Dual Hesitant Fuzzy Sets

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Introduction
Since Zadeh 1 introduced fuzzy sets FSs and gave intensive research 2-5 , several famous extensions have been developed, such as intuitionistic fuzzy sets IFSs 6 , type 2 fuzzy sets T2FSs 3, 7 , type n fuzzy sets TnFSs , fuzzy multisets FMSs 8-14 , interval-valued fuzzy sets IVFSs 3, 15 , interval-valued intuitionistic fuzzy sets IVIFSs 16 , and hesitant fuzzy sets HFSs 17-20 .Actually, these sets have given various ways to assign the membership degree or the nonmembership degree of an element to a given set characterized by different properties.
IFSs, also known as IVFSs from a mathematical point of view, can be modeled with two functions that define an interval to reflect some uncertainty on the membership function of the elements.IVFSs are the generalization of FSs and can model uncertainty due to the lack of information, in which a closed subinterval of 0, 1 is assigned to the membership degree.Atanassov and Gargov 16 proved that IFSs and IVFSs are equipollent generalizations of FSs, and proposed the notion of IVIFS, which has been studied and used extensively 21-25 .T2FSs, described by membership functions that are characterized by more parameters, permit the fuzzy membership as a fuzzy set improving the modeling capability than the original one.Mathematically, IFSs can be seen as a particular case of T2FSs, where the membership function returns a set of crisp intervals.Despite the wide applications of T2FSs 26-29 , they have difficulties in establishing the secondary membership functions and difficulties in manipulation 30-32 .FMSs are another generalization of FSs that permit multiple occurrences of an element, and correspond to the case where the membership degrees to the multisets are not Boolean but fuzzy.Note that although the features of FMSs allow the application to information retrieval on the world wide web, where a search engine retrieves multiple occurrences of the same subjects with possible different degrees of relevance 10 , they have problems with the basic operations, such as the definitions for union and intersection, which do not generalize the ones for FSs.Miyamoto 13 gave an alternative definition that emphasizes the usefulness of a commutative property between a set operation and an α-cut, resolving this problem 8-13 .
HFSs were originally introduced by Torra 17,18 .The motivation to propose the HFSs is that when defining the membership of an element, the difficulty of establishing the membership degree is not a margin of error as in IFSs , or some possibility distribution as in T2FSs on the possible values, but a set of possible values.Torra 17 reviewed IFSs and FMSs, drew comparisons, and created inherent connections among them.He pointed out that the operations for FMSs do not apply correctly to HFSs, although in some situations we can use FMSs as a model.HFSs were deemed IFSs when the HFS is a nonempty closed interval.Based on the relationships between IFSs and HFSs, Torra 17 gave a definition corresponding to the envelope of HFS.Xu and Xia 19, 20 investigated the aggregation operators, distance, and similarity measures for HFSs and applied them to decision making.
In this paper, we introduce dual hesitant fuzzy set DHFS , which is a new extension of FS.As we know, in natural language, many categories cannot be distinguished clearly, but can be represented by a matter of degree in the notion of fuzziness.For example, when we talk about fish and monkeys, clear separation can be recognized between them.However, the borderline may not be easy to be distinguished with respect to starfish or bacteria.Although sometimes humans cannot recognize an object clearly, this class of fuzzy recognition against preciseness plays a vital role in human thinking, pattern recognition, and communication of information.The FS, which is stuck into the transition between the membership and the nonmembership, is the gradualness of predication.Zadeh's original intuition 1 is to show the objectivity of truth as "gradual rather than abrupt."Atanassov's IFS 6 used two functions to handle the membership and the nonmembership separately, as it seems to be the case in the human brain, which is limited by the perception of shades.The membership and the nonmembership represent the opposite epistemic degrees, apparently, the membership comes to grips with epistemic certainty, and the nonmembership comes to grips with epistemic uncertainty, they can reflect the gradual epistemic degrees respectively to be the bipolar notions.Similar to HFSs, we can also use the nonmembership to deal with a set of possible values manifesting either a precise gradual composite entity or an epistemic construction refereeing to an ill-known object.
Furthermore, DHFSs consist of two parts, that is, the membership hesitancy function and the nonmembership hesitancy function, supporting a more exemplary and flexible access to assign values for each element in the domain, and we have to handle two kinds of hesitancy in this situation.The existing sets, including FSs, IFSs, HFSs, and FMSs, can be regarded as special cases of DHFSs; we do not confront an interval of possibilities as in IVFSs or IVIFSs , or some possibility distributions as in T2FSs on the possible values, or multiple occurrences of an element as in FMSs , but several different possible values indicate the epistemic degrees whether certainty or uncertainty.For example, in a multicriteria decision-making problem, some decision makers consider as possible values for the membership degree of x into the set A a few different values 0.1, 0.2, and 0.3, and for the nonmembership degrees 0.4, 0.5 and 0.6 replacing just one number or a tuple.So, the certainty and uncertainty on the possible values are somehow limited, respectively, which can reflect the original information given by the decision makers as much as possible.Utilizing DHFSs can take much more information into account, the more values we obtain from the decision makers, the greater epistemic certainty we have, and thus, compared to the existing sets mentioned above, DHFS can be regarded as a more comprehensive set, which supports a more flexible approach when the decision makers provide their judgments.
We organize the remainder of the paper as follows.In Section 2, we review some basic knowledge of the existing sets.Section 3 proposes DHFSs and investigates some of their basic operations and properties.Then, in Section 4, we present an extension principle of DHFSs, and give some examples to illustrate our results.Section 5 ends the paper with the concluding remarks.

FSs, T2FSs, TnFSs, and FMSs
In this section, we review some basic definitions and operations, necessary to understand the proposal of the DHFS and its use.
Definition 2.2 see 18 .Let M be the set of all fuzzy sets on μ: 0, 1 → 0, 1 .Definition 2.3 see 18 .Let M i be the set of all fuzzy sets of type i on 0, 1 .That is, the set of all μ i : 0, 1 → M i−1 , where M 1 is defined as M.
Apparently, if we use the functions μ 2 or μ n to replace the membership function μ, and returns an FS, then we obtain the notions of type 2 fuzzy sets T2FSs and type n fuzzy sets TnFSs .
Yager 14 and Miyamoto 8-13 first studied FMSs and defined several basic operations.FMSs generalize the multisets, which are also known as bags allowing multiple occurrences of elements, associating with the membership degrees.Definition 2.4 see 14 .Let A and B be two multisets, and a the element in the reference set, then However, this definition comes into conflict with FSs.Miyamoto 8-13 gave the corresponding solutions.He proposed an alternative definition for FMSs.Using the membership sequence Seq A a the membership values a in a fuzzy multiset A in decreasing order to define the union and intersection operators.Atanassov 6 gave the definition of IFSs as follows.

IFSs
Definition 2.5 see 6 .Let X be a fixed set, an intuitionistic fuzzy set IFS A on X is represented in terms of two functions μ: X → 0, 1 and ν: X → 0, 1 , with the condition 0 ≤ μ x ν x ≤ 1, for all x ∈ X.
We use x, μ A , ν A for all x ∈ X to represent IFSs considered in the rest of the paper without explicitly mentioning it.
Furthermore, π x 1 − μ x − ν x is called a hesitancy degree or an intuitionistic index of x in A. In the special case π x 0, that is, μ x ν x 1, the IFS A reduces to an FS.Atanassov 6 and De et al. 33 gave some basic operations on IFSs, which ensure that the operational results are also IFSs.Definition 2.6 see 6 .Let a set X be fixed, and let A represented by the functions μ A and ν A , A 1 μ A 1 and ν A 1 , A 2 μ A 2 and ν A 2 , be three IFSs.Then the following operations are valid: De et al. 33 further gave another two operations of IFSs: where n is a positive integer, Atanassov and Gargov 16 used the following: 2 the map g assigns to every IFS B μ A x , ν A x an IVFS A f B given by μ A x μ A x , 1 − ν A x , to prove that IFSs and IVFSs are equipollent generalizations of the notion of FSs.
Xu and Yager 34 called each pair μ A x , ν A x an intuitionistic fuzzy number IFN , and, for convenience, denoted an IFN by α μ α , ν α .Moreover, they gave a simple method to rank any two IFNs, and introduced some of their operational laws as follows.

HFSs
Torra 17 defined the HFS in terms of a function that returns a set of membership values for each element in the domain and in terms of the union of their memberships.
Definition 2.8 see 17 .Let X be a fixed set, then we define hesitant fuzzy set HFS on X is in terms of a function h applied to X returns a subset of 0, 1 , and h x a hesitant fuzzy element HFE .Then, Torra 17 gave an example to show several special sets for all x in X: 1 empty set: h x {0}; 2 full set: h x {1}; 3 complete ignorance: h x 0, 1 ; 4 nonsense set: Apparently, this definition encompasses IFSs as a particular case in the form of a nonempty closed interval, and also a particular case of T2FSs from a mathematical point of view.
Torra and Narukawa 18 and Torra 17 showed that the envelop of a HFE is an IFN, expressed in the following definition.According to this definition, IFSs can also be represented by HFSs, that is, for a given IFS { x, μ A x , ν A x }, the corresponding HFS is h Torra 17 gave the complement of a HFS as the following.
Definition 2.10 see 17 .Given an HFS represented by its membership function h, we define its complement as h c x ∪ γ∈h x {1 − γ}.
Additionally, Torra 17 considered the relationships between HFSs and fuzzy multisets FMSs .He proved that a HFS can be represented a FMS.Definition 2.11 see 17 .Given a HFS A on X, and h x for all x in X, then the HFS can be defined as a FMS: Torra 17 also proved that the union and the intersection of two corresponding FMSs do not correspond to the union and the intersection of two HFSs.
Xia and Xu 19 gave a method to rank any two HFEs as the following.
Definition 2.12 see 19 .For a HFE h, s h 1/#h γ∈h γ is called the score function of h, where #h is the number of the elements in h.Moreover, for two HFEs h 1 and h 2 , if s h 1 > s h 2 , then h 1 > h 2 ; if s h 1 s h 2 , then h 1 h 2 .

The Notion of DHFS
We now define dual hesitant fuzzy set in terms of two functions that return two sets of membership values and nonmembership values, respectively, for each element in the domain as follows.
Definition 3.1.Let X be a fixed set, then a dual hesitant fuzzy set DHFS D on X is described as: in which h x and g x are two sets of some values in 0, 1 , denoting the possible membership degrees and nonmembership degrees of the element x ∈ X to the set D, respectively, with the conditions: where γ ∈ h x , η ∈ g x , γ ∈ h x ∪ γ∈h x max{γ}, and η ∈ g x ∪ η∈g x max{η} for all x ∈ X.For convenience, the pair d x h x , g x is called a dual hesitant fuzzy element DHFE denoted by d h, g , with the conditions: First we define some special DHFEs.Given a DHFE, d, then we have 1 complete uncertainty: d {{0}, {1}}; 2 complete certainty: d {{1}, {0}}; 3 complete ill-known all is possible : d 0, 1 ; Based on the background knowledge introduced in Section 2, we can obtain some results in special cases.For a given d / ∅, if h and g have only one value γ and η, respectively, and γ η < 1, then the DHFS reduces to an IFS.If h and g have only one value γ and η respectively, and γ η 1, or h owns one value, and g ∅, then the DHFS reduces to an FS also can be regarded as HFSs .If g ∅ and h / ∅, then the DHFS reduces to a HFS, and according to Definition 2.11, DHFSs can be defined as FMSs.Thus the definition of DHFSs encompasses these fuzzy sets above.Next we will discuss the DHFS in detail and use x min{η} , η in the rest of the paper without explicitly mentioning it.
Actually, for a typical DHFS, h and g can be represented by two intervals as: Based on Definition 2.9, there is a transformation between IFSs and HFSs, we can also transform g to h 2 , that is, the number 2 HFE h 2 x 1 − η , 1 − η − denoting the possible membership degrees of the element x ∈ X.Thus, both h and h 2 indicate the membership degrees, we can use a "nested interval" to represent d x as: The common ground of these sets is to reflect fuzzy degrees to an object, according to either fuzzy numbers or interval fuzzy numbers.Therefore, we use nonempty closed interval as a uniform framework to indicate a DHFE d, which is divided into different cases as follows: which reflects the connections among all the sets mentioned above, and the merit of DHFS is more flexible to be valued in multifold ways according to the practical demands than the existing sets, taking much more information given by decision makers into account.

3.6
Apparently, the complement is involutive represented as d c c d.
We now define the union and the intersection of DHFSs.For two DHFSs d 1 and d 2 , it is clear that the corresponding lower and upper bounds to h and g are h − , h , g − and g , respectively, where h − ∪ γ∈h min{γ}, h ∪ γ∈h max{γ}, g − ∪ η∈g min{η}, and g ∪ η∈g max{η} represent this group notations and no confusion will arise in the rest of this paper.Definition 3.3.Let X be a fixed set, d 1 and d 2 two DHFEs, we define their union and intersection, respectively, as: 1 , the following operations are valid: We can easily prove the following theorem according to Definition 3.3 Theorem 3.5.Let d, d 1 , and d 2 be any three DHFEs, λ ≥ 0, then To compare the DHFEs, and based on Definitions 2.7 and 2.12, we give the following comparison laws.and p , where #h and #g are the numbers of the elements in h and g, respectively, then

Extension Principle
Torra and Narukawa 18 introduced an extension principle applied to HFSs, which permits us to export operations on fuzzy sets to new types of sets.The extension of an operator O on a set of HFSs considers all the values in such sets and the application of O on them.The definition is as the following.Definition 4.1 see 18 .Let O be a function O: 0, 1 N → 0, 1 , and H a set of N HFSs on the reference set X i.e., H {h 1 , h 2 , . . ., h N } is a HFS on X .Then, the extension of O on H is defined for each x in X by Mesiar and Mesiarova-Zemankova 35 investigated the ordered modular average OMA , which generalizes the ordered weighted average OWA operator, with the replacement of the additivity property by the modularity.The linear interpolating functions of the OWA operator were replaced by rather general nondecreasing functions in the OMA, which can be used to aggregate any finite number of input arguments.Definition 4.2 see 35 .Let A: 0, 1 N → 0, 1 be a modular aggregation function modular average , A x n i 1 f i x i .Then, its symmetrization S A is called the OMA, that is, S A OMA is given by where f 1 , f 2 , . . ., f n : 0, 1 → 0, 1 are the nondecreasing functions satisfying n i 1 f i id identity on 0, 1 .
Motivated by the extension principle of HFSs and the OMA, we propose an extension principle based on the OMA so as to develop basic operators and aggregation operations of DHFS.With respect to DHFSs, the new extension of an operator Z considers all the values in the DHFSs and the application of Z on them, which is defined as follows.

4.7
Furthermore, we can get lots of aggregation operators by this extension principle.For example, if we let f i x i ω i x i , n i 1 ω i 1 and C f 1 γ 1 , f 2 γ 2 , . . ., f n γ n ⊕ n i 1 ω i γ i , then we can obtain the weighted dual hesitant fuzzy averaging WDHFA operator, which will not be discussed in this paper.We now give another example, an application of DHFSs to group forecasting.In a traditional forecasting problem, the probability of uncertain event is often used to obtain expectations, however, it cannot reflect opinions from all decision makers, nor can it depict epistemic degrees of certainty and uncertainty in the same time.So, the DHFSs are employed to replace the probability in next example.
Example 4.5.Several directors of a pharmaceutical company need to decide the additional investment priorities to three subsidiaries in the next quarter based on the net income foresting of them.Assume that the epistemic degrees of three subsidiaries y i i 1, 2, 3 with respect to the predictive values of the net incomes c j j 1, 2, 3 are represented by the DHFEs where γ ij indicates the degree that the alternative y i satisfies the criterion c j , η ij indicates the degree that the alternative y i does not satisfy the criterion c j , such that γ ij ∈ 0, 1 , η ij ∈ 0, 1 , γ ij η ij ≤ 1, for details see Table 1.In what follows, we give an approach for group foresting in terms of DHFEs as follows.
Step 1. Utilize the score function of DHFEs Definition 3.6 to obtain the score of each DHFE, and transform the results into the normalizations by the method given as f i 3 j 1 s d ij 1 /2, b ij s d ij 1 /2f i where b ij ∈ 0, 1 , i, j 1, 2, 3as shown in Table 2.
Step 2. Use e i c ij × b ij i, j 1, 2, 3 to obtain the expectations of net incomes as shown in Table 3.
Thus, e 1 > e 3 > e 2 , and then y 1 y 3 y 2 , that is, y 1 is the optimal choice of additional investment.Obviously, the DHFS is an effective and convenient tool applied to group forecasting.A transparent result can be obtained by utilizing the DHFS, which reflects the epistemic degree to the predictive values of net incomes.Comparing with other types of FSs, the DHFSs can take the information from the decision makers directors into account as much as possible, and it is more flexible in practical applications.

Concluding Remarks
In this paper, we have introduced the dual hesitant fuzzy set DHFS , which is a comprehensive set encompassing several existing sets, and whose membership degrees and nonmembership degrees are represented by a set of possible values.The common ground on the existing sets and the DHFS has been found out.Although in special cases, the DHFS can be reduced to some existing ones, it has the desirable characteristics and advantages of its own and appears to be a more flexible method to be valued in multifold ways according to the practical demands than the existing fuzzy sets, taking much more information given by decision makers into account.We have investigated some basic operations and properties of DHFSs, and an extension principle for DHFSs has also been developed for further study of basic operations and aggregation operators.Our results have been illustrated by a practical example of group forecasting.
Atanassov 6and Torra 17 gave the complements of the IFSs and the HFSs, respectively, according to Definitions 2.6 and 2.10.In the following, we define the complement of the DHFS depending on different situations.
Definition 3.2.Given a DHFE represented by the function d, and d / ∅, its complement is defined as: where n is a positive integral and all the results are also DHFEs.

Table 1 :
The DHFEs for group forecasting.