#### 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, type 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.

#### 1. Introduction

Since Zadeh [1] introduced fuzzy sets (FSs) and gave intensive research [2β5], several famous extensions have been developed, such as intuitionistic fuzzy sets (IFSs) [6], type 2 fuzzy sets (T2FSs) [3, 7], type fuzzy sets (TnFSs), fuzzy multisets (FMSs) [8β14], interval-valued fuzzy sets (IVFSs) [3, 15], interval-valued intuitionistic fuzzy sets (IVIFSs) [16], and hesitant fuzzy sets (HFSs) [17β20]. 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 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 [21β25].

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 [26β29], they have difficulties in establishing the secondary membership functions and difficulties in manipulation [30β32].

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 -*cut*, resolving this problem [8β13].

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. Zadehβs original intuition [1] is to show the objectivity of truth as βgradual rather than abrupt.β Atanassovβs 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 into the set 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.

#### 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 set , a fuzzy set (FS) on is in terms of the function : .

*Definition 2.2 (see [18]). *Let be the set of all fuzzy sets on : .

*Definition 2.3 (see [18]). *Let be the set of all fuzzy sets of type on . That is, the set of all : , where is defined as .

Apparently, if we use the functions or to replace the membership function , and returns an FS, then we obtain the notions of type 2 fuzzy sets (T2FSs) and type fuzzy sets (TnFSs).

Yager [14] and Miyamoto [8β13] 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 and be two multisets, and the element in the reference set, then(1)addition: ;(2)union: ;(3)intersection: .

However, this definition comes into conflict with FSs. Miyamoto [8β13] gave the corresponding solutions. He proposed an alternative definition for FMSs. Using the membership sequence (the membership values in a fuzzy multiset ) 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]). *Let be a fixed set, an intuitionistic fuzzy set (IFS) on is represented in terms of two functions : and : , with the condition , .

We use for all to represent IFSs considered in the rest of the paper without explicitly mentioning it.

Furthermore, is called a hesitancy degree or an intuitionistic index of in . In the special case , that is, , the IFS 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 be fixed, and let (represented by the functions and ), ( and ), ( and ), be three IFSs. Then the following operations are valid:(1) complement: ;(2)union: ;(3)intersection: ;(4)-union: ;(5)-intersection: .

De et al. [33] further gave another two operations of IFSs:(6);
(7), where is a positive integer.

Atanassov and Gargov [16] used the following:(1)the map assigns to every IVFS = an IFS, given by , ;(2)the map assigns to every IFS = an IVFS given by , to prove that IFSs and IVFSs are equipollent generalizations of the notion of FSs.

Xu and Yager [34] called each pair an intuitionistic fuzzy number (IFN), and, for convenience, denoted an IFN by . 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]). *Let be any two IFNs, the scores of , respectively, and the accuracy degrees of , respectively, then(i)if , then is larger than , denoted by ;(ii)if , then(1)if , then and represent the same information, that is, and , denoted by ;(2)if , then is larger than , denoted by .

##### 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]). *Let be a fixed set, then we define hesitant fuzzy set (HFS) on is in terms of a function applied to returns a subset of , and a hesitant fuzzy element (HFE).

Then, Torra [17] gave an example to show several special sets for all in : (1) empty set: ; (2) full set: ; (3) complete ignorance: ; (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.

*Definition 2.9 (see [17, 18]). *Given an HFE , the pair of functions and define an intuitionistic fuzzy set , denoted by .

According to this definition, IFSs can also be represented by HFSs, that is, for a given IFS , the corresponding HFS is , if .

Torra [17] gave the complement of a HFS as the following.

*Definition 2.10 (see [17]). *Given an HFS represented by its membership function , we define its complement as .

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 on , and for all in , 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 , is called the score function of , where is the number of the elements in . Moreover, for two HFEs and , if , then ; if , then .

#### 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. *Let be a fixed set, then a dual hesitant fuzzy set (DHFS) on is described as:
in which and are two sets of some values in , denoting the possible membership degrees and nonmembership degrees of the element to the set , respectively, with the conditions:
where , , , and for all . For convenience, the pair is called a dual hesitant fuzzy element (DHFE) denoted by , with the conditions: , , , , , and .

First we define some special DHFEs. Given a DHFE, , then we have(1)complete uncertainty: ;(2)complete certainty: ;(3)complete ill-known (all is possible): ;(4)nonsensical element: .

Based on the background knowledge introduced in Section 2, we can obtain some results in special cases. For a given , if and have only one value and , respectively, and , then the DHFS reduces to an IFS. If and have only one value and respectively, and , or owns one value, and , then the DHFS reduces to an FS (also can be regarded as HFSs). If and , 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 , , , in the rest of the paper without explicitly mentioning it.

Actually, for a typical DHFS, and can be represented by two intervals as:

Based on Definition 2.9, there is a transformation between IFSs and HFSs, we can also transform to , that is, the number 2 HFE denoting the possible membership degrees of the element . Thus, both and indicate the membership degrees, we can use a βnested intervalβ to represent 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 , 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.

##### 3.2. 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 function , and , its complement is defined as:
Apparently, the complement is involutive represented as .

We now define the union and the intersection of DHFSs. For two DHFSs and , it is clear that the corresponding lower and upper bounds to and are , , and , respectively, where , , , and represent this group notations and no confusion will arise in the rest of this paper.

*Definition 3.3. *Let be a fixed set, and two DHFEs, we define their union and intersection, respectively, as:

(1);(2),
the following operations are valid:-union: ;(2)-intersection: ;(3);
(4), where is a positive integral and all the results are also DHFEs.

*Example 3.4. *Let and be two DHFEs, then we have(1)complement: ;(2)union: ;(3)intersection: .

We can easily prove the following theorem according to Definition 3.3

Theorem 3.5. *Let , , and be any three DHFEs, , then*(1)*;*(2)*;*(3)*;*(4)*.*

To compare the DHFEs, and based on Definitions 2.7 and 2.12, we give the following comparison laws.

*Definition 3.6. *Let be any two DHFEs, the score function of , and the accuracy function of , where and are the numbers of the elements in and , respectively, then(i)if , then is superior to , denoted by ;(ii)if , then(1)if , then is equivalent to , denoted by ;(2)If , then is superior than , denoted by .

*Example 3.7. *Let and be two DHFEs, then based on Definition 3.1, we obtain , , and thus, .

#### 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 on a set of HFSs considers all the values in such sets and the application of on them. The definition is as the following.

*Definition 4.1 (see [18]). *Let be a function : , and a set of HFSs on the reference set (i.e., is a HFS on ). Then, the extension of on is defined for each in 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 : be a modular aggregation function (modular average), . Then, its symmetrization is called the OMA, that is, is given by
where : are the nondecreasing functions satisfying (identity on ).

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 considers all the values in the DHFSs and the application of on them, which is defined as follows.

*Definition 4.3. *Let the functions : and : , and let be a set of dual hesitant fuzzy sets on the reference set represented as . Then, the extension of on is defined for each in by
where : , .

If we let , some basic operations can be obtained according to this extension principle in Example 4.4.

*Example 4.4. *Let and be two DHFEs such that
If and , then

Similarly, we have

In particular, if , then is a real number; if , then is a real number:

Furthermore, we can get lots of aggregation operators by this extension principle. For example, if we let , and , 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 with respect to the predictive values of the net incomes are represented by the DHFEs , where indicates the degree that the alternative satisfies the criterion , indicates the degree that the alternative does not satisfy the criterion , such that , , , 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 , where , as shown in Table 2.

*Step 2. *Use to obtain the expectations of net incomes as shown in Table 3.

Thus, , and then , that is, 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.

#### 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.

#### Acknowledgments

The author, Bin Zhu, is very grateful to the anonymous reviewers for their insightful and constructive comments and suggestions that have led to an improved version of this paper. The work was supported by the National Natural Science Foundation of China (No.71071161).