on intuitionism and fuzziness of IFS, Mao et al. [23] constructed a new cross-entropy to measure discrimination uncertain information between IFSs.Peng et al. [32] proposed the cross-entropy of intuitionistic hesitant fuzzy sets (IHFSs) which was developed by integrating the cross-entropy of IFSs and HFSs.By transforming IvIFS into FS, Ye [33] structured a fuzzy cross-entropy of IvIFSs.
Previous works only reflect one or two aspects of the uncertainty information of IvIFS.In fact, there are three types of uncertainty factors for IvIFS, including intuitionistic factor, fuzzy factor, and newly proposed span factor which can depict the extent of variation for the interval values of membership degree and nonmembership degree.Based on these three kinds of uncertainty factors, the main purpose of this paper is to construct a new compositive entropy which can measure uncertain information of IvIFS accurately.First, we put forward three kinds of entropy of IvIFS, including interval-valued intuitionistic entropy, interval-valued fuzzy entropy, and interval-valued span entropy.The intervalvalued span entropy is studied in particular and general measure formula for the interval-valued span entropy is presented.Then by integrating these three kinds of entropy of IvIFS, we give a measure formula of compositive entropy which meets the axiomatic definition of compositive entropy of IvIFS.Finally, we also make comparisons with other existing formulas and apply the compositive entropy measure in decision-making and pattern recognition to demonstrate its efficiency.The rest of the paper is organized as follows.In Section 2, some definitions about IFS and IvIFS are shown.In Section 3, firstly, the intuitionistic fuzzy cross-entropy and intuitionistic fuzzy entropy are presented.Then we propose another three kinds of entropy for IvIFS, including interval-valued intuitionistic entropy, interval-valued fuzzy entropy, and interval-valued span entropy.Finally, based on intuitionistic factor, fuzzy factor, and span factor, we show the axiomatic conditions of the compositive entropy and construct a measure formula.In Section 4, we compare the compositive entropy measure with other existing entropy measures.And the proposed formula is applied to multiple attributes' decision-making and pattern recognition.Conclusions are presented in Section 5.

Preliminaries
In this section, some basic concepts are illustrated, which will be needed in the following analysis.
The numbers   (), ]  () are called the degree of membership and nonmembership of  to , respectively.We call   () = 1 −   () − ]  () the intuitionistic index of  in , which denotes the hesitancy degree of  to .And it is evident that 0 ≤   () ≤ 1 for all  ∈ .For convenience, we abbreviate the intuitionistic fuzzy set to IFS and denote the set of all IFSs on  by IFS().
Sometimes, we can not give the accurate values of membership degree and nonmembership degree, but a value range.In such cases, Atanassov and Gargov [7] introduced the following notion of the interval-valued intuitionistic fuzzy set, which generalized IFS.
The interval numbers   (), ]  () are called the degree of membership and nonmembership of  to , respectively.For convenience, we denote where the correlation of two IvIFSs  and  is given by and the informational intuitionistic energies of two IvIFSs  and  are given by The correlation coefficient of two IvIFSs  and  satisfies the following properties [9]:

The Uncertain Information of IvIFS
In this section, we give some kinds of entropy to measure different types of uncertain information of IvIFS and show an efficient compositive entropy measure of IvIFS.We only discuss the case where the universe has finite objects; that is,  = { 1 ,  2 , . . .,   }.
The symmetric intuitionistic fuzzy cross-entropy (, ) is used to describe discrimination uncertain information which includes intuitionism and fuzziness.So when  is a crisp set in (, ), that is,   () = 0, Δ  () = 1 for all  ∈ , then (, ) can describe intuitionistic and fuzzy information of .Based on this, Mao et al. [23] construct an intuitionistic fuzzy entropy measure which satisfies four axiomatic principles of the intuitionistic fuzzy entropy in [23].

Interval-Valued Intuitionistic Fuzzy Cross-Entropy and
Interval-Valued Intuitionistic Fuzzy Entropy.In order to reflect the intuitionism and fuzziness of IvIFS, we will introduce two factors Ψ  (), Δ  ().
The intuitionistic factor Ψ  () is defined as the arithmetic average of  −  () and  +  (); that is, which can depict the average hesitancy degree of  to .

Mathematical Problems in Engineering
The fuzzy factor Δ  () is defined as the distance between which can describe the balance of power between membership degree and nonmembership degree of  to .
Example 7. Assume that there is an expert evaluating an event, where the support degree is given by interval [0.6, 0.8] and the opposition degree is the interval [0, 0.2]; then the hesitancy degree is the interval [0, 0.4].We can calculate the intuitionistic factor by Ψ = (0 + 0.4)/2 = 0.2 and the fuzzy factor by Now we will define the interval-valued intuitionistic fuzzy cross-entropy of IvIFS based on the intuitionistic fuzzy crossentropy of IFS [23] as follows.
(c) The relation reaches maximum ln 2. For all   ∈ , property (c) can be verified.Now, we proceed to the symmetric interval-valued intuitionistic fuzzy cross-entropy (, ) = (, ) + (, ).It is easy to verify that (, ) have the following properties: (a) (, ) = (, ); Based on intuitionistic factor and fuzzy factor of IvIFS, we propose the axiomatic conditions of the interval-valued intuitionistic fuzzy entropy, which can measure the uncertain information of fuzziness and intuitionism of IvIFS.
Definition 10.For an IvIFS  defined on , the intervalvalued intuitionistic fuzzy entropy is a real-valued function () = (Ψ  , Δ  ) : IvIFS() → [0, 1], which satisfies the following axiomatic conditions: ) is a real-valued continuous function being increasing with respect to the variable Ψ  and decreasing with the variable Δ  , where Ψ  , Δ  are in Definition 6.
Theorem 11.Let  ∈ IvIFS() and  is a crisp set; then Mathematical Problems in Engineering 5 is an interval-valued intuitionistic fuzzy entropy measure, where (, ) is the symmetric interval-valued intuitionistic fuzzy cross-entropy.
Proof.For (), in order to be qualified as the intervalvalued intuitionistic fuzzy entropy measure, it must satisfy conditions (a)-(d) in Definition 10.

Interval-Value Intuitionistic Entropy and Interval-Valued
Fuzzy Entropy.Mao et al. [23] presented that the intuitionistic fuzzy entropy for IFS could measure intuitionism and fuzziness of IFS.And they rewrote the intuitionistic fuzzy entropy measure as () = (1/2)  () + (1/2)  (), where   (),   () are the intuitionistic entropy and fuzzy entropy measure of  of IFS, respectively.For IvIFS, the intervalvalued intuitionistic fuzzy entropy has similar characteristic.In fact, for the interval-valued intuitionistic fuzzy entropy measure (), we have where Now, we will extend   () and   () of IFS to counterparts of IvIFS.
Proof.For   (), in order to be qualified as an intervalvalued intuitionistic entropy measure, it must satisfy conditions  (  ) + ] +  (  ) = 1 for all   ∈ ; we know that  is a fuzzy set.Conversely, when  is a fuzzy set, it is easy to verify that is an interval-value fuzzy entropy measure.
Proof.For   (), in order to be qualified as an intervalvalued fuzzy entropy measure, it must satisfy conditions (a)-(d) in Definition 14.
(d) From (d) in Theorem 11, we know that condition (d) in Definition 14 is satisfied.
Based on the above analysis, the interval-valued intuitionistic fuzzy entropy measure can be seen as arithmetic average of the interval-valued intuitionistic entropy measure and the interval-valued fuzzy entropy measure.

Interval-Valued Span Entropy.
In this section, we will introduce another entropy of the interval-value span entropy to measure uncertain information caused by the uncertainty of membership and nonmembership degree of IvIFS.
Here, we first put forward the axiomatic conditions of the span entropy.Definition 16.For an IvIFS , the interval-valued span entropy is a real-valued function   () : IvIFS() → [0, 1], which satisfies the following axiomatic conditions: Since the interval-valued span entropy is a magnitude which allows us to measure the extent of variation for the interval values of membership degree and nonmembership degree, this four conditions can be understood as follows: for an IvIFS , one has the following.
(a) The interval-valued span entropy is zero iff the values of membership degree and nonmembership degree are certain; that is,  is IFS.
(b) When the interval-valued span entropy reaches maximum, it means that the total span of   (), ]  () is the largest.Since  +  ()+] +  () ≤ 1, we can intuitively understand that the interval-valued span entropy reaches maximum iff ),  +  ()]⟩ |  ∈ }, for  and   , we know that the extent of variation for the interval values of membership degree and nonmembership degree is the same.
Using the previous function formulas of Φ(, ), we can construct the following interval-valued span entropy measures:

The Compositive
which can depict the extent of variation for the interval values of membership degree and nonmembership degree of  to .
Based on intuitionistic factor, fuzzy factor, and span factor, in the following, we will give the axiomatic definition of a kind of compositive entropy.Inspired by equation () = (1/2)  () + (1/2)  (), now we will construct a compositive entropy measure.

Theorem 20. Consider 𝐴 ∈ IvIFS(𝑋); then
is a compositive entropy measure of IvIFS, which is also called the interval-valued intuitionistic fuzzy span entropy measure.
Proof.For (), in order to be qualified as an interval-valued intuitionistic fuzzy span entropy measure, it must satisfy conditions (a)-(d) in Definition 19.
Based on the results of Table 1, we can see only  meets the relation () > () > () > ().In addition, four entropy values of , , , and  only calculated by where In what follows, we only discuss the case where the weight vector of attributes  is completely unknown.
According to the entropy theory [34], if the entropy value for a criterion is smaller across alternatives, it should provide decision-makers with the useful information.Therefore, the criterion should be assigned a bigger weight; otherwise, such a criterion will be judged unimportant by the decision-maker.In other words, such a criterion should be evaluated as a very small weight.Now we can establish an exact model of entropy weights [34]: where ∑  1   = 1 and   ∈ [0, 1] and (  ) is calculated by where (  ) is the compositive entropy of the IvIFS   in (17).
In multiple attributes' decision-making environment, the concept of positive point has been used to identify the best alternative.Although the positive alternative does not exist in real world, it does provide a useful theoretical evaluating standard for all the criteria.Here we define the positive alternative  + = {⟨  , [1, 1], [0, 0]⟩ |   ∈ }.Then based on the correlation coefficient equation (1) between IvIFSs and the weight vector  on attribute vector, we can define the weighted correlation coefficient   (  ,  + ),  = 1, 2, . . ., , between an alternative   and the positive alternative  + as follows: We can check that the weighted correlation coefficient   (  ,  + ) has the following properties: The larger the value of weighted correlation coefficient   (  ,  + ), the better the alternative   , as the alternative   is closer to the ideal alternative  + .Therefore, all the alternatives can be ranked according to the value of the weighted correlation coefficients so that the best alternative can be selected.The decision procedure for the proposed method can be summarized as follows.
Step 3. Rank the alternatives according to the obtained correlation coefficients, and then obtain the best choice.
Example 22. Assume that a fund manager in a wealth management firm is assessing four potential investment opportunities; that is, the set of alternatives is ( 1 ,  2 ,  3 ,  4 ).The firm mandates that the fund manager has to evaluate each investment against four criteria: risk ( 1 ), growth ( 2 ), sociopolitical issues ( 3 ), and environmental impacts ( 4 ).In addition, the fund manager is only comfortable with providing his/her assessment of each alternative on each criterion as an IvIFS and the decision matrix is as follows: Each element of this matrix is an IvIFS, representing the fund managers assessment as to what degree an alternative is and is not an excellent investment as per criterion.For instance, the top-left cell, ([0.4,0.5], [0.3, 0.4]), reflects the fund managers belief that alternative  1 is an excellent investment from a risk perspective ( 1 ) with a margin of 40-50% and  1 is not an excellent choice given its risk profile ( 1 ) with a chance between 30% and 40%.The proposed method is applied to solve this problem according to the following computational procedure.

(25)
Step 3. From the weighted correlation coefficients between the alternatives and the positive alternative, the ranking order is  2 ⪰  4 ⪰  1 ⪰  3 .And the alternative  2 is the best choice.
Wei [35] and Zhang et al. [26] have introduced two kinds of efficient methods for multiple attribute decision-making with completely unknown attribute weights.By applying Wei's method [35] to Example 22, the ranking order of all the alternatives is  2 ⪰  4 ⪰  1 ⪰  3 , and the most desirable alternative is  2 .By applying the method proposed by Zhang et al. [26] to Example 22, the ranking order of all the alternatives is  2 ⪰  4 ⪰  1 ⪰  3 , and the most desirable alternative is  2 .We can see that all results are uniform.

The Application of the Compositive Entropy Measure to
Pattern Recognition.First, we present a similarity measure of IvIFS via the relationship between similarity measure and entropy of IvIFS.Hu and Li [36] give transformation methods from entropy to similarity of IvIFS.Utilizing the compositive entropy measure above, we propose a new similarity measure which can be used for pattern recognition.
in , which denotes the hesitancy degree of  to .Similarly, we abbreviate the interval-valued intuitionistic fuzzy set to IvIFS and denote the set of all the IvIFSs on  by IvIFS().
Entropy of IvIFS.We already know that the proposed interval-valued intuitionistic fuzzy entropy in Theorem 11 can only express fuzziness and intuitionism of IvIFS and the interval-valued span entropy can reflect another kind of uncertain information of IvIFS.Naturally, we want to propose a sensible entropy of IvIFS which can be concerned with interval-valued intuitionistic entropy, intervalvalued fuzzy entropy, and interval-valued span entropy.First, we give another type of uncertainty factor.