# Dual Hesitant Fuzzy Sets

Authors: Bin Zhu; Zeshui Xu; Meimei Xia
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

In recent decades, several types of sets, such as fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, type 2 fuzzy sets, typen fuzzy sets, and hesitant fuzzy sets, have been introduced and investigated widely. In this paper, we propose dual hesitant fuzzy sets (DHFSs), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as special cases. Then we investigate the basic operations and properties of DHFSs. We also discuss the relationships among the sets mentioned above, use a notion of nested interval to reflect their common ground, then propose an extension principle of DHFSs. Additionally, we give an example to illustrate the application of DHFSs in group forecasting.


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## Body

## 1. 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 ofx 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 Section2, 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.


## 2. Preliminaries

### 2.1. 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 2.1 (see [1]).
Given a reference setX, a fuzzy set (FS) A on X is in terms of the function μ: X→[0,1].Definition 2.2 (see [18]).
LetM be the set of all fuzzy sets on μ: [0,1]→[0,1].Definition 2.3 (see [18]).
LetMi be the set of all fuzzy sets of type i on [0,1]. That is, the set of all μi: [0,1]→Mi-1, where M1 is defined as M.Apparently, if we use the functionsμ2 or μn 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]).
LetA and B be two multisets, and a the element in the reference set, then(1)
addition:countA⊕B(a)=countA(a)+countB(a);(2)
union:countA∪B(a)=max(countA(a),countB(a));(3)
intersection:countA∩B(a)=min(countA(a),countB(a)).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 SeqA(a) (the membership values a in a fuzzy multiset A) in decreasing order to define the union and intersection operators.


### 2.2. IFSs

Atanassov [6] gave the definition of IFSs as follows.Definition 2.5 (see [6]).
LetX 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, forallx∈X.We use〈x,μA,νA〉 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 setX be fixed, and let A (represented by the functions μA and νA), A1 (μA1 and νA1), A2 (μA2 and νA2), be three IFSs. Then the following operations are valid:(1)
complement:A̅={〈x,νA(x),μA(x)〉∣x∈X};(2)
union:A1∩A2={〈x,min{μA1(x),μA2(x)},max{νA1(x),νA2(x)}〉∣x∈X};(3)
intersection:A1∪A2={〈x,max{μA1(x),μA2(x)},min{νA1(x),νA2(x)}〉∣x∈X};(4)
⊕-union: A1⊕A2={〈x,μA1(x)+μA2(x)-μA1(x)μA2(x),νA1(x)νA2(x)〉∣x∈X};(5)
⊗-intersection: A1⊗A2={〈x,μA1(x)μA2(x),νA1(x)+νA2(x)-νA1(x)νA2(x)〉∣x∈X}.
De et al. [33] further gave another two operations of IFSs:(6)
nA={〈x,1-(1-μA(x))n,(νA(x))n〉∣x∈X};(7)
An={〈x,(μA(x))n,1-(1-νA(x))n〉∣x∈X}, where n is a positive integer.
Atanassov and Gargov [16] used the following:(1)
the mapf assigns to every IVFS A(=[μAL(x),μAU(x)]) an IFS, B=f(A) given by μB(x)=μAL(x), νA(x)=1-μAU(x);(2)
the mapg assigns to every IFS B(=〈μA(x),νA(x)〉) an IVFS A=f(B) given by μA(x)=[μA(x),1-νA(x)], to prove that IFSs and IVFSs are equipollent generalizations of the notion of FSs.Xu and Yager [34] called each pair (μA(x),νA(x)) an intuitionistic fuzzy number (IFN), and, for convenience, denoted an IFN by α=(μα,να). Moreover, they gave a simple method to rank any two IFNs, and introduced some of their operational laws as follows.Definition 2.7 (see [34]).
Letαi=(μαi,ναi)(i=1,2) be any two IFNs, sαi=μαi-ναi(i=1,2) the scores of αi(i=1,2), respectively, and hαi=μαi+ναi(i=1,2) the accuracy degrees of αi(i=1,2), respectively, then(i)
ifsα1>sα2, then α1 is larger than α2, denoted by α1>α2;(ii)
ifsα1=sα2, then
(1)
ifhα1=hα2, then α1 and α2 represent the same information, that is, μα1=μα2 and να1=να2, denoted by α1=α2;(2)
ifhα1>hα2, then α1 is larger than α2, denoted by α1>α2.


### 2.3. 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]).
LetX 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: h(x)=∅.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.Definition 2.9 (see [17, 18]).
Given an HFEh, the pair of functions h-=μ(h-=min{γ∣γ∈h}) and 1-h+=ν(h+=max{γ∣γ∈h}) define an intuitionistic fuzzy set (〈x,μ,ν〉), denoted by Aenv(h).According to this definition, IFSs can also be represented by HFSs, that is, for a given IFS{〈x,μA(x),νA(x)〉}, the corresponding HFS is h(x)=[μA(x),1-νA(x)], if μA(x)≠1-νA(x).Torra [17] gave the complement of a HFS as the following.Definition 2.10 (see [17]).
Given an HFS represented by its membership functionh, we define its complement as hc(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 HFSA on X, and h(x) for all x in X, then the HFS can be defined as a FMS: FMSA=⊕x∈X⊕γ∈h(x){(x,γ)}.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 HFEh, s(h)=(1/#h)∑γ∈hγ is called the score function of h, where #h is the number of the elements in h. Moreover, for two HFEs h1 and h2, if s(h1)>s(h2), then h1>h2; if s(h1)=s(h2), then h1=h2.


## 2.1. 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 2.1 (see [1]).
Given a reference setX, a fuzzy set (FS) A on X is in terms of the function μ: X→[0,1].Definition 2.2 (see [18]).
LetM be the set of all fuzzy sets on μ: [0,1]→[0,1].Definition 2.3 (see [18]).
LetMi be the set of all fuzzy sets of type i on [0,1]. That is, the set of all μi: [0,1]→Mi-1, where M1 is defined as M.Apparently, if we use the functionsμ2 or μn 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]).
LetA and B be two multisets, and a the element in the reference set, then(1)
addition:countA⊕B(a)=countA(a)+countB(a);(2)
union:countA∪B(a)=max(countA(a),countB(a));(3)
intersection:countA∩B(a)=min(countA(a),countB(a)).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 SeqA(a) (the membership values a in a fuzzy multiset A) in decreasing order to define the union and intersection operators.


## 2.2. IFSs

Atanassov [6] gave the definition of IFSs as follows.Definition 2.5 (see [6]).
LetX 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, forallx∈X.We use〈x,μA,νA〉 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 setX be fixed, and let A (represented by the functions μA and νA), A1 (μA1 and νA1), A2 (μA2 and νA2), be three IFSs. Then the following operations are valid:(1)
complement:A̅={〈x,νA(x),μA(x)〉∣x∈X};(2)
union:A1∩A2={〈x,min{μA1(x),μA2(x)},max{νA1(x),νA2(x)}〉∣x∈X};(3)
intersection:A1∪A2={〈x,max{μA1(x),μA2(x)},min{νA1(x),νA2(x)}〉∣x∈X};(4)
⊕-union: A1⊕A2={〈x,μA1(x)+μA2(x)-μA1(x)μA2(x),νA1(x)νA2(x)〉∣x∈X};(5)
⊗-intersection: A1⊗A2={〈x,μA1(x)μA2(x),νA1(x)+νA2(x)-νA1(x)νA2(x)〉∣x∈X}.
De et al. [33] further gave another two operations of IFSs:(6)
nA={〈x,1-(1-μA(x))n,(νA(x))n〉∣x∈X};(7)
An={〈x,(μA(x))n,1-(1-νA(x))n〉∣x∈X}, where n is a positive integer.
Atanassov and Gargov [16] used the following:(1)
the mapf assigns to every IVFS A(=[μAL(x),μAU(x)]) an IFS, B=f(A) given by μB(x)=μAL(x), νA(x)=1-μAU(x);(2)
the mapg assigns to every IFS B(=〈μA(x),νA(x)〉) an IVFS A=f(B) given by μA(x)=[μA(x),1-νA(x)], to prove that IFSs and IVFSs are equipollent generalizations of the notion of FSs.Xu and Yager [34] called each pair (μA(x),νA(x)) an intuitionistic fuzzy number (IFN), and, for convenience, denoted an IFN by α=(μα,να). Moreover, they gave a simple method to rank any two IFNs, and introduced some of their operational laws as follows.Definition 2.7 (see [34]).
Letαi=(μαi,ναi)(i=1,2) be any two IFNs, sαi=μαi-ναi(i=1,2) the scores of αi(i=1,2), respectively, and hαi=μαi+ναi(i=1,2) the accuracy degrees of αi(i=1,2), respectively, then(i)
ifsα1>sα2, then α1 is larger than α2, denoted by α1>α2;(ii)
ifsα1=sα2, then
(1)
ifhα1=hα2, then α1 and α2 represent the same information, that is, μα1=μα2 and να1=να2, denoted by α1=α2;(2)
ifhα1>hα2, then α1 is larger than α2, denoted by α1>α2.


## 2.3. 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]).
LetX 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: h(x)=∅.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.Definition 2.9 (see [17, 18]).
Given an HFEh, the pair of functions h-=μ(h-=min{γ∣γ∈h}) and 1-h+=ν(h+=max{γ∣γ∈h}) define an intuitionistic fuzzy set (〈x,μ,ν〉), denoted by Aenv(h).According to this definition, IFSs can also be represented by HFSs, that is, for a given IFS{〈x,μA(x),νA(x)〉}, the corresponding HFS is h(x)=[μA(x),1-νA(x)], if μA(x)≠1-νA(x).Torra [17] gave the complement of a HFS as the following.Definition 2.10 (see [17]).
Given an HFS represented by its membership functionh, we define its complement as hc(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 HFSA on X, and h(x) for all x in X, then the HFS can be defined as a FMS: FMSA=⊕x∈X⊕γ∈h(x){(x,γ)}.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 HFEh, s(h)=(1/#h)∑γ∈hγ is called the score function of h, where #h is the number of the elements in h. Moreover, for two HFEs h1 and h2, if s(h1)>s(h2), then h1>h2; if s(h1)=s(h2), then h1=h2.


## 3. DHFSs

### 3.1. 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.
LetX be a fixed set, then a dual hesitant fuzzy set (DHFS) D on X is described as:
(3.1)D={〈x,h(x),g(x)〉x∈X},
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:
(3.2)0≤γ,η≤1,0≤γ++η+≤1,
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: γ∈h, η∈g, γ+∈h+=∪γ∈hmax{γ}, η+∈g+=∪η∈gmax{η}, 0≤γ,η≤1, and 0≤γ++η+≤1.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];(4)
nonsensical element:d=∅(h=∅,g=∅).Based on the background knowledge introduced in Section2, 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 γ-(γ-∈h-=∪γ∈h(x)min{γ}), γ+, η-(η-∈g-=∪η∈g(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:(3.3)h=[γ-,γ+],g=[η-,η+].Based on Definition2.9, there is a transformation between IFSs and HFSs, we can also transform g to h2, that is, the number 2 HFE h2(x)=[1-η+,1-η-] denoting the possible membership degrees of the element x∈X. Thus, both h and h2 indicate the membership degrees, we can use a “nested interval” to represent d(x) as:(3.4)d=[[γ-,γ+],[1-η+,1-η-]].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 DHFEd, which is divided into different cases as follows:(3.5)d={∅,ifg=∅,h=∅,(γ),{ifg=∅,h≠∅,γ-=γ+=γ,ifg≠∅,h≠∅,γ-=γ+=γ=1-η-=1-η+=1-η,(1-η),ifg≠∅,h=∅,η-=η+=η,[γ-,γ+],ifg=∅,h≠∅,γ-≠γ+,[1-η+,1-η-],ifg≠∅,h=∅,η-≠η+,[γ,[1-η+,1-η-]],ifg≠∅,h≠∅,η-≠η+,γ-=γ+=γ,[[γ-,γ+],η],ifg≠∅,h≠∅,γ-≠γ+,η-=η+=η,[[γ-,γ+],[1-η+,1-η-]],ifg≠∅,h≠∅,η-≠η+,γ-≠γ+,
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.


### 3.2. Basic Operations and Properties of DHFSs

Atanassov [6] and 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 functiond, and d≠∅, its complement is defined as:
(3.6)dc={∪γ∈h,η∈g{{η},{γ}},ifg≠∅,h≠∅,∪γ∈h{{1-γ},{∅}},ifg=∅,h≠∅,∪η∈g{{∅},{1-η}},ifh=∅,g≠∅.
Apparently, the complement is involutive represented as (dc)c=d.We now define the union and the intersection of DHFSs. For two DHFSsd1 and d2, it is clear that the corresponding lower and upper bounds to h and g are h-, h+, g- and g+, respectively, where h-=∪γ∈hmin{γ}, h+=∪γ∈hmax{γ}, g-=∪η∈gmin{η}, and g+=∪η∈gmax{η} represent this group notations and no confusion will arise in the rest of this paper.Definition 3.3.
LetX be a fixed set, d1 and d2 two DHFEs, we define their union and intersection, respectively, as:
(1)
d1∪d2={h∈(h1∪h2)∣h≥max(h1-,h2-),g∈(g1∩g2)∣g≤min(g1+,g2+)};(2)
d1∩d2={h∈(h1∩h2)∣h≤min(h1+,h2+),g∈(g1∪g2)∣g≥max(g1-,g2-)},
the following operations are valid:(1)
⊕-union: d1⊕d2={hd1⊕hd2,gd1⊗gd2}=∪γd1∈hd1,ηd1∈gd1,γd2∈hd2,ηd2∈gd2{{γd1+γd2-γd1γd2},{ηd1ηd2}};(2)
⊕-intersection: d1⊗d2={hd1⊗hd2,gd1⊕gd2}=∪γd1∈hd1,ηd1∈gd1,γd2∈hd2,ηd2∈gd2{{γd1γd2},{ηd1+ηd2-ηd1ηd2}};(3)
nd=∪γd∈hd,ηd∈gd{1-(1-γd)n,(ηd)n};(4)
dn=∪γd∈hd,ηd∈gd{(γd)n,1-(1-ηd)n}, where n is a positive integral and all the results are also DHFEs.Example 3.4.
Letd1={{0.1,0.3,0.4},{0.3,0.5}} and d2={{0.2,0.5},{0.1,0.2,0.4}} be two DHFEs, then we have(1)
complement:d1c={{0.3,0.5},{0.1,0.3,0.4}};(2)
union:d1∪d2={{0,2,0.3,0.4,0.5},{0.1,0.2,0.3,0.4}};(3)
intersection:d1∩d2={{0,1,0.2,0.3,0.4},{0.3,0.4,0.5}}.We can easily prove the following theorem according to Definition3.3Theorem 3.5.
Letd, d1, and d2 be any three DHFEs, λ≥0, then(1)
d1⊕d2=d2⊕d1;(2)
d1⊗d2=d2⊗d1;(3)
λ(d1⊗d2)=λd1⊗λd2;(4)
(d1⊗d2)λ=d1λ⊗d2λ.To compare the DHFEs, and based on Definitions2.7 and 2.12, we give the following comparison laws.Definition 3.6.
Letdi={hdi,gdi}(i=1,2) be any two DHFEs, sdi=(1/#h)∑γ∈hγ-(1/#g)∑η∈gη(i=1,2) the score  function of di(i=1,2), and pdi=(1/#h)∑γ∈hγ+(1/#g)∑η∈gη(i=1,2) the accuracy function of di(i=1,2), where #h and #g are the numbers of the elements in h and g, respectively, then(i)
ifsd1>sd2, then d1 is superior to d2, denoted by d1≻d2;(ii)
ifsd1=sd2, then
(1)
ifpd1=pd2, then d1 is equivalent to d2, denoted by d1~d2;(2)
Ifpd1>pd2, then d1 is superior than d2, denoted by d1≻d2.Example 3.7.
Letd1={{0.1,0.3},{0.3,0.5}} and d2={{0.2,0.4},{0.4,0.6}} be two DHFEs, then based on Definition 3.1, we obtain sd1=sd2=0, pd2(0.8)>pd1(0.6), and thus, d2≻d1.


## 3.1. 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.
LetX be a fixed set, then a dual hesitant fuzzy set (DHFS) D on X is described as:
(3.1)D={〈x,h(x),g(x)〉x∈X},
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:
(3.2)0≤γ,η≤1,0≤γ++η+≤1,
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: γ∈h, η∈g, γ+∈h+=∪γ∈hmax{γ}, η+∈g+=∪η∈gmax{η}, 0≤γ,η≤1, and 0≤γ++η+≤1.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];(4)
nonsensical element:d=∅(h=∅,g=∅).Based on the background knowledge introduced in Section2, 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 γ-(γ-∈h-=∪γ∈h(x)min{γ}), γ+, η-(η-∈g-=∪η∈g(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:(3.3)h=[γ-,γ+],g=[η-,η+].Based on Definition2.9, there is a transformation between IFSs and HFSs, we can also transform g to h2, that is, the number 2 HFE h2(x)=[1-η+,1-η-] denoting the possible membership degrees of the element x∈X. Thus, both h and h2 indicate the membership degrees, we can use a “nested interval” to represent d(x) as:(3.4)d=[[γ-,γ+],[1-η+,1-η-]].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 DHFEd, which is divided into different cases as follows:(3.5)d={∅,ifg=∅,h=∅,(γ),{ifg=∅,h≠∅,γ-=γ+=γ,ifg≠∅,h≠∅,γ-=γ+=γ=1-η-=1-η+=1-η,(1-η),ifg≠∅,h=∅,η-=η+=η,[γ-,γ+],ifg=∅,h≠∅,γ-≠γ+,[1-η+,1-η-],ifg≠∅,h=∅,η-≠η+,[γ,[1-η+,1-η-]],ifg≠∅,h≠∅,η-≠η+,γ-=γ+=γ,[[γ-,γ+],η],ifg≠∅,h≠∅,γ-≠γ+,η-=η+=η,[[γ-,γ+],[1-η+,1-η-]],ifg≠∅,h≠∅,η-≠η+,γ-≠γ+,
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.


## 3.2. Basic Operations and Properties of DHFSs

Atanassov [6] and 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 functiond, and d≠∅, its complement is defined as:
(3.6)dc={∪γ∈h,η∈g{{η},{γ}},ifg≠∅,h≠∅,∪γ∈h{{1-γ},{∅}},ifg=∅,h≠∅,∪η∈g{{∅},{1-η}},ifh=∅,g≠∅.
Apparently, the complement is involutive represented as (dc)c=d.We now define the union and the intersection of DHFSs. For two DHFSsd1 and d2, it is clear that the corresponding lower and upper bounds to h and g are h-, h+, g- and g+, respectively, where h-=∪γ∈hmin{γ}, h+=∪γ∈hmax{γ}, g-=∪η∈gmin{η}, and g+=∪η∈gmax{η} represent this group notations and no confusion will arise in the rest of this paper.Definition 3.3.
LetX be a fixed set, d1 and d2 two DHFEs, we define their union and intersection, respectively, as:
(1)
d1∪d2={h∈(h1∪h2)∣h≥max(h1-,h2-),g∈(g1∩g2)∣g≤min(g1+,g2+)};(2)
d1∩d2={h∈(h1∩h2)∣h≤min(h1+,h2+),g∈(g1∪g2)∣g≥max(g1-,g2-)},
the following operations are valid:(1)
⊕-union: d1⊕d2={hd1⊕hd2,gd1⊗gd2}=∪γd1∈hd1,ηd1∈gd1,γd2∈hd2,ηd2∈gd2{{γd1+γd2-γd1γd2},{ηd1ηd2}};(2)
⊕-intersection: d1⊗d2={hd1⊗hd2,gd1⊕gd2}=∪γd1∈hd1,ηd1∈gd1,γd2∈hd2,ηd2∈gd2{{γd1γd2},{ηd1+ηd2-ηd1ηd2}};(3)
nd=∪γd∈hd,ηd∈gd{1-(1-γd)n,(ηd)n};(4)
dn=∪γd∈hd,ηd∈gd{(γd)n,1-(1-ηd)n}, where n is a positive integral and all the results are also DHFEs.Example 3.4.
Letd1={{0.1,0.3,0.4},{0.3,0.5}} and d2={{0.2,0.5},{0.1,0.2,0.4}} be two DHFEs, then we have(1)
complement:d1c={{0.3,0.5},{0.1,0.3,0.4}};(2)
union:d1∪d2={{0,2,0.3,0.4,0.5},{0.1,0.2,0.3,0.4}};(3)
intersection:d1∩d2={{0,1,0.2,0.3,0.4},{0.3,0.4,0.5}}.We can easily prove the following theorem according to Definition3.3Theorem 3.5.
Letd, d1, and d2 be any three DHFEs, λ≥0, then(1)
d1⊕d2=d2⊕d1;(2)
d1⊗d2=d2⊗d1;(3)
λ(d1⊗d2)=λd1⊗λd2;(4)
(d1⊗d2)λ=d1λ⊗d2λ.To compare the DHFEs, and based on Definitions2.7 and 2.12, we give the following comparison laws.Definition 3.6.
Letdi={hdi,gdi}(i=1,2) be any two DHFEs, sdi=(1/#h)∑γ∈hγ-(1/#g)∑η∈gη(i=1,2) the score  function of di(i=1,2), and pdi=(1/#h)∑γ∈hγ+(1/#g)∑η∈gη(i=1,2) the accuracy function of di(i=1,2), where #h and #g are the numbers of the elements in h and g, respectively, then(i)
ifsd1>sd2, then d1 is superior to d2, denoted by d1≻d2;(ii)
ifsd1=sd2, then
(1)
ifpd1=pd2, then d1 is equivalent to d2, denoted by d1~d2;(2)
Ifpd1>pd2, then d1 is superior than d2, denoted by d1≻d2.Example 3.7.
Letd1={{0.1,0.3},{0.3,0.5}} and d2={{0.2,0.4},{0.4,0.6}} be two DHFEs, then based on Definition 3.1, we obtain sd1=sd2=0, pd2(0.8)>pd1(0.6), and thus, d2≻d1.


## 4. 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]).
LetO be a function O: [0,1]N→[0,1], and H a set of N HFSs on the reference set X (i.e., H={h1,h2,…,hN} is a HFS on X). Then, the extension of O on H is defined for each x in X by
(4.1)OH(x)=∪γ∈{h1(x)×h2(x)×⋯×hN(x)}{O(γ)}.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]).
LetA: [0,1]N→[0,1] be a modular aggregation function (modular average), A(x)=∑i=1nfi(xi). Then, its symmetrization SA is called the OMA, that is, SA=OMA is given by
(4.2)OMA(x)=∑i=1nfi(xσ(i)),
where f1,f2,…,fn: [0,1]→[0,1] are the nondecreasing functions satisfying ∑i=1nfi=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 operatorℤ considers all the values in the DHFSs and the application of ℤ on them, which is defined as follows.Definition 4.3.
Let the functionsℂ: [0,1]N→[0,1] and ℕ: [0,1]N→[0,1], and let D be a set of ndual hesitant fuzzy sets on the reference set X represented as D={d1,d2,…,dn}={〈x,{h(x)},{g(x)}〉}. Then, the extension of ℤ on D is defined for each x in X by
(4.3)ZD(x)=∪γ1∈h1(x),γ2∈h2(x),…,γn∈hn(x);η1∈g1(x),η2∈g2(x),…,ηn∈gn(x)={C(f1(γ1),f2(γ2),…,fn(γn)),N(f1(η1),f2(η2),…,fn(ηn))},
where fi(x): [0,1]→[0,1], i=1,2,…n.If we letfi(x)=x(i=1,2,…n), some basic operations can be obtained according to this extension principle in Example 4.4.Example 4.4.
Letd1 and d2 be two DHFEs such that
(4.4)d1={{0.1,0.3,0.4},{0.3,0.5}},d2={{0.2,0.5},{0.1,0.2,0.4}}.
If ℂ=(γ1,γ2)max(γ1,γ2) and ℕ(η1,η2)=min(η1,η2), then
(4.5)d1∪d2=Zd1∪d2=∪γ1∈h1,γ2∈h2,η1∈g1,η2∈g2{max⁡(γ1,γ2),min⁡(η1,η2)}={{0,2,0.3,0.4,0.5},{0.1,0.2,0.3,0.4}}.
Similarly, we have(4.6)d1∩d2=Zd1∩d2=∪γ1∈h1,γ2∈h2,η1∈g1,η2∈g2{min⁡(γ1,γ2),max⁡(η1,η2)}={{0,1,0.2,0.3,0.4},{0.3,0.4,0.5}}.
In particular, ifγ1=γ2, then min(γ1,γ2) is a real number; if η1=η2, then max(η1,η2) is a real number:
(4.7)d1c=Zd1c=∪γ1∈h1,η1∈g1{{η},{γ}}={{0.3,0.5},{0.1,0.3,0.4}};d1⊕d2=Zd1⊕d2=∪γ1∈h1,γ2∈h2,η1∈g1,η2∈g2{(γ1⊕γ2),(η1⊗η2)}={{0.28,0.55,0.44,0.65,0.52,0.70},{0.05,0.10,0.20,0.03,0.06,0.12}};d1⊗d2=Zd1⊗d2=∪γ1{(γ1⊗γ2),(η1⊕η2)}={{0.02,0.05,0.06,0.15,0.08,0.20},{0.37,0.44,0.05,0.55,0.03,0.19}}.Furthermore, we can get lots of aggregation operators by this extension principle. For example, if we letfi(xi)=ωixi, ∑i=1nωi=1 and ℂ(f1(γ1),f2(γ2),…,fn(γn))=⊕i=1nωiγi, 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 subsidiariesyi(i=1,2,3) with respect to the predictive values of the net incomes cj(j=1,2,3) are represented by the DHFEs dij=∪γij∈hij,ηij∈gij{{γij},{ηij}}, where γij indicates the degree that the alternative yi satisfies the criterion cj, ηij indicates the degree that the alternative yi does not satisfy the criterion cj, such that γij∈[0,1], ηij∈[0,1], γij++ηij+≤1, for details see Table 1.Table 1
The DHFEs for group forecasting.
SubsidiariescjPredictive values (million)dijy1c190{{0.3,0.4},{0.5}}c2100{{0.3},{0.4,0.5,0.6}}c350{{0.4},{0.4}}y2c1100{{0.5},{∅}}c270{{0.3,0.4},{0.4,0.5,0.6}}c360{{0.7},{0.1,0,2}}y3c170{{∅},{0.5,0.6}}c2100{{0.5,0.6,0.7},{∅}}c380{{0.4},{0.3,0.4}}In what follows, we give an  approach for group foresting in terms of DHFEs as follows.Step 1.
Utilize the score function of DHFEs (Definition3.6) to obtain the score of each DHFE, and transform the results into the normalizations by the method given as fi=∑j=13(s(dij)+1)/2, bij=(s(dij)+1)/2fi where bij∈[0,1], i,j=1,2,3as shown in Table 2.Table 2b11b12b13b21b22b23b31b32b330.5150.48500.3850.3180.2970.1450.5160.339Step 2.
Useei=cij×bij(i,j=1,2,3) to obtain the expectations of net incomes as shown in Table 3.
Thus,e1>e3>e2, and then y1≻y3≻y2, that is, y1 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.Table 3e1e2e394.8578.5888.87


## 5. 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.



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*Source: 879629-2012-05-10.xml*