Some Aggregation Operators Based on Einstein  Operations under Interval-Valued Dual Hesitant Fuzzy  Setting and Their Application

Zhang, Wenkai; Li, Xia; Ju, Yanbing

doi:https://doi.org/10.1155/2014/958927

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 958927 | https://doi.org/10.1155/2014/958927

Some Aggregation Operators Based on Einstein Operations under Interval-Valued Dual Hesitant Fuzzy Setting and Their Application

Wenkai Zhang,¹Xia Li,¹and Yanbing Ju¹

Academic Editor: Wudhichai Assawinchaichote

Received28 Apr 2014

Accepted07 Aug 2014

Published04 Nov 2014

Abstract

We investigate the multiple attribute decision making (MADM) problems in which attribute values take the form of interval-valued dual hesitant fuzzy information. Firstly, some operational laws for interval-valued dual hesitation fuzzy elements (IVDHFEs) based on Einstein operations are developed. Then we develop some aggregation operators based on Einstein operations: the interval-valued dual hesitant fuzzy Einstein weighted averaging (IVDHFEWA) operator, interval-valued dual hesitant fuzzy Einstein ordered weighted averaging (IVDHFEOWA) operator, interval-valued dual hesitant fuzzy Einstein hybrid averaging (IVDHFEHA) operator, interval-valued dual hesitant fuzzy Einstein weighted geometric (IVDHFEWG) operator, interval-valued dual hesitant fuzzy Einstein ordered weighted geometric (IVDHFEOWG) operator, and interval-valued dual hesitant fuzzy Einstein hybrid geometric (IVDHFEHG) operator. Furthermore, we discuss some desirable properties of these operators, and investigate the relationship between the developed operators and the existing ones. Based on the IVDHFEWA operator, an approach to MADM problems is proposed under the interval-valued dual hesitant fuzzy environment. Finally, a numerical example is given to show the application of the developed method, and a comparison analysis is conducted to demonstrate the effectiveness of the proposed approach.

1. Introduction

The fuzzy set [1] has received increasing attention since its introduction by Zadeh. Various extensions of this theory have been developed, including the interval-valued fuzzy set [2], type-2 fuzzy set [3], intuitionistic fuzzy set [4], interval-valued intuitionistic fuzzy set [5], and linguistic fuzzy set [6]. However, the aforementioned extensions cannot deal with the situation where it is difficult to determine the membership of an element to a set owing to ambiguity among several different values; that is, the difficulty in establishing the membership of an element to a set does not arise from a margin of error (as in intuitionistic or interval-valued fuzzy sets) or a specified possibility distribution of the possible values (as in type-2 fuzzy set) but instead arises from our hesitation among a few different values. Recently, Torra and Narukawa [7] and Torra [8] introduced the concept of hesitant fuzzy sets (HFSs) to handle such cases. HFSs permit the membership degree of an element to a set to be represented by a set of possible values. Hesitant fuzzy aggregation operators have received increasing attention from researchers recently. Xia and Xu [9] defined some hesitant fuzzy operational rules and discussed a series of operators under various conditions. Furthermore, Xia et al. [10] developed some quasiarithmetic aggregation operators and some induced aggregation operators for hesitant fuzzy information. Zhu et al. [11] defined the hesitant fuzzy geometric Bonferroni mean (HFGBM), the hesitant fuzzy Choquet geometric Bonferroni mean (HFCGBM) and then applied them to MADM problems. Motivated by the idea of prioritized aggregation operators, Wei [12] developed some prioritized aggregation operators for aggregating hesitant fuzzy information. Zhang [13] extended the classical power aggregation operators to hesitant fuzzy environment and then developed the hesitant fuzzy power averaging (HFPA) operator, generalized hesitant fuzzy power averaging (GHFPA) operator, weighted generalized hesitant fuzzy power averaging (WGHFPA) operator, hesitant fuzzy power ordered weighted averaging (HFPOWA) operator, and generalized hesitant fuzzy power ordered weighted averaging (GHFPOWA) operator. Lin et al. [14] proposed hesitant fuzzy linguistic set (HFLS) and developed some hesitant fuzzy linguistic aggregation operators. In addition to the aforementioned aggregation operators for hesitant fuzzy information, many other research topics have also been discussed with the help of HFSs [15–26].

Recently, Zhu et al. [27] proposed dual hesitant fuzzy sets (DHFSs), which consists of two parts: the membership hesitancy function and the nonmembership hesitancy function. They have investigated some basic operations and properties of DHFS. Furthermore, Wang et al. [28] developed some aggregation operators based on dual hesitant fuzzy elements (DHFEs), such as the dual hesitant fuzzy weighted averaging (DHFWA) operator, the dual hesitant fuzzy weighted geometric (DHFWG) operator, the dual hesitant fuzzy ordered weighted averaging (DHFOWA) operator, the dual hesitant fuzzy ordered weighted geometric (DHFOWG) operator, the dual hesitant fuzzy hybrid averaging (DHFHA) operator, and the dual hesitant fuzzy hybrid geometric (DHFHG) operator, and then studied some properties of these operators. Ju et al. [29] developed some aggregation operators with interval-valued dual hesitant fuzzy information.

It is obvious that the aforementioned aggregation operators all built on the basic algebraic product and algebraic sum, which are not the unique operations that can be chosen to model the intersection and union of IVDHFEs. Einstein operations include Einstein product and Einstein sum, which are good alternatives to the algebraic product and algebraic sum, respectively. Moreover, it seems that there are some investigations on aggregation techniques using the Einstein operations on IFSs (or HFSs) for aggregating a collection of IFVs (or HFEs). Zhao and Wei [30] applied the intuitionistic fuzzy Einstein hybrid averaging operator and intuitionistic fuzzy Einstein hybrid geometric operator to deal with MADM problems. Wang and Liu [31, 32] developed some arithmetic aggregation operators and geometric aggregation operators by using Einstein operations to aggregate intuitionistic fuzzy information. Wang and Liu [33, 34] further investigated the Einstein operators under interval-valued intuitionistic fuzzy environments. Zhang and Yu [35] proposed some geometric Choquet aggregation operators using Einstein operations to deal with MADM problems. Zhao et al. [36] utilized Einstein operations to develop some hesitant fuzzy correlated aggregation operators. Wei and Zhao [37] developed some induced hesitant interval-valued fuzzy Einstein aggregation operators to deal with MADM problems with hesitant interval-valued fuzzy information. Zhao et al. [38] developed some hesitant triangular fuzzy aggregation operators based on the Einstein operations.

Based on the above analysis, we find that how to extend the Einstein operations to aggregate the interval-valued dual hesitation fuzzy information is a meaningful work. Therefore, we will develop some aggregation operators based on Einstein operations under interval-valued dual hesitant fuzzy setting. To do that, the remainder of the paper is organized as follows: Section 2 reviews some basic concepts related to HFSs, DHFSs, and IVDHFSs. In Section 3, we define some operational laws for interval-valued dual hesitant fuzzy element (IVDHFE) based on Einstein operations and develop some aggregation operators for aggregating interval-valued dual hesitant fuzzy information based on Einstein operations of IVDHFE. Section 4 proposes a method to MADM problems under interval-valued dual hesitant fuzzy setting. A numerical example is developed to illustrate how to apply the proposed approach in Section 5, followed by concluding remarks in Section 6.

2. Preliminaries

In this section, we briefly review some basic notations and definitions regarding hesitant fuzzy sets, dual hesitant fuzzy sets, and interval-valued hesitant fuzzy sets.

2.1. Hesitant Fuzzy Sets

Torra and Narukawa [7] and Torra [8] firstly proposed the hesitant fuzzy set. The concept of hesitant fuzzy set (HFS) and some operation laws of hesitant fuzzy elements are given as follows.

Definition 1 (see [9]). 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 [9] expressed the HFS by a mathematical symbol: where is a set of some values in , denoting the possible membership degrees of the element to the set . For convenience, Xia and Xu [9] called a hesitant fuzzy element (HFE) and the set of all hesitant fuzzy elements (HFEs).

Definition 2 (see [9]). Let , , and be any three HFEs; then some operation laws about HFEs are defined as follows:

Definition 3 (see [9]). Let be a HFE; then the score function of is determined as follows: where is the number of the elements in . For two HFEs, and , if , then ; if , then .

2.2. Dual Hesitant Fuzzy Sets

As an extension of HFS, Zhu et al. [27] developed the concept of dual hesitant fuzzy sets (DHFSs), 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 4 (see [27]). Let be a fixed set; then a dual hesitant fuzzy set (DHFS) on is described as follows: in which and are two sets of some values in , denoting the possible membership degrees and nonmembership degrees of the element to the set , with the conditions: and , where , , , and for all . For convenience, the pair is called a dual hesitant fuzzy element (DHFE) denoted by .

To compare the DHESs, Zhu et al. [27] gave the following comparison laws.

Definition 5 (see [27]). Let and be any two DHFSs; then the score function of () is () and the accuracy function of () is (), where and are the numbers of the elements in and , respectively; then consider the following:(1)if , then is superior to , denoted by ;(2)if , then consider the following:(a)if , then is superior to , denoted by ;(b)if , then is equivalent to , denoted by .

2.3. Interval-Valued Dual Hesitant Fuzzy Set

In some real-life decision making problems, decision makers may find it hard to express their evaluation about an alternative under a specific attribute with exact and crisp values. Since the interval-valued fuzzy set is usually more adequate or sufficient to model real-life decision problems than real numbers, Ju et al. [29] developed the interval-valued dual hesitant fuzzy set.

Definition 6 (see [29]). Let be a fixed set; then an interval-valued dual hesitant fuzzy set (IVDHFS) on is defined as where and are two sets of some interval values in , denoting the possible membership degrees and nonmembership degrees of the element to the set , respectively, with the conditions: and , where , , , and for all . For convenience, we call the pair an interval-valued dual hesitant fuzzy element (IVDHFE) denoted by and the set of all IVDHFEs.

Especially, if and , then reduces to a dual hesitant fuzzy set.

To compare the IVDHFEs, Ju et al. [29] give the following comparison laws.

Definition 7 (see [29]). Let be an interval-valued dual hesitant fuzzy element; then is called the score function of , and is called the accuracy function of , where and are the numbers of interval values in and , respectively.

Theorem 8 (see [29]). Let and be any two IVDHFEs; then one can compare them in terms of the following rules:(1)if , then ;(2)if , then(a)if , then ;(b)if , then ;(c)if , then .

Based on algebraic operations of IVDHFEs, some interval-valued dual hesitant fuzzy aggregation operators can be defined as follows.

Motivated by the intuitionistic fuzzy aggregation operators developed by Xu [39] and Xu and Yager [40], some interval-valued dual hesitant fuzzy aggregation operators can be defined as follows based on algebraic operations of IVDHFEs.

Definition 9 (see [29]). Let () be a collection of IVDHFEs and let be the weight vector of (), with and ; then consider the following.(1)An interval-valued dual hesitant fuzzy weighted average (IVDHFWA) operator is defined as follows: (2)An interval-valued dual hesitant fuzzy weighted geometric (IVDHFWG) operator is defined as follows:

Definition 10 (see [29]). Let () be a collection of IVDHFEs, let be the th largest of them, and let be the aggregation-associated weight vector with and ; then consider the following.(1)An interval-valued dual hesitant fuzzy ordered weighted averaging (IVDHFOWA) operator is defined as follows: (2)An interval-valued dual hesitant fuzzy ordered weighted geometric (IVDHFOWG) operator is defined as follows:

Definition 11 (see [29]). Let () be a collection of IVDHFEs, let be the weight vector of (), with and , and let be the balancing coefficient which plays a role of balance; then based on the location weighted vector , such that and , some interval-valued dual hesitant fuzzy hybrid aggregation operators are defined as follows.(1)An interval-valued dual hesitant fuzzy hybrid average (IVDHFHA) operator is defined as follows:(2)An interval-valued dual hesitant fuzzy hybrid geometric (IVDHFHG) operator is defined as follows: in which is the th largest of interval-valued dual hesitant fuzzy weighted arguments (, ), and is the th largest of interval-valued dual hesitant fuzzy weighted arguments (, ).

3. Interval-Valued Dual Hesitant Fuzzy Aggregation Operators Based on Einstein Operations

In this section, we will develop some aggregation operators to aggregate interval-valued dual hesitant fuzzy information based on Einstein operations. Einstein operations include the Einstein product and Einstein sum. Einstein product is a -norm and Einstein sum is a -conorm [41], where(1);(2), .

3.1. Operational Laws for IVDHFEs Based on Einstein Operations

Let , , and be any three interval-valued dual hesitant fuzzy elements, where , , , and ; then some operational laws of the IVDHFEs can be defined based on Einstein operations.

Definition 12. Let , , and be three IVDHFEs, then we have the following operational rules:Obviously, the above operational rules are still IVDHFEs. Some relationships can be further established for these operations on IVDHFEs.

Theorem 13. Let , , and be any three IVDHFEs; then one has (1);(2);(3), ;(4), .

3.2. Interval-Valued Dual Hesitant Fuzzy Einstein Weighted Aggregation Operators

Based on the above operational laws, we develop a new operator, which is defined as follows.

Definition 14. Let () be a collection of IVDHFEs; an interval-valued dual hesitant fuzzy Einstein weighted averaging (IVDHFEWA) operator is a mapping IVDHFEWA: , such that where is the weight vector of with and .

Theorem 15. Let () be a collection of IVDHFEs; then their aggregation value by using the IVDHFEWA operator is also an IVDHFE, andwhere is the weight vector of with and .

Proof. The first result follows quickly from Definition 14. In what follows, we prove (16) using mathematical induction on .(1)When , it is easy to conclude that (16) holds according to the Einstein operational law in Definition 12: (2)Assume that (16) holds for (); namely,When , we getLet then According to the Einstein operational law in Definition 12, we havethat is, (16) holds for .
According to steps and , we know that (16) holds for any positive integer .
The proof is completed.

Especially, if , then the IVDHFEWA operator is reduced to an interval-valued dual hesitant fuzzy Einstein averaging (IVDHFEA) operator, which is shown as follows:

Theorem 16. Let () be a collection of IVDHFEs, then we have the following properties:(1) Idempotency. If all () are equal and , for all , then

Proof. Since , for all , thenThus, .

The proof is completed.(2) Boundedness. If , , , , , , , and , for all , then we can obtain

Proof. Let , ; then ; that is, is a decreasing function. Since , then, for all , we have ; that is, Let be the weight vector of , such that and . Then, for all , we have Thus that is, Similarly, we have For all , we have where is the number of interval values in the membership degrees of .

Let , ; then ; that is, is a decreasing function. Since , then, for all , we have ; that is, . Let be the weight vector of , such that and . Then, for all , we have Thus that is, Similarly, we have That is, where is the number of interval values in the nonmembership degrees of .

So we can get Therefore, according to Theorem 8, we have The proof is completed.

Lemma 17 (see [42]). Let , , , and ; then with equality if and only if .

To compare the aggregated values between the IVDHFEWA operator and IVDHFWA operator in (8), we give the following theorem.

Theorem 18. Let () be a collection of IVDHFEs and let be the weight vector of with and ; then

Proof. According to Lemma 17, for any , , we have Thus, we have Similarly, for any , , we have That is, that is, where and are the numbers of interval values in the membership degrees and nonmembership degrees of , respectively; and are the numbers of interval values in the membership degrees and nonmembership degrees of , respectively. Therefore, according to Theorem 8, we obtain The proof is completed.

Definition 19. Let () be a collection of IVDHFEs; an interval-valued dual hesitant fuzzy Einstein weighted geometric (IVDHFEWG) operator is a mapping IVDHFEWG: , such thatwhere is the weight vector of with and .

Especially, if , then the IVDHFEWG operator is reduced to an interval-valued dual hesitant fuzzy Einstein geometric (IVDHFEG) operator. Consider

Similar to the IVDHFEWA operator, the IVDHFEWG operator also has the properties of idempotency and boundedness.

Theorem 20. Let () be a collection of IVDHFEs and let be the weight vector of with and ; then

This theorem can be proved similar to Theorem 18.

3.3. Interval-Valued Dual Hesitant Fuzzy Einstein Ordered Weighted Aggregation Operators

Motivated by the idea of the ordered weighted averaging (OWA) [43] and the ordered weighted geometric [44] operators, we develop some interval-valued dual hesitant fuzzy Einstein ordered weighted aggregation operators.

Definition 21. Let () be a collection of IVDHFEs, let be the th largest of them, and let be the aggregation-associated weight vector, such that and ; then an interval-valued dual hesitant fuzzy Einstein ordered weighted averaging (IVDHFEOWA) operator is a mapping IVDHFEOWA: , wherewhere is a permutation of (), such that for all .

Especially, if , then the IVDHFEOWA operator reduces to the IVDHFEA operator in (23).

Similar to IVDHFEWA and IVDHFEWG operators, the IVDHFEOWA operator also has the properties of idempotency and boundedness. In addition, it has the property of commutativity shown as follows.

Theorem 22 (commutativity). Let be a collection of IVDHFEs and let be any permutation of ; then

Proof. Since is a permutation of , we have for all . Then, based on Definition 21, we obtain The proof is completed.

Theorem 23. Let () be a collection of IVDHFEs and let be the aggregation-associated weight vector, such that and ; then

This theorem can be proved similar to Theorem 18.

Definition 24. Let () be a collection of IVDHFEs, let be the th largest of them, and let be the aggregation-associated weight vector, such that and ; then an interval-valued dual hesitant fuzzy Einstein ordered weighted geometric (IVDHFEOWG) operator is a mapping IVDHFEOWG: , wherewhere is a permutation of (), such that for all .

Especially, if , then the IVDHFEOWG operator reduces to the IVDHFEG operator in (49).

Theorem 25. Let () be a collection of IVDHFEs and let be the aggregation-associated weight vector, such that and ; then

This theorem can be proved similar to Theorem 18.

3.4. Interval-Valued Dual Hesitant Fuzzy Hybrid Aggregation Operators Based on Einstein Operations

From Definitions 14–24, we can see that the IVDHFEWA and IVDHFEWG operators only weight the importance of interval-valued dual hesitant fuzzy argument itself, while the IVDHFEOWA and IVDHFEOWG operators only weight the importance of ordered position of each argument. Therefore, weights represent different aspects in both weighted aggregation (IVDHFEWA and IVDHFEWG) operators and ordered weighted aggregation (IVDHFEOWA and IVDHFEOWG) operators. To solve this drawback, in what follows, we will propose some interval-valued dual hesitant fuzzy Einstein hybrid aggregation operators, which weight both the given interval-valued dual hesitant fuzzy arguments and their ordered positions. Motivated by the hybrid aggregation operators [45], which consider both the given arguments and their ordered positions, in what follows, we will propose some interval-valued dual hesitant fuzzy Einstein hybrid aggregation operators.

Definition 26. Let () be a collection of IVDHFEs, let be the weight vector of () with , , and let be the balancing coefficient which plays a role of balance; then an interval-valued dual hesitant fuzzy Einstein hybrid averaging (IVDHFEHA) operator is a mapping with the aggregation-associated weight vector , such that and :where is the th largest of interval-valued dual hesitant fuzzy weighted arguments (), (). is the weight vector of and is the aggregation-associated weight vector, such that and .

Especially, if , then the IVDHFEHA operator reduces to the IVDHFEWA operator in (16). If , then the IVDHFEHA operator reduces to the IVDHFEOWA operator in (51).

Theorem 27. Let () be a collection of IVDHFEs, let be the weight vector of (), with , , and let be the aggregation-associated weight vector, such that and ; then

This theorem can be proved similar to Theorem 18.

Definition 28. Let () be a collection of IVDHFEs, let be the weight vector of (), with , , and let be the balancing coefficient which plays a role of balance; then an interval-valued dual hesitant fuzzy Einstein hybrid geometric (IVDHFEHG) operator is a mapping with the aggregation-associated weight vector , such that and :where is the th largest of interval-valued dual hesitant fuzzy weighted arguments (), (). is the weight vector of and is the aggregation-associated weight vector, such that and .

Especially, if , then the IVDHFEHG operator reduces to the IVDHFEWG operator in (49). If , then the IVDHFEHG operator reduces to the IVDHFEOWG operator in (55).

Theorem 29. Let () be a collection of IVDHFEs, let be the weight vector of (), with , , and let be the aggregation-associated weight vector, such that and ; then

This theorem can be proved similar to Theorem 18.

4. An Approach to MADM with Interval-Valued Dual Hesitant Fuzzy Information

In this section, we apply the aggregation operators proposed above to multiattribute decision making with interval-valued dual hesitant fuzzy information. Let be a finite set of alternatives, let be the set of attributes, and let be the weight vector of attributes () with and . Suppose that is an interval-valued dual hesitant fuzzy matrix, where is in the form of IVDHFE given for alternative () with respect to attribute (), with and . Then, to determine the most desirable alternative(s), the IVDHFEWA operator is utilized to develop a multiattribute decision making method with interval-valued dual hesitant fuzzy information by the following steps.

Step 1. Obtain the interval-valued dual hesitant fuzzy matrix. The decision makers provide their evaluations about alternative under attribute , denoted by the interval-valued dual hesitant fuzzy elements , (; ).

Step 2. Compute overall assessments of alternatives. Utilize the IVDHFEWA operator to aggregate all the rating values () of the th line and get the overall rating value corresponding to alternative (); that is,where is the weight vector of attributes (), such that and . Note that is in the forms of IVDHFEs, and it can be denoted by , , and .

Step 3. Compare the score values of overall assessments values () using score function by where and are the numbers of interval values in and , respectively. If , then we need to calculate the accuracy values and of alternatives and () by the following:

Step 4. Rank all feasible alternatives () according to Theorem 8 and select the most desirable alternative(s).

Step 5. End.

5. Illustrative Example

5.1. An Example

In this section, a MADM problem adapted from [34] is used to illustrate the developed procedure. In [34], Wang and Lee considered a software selection problem in which the alternatives are the software packages to be selected and the criteria are the attributes under consideration. The manger of a computer center at a university wishes to select a new information system to improve work productivity. After preliminary screening, four alternatives () remain on the candidate list. And three attributes are under consideration: the cost of the hardware/software investment (), the contribution to the performance of the organization (), and the effort to transfer from the current system (). The weight vector of attributes () is . The experts evaluate the software packages () with respect to attributes () and the evaluations are expressed in the form of IVDHFE. In what follows, we use the MADM method proposed in Section 4 to select the most desirable software package(s).

Step 1. Determine the interval-valued dual hesitant fuzzy matrix shown in Table 1, where is the evaluation value about the alternative with respect to the attribute and it is in the form of IVDHFE.

Step 2. Utilize (61) to aggregate all the rating values of alternative () on all attributes () into overall assessment values (), which are shown as follows:

Step 3. Compare the magnitude of the different overall assessments values () using score function according to (62):

Step 4. Rank all the alternatives () according to Theorem 8. Since , then the ranking of the alternatives is shown as follows: . Therefore, the most desirable alternative is .

5.2. Comparison with Other Methods

To test the validity of the proposed method, the evaluation results in Section 5.1 are compared with that from the other method proposed by Ju et al. [29]. The main difference between the two methods is due to the aggregation process. Specifically, in the process of aggregating the attributes’ values, the method proposed in this paper uses the IVDHFEWA operator that is based on the Einstein -norms and -conorms, while the method presented in [29] uses the IVDHFWA operator that is based on the algebraic -norms and -conorms.

Therefore, if we use the IVDHFWA operator instead of the IVDHFEWA operator in Step 2 in Section 5.1, the aggregated results () with respect to the rating values (, ) are shown as follows:

Similarly, the score function values of the () can be calculated according to (62); the results are shown as follows:

Obviously, the ranking order of the four alternatives is , which is exactly the same as that obtained in Section 5.1.

It is interesting to point out that the score values obtained by the IVDHFEWA operator are smaller than those obtained by the IVDHFWA operator, which is consistent with Theorem 18.

From the above analysis, we can clearly find that the proposed approach is effective. In addition, when the decision makers show some kind of pessimistic attitude towards the decision making problems, they can choose the IVDHFEWA operator, which has more merits in characterizing the pessimistic attitude than the IVDHFWA operator.

6. Conclusions

The traditional dual hesitant fuzzy aggregation operators are generally suitable for aggregating information taking the form of numerical numbers, and yet they will fail in dealing with interval-valued dual hesitant fuzzy information. In this paper, we investigate the MADM problems in which the attribute values take the form of interval-valued dual hesitant fuzzy information. Firstly, we propose some operational laws for IVDHFEs based on Einstein operations. Then, we develop some interval-valued dual hesitant fuzzy Einstein aggregation operators: the IVDHFEWA operator, IVDHFEWG operator, IVDHFEOWA operator, and IVDHFEOWG operator. Some desirable properties of these operators and the relationship between the developed operators and the existing ones are investigated. To emphasize the importance of ordered position of each argument and the importance of the argument itself, we also proposed the IVDHFEHA operator and IVDHFEHG operator, respectively. In addition, we put forward an approach to deal with MADM problems under interval-valued dual hesitant fuzzy setting. Finally an illustrated example is given to show the developed method, and a comparison analysis is also conducted to demonstrate the effectiveness and superiority of the proposed approach. All the aggregation operators proposed in this paper are based on the assumption that the attributes in a given set are independent; that is, we only consider the addition of the importance of individual elements. However, in many practical situations, the elements in a set are usually correlative. Therefore, how to deal with the situations in which the arguments in a question are correlative is our future work.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express appreciation to the anonymous reviewers and the editor Wudhichai Assawinchaichote for their very helpful comments that improved the paper. This research is supported by Program for New Century Excellent Talents in University (NCET-13-0037), Natural Science Foundation of China (nos. 70972007, 71271049), and Beijing Municipal Natural Science Foundation (nos. 9102015, 9133020).

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Copyright © 2014 Wenkai Zhang 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.

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