#### Abstract

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 to multiple attributesâ€™ decision-making by using the weighted correlation coefficient between IvIFSs and pattern recognition by a similarity measure transformed from the compositive entropy.

#### 1. 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 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 interval-valued 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 attributesâ€™ decision-making and pattern recognition. Conclusions are presented in Section 5.

#### 2. Preliminaries

In this section, some basic concepts are illustrated, which will be needed in the following analysis.

*Definition 1 (see [2]). *Let a nonempty set be the universe of discourse; an intuitionistic fuzzy set on is defined by Atanassov as , where , with the condition for all .

The numbers are called the degree of membership and nonmembership of to , respectively. We call the intuitionistic index of in , which denotes the hesitancy degree of to . And it is evident that 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.

*Definition 2 (see [7]). *Let a nonempty set be the universe of discourse and let be all closed subintervals of the interval . An interval-valued intuitionistic fuzzy set is defined by Atanassov and Gargov as , where , with the condition for all .

The interval numbers are called the degree of membership and nonmembership of to , respectively. For convenience, we denote , , and then with condition and call the interval abbreviated by the interval-valued intuitionistic index of 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().

*Definition 3 (see [18, 24]). *Let ; we define(a), iff , ;(b), iff , ;(c), iff , .

*Definition 4 (see [18, 24]). *For two IvIFSs , , we define the following relations and operation:(a), iff , , for all ;(b), iff , , for all ;(c), iff , , for all ;(d).

*Definition 5 (see [9]). *Let , , , and ; then the correlation coefficient of and is defined bywhere the correlation of two IvIFSs and is given byand the informational intuitionistic energies of two IvIFSs and are given by

The correlation coefficient of two IvIFSs and satisfies the following properties [9]:(a);(b);(c) iff .

#### 3. 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, .

In order to describe the discrimination information of IFSs, Mao et al. [23] define a suitable intuitionistic fuzzy cross-entropy. Let , ; then the intuitionistic fuzzy cross-entropy of against based on uncertain information is defined aswhere shows degree of hesitancy and describe the balance of power between membership degree and nonmembership degree of to .

The symmetric intuitionistic fuzzy cross-entropy is , and it has been proved that , .

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, , for all , then can describe intuitionistic and fuzzy information of . Based on this, Mao et al. [23] construct an intuitionistic fuzzy entropy measurewhich satisfies four axiomatic principles of the intuitionistic fuzzy entropy in [23].

##### 3.1. 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 , .

*Definition 6. *Let , , . The intuitionistic factor is defined as the arithmetic average of and ; that is,which can depict the average hesitancy degree of to .

The fuzzy factor is defined as the distance between and ; that is,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 and the opposition degree is the interval ; then the hesitancy degree is the interval . We can calculate the intuitionistic factor by and the fuzzy factor by .

For a crisp set , that is, , or , for all , we can calculate , for all .

Now we will define the interval-valued intuitionistic fuzzy cross-entropy of IvIFS based on the intuitionistic fuzzy cross-entropy of IFS [23] as follows.

*Definition 8. *Let , . Based on intuitionistic factor and fuzzy factor, the interval-valued intuitionistic fuzzy cross-entropy of against is defined aswhere

Proposition 9. *Let ; satisfies following properties:*(a)*;*(b)* iff or ;*(c)*.*

*Proof. *(a) One has . Based on formulas (6) and (7) in Definition 6, we can see , , , and for all . From equation (8), it is easy to verify property (a).

(b) First we have = + â‰¤ , for all . According to Shannonâ€™s inequality, we can verify and iff or .

(c) The relation + is true, and when , or , , reaches maximum . 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);(b);(c) iff or ;(d).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 interval-valued intuitionistic fuzzy entropy is a real-valued function , which satisfies the following axiomatic conditions:(a) iff is a crisp set; that is, or , for all ;(b) iff , for all ;(c);(d) is a real-valued continuous function being increasing with respect to the variable and decreasing with the variable , where are in Definition 6.Analogously, we know when is a crisp set, that is, , for all , can measure the uncertain information of IvIFS, including fuzziness and intuitionism.

Theorem 11. *Let and is a crisp set; thenis 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 interval-valued intuitionistic fuzzy entropy measure, it must satisfy conditions (a)â€“(d) in Definition 10.

(a) Let be a crisp set; that is, , for all ; it is easy to verify that . Conversely, suppose ; that is, ; from the properties of , we know or , which means is also a crisp set.

(b) For an IvIFS , when , for all , we can calculate , . It is easy to verify that . Conversely, when , we only have iff , iff , for all .

(c) From the properties of , we know , so .

(d) In order to prove that satisfies condition (d) in Definition 10, it suffices to prove that the function , where , is increasing with its argument and decreasing for its argument . Taking the partial derivative of with and , respectively, we yieldSince , we have , , so .

##### 3.2. 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 , where are the intuitionistic entropy and fuzzy entropy measure of of IFS, respectively. For IvIFS, the interval-valued intuitionistic fuzzy entropy has similar characteristic.

In fact, for the interval-valued intuitionistic fuzzy entropy measure , we havewhere and .

Now, we will extend and of IFS to counterparts of IvIFS.

*Definition 12. *For an IvIFS , the interval-valued intuitionistic entropy is a real-valued function , which satisfies the following axiomatic conditions:(a) iff is a fuzzy set;(b) iff , for all ;(c);(d) is a real-valued continuous function being increasing with respect to the variable .

Theorem 13. *Let ; thenis an interval-value intuitionistic entropy measure.*

*Proof. *For , in order to be qualified as an interval-valued intuitionistic entropy measure, it must satisfy conditions (a)â€“(d) in Definition 12.

(a) When , we only have for all , , . Subtracting the latter two equations, we have , . Additionally, for all ; we know that is a fuzzy set. Conversely, when is a fuzzy set, it is easy to verify that for ; then .

(b) When , we only have , . That is to say, , for all . Conversely, when , for all , we have ; then .

(c) Since for all , .

(d) Condition (d) in Definition 12 is apparent.

*Definition 14. *For IvIFS , the interval-valued fuzzy entropy is a real-valued function , which satisfies the following axiomatic conditions:(a) iff is a crisp set;(b) iff for all ;(c);(d) is a real-valued continuous function being decreasing with respect to the variable .

Theorem 15. *Let ; thenis an interval-value fuzzy entropy measure.*

*Proof. *For , in order to be qualified as an interval-valued fuzzy entropy measure, it must satisfy conditions (a)â€“(d) in Definition 14.

(a) When , we only have , , or , for all ; that is, is a crisp set. Conversely, when is a crisp set, that is, , it is easy to verify that .

(b) When , we only have , for all ; that is, . Conversely, when , that is, , it is easy to verify that .

(c) Since for all , .

(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.

##### 3.3. 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 , which satisfies the following axiomatic conditions:(a) iff is IFS;(b) iff , for all ;(c);(d)let , , if and for all .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 , we can intuitively understand that the interval-valued span entropy reaches maximum iff , .

(c) As , for and , we know that the extent of variation for the interval values of membership degree and nonmembership degree is the same.

(d) The spans of both and are bigger; we know that the interval values of membership degree and nonmembership degree are more uncertain; that is, the interval-valued span entropy is bigger.

Now our goal is to give the specific interval-valued span entropy measures. To construct the entropy on IFS, Burillo and Bustince [19] introduced function , where . Meanwhile, function satisfies the following conditions:(a) iff ;(b) iff ;(c);(d)if and , then .We can verify the following functions are abbreviated by :(a);(b), ;(c);(d).

Theorem 17. *For an IvIFS , the real-valued function is an interval-valued span entropy measure, where the function of satisfies the previous four conditions (a)â€“(d), is the span of membership degree, and is the span of nonmembership degree.*

*Proof. *For , in order to be qualified as an interval-valued span entropy measure, it must satisfy conditions (a)â€“(d) in Definition 16.

(a) Based on condition (b) of , we know iff iff iff , for all iff is IFS.

(b) Based on condition (a) of , we know iff iff iff iff and for all .

(c) One has . We know , . From condition (c) of , we can obtain .

(d) If and , we can obtain and for all . From condition (d) of , we know that condition (d) in Definition 16 is suited.

Using the previous function formulas of , we can construct the following interval-valued span entropy measures:(a);(b), ;(c);(d).

##### 3.4. The Compositive 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, interval-valued fuzzy entropy, and interval-valued span entropy. First, we give another type of uncertainty factor.

*Definition 18. *Let , . The span factor is defined aswhich 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.

*Definition 19. *For an IvIFS , the compositive entropy of IvIFS also called interval-valued intuitionistic fuzzy span entropy is a real-valued function , which satisfies the following axiomatic conditions:(a) iff is a crisp set;(b) iff , for all ;(c);(d) is a real-valued continuous function being increasing with respect to the variables , and decreasing with the variable ., are the intuitionistic factor and fuzzy factor in Definition 6 and is the span factor in Definition 18.

Inspired by equation , now we will construct a compositive entropy measure.

Theorem 20. *Consider ; thenis 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)â€“(d) in Definition 19.

(a) It is easy to verify that iff iff is a crisp set.

(b) Taking , , and into , we haveWe want to prove iff , for all . Sufficient condition is obvious. As for necessary condition, we can see reaches maximum when , and reaches maximum when . Ultimately, we have , for all when .

(c) Since , , and , it is easy to verify .

(d) Considering the function: , where , similar to the process of proof in Theorem 11(d), we can verify that is increasing with the arguments and decreasing with its argument .

Let us now consider an IvIFS in . Using the color of each point to represent the value of compositive entropy, Figures 1, 2, and 3 can reflect the relation between the value of compositive entropy and , , and .

#### 4. The Application of the Compositive Entropy Measure

##### 4.1. The Effectiveness of the Compositive Entropy Measure

In fact, there have been some research results about the entropy of IvIFS; for example, Zhang et al. [24] defined , Wei et al. [25] put forward , Jin et al. [27] presented , Ye [28] gave , Zhang et al. [26] proposed , and Qu et al. [29] showed , which can be listed as follows:

(a) /;

(b) /;

(c) /, where , . is the attitudinal character of BUM function with ;

(d) + , where and here we choose ;

(e) , where , , . Here we choose ;

(f) + .

We will compare these entropy measures with our proposed entropy measures, including the interval-valued intuitionistic fuzzy entropy measure and the compositive entropy measure , by an example.

*Example 21. *Consider , . For short, we denote , = , = , and = . Based on intuitionistic factor, fuzzy factor, and span factor of IvIFS, we can analyze the uncertain relationships of , , and as follows: comparing with , intuitionistic factor and fuzzy factor of , are the same, but span factor of has a greater impact on uncertainty than , so we have . For and , fuzzy factor and span factor of and are the same, but intuitionistic factor of has a greater impact on uncertainty than , so we have . Analogously, for and , considering fuzzy factor, we have . So we have the relation .

According to entropy formulas above, we can obtain calculation results presented in Table 1.

Based on the results of Table 1, we can see only meets the relation . In addition, four entropy values of , , , and only calculated by are different, which means only can distinguish , , , and well. Although it is possible that two or more IvIFSs have the same entropy, the proposed entropy measure is more precise and detailed than other entropy measures for presenting the degree of the uncertain information under the new framework of Definition 19.

##### 4.2. The Application of the Compositive Entropy Measure to Decision-Making

Since the compositive entropy measure not only has theoretical advantage, which is taking three kinds of entropy measures into account, but also has practical excellent effect, now we will show the application on multiple attributesâ€™ decision-making with unknown information of criteria weights for alternatives.

Let be a set of criteria and let be a set of alternatives which consists of noninferior alternatives (an alternative is noninferior if there exists no other alternative which can yield an improvement in one attribute, without causing a degradation in another). The evaluation value of criteria on an alternative is represented by the following IvIFS:where , , , . For convenience, we denote