The Scientific World Journal

Volume 2014 (2014), Article ID 897304, 20 pages

http://dx.doi.org/10.1155/2014/897304

## Continuous Hesitant Fuzzy Aggregation Operators and Their Application to Decision Making under Interval-Valued Hesitant Fuzzy Setting

^{1}Institute of Quality Development, Kunming University of Science and Technology, Kunming 650093, China^{2}School of Management, Harbin University of Science and Technology, Harbin 150040, China

Received 5 February 2014; Revised 7 April 2014; Accepted 17 April 2014; Published 25 May 2014

Academic Editor: Wlodzimierz Ogryczak

Copyright © 2014 Ding-Hong Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Interval-valued hesitant fuzzy set (IVHFS), which is the further generalization of hesitant fuzzy set, can overcome the barrier that the precise membership degrees are sometimes hard to be specified and permit the membership degrees of an element to a set to have a few different interval values. To efficiently and effectively aggregate the interval-valued hesitant fuzzy information, in this paper, we investigate the continuous hesitant fuzzy aggregation operators with the aid of continuous OWA operator; the C-HFOWA operator and C-HFOWG operator are presented and their essential properties are studied in detail. Then, we extend the C-HFOW operators to aggregate multiple interval-valued hesitant fuzzy elements and then develop the weighted C-HFOW (WC-HFOWA and WC-HFOWG) operators, the ordered weighted C-HFOW (OWC-HFOWA and OWC-HFOWG) operators, and the synergetic weighted C-HFOWA (SWC-HFOWA and SWC-HFOWG) operators; some properties are also discussed to support them. Furthermore, a SWC-HFOW operators-based approach for multicriteria decision making problem is developed. Finally, a practical example involving the evaluation of service quality of high-tech enterprises is carried out and some comparative analyses are performed to demonstrate the applicability and effectiveness of the developed approaches.

#### 1. Introduction

As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [1, 2] introduced by Torra and Narukawa have been successfully used in the decision making field as a powerful tool for processing uncertain and vague information. Different from the other generalizations of fuzzy sets, HFSs permit the membership degree of an element to a set to be represented as several possible values between 0 and 1, which are very useful in dealing with the situations where people hesitate between several values to express their judgments [3–5] or their opinions with incongruity [6–8], particularly, the group decision making with anonymity [9–12]. Meanwhile, HFSs can also avoid performing information aggregation and can directly reflect the differences of the opinions of different experts [1, 13, 14]. Furthermore, as Torra reported that the envelope of HFS is an intuitionistic fuzzy set (IFS), all HFSs are type-2 fuzzy sets, and HFSs and fuzzy multisets (FMSs) have the same form, but their operations are different [2]. Thus HFSs open new views for further research on decision making under hesitant environments and have received much attention from many authors. Torra and Narukawa [1, 2] proposed some set theoretic operations such as union, intersection and complement, and the extension principle on HFSs. Subsequently, Xia and Xu [6] defined some new operations on HFSs based on the interconnection between HFSs and IFSs and then made an intensive study of hesitant fuzzy information aggregation techniques and their applications in decision making. Xu and Xia [7] investigated some distance measures for HFSs drawing on the well-known Hamming distance, the Euclidean distance, the Hausdorff metric, and their generalizations. Following these pioneering studies, many subsequent studies on the basic theory [9, 15], the aggregation operators [8, 9, 12–17], the discrimination measures [18] (including distance measures [3–5, 7], similarity measures [7], correlation measures [3, 19], entropy, and cross-entropy [20], etc.) for HFSs, and the further extensions of the HFSs such as the interval-values HFSs (IVHFSs) [11, 19], the dual (or generalized) HFSs (DHFSs) [10, 21, 22], and the hesitant fuzzy linguistic term sets (HFLTSs) [23, 24] have been conducted.

In some practical decision making problems, however, the precise membership degrees of an element to a set are sometimes hard to be specified. To overcome the barrier, Chen et al. [11, 19] proposed the concept of interval-valued hesitant fuzzy sets (IVHFSs) that represent the membership degrees of an element to a set with several possible interval values and then presented some interval-valued hesitant fuzzy aggregation operators. Although the concept of IVHFSs is very recent, it has received a lot of attention by other researchers in the community; Wei [25] presented some interval-valued hesitant fuzzy Choquet ordered averaging operators, interval-valued hesitant fuzzy prioritized aggregation operators, and interval-valued hesitant fuzzy power aggregation operators. Wei and Zhao [26] presented some interval-valued hesitant fuzzy Einstein aggregation operators and induced interval-valued hesitant fuzzy Einstein aggregation operators. Li and Peng [27] presented some interval-valued hesitant fuzzy Hamacher synergetic weighted aggregation operators to select shale gas areas. Peng and Wang [17] presented some dynamic interval-valued hesitant fuzzy aggregation operators to aggregate the interval-valued hesitant fuzzy information collected at different periods in multiperiod decision making. Nevertheless, the above-mentioned operators are straightforward extensions of their respective proposals for the case of HFEs; they only focus on the endpoints of the closed intervals of interval-valued hesitant fuzzy elements (IVHFEs) on the basis of the characteristics of interval numbers and therefore are not rich enough to capture all the information contained in IVHFEs. Additionally, due to the fact that decision making problems are essentially humanistic and subjective in nature, decision makers’ (DMs’) risk preferences play an important role. How to reflect DMs’ risk preferences in decision making is a crucial problem. Yet the above-mentioned operators do not consider the problem. Thus, it is necessary to explore some new techniques for aggregating interval-valued hesitant fuzzy information in accordance with DMs’ risk preferences. The continuous ordered weighted averaging (C-OWA) operator was formally presented by Yager [28] (which was previously introduced by Torra and Godo [29, 30] in 1997, see also [31]) and is appropriate for aggregating decision information which is given in the forms of valued interval. A distinguished advantage of C-OWA operator is that it can lead to every value in the interval being aggregated and aggregate the valued interval to a precise value based on decision attitudes of DMs. Thus, based on the C-OWA operator, some extended continuous aggregation operators are further developed, such as the continuous ordered weighted geometric (C-OWG) operator [32], the continuous generalized OWA (C-GOWA) operator [33], the continuous quasi-OWA (C-QOWA) operator [34], and the induced generalized continuous OWA (IGCOWA) operator [35]. in view of the predominant advantages of C-OWA operator, in this paper, we investigate the continuous hesitant fuzzy aggregation operators to efficiently and effectively aggregate the interval-valued hesitant fuzzy information and apply them to the multiple criteria decision making.

To do so, the remainder of this paper is set out as follows. Section 2 introduces some preliminary concepts including hesitant fuzzy sets, interval-valued hesitant fuzzy sets, and continuous OW (C-OWA and C-OWG) operators. In Section 3, we propose the continuous HFOWA operator; the continuous HFOWG operator and their essential properties are studied in detail. In Section 4, we extend the C-HFOW operators to efficiently and effectively aggregate multiple interval-valued hesitant fuzzy elements and then develop the weighted C-HFOWA operator, the weighted C-HFOWG operator, the ordered weighted C-GOWA operator, the ordered weighted C-GOWG operator, the synergetic weighted C-GOWA operator, and the synergetic weighted C-GOWG operator; some properties are also discussed to support them. In Section 5, we develop an approach based on the SWC-HFOW operators to multicriteria decision-making under interval-valued hesitant fuzzy environments and in Section 6 a practical example involving the evaluation of service quality of high-tech enterprises is carried out and some comparative analyses are performed to demonstrate the applicability and effectiveness of the developed approaches. Finally, we summarize the main conclusions of the paper in Section 7.

#### 2. Preliminaries

In this section, we introduce some basic notions related to hesitant fuzzy sets, interval-valued hesitant fuzzy sets, and continuous OW operators.

##### 2.1. Hesitant Fuzzy Sets

Hesitant fuzzy sets (HFSs) are quite suited for the situation where we have a set of possible values, rather than a margin of error or some possibility distribution on the possible values. Thus, HFSs can be considered as a powerful tool to express uncertain information in the process of decision making with hesitancy and incongruity.

*Definition 1 (see [2]). *Let be a fixed set; a hesitant fuzzy set (HFS) on is in terms of a function that when applied to returns a subset of [].

To be easily understood, Xia and Xu [6] expressed the HFS as the following mathematical symbol:
where is a set of values in [], denoting the possible membership degrees of the element to the set . For convenience, we call a hesitant fuzzy element (HFE).

*Definition 2 (see [6]). *Let , , and be three HFEs; then(1)(2)(3)(4).

To compare the HFEs, Xia and Xu [6] defined the following comparison laws.

*Definition 3 (see [6]). *For a HFE , is called the score function of , where is the number of the values in . Moreover, for two HFEs and , if , then ; if , then .

It is noted that the numbers of values in different HFEs may be different, and thus the traditional operations and operators cannot be used. For the aggregation of hesitant fuzzy information, Torra and Narukawa [1] proposed the following extension principle that extends functions to HFEs.

*Definition 4 (see [1]). *Let be a set of HFEs and let be a function on , ; then

Through the extension principle, one can not only realize the synthesis of HFEs with different numbers of values but also utilize properly all information in HFEs, and it can guarantee that the properties on lead to related properties on , which is also an essential difference between the operations of HFSs and the ones of the FMSs.

Based on the above extension principle, Xia and Xu [6] developed a series of specific aggregation operators for HFEs.

*Definition 5 (see [6]). *Let be a collection of HFEs, is the th largest of them, and is the associated (order) weight vector with and , then consider the following.(1)A hesitant fuzzy ordered weighted averaging (HFOWA) operator is a mapping , such that
(2)A hesitant fuzzy ordered weighted geometric (HFOWG) operator is a mapping , such that
The results of the hesitant fuzzy aggregation operators are also HFEs.

##### 2.2. Interval-Valued Hesitant Fuzzy Sets

To overcome the barrier that the precise membership degrees of an element to a set are sometimes hard to be specified, Chen et al. [11, 19] introduced the interval-valued hesitant fuzzy sets (IVHFSs) which permit the membership degrees of an element to a set to be several possible interval values.

*Definition 6 (see [11]). *Let be a reference set, and let be the set of all closed subintervals of ; then an IVHFS on is defined as
where : denotes all possible interval-valued membership degrees of the element to the set . For convenience, we call an interval-valued hesitant fuzzy element (IVHFE), which reads

*Definition 7 (see [11]). *Let , , and be three IVHFEs; then(1),(2),(3),(4).

Chen et al. [11] defined the score function of IVHFE and utilized the possibility degree formula to compare the score values of two IVHFEs.

*Definition 8. *For an IVHFE , is called the score function of . Moreover, for two IVHFEs and , if
then ; if , then .

*Definition 9 (see [11]). *Let be a collection of IVHFEs, be the th largest of them, be the associated weight vector with , and , then consider the following.(1)An interval-valued hesitant fuzzy ordered weighted averaging (IVHFOWA) operator is a mapping , where
(2)An interval-valued hesitant fuzzy ordered weighted geometric (IVHFOWG) operator is a mapping , where

The results of the IVHFOWA and IVHFOWG operators are also IVHFEs; that is, the results consist of some interval values. Meanwhile, as the analysis above, the operators only focus on the endpoints of the closed intervals of IVHFEs and therefore are not rich enough to capture all the information contained in IVHFEs. Furthermore, they do not consider the DMs’ risk preferences in aggregation process.

##### 2.3. Continuous Ordered Weighted Aggregation Operators

*Definition 10 (see [36]). *An OWA operator of dimensions is a mapping that has an associated weight vector with the properties and , such that
where defines a permutation of such that for all .

The OWA operator is bounded, idempotent, commutative, and monotonic. Note that the weights are assigned according to the positions of argument variables in OWA operator, that is, each argument value and its corresponding associated weight existing one-to-one relative relations [4, 27, 37, 38]; thus we can find a permutation , which is the inverse permutation of ; that is, , and the OWA operator can be alternatively defined as

Proposition 11. * is the associated weight vector with , , and are two permutations of , if , then
*

Proposition 11 shows the equivalence between the original definition and the alternative definition of the OWA operator.

In order to aggregate all the values in a closed interval , Yager [28] presented a continuous ordered weighted averaging (C-OWA) operator based on OWA operator and the basic unit-interval monotonic (BUM) function [39].

*Definition 12 (see [28]). *A continuous ordered weighted averaging (C-OWA) operator is a mapping which is defined as follows:
where is a BUM function and is monotonic with the properties , , and if . .

The C-OWA operator is not only bounded but also monotonic and associated with both the argument values and [28].

Subsequently, Yager and Xu [32] proposed the continuous ordered weighted geometric (C-OWG) operator based on the C-OWA operator and the geometric mean.

*Definition 13 (see [32]). *A continuous ordered weighted geometric (C-OWG) operator is a mapping which is defined as follows:
where is a BUM function and is monotonic with the properties , , and if . .

Lemma 14 (see [40]). *Let , , , and ; then
*

Proposition 15. *For a closed interval , is a BUM function; then
*

*Proof. *Since
according to Lemma 14, we have
thus

#### 3. Continuous Hesitant Fuzzy Ordered Weighted Aggregation Operators

In this section, some novel continuous ordered weighted aggregation operators are proposed to aggregate an IVHFE, such as the continuous hesitant fuzzy ordered weighted averaging (C-HFOWA) operator and the continuous hesitant fuzzy ordered weighted geometric (C-HFOWG) operator. Some essential properties of these operators are also studied in detail.

##### 3.1. Continuous Hesitant Fuzzy Ordered Weighted Averaging Operator

*Definition 16. *A continuous HFOWA (C-HFOWA) operator is a mapping , which has associated with it a BUM function having the properties , , and if , such that

The motivation behind the above definition is as follows. In fact, since is an interval whose arguments are preordered thus we do not need a reordering step, is a BUM function, and , which satisfy the conditions and . Based on Definition 5, (3), we have Let ; we get

When , denote , and ranges from 0 to , then we have , and thus .

From Definition 16 and the above analysis, we know that the aggregated result of the C-HFOWA operator is a HFE and the number of its possible membership values is the same as the one of the IVHFE to be aggregated; that is, .

*Example 17. *Let be an IVHFE, and ; then

The C-HFOWA operator has the following essential properties.

Proposition 18 (Bounded). *For an IVHFE , then
*

*Proof. *For any , when , we have . Since if , then , we have According to , we can obtain

According to the extension principle of HFS, we have
thus .

Proposition 19 (Idempotency). *For an IVHFE , if all , then is reduced to a HFE , and thus .*

*Proof. *Consider

Proposition 20 (Monotonicity for ). *For any two IVHFEs and , if for all , , then .*

*Proof. *Since , we have

According to the extension principle of HFS, we have
thus .

Proposition 21 (Monotonicity for ). *For an IVHFE , and for all , then .*

*Proof. *Since
and for all , so when for all , we have
furthermore, the relation
holds, and thus we can get
that is, .

Proposition 22. *For an IVHFE , and , then,
*

*Proof. *Since

When , denote , and ranges from 0 to , then we have , and then . Thus .

##### 3.2. Continuous Hesitant Fuzzy Ordered Weighted Geometric Operator

*Definition 23. *A continuous HFOWG (C-HFOWG) operator is a mapping , which has associated with it a BUM function: having the properties , , and if , such that

Now let us investigate how we can obtain Definition 23. In fact, since is an interval whose arguments are preordered thus we do not need a reordering step; is a BUM function and , which satisfy the conditions and . Based on Definition 5, (4), we have
Let ; we get

When , denote , and ranges from 0 to , then we have , and thus

From Definition 23 and the above analysis, we know that the aggregated result of the C-HFOWG operator is a HFE and the number of its possible membership values is the same as the one of the IVHFE to be aggregated; that is, .

*Example 24. *Let be an IVHE and , , and , then

From Example 24, we can see that the aggregated results are different when different BUM functions are adopted in the example, which indicates that the C-HFOWG operator can reflect the decision maker's risk preferences by using different BUM functions. Moreover, the aggregated results derived by the C-HFOWG operator become smaller as the BUM function values decrease.

Similar to the C-HFOWA operator, the C-HFOWG operator has the following essential properties.

Proposition 25 (Bounded). *For an IVHFE , then
*

*Proof. *For any , when , we have . Since if , then , we have According to , we can obtain furthermore,
According to the extension principle of HFS, we have
thus .

Proposition 26 (Idempotency). *For an IVHFE , if all , then is reduced to a hesitant set , and thus .*

*Proof. *Consider

Proposition 27 (Monotonicity for ). *For any two IVHFEs and , if for all , , then .*

*Proof. *Since , we have
According to the extension principle of HFS, we have
thus, .

Proposition 28 (Monotonicity for ). *For an IVHFE , and for all , then .*

*Proof. *Since
and for all , when for all , we have furthermore, the relation
holds, and thus we can get
that means .

Proposition 29. *For an IVHFE , and , then
*

*Proof. *Since

Thus, .

#### 4. Extended C-HFOW Operators

In order to aggregate multiple IVHFEs, we extend the C-HFOW (C-HFOWA and C-HFOWG) operators to the case where the given inputs are multiple IVHFEs of dimension and develop some extended C-HFOW operators.

##### 4.1. Weighted C-HFOW Operators

*Definition 30. *Let be a collection of IVHFEs, and let be the relative weight vector of , with and . A weighted C-HFOWA (WC-HFOWA) operator is a mapping , according to the following expression:

It is natural that the aggregated result derived from the WC-HFOWA operator is a HFE and the number of the possible aggregated values satisfies the following inequality:

Clearly, if all possible aggregated values in the derived HFE are identical, then , on the contrary, if the all possible aggregated values in the derived HFE are different, then .

On the basis of the properties of the C-HFOWA operator, we can further obtain some properties of WC-HFOWA operator.

Proposition 31 (Idempotency). *Let be a collection of IVHFEs, if , then
*

*Proof. *Since , we have

Proposition 32 (Bounded 1). *Let be a collection of IVHFEs, , and , then
*

*Proof. *Let and . We have . Therefore, .

Proposition 33 (Bounded 2). *Let be a collection of IVHFEs; then
*

*Proof. *Without loss of generality, assume that