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.
As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [
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. [
To do so, the remainder of this paper is set out as follows. Section
In this section, we introduce some basic notions related to hesitant fuzzy sets, interval-valued hesitant fuzzy sets, and continuous OW operators.
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.
Let
To be easily understood, Xia and Xu [
Let
To compare the HFEs, Xia and Xu [
For a HFE
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 [
Let
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
Based on the above extension principle, Xia and Xu [
Let A hesitant fuzzy ordered weighted averaging (HFOWA) operator is a mapping A hesitant fuzzy ordered weighted geometric (HFOWG) operator is a mapping
To overcome the barrier that the precise membership degrees of an element to a set are sometimes hard to be specified, Chen et al. [
Let
Let
Chen et al. [
For an IVHFE
Let An interval-valued hesitant fuzzy ordered weighted averaging (IVHFOWA) operator is a mapping An interval-valued hesitant fuzzy ordered weighted geometric (IVHFOWG) operator is a mapping
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.
An OWA operator of dimensions
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 [
Proposition
In order to aggregate all the values in a closed interval
A continuous ordered weighted averaging (C-OWA) operator is a mapping
The C-OWA operator is not only bounded but also monotonic and associated with both the argument values and
Subsequently, Yager and Xu [
A continuous ordered weighted geometric (C-OWG) operator is a mapping
Let
For a closed interval
Since
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.
A continuous HFOWA (C-HFOWA) operator is a mapping
The motivation behind the above definition is as follows. In fact, since
When
From Definition
Let
The C-HFOWA operator has the following essential properties.
For an IVHFE
For any
According to the extension principle of HFS, we have
For an IVHFE
Consider
For any two IVHFEs
Since
According to the extension principle of HFS, we have
For an IVHFE
Since
For an IVHFE
Since
When
A continuous HFOWG (C-HFOWG) operator is a mapping
Now let us investigate how we can obtain Definition
When
From Definition
Let
From Example
Similar to the C-HFOWA operator, the C-HFOWG operator has the following essential properties.
For an IVHFE
For any
For an IVHFE
Consider
For any two IVHFEs
Since
For an IVHFE
Since
For an IVHFE
Since
Thus,
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
Let
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 basis of the properties of the C-HFOWA operator, we can further obtain some properties of WC-HFOWA operator.
Let
Since
Let
Let
Let
Without loss of generality, assume that
Let
Since
Let
It is natural that the aggregated result derived from the WC-HFOWG 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 the identical, then
The WC-HFOWG operator has similar properties with the WC-HFOWA operator.
Let
Let
Let
Let
Let
Let
It is natural that the aggregated result derived from the OWC-HFOW (OWC-HFOWA or OWC-HFOWG) operators 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
The OWC-HFOW operators have similar properties with the WC-HFOW operators; they are idempotent, bounded, monotonic, and so forth, and the proofs of them are omitted here for saving space.
From the above definitions, we know that the WC-HFOW (WC-HFOWA or WC-HFOWG) operators focus solely on the weight of the individual argument variable itself and ignore the associated (position) weight with respect to the individual argument variable value. However, the OWC-HFOW (OWC-HFOWA or OWC-HFOWG) operators focus on the associated (position) weight with respect to the individual argument variable value and ignore the weight of the individual argument variable itself. To generalize the WC-HFOW operators and the OWC-HFOW operators, motivated by the idea of the weighted OWA operator [
Let
Alternatively, according to the Proposition
Let
It is natural that the result derived from the SWC-HFOW operators (AWC-HFOWA or SWC-HFOWG) 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 the identical, then
With regard to the SWC-HFOW operators, we have the following propositions.
If the relative weight vector
If the associated weight vector
If the relative weight vector
Concretely, consider the averaging C-HFOWA operator:
The proofs of them are intuitional and omitted here.
Let
First,
Furthermore, according to the score function of HFE, we can derive the score values of the aggregated results:
From Propositions
In this section, we consider the multiple criteria decision making (MCDM) problem where all the criteria values are expressed in interval-valued hesitant fuzzy information. The following notations are used to depict the considered problem. Let
In general, there are benefit criteria (the bigger the criteria values, the better) and cost criteria (the smaller the criteria values, the better) in MCDM problems. In order to measure all criteria in dimensionless units and to facilitate intercriteria comparisons, in the following we normalize the decision matrix
In the following, we apply the above synergetic weighted C-HFOW operators to multiple criteria decision making under interval-valued hesitant fuzzy setting.
Normalize the original interval-valued hesitant fuzzy decision matrix
Select a BUM function
Aggregate decision information of
Calculate the score values
Rank all the alternatives
Service activities have become the fundamental and dominant factors of the economic system over the past decades and the significance and influence of service quality have been recognized through the great effect on customer satisfaction and loyalty. Relevant studies indicated that service quality is a key factor for survival and development in today’s keen competition. Thus, the evaluation of service quality has become an important issue. Suppose that there are five alternatives (high-tech enterprises)
Interval-valued hesitant fuzzy decision matrix.
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Since all the criteria are the benefit criteria, then the criteria values do not need normalization.
Select a BUM function
Utilize the SWC-HFOWA operator, (
First, we use the C-HFOWA to aggregate each IVHFE, and the aggregated results are listed in Table
Since
Thus
Similarly, we can obtain
Aggregated results derived by C-HFOWA operator.
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Calculate the scores values of
Rank the alternatives
In the following, we use the SWC-HFOWG operator to solve the same problem.
First, utilize C-HFOWG operator to aggregate each IVHFE, and the aggregated results are listed in Table
Aggregated results derived by C-HFOWG operator.
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Then, obtain the aggregated results
Obviously, the identical ranking results can be obtained through the C-HFOWA operator based and the C-HFOWG operator based approaches, which implies the two proposed approaches all are feasible and effective.
Moreover, to understand more the effect of different types of weights in aggregation, we use the WC-HFOWA, OWC-HFOWA, and SWC-HFOWA operators to the example above and their final score values and ranking results are listed in Table
Score values and ranking results derived by WC-HFOWA, OWC-HFOWA, and SWC-HFOWA operators.
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Ranking results | |
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WC-HFOWA | 0.409 | 0.601 | 0.463 | 0.619 | 0.523 |
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SWC-HFOWA | 0.372 | 0.504 | 0.437 | 0.569 | 0.46 |
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OWC-HFOWA | 0.390 | 0.532 | 0.450 | 0.594 | 0.492 |
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From Table
Furthermore, we compare our proposed operators with the existing interval-valued hesitant fuzzy aggregation operators; here we use the IVHFWA operator [
Then we calculate the score values of
Similarly, we calculate the rest possibility degrees and then obtain a possibility degree matrix
Finally, we average all elements in each line of the possibility degree matrix and then get the relative possibility degrees
By the above analysis, we can find that the final decision results (score values) are different and yet the ranking results of alternatives derived from the WC-HFOWA, OWC-HFOWA, SWC-HFOWA, WC-HFOWG, and IVHFWA operators are identical, which further indicate that they all are effective and reasonable. The IVHFWA operator is straightforward extensions of HFWA operator; it only focuses on the endpoints of the closed intervals of IVHFEs and therefore is not rich enough to capture all the information contained in IVHFEs and much useful information may be lost. However our operators aggregate all the information over closed intervals of IVHFEs and thus can effectively avoid the information loss. In decision making with the IVHFWA operator, the score values of aggregated results (alternatives) are still interval-valued. In order to rank the alternatives, we have to first use the possibility degree formula to compare each pair of score values of the alternatives and then calculate the relative possibility degrees of the alternatives. Such procedure needs a large amount of computational efforts and takes a lot of time to be accomplished, especially, with the increases of the number of alternatives. Moreover, if use the IVHFOWA operator [ The aggregation of IVHFWA operator does not consider the DMs’ risk preferences, which implies that the importance of all information in the closed intervals of IVHFEs is the same. However, the decision result needs usually to reflect the DMs’ risk preferences, that is to say, the DMs’ risk preferences should be added to the aggregation of each possible interval of IVHFEs but the endpoints of the possible intervals of IVHFEs should not be simply regarded as the same. Our operators and approaches consider the DMs’ risk preferences via the basic unit-interval monotonic (BUM) function, which are very suitable for the practical decision making situations.
To efficiently and effectively aggregate the interval-valued hesitant fuzzy information, in this paper, we have presented some continuous hesitant fuzzy aggregation operators, that is, the continuous hesitant fuzzy ordered weighted averaging (C-HFOWA) operator and the continuous hesitant fuzzy ordered weighted geometric (C-HFOWG) operator, and their fundamental properties are studied in detail. Then, we extended the operators to aggregate multiple interval-valued hesitant fuzzy elements and then developed the weighted C-HFOW (WC-HFOWA and WC-HFOWG), ordered weighted C-HFOW (OWC-HFOWA and OWC-HFOWG), and synergetic weighted C-HFOW (SWC-HFOWA and SWC-HFOWG) operators; some properties of them are also discussed. Based on the SWC-HFOW operators, we developed an approach for multicriteria decision making under interval-valued hesitant fuzzy setting. Finally, a practical example involving the evaluation of service quality of high-tech enterprises is carried out and some comparative analysis are performed to illustrate the applicability and effectiveness of the developed approach. In the future, we will further investigate the continuous hesitant fuzzy aggregation operators that there is some degree of interdependent characteristics between argument variables with the help of the continuous Choquet integral [
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are very grateful to the Editor, Professor Wlodzimierz Ogryczak, and the anonymous referees for their insightful and constructive comments and suggestions which have helped to improve the paper. This work was supported in part by the National Natural Science Funds of China (no. 61364016), the China Postdoctoral Science Foundation (no. 2014M550473), the Scientific Research Fund Project of Educational Commission of Yunnan Province, China (no. 2013Y336), the Science and Technology Planning Project of Yunnan Province, China (no. 2013SY12), and the Natural Science Funds of KUST (no. KKSY201358032).