On Uncertainty Measures of the Interval-Valued Hesitant Fuzzy Set

. Interval-valued hesitant fuzzy sets (IVHFS), as a kind of decision information presenting tool which is more complicated and more scientifc and more elastic, have an important practical value in multiattribute decision-making. Tere is little research on the uncertainty of IVHFS. Te existing uncertainty measure cannot distinguish diferent IVHFS in some contexts. In my opinion, for an IVHFS, there should exist two types of uncertainty: one is the fuzziness of an IVHFS and the other is the nonspecifcity of the IVHFS. To the best of our knowledge, the existing index to measure the uncertainty of IVHFS all are single indexes, which could not consider the two facets of an IVHFS. First, a review is given on the entropy of the interval-valued hesitant fuzzy set, and the fact that existing research cannot distinguish diferent interval-valued hesitant fuzzy sets in some circumstances is pointed out. With regard to the uncertainty measures of the interval-valued hesitant fuzzy set, we propose a two-tuple index to measure it. One index is used to measure the fuzziness of the interval-valued hesitant fuzzy set, and the other index is used to measure the nonspecifcity of it. Te method to construct the index is also given. Te proposed two-tuple index can make up the fault of the existing interval-valued hesitant fuzzy set’s entropy measure.


Introduction
In some real-life scenarios, we often need to make multicriteria decision-making, which is to sort some plans with several criteria and select the best one. One important stage in multicriteria decision-making is determining the membership degree of one alternative in regard to one certain evaluation term. Te traditional method is a black and white problem. Tat is if the alternative meets the requirement of the evaluation term, then the membership degree is one; otherwise, the membership degree is zero. Tis kind of rule is simple to operate but is too absolute to lose a lot of information. In fact, the membership degree in a lot of circumstances is not a clear distinction between black and white, which otherwise have a certain degree of grey. In order to describe membership degree more perfectly, Zadeh creatively proposed the fuzzy sets (FS) theory based on sets theory [1]. In fuzzy sets, the information has some kind of uncertainty which has two dimensions. Te frst dimension is fuzziness which states that we cannot clearly defne the degrees that one element is belonging to and not to a certain fuzzy set. De Luca and Termini proposed an entropy measure for FS which is not based on probability theory [2], and Liu developed the axiomatic defnition of entropy for FS [3], both of which are important research studies on the fuzziness of FS. Fan and Ma had given some general results of the fuzzy entropy of FS based on the axiomatic defnition of the fuzzy entropy of FS and distance measure of FS, and they generalized the fuzzy entropy formulation of FS proposed by De Luca and Termini [2]. Te other aspect of the uncertainty of the FS is nonspecifcity which measures the amount of information contained in the FS. Yager proposed several nonspecifcity indexes to measures the degree that the FS only contains one element [5]. Garmendia et al. gave the general formulation for the nonspecifcity measure of FS based on T-norms and negation operator [6].
Tere is one membership degree and nonmembership degree for each element in the FS. However, in some circumstances, it is more suitable to consider the hesitation degree. We assume that a committee is composed of ten experts, and the attitude of fve of them is positive, that of three of them is negative, and two are abstained from voting. Ten, the membership degree for the alternative to the feasible alternative set may be defned as 0.5, the nonmembership degree may be defned as 0.3, and the hesitation degree may be defned as 0.2. FS is not suited to be used in this kind of cases. Because of the universality of this kind of cases, Atanassov generalized FS to the intuitionistic fuzz set (IFS) [7]. Each element in the IFS has a membership degree, a nonmembership degree, and a hesitation degree, thus making IFS more suitable to deal with problems of fuzziness and uncertainty. Some research studies have been conducted on the quantifcation of the uncertainty of IFS. Xia and Xu proposed a new entropy and a new crossentropy of IFS, and they discussed the relation between them [8]. Huang developed two entropy measures for IFS based on the distance between two IFSs, which is simple to calculate and can give reliable results [9]. Huang and Yang gave the defnition of fuzzy entropy based on probability theory [10]. Pal et al. pointed that there are two aspects associated with the uncertainty of IFS, which are fuzziness and nonspecifcity, and existing studies cannot distinguish them [11].
Sometimes, in real decision-making, there is a hesitation among several membership degree values. We assume that several experts evaluate a plan on one attribute. Expert A thinks the membership degree of the plan that belongs to the attribute is 0.4, expert B thinks that the membership degree is 0.6, expert C thinks that the membership degree is 0.8, and they cannot reach an agreement, so how do we describe the evaluation result? FS and IFS both cannot be used in this circumstance. Hesitant fuzzy sets (HFS), proposed by Torra and Narukawa [12] and Torra [13], are more suitable in this kind of circumstance. Te membership degree of every element in an HFS is a set, called the hesitant fuzzy element (HFE). HFS is an efective tool describing the hesitance degree of the decision maker, which is widely used in practical decision-making problems [14], so it is important to study uncertainty problems associated to the HFS. HFS is a new kind of information presentation tool, and there is little research on the uncertainty of it. Xu and Xia gave the axiom defnition of the entropy for HFE, and they proposed several entropy formulations to measure the fuzziness degree of HFE [15]. Farhadina [16] pointed out that the entropy formulation of HFE proposed by Xu and Xia [15] gave the same value to several HFEs with diferent uncertainties intuitively. Singh and Ganie thought that the entropy formulation developed by Xu and Xia [15] cannot distinguish diferent HFEs in some circumstances and gave the same weights to attributes having diferent importance obviously, and they constructed creatively generalized hesitant fuzzy knowledge measure formulation which can be used to handle these two problems [17]. Zhao et al. [18] think that the entropy formula for the HFE introduced by Farhadinia [16] cannot diferentiate diferent HFEs in some circumstances such as when two HFEs have the same distance to HFE {0.5}, and they gave the defnition of binary entropy for HFS, with one entropy measuring the fuzziness of the HFE and the other measuring the nonspecifcity. Wei et al. investigated the problem of how to apply diferent uncertainty facets of hesitant fuzzy linguistic term sets in diferent decision-making settings [19]. Xu et al. established the axiomatic defnitions of fuzzy entropy and hesitancy entropy of weak probabilistic hesitant fuzzy elements [20]. Fang revisited the concept of uncertainty measures for probabilistic hesitant fuzzy information by comprehensively considering their fuzziness and hesitancy and proposed some novel entropy and cross-entropy measures for them [21]. Wei et al. focused on studying how to measure the uncertainty presented by the information of an extended hesitant fuzzy linguistic term set [22]. Fang developed some hybrid entropy and crossentropy measures of probabilistic linguistic term sets [23]. Wang et al. proposed an entropy measure of the Pythagorean fuzzy set by taking into account both Pythagorean fuzziness entropy in terms of membership and nonmembership degrees and Pythagorean hesitation entropy in terms of the hesitant degree [24]. Xu et al. modifed the axiomatic defnition of fuzzy entropy fuzzy sets (FSs), and the axiomatic defnitions of fuzzy entropy and hesitancy entropy of intuitionist fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs) are also revised [25]. In order to measure the uncertainly for type-2 fuzzy sets (T2FSs), the axiomatic framework of fuzzy entropy of T2FSs is established [26].
Chen et al. introduced the thought of interval number into HFS and proposed the defnition of interval-valued hesitant fuzzy sets (IVHFS) which is a kind of generalization of HFS [27]. IVHFSs, as a kind of decision information presenting tool which is more complicated and more scientifc and more elastic, have an important practical value in multiattribute decision-making [28,29]. Tere is little research on the uncertainty of IVHFS. Farhadinia proposed the defnition of entropy for IVHFS based on the distance between two IVHFSs, but the entropy formula of IVHFS cannot distinguish diferent IVHFSs in some contexts [16]. Pal et al. pointed out that there exist two types of uncertainty for an IFS, fuzzy-type uncertainty and nonspecifcity-type uncertainty [11]. Zhao et al. thought that for an HFS, except for the fuzziness, there exists another kind of uncertainty, which is nonspecifcity [18]. In my opinion, for an IVHFS, there exist two types of uncertainty, one is the fuzziness of an IVHFS, which is related to the departure of the IVHFS from its nearest script set, and the other is the nonspecifcity of the IVHFS, which is related to the imprecise knowledge contained in the IVHFS. To the best of our knowledge, the existing index to measure the uncertainty of IVHFS are all single indexes, which could not consider the two facets of an IVHFS. Pal et al. stated that we cannot put forward any total measure of uncertainty for an HFE as we do not know how exactly these two types of uncertainty interact, so we also cannot put forward any total measure of uncertainty for an IVHFS as we do not know how exactly these two types of uncertainty of an IVHFS interact [13]. In view of that, this paper proposes an axiom frame which uses two-tuple entropy indexes to measure the uncertainty of the IVHFS. One entropy index is used to measure the IVHFS' fuzzy degree and the other to measure its unspecifcity. Te approaches to construct the two kinds of uncertainty measure are also given, and the two-tuple indexes can make up for the shortcomings of the existing entropy measures. Te novelty of the paper lies in that, and to my knowledge, this is the frst paper studying on the uncertainty measure of the intervalvalued hesitant fuzzy set. Based on the two-tuple index, we can defne distance measure, similarity measure, and design clustering algorithm to classify a set of interval-valued hesitant fuzzy sets. Distance measure has wide applications in decision-making, such as developing methods to reach consensus in a group, pattern recognition, and image processing [16]. So, this paper lays the foundation to develop distance measure and similarity measure between intervalvalued hesitant fuzzy sets.
Te paper is organized as follows: Section 2 introduces the concept of HFS, IVHFS, and the existing uncertainty measure of IVHFS. Section 3 proposes a two-tuple index and approaches to construct the two kinds of uncertainty measure and some theorems. Te paper is concluded in Section 4, including the trends and directions of the IVHFS.

Preliminaries
Defnition 1 (see [13]). An HFS M on the reference set X is defned in terms of a function h M (X) as follows: where h M (x) is a set of several values in [0, 1], which represent possible membership degrees of the elements x of X to the set M. Based on the practical need, Chen et al. integrated the thought of interval number into HFS and proposed the interval-valued hesitant fuzzy sets [27].
Defnition 2 An internal-valued hesitant fuzzy set M on the reference set X is defned as follows: where h M (x) is a set of several intervals of [0, 1], representing the membership degree of the element x in the reference set X to the IVHFS M. Chen et al. called M as the interval-valued hesitant fuzzy element (IVHFE) [27].
is an IVHFS. Let H be the set of all IVHFEs. Based on the defnition of the complement to an HFE α proposed by Torra and Narukawa [12], this paper defnes the complement of an IVHFE α as Defnition 3 (see [15]). Given two IVHFSs M and N, h h N (x i ), respectively. Te interval-valued hesitant normalized Hamming distance is defned as follows: Te generalized hybrid interval-valued hesitant weighted distance is defned as follows:

Advances in Fuzzy Systems
where w i (i � 1, 2, ..., n) is the weight of the element x i with w i ∈ [0, 1] and n i�1 w i � 1.
Theorem 4 (see [15]) Let Z: [0, 1] ⟶ [0, 1] be a strictly monotone decreasing real function and d be a distance between IVHFSs. Ten, for any IVHFS, M and N Are, respectively, a similarity measure and two entropies for IVHFSs based on the corresponding distance d.
It is obvious that, for any , so we cannot diferentiate M and N in this case. What is more, uncertainty can be considered of diferent types such as fuzziness and nonspecifcity [30], and the index to measure the uncertainty of the interval-valued hesitant fuzzy set proposed by Farhadinia is a single index, which could not consider all facets of an interval-valued hesitant fuzzy set.
In view of that, this paper proposes an axiom frame in Section 3 which uses two-tuple entropy indexes to measure the uncertainty of the IVHFS. One entropy index is used to measure the IVHFS' fuzzy degree, and the other is used to measure its unspecifcity.

Two-Tuple Entropy Measures for HFE
Note. Te relationship of ″≺″ and ″≻″ in (E F 4) is complemented according to the calculation rules of the interval number [28]. For example, we assume that a and b are two interval numbers, and if the possibility degree of a bigger than b is larger or equates to 0.5, we call a ≻ b is true; otherwise, if the possibility degree of b bigger than a is larger or equates to 0.5, we call b ≻ a is true. (2) In (E NS 4), α σ(i) − α σ(j) and β σ(i) − β σ(j) are two interval numbers. If interval number a is [0.2,0.5], then we have In Defnition 3, a two-tuple (E F , E NS ) is utilized to measure the uncertainty of IVHFE α. Te fuzzy entropy E F is used to measure the fuzziness degree of α, that is, the distance between α and the crisp value which is closest to α. Te nonspecifcity entropy E NS is used to measure the nonspecifc degree of α, that is, the degree of which only contains one interval. Terefore, the two-tuple (E F , E NS ) not only considers the fuzziness of a set which traditional entropy can measure, but it also quantifes the nonspecifcity of a set, which is more reasonable [11].

Te Fuzzy Entropy E F of IVHFE.
Te uncertainty of an IVHFE α comprised fuzziness and nonspecifcity. First, we study how to measure the fuzziness degree of an IVHFE. We will give some methods to construct the measure that can be used to quantify the fuzziness degree of an IVHFE. First, a general result is given as follows: Proof. Te proof of Teorem 6 is provided in Appendix A. Note. From the proof of Teorem 4, we have E F (α) defned in (7) Advances in Fuzzy Systems satisfying axiom (E F 1) − (E F 4). Ten, R possesses the following properties: Proof. Te proof of Teorem 7 is provided in Appendix B.

Te Nonspecifcity Entropy of IVHFE E NS .
In this section, we investigate the other aspect of the uncertainty of the IVHFE, that is, nonspecifcity. First, we proposed a new measure called nonspecifcity used to measure the other aspect of uncertainty of the IVHFE. If l α is one, let 〈l α 〉 take the value two; otherwise, if l α is equal or larger than two, let 〈l α 〉 take the value l α (l α − 1). We give a general result as follows: Which meets axiom (E NS 1) − (E NS 4). Ten, F has the properties as follows: Proof. Te proof of Teorem 9 is provided in Appendix D.
Note. We assume that h: [− 1, 1] ⟶ I is a function which can generate a new interval by taking the absolution of the two endpoints and sorting the two numbers. Proof. Te proof of Teorem 10 is provided in Appendix E.
We assume that [a 1 , a 2 ] � x≺y � [b 1 , b 2 ], which is the possibility of x which is smaller than y and is larger than 0.5, and from Xu and Da [30], we obtain that b 1 + b 2 > a 1 + a 2 . Ten, we have.

Conclusions
To the best of our knowledge, there are a few research studies on the uncertainty of IVHFE and most of them cannot diferentiate diferent IVHFE in some situations. Tis paper proposed a two-tuple entropy model to quantify the uncertainty of IVHFE. We use one index to measure its fuzziness degree and the other index to measure its nonspecifcity. For nonspecifcity entropy, we gave some methods to construct this index and represent some examples to illustrate the efectiveness of it. With regard to fuzzy entropy, due to the difculty in the comparison of the interval number, we failed to give construction approaches, which is the important problem that we are going to focus in the near future. Furthermore, the theoretical frame of this paper can be used to quantify the uncertainty of more generalized fuzzy sets. For example, Fu and Zhao proposed the concept of the hesitant intuition fuzzy set, integrating the advantages of both hesitant fuzzy set and intuition fuzzy set [31], and Zhu et al. proposed the concept of the dual hesitant fuzzy set (DHFS) [32], as an extension of HFS to deal with the hesitant fuzzy set both for membership degree and nonmembership degree. Ren et al. introduced the normal wiggly hesitant fuzzy sets (NWHFS) as an extension of the hesitant fuzzy set [33]. So, how to apply the theoretical frame of this paper in the hesitant intuition fuzzy situation, the dual hesitant fuzzy situation, and the normal wiggly hesitant fuzzy information environment is an important topic. Based on the proposed two-tuple index, the fuzzy knowledge measure and accuracy measure can be developed further which can be used in pattern analysis and multiple attribute decision-making [34].
Based on the two-tuple entropy measure, the experts can construct interval-valued hesitant fuzzy preference relations in group decision-making problems. In order to guarantee that decision makers are nonrandom and logical and obtain reasonable decision results that are accepted by most decision makers, we can consider individual consistency control in consensus reaching processes for group decisionmaking problems [35]. Due to increasingly complicated decision conditions and relatively limited knowledge of decision makers, decision makers may provide incomplete interval-valued hesitant fuzzy preference relations, so how to apply the new two-tuple measure in an incomplete environment is also an important research topic. To fully consider the properties of social network evolution and improve the efciency of consensus reaching process in group decision-making, Dong et al. introduced the concept of the local world opinion derived from individuals' common friends and then proposed an individual and local world opinion-based opinion dynamics (OD) model [36]. As future work, the study of the OD model based on social network could be extended to interval-valued hesitant fuzzy preference relations in group decision-making problems [37]. Besides, how to apply the two-tuple index proposed in this paper to the OD model is an interesting research topic. Let l α be the number of intervals in α, then the mapping E F : H ⟶ [0, 1] defned as follows meets the axioms (E F 1) − (E F 4): Proof. We assume that E F (α) is defned as equation (A.1).

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te author declares that there are no conficts of interest.