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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 745617, 14 pages
Multiple Attribute Decision Making Based on Hesitant Fuzzy Einstein Geometric Aggregation Operators
1College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
2College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China
Received 25 June 2013; Accepted 12 November 2013; Published 16 January 2014
Academic Editor: Yi-Chi Wang
Copyright © 2014 Xiaoqiang Zhou and Qingguo Li. 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.
We first define an accuracy function of hesitant fuzzy elements (HFEs) and develop a new method to compare two HFEs. Then, based on Einstein operators, we give some new operational laws on HFEs and some desirable properties of these operations. We also develop several new hesitant fuzzy aggregation operators, including the hesitant fuzzy Einstein weighted geometric (HFEWGε) operator and the hesitant fuzzy Einstein ordered weighted geometric (HFEWGε) operator, which are the extensions of the weighted geometric operator and the ordered weighted geometric (OWG) operator with hesitant fuzzy information, respectively. Furthermore, we establish the connections between the proposed and the existing hesitant fuzzy aggregation operators and discuss various properties of the proposed operators. Finally, we apply the HFEWGε operator to solve the hesitant fuzzy decision making problems.
Atanassov [1, 2] introduced the concept of intuitionistic fuzzy set characterized by a membership function and a nonmembership function. It is more suitable to deal with fuzziness and uncertainty than the ordinary fuzzy set proposed by Zadeh  characterized by one membership function. Information aggregation is an important research topic in many applications such as fuzzy logic systems and multiattribute decision making as discussed by Chen and Hwang . Research on aggregation operators with intuitionistic fuzzy information has received increasing attention as shown in the literature. Xu  developed some basic arithmetic aggregation operators based on intuitionistic fuzzy values , such as the intuitionistic fuzzy weighted averaging operator and intuitionistic fuzzy ordered weighted averaging operator, while Xu and Yager  presented some basic geometric aggregation operators for aggregating IFVs, including the intuitionistic fuzzy weighted geometric operator and intuitionistic fuzzy ordered weighted geometric operator. Based on these basic aggregation operators proposed in  and , many generalized intuitionistic fuzzy aggregation operators have been investigated [5–30]. Recently, Torra and Narukawa  and Torra  proposed the hesitant fuzzy set , which is another generalization form of fuzzy set. The characteristic of is that it allows membership degree to have a set of possible values. Therefore, is a very useful tool in the situations where there are some difficulties in determining the membership of an element to a set. Lately, research on aggregation methods and multiple attribute decision making theories under hesitant fuzzy environment is very active, and a lot of results have been obtained for hesitant fuzzy information [33–43]. For example, Xia et al.  developed some confidence induced aggregation operators for hesitant fuzzy information. Xia et al.  gave several series of hesitant fuzzy aggregation operators with the help of quasiarithmetic means. Wei  explored several hesitant fuzzy prioritized aggregation operators and applied them to hesitant fuzzy decision making problems. Zhu et al.  investigated the geometric Bonferroni mean combining the Bonferroni mean and the geometric mean under hesitant fuzzy environment. Xia and Xu  presented some hesitant fuzzy operational laws based on the relationship between the and the . They also proposed a series of aggregation operators, such as hesitant fuzzy weighted geometric operator and hesitant fuzzy ordered weighted geometric operator. Furthermore, they applied the proposed aggregation operators to solve the multiple attribute decision making problems.
Note that all aggregation operators introduced previously are based on the algebraic product and algebraic sum of (or ) to carry out the combination process. However, the algebraic operations include algebraic product and algebraic sum, which are not the unique operations that can be used to perform the intersection and union. There are many instances of various t-norms and t-conorms families which can be chosen to model the corresponding intersections and unions, among which Einstein product and Einstein sum are good alternatives for they typically give the same smooth approximation as algebraic product and algebraic sum, respectively. For intuitionistic fuzzy information, Wang and Liu [10, 11, 44] and Wei and Zhao  developed some new intuitionistic fuzzy aggregation operators with the help of Einstein operations. For hesitant fuzzy information, however, it seems that in the literature there is little investigation on aggregation techniques using the Einstein operations to aggregate hesitant fuzzy information. Therefore, it is necessary to develop some hesitant fuzzy information aggregation operators based on Einstein operations.
The remainder of this paper is structured as follows. In Section 2, we briefly review some basic concepts and operations related to and . we also define an accuracy function of to distinguish the two having the same score values, based on which we give the new comparison laws on . In Section 3, we present some new operations for and discuss some basic properties of the proposed operations. In Section 4, we develop some novel hesitant fuzzy geometric aggregation operators with the help of Einstein operations, such as the operator and the operator, and we further study various properties of these operators. Section 5 gives an approach to solve the multiple attribute hesitant fuzzy decision making problems based on the operator. Finally, Section 6 concludes the paper.
In this section, we briefly introduce Einstein operations and some notions of and . Meantime, we define an accuracy function of and redefine the comparison laws between two .
2.1. Einstein Operations
Since the appearance of fuzzy set theory, the set theoretical operators have played an important role and received more and more attention. It is well known that the t-norms and t-conorms are the general concepts including all types of the specific operators, and they satisfy the requirements of the conjunction and disjunction operators, respectively. There are various t-norms and t-conorms families that can be used to perform the corresponding intersections and unions. Einstein sum and Einstein product are examples of t-conorms and t-norms, respectively. They are called Einstein operations and defined as 
2.2. Intuitionistic Fuzzy Set
Definition 1. Let be fixed an on is given by; where and , with the condition for all . Xu  called an .
For , Wang and Liu  introduced some operations as follows.
Let , and be two ; then
2.3. Hesitant Fuzzy Set
Definition 2. Let be a reference set; an on is in terms of a function that when applied to returns a subset of .
To be easily understood, Xia and Xu use the following mathematical symbol to express the : where is a set of some values in , denoting the possible membership degrees of the element to the set . For convenience, Xu and Xia  called a hesitant fuzzy element .
Let , and be two . Xia and Xu  defined some operations as follows:
In , Xia and Xu defined the score function of an to compare the and gave the comparison laws.
Definition 3. Let be an ; is called the score function of , where is the number of values of . For two and , if , then ; if , then .
From Definition 3, it can be seen that all are regarded as the same if their score values are equal. In hesitant fuzzy decision making process, however, we usually need to compare two for reordering or ranking. In the case where two have the same score values, they can not be distinguished by Definition 3. Therefore, it is necessary to develop a new method to overcome the difficulty.
For an , Hong and Choi  showed that the relation between the score function and the accuracy function is similar to the relation between mean and variance in statistics. From Definition 3, we know that the score value of is just the mean of the values in . Motivated by the idea of Hong and Choi , we can define the accuracy function of by using the variance of the values in .
Definition 4. Let be an HFE; is called the accuracy function of , where is the number of values in and is the score function of .
It is well known that an efficient estimator is a measure of the variance of an estimate’s sampling distribution in statistics: the smaller the variance, the better the performance of the estimator. Motivated by this idea, it is meaningful and appropriate to stipulate that the higher the accuracy degree of , the better the . Therefore, in the following, we develop a new method to compare two , which is based on the score function and the accuracy function, defined as follows.
Definition 5. Let and be two HFEs and let and be the score function and accuracy function of , respectively. Then(1)if , then is smaller than , denoted by ;(2)if , then(i)if , then is smaller than , denoted by ;(ii)if , then and represent the same information, denoted by . In particular, if for any and , then is equal to , denoted by .
Example 6. Let , , , , , and ; then , , , , , , and . By Definition 5, we have .
3. Einstein Operations of Hesitant Fuzzy Sets
In this section, we will introduce the Einstein operations on and analyze some desirable properties of these operations. Motivated by the operational laws (1)–(3) on and based on the interconnection between and , we give some new operations of as follows.
Let , , , and be three ; then
Proposition 7. Let , , , , and be three HFEs; then(1),(2),(3),(4);(5),(6).
Proof. It is trivial.
By the operational law (9), we have
Let ; then Since and , then Thus .
Since , then
By the definition of the envelope of an and the operation laws and , we have Thus, .
By the definition of the envelope of an and the operation laws and , we have Thus, .
Remark 8. Let , , and be an . It is worth noting that does not hold necessarily in general. To illustrate that, an example is given as follows.
Example 9. Let , ; then , and . Clearly, . Thus .
However, if the number of the values in is only one, that is, is reduced to a fuzzy value, then the above result holds.
Proposition 10. Let , , and be an , in which the number of the values is only one, that is, ; then .
Proof. Since and , then
4. Hesitant Fuzzy Einstein Geometric Aggregation Operators
The weighted geometric operator  and the ordered weighted geometric operator  are two of the most common and basic aggregation operators. Since their appearance, they have received more and more attention. In this section, we extend them to aggregate hesitant fuzzy information using Einstein operations.
4.1. Hesitant Fuzzy Einstein Geometric Weighted Aggregation Operator
Based on the operational laws and on , Xia and Xu  developed some hesitant fuzzy aggregation operators as listed below.
Let be a collection of ; then.
the hesitant fuzzy weighted geometric (HFWG) operator where is the weight vector of with and .
the hesitant fuzzy ordered weighted geometric operator where is a permutation of , such that for all and is aggregation-associated vector with and .
For convenience, let be the set of all . Based on the proposed Einstein operations on , we develop some new aggregation operators for and discuss their desirable properties.
Definition 11. Let be a collection of . A hesitant fuzzy Einstein weighted geometric operator of dimension is a mapping defined as follows: where is the weight vector of and , . In particular, when , , the operator is reduced to the hesitant fuzzy Einstein geometric operator:
From Proposition 10, we easily get the following result.
Corollary 12. If all are equal and the number of values in is only one, that is, for all , then
Note that the operator is not idempotent in general; we give the following example to illustrate this case.
Example 13. Let , ; then . By Definition 3, we have . Hence .
Theorem 15. Let be a collection of and the weight vector of with and . Then where the equality holds if only if all are equal and the number of values in is only one.
Theorem 15 tells us the result that the operator shows the decision maker’s more optimistic attitude than the operator proposed by Xia and Xu  (i.e., (15)) in aggregation process. To illustrate that, we give an example adopted from Example 1 in  as follows.
Based on Definition 11 and the proposed operational laws, we can obtain the following properties on operator.
Theorem 17. Let , , be a collection of HFEs and the weight vector of with and . Then
Proof. Since for all , by the definition of , we have Since , then
Theorem 18. Let be an HFE, a collection of HFEs, and the weight vector of with and . Then
Proof. By the definition of and Einstein product operator of , we have Since for all , by the definition of , we have
Theorem 19. Let , be an HFE, let be a collection of HFEs, and let be the weight vector of with and . Then
Theorem 20. Let and be two collections of HFEs and the weight vector of with and . Then
Proof. By the definition of and Einstein product operator of , we haveSince for all , by the definition of , we have
Theorem 21. Let be a collection of HFEs, , and , and let be the weight vector of with and . Then where the equality holds if only if all are equal and the number of values in is only one.
Proof. Let , . Then . Hence is a decreasing function. Since for any , then ; that is, . Then for any , we have It follows that . Thus we have .
Remark 22. Let and be two collections of HFEs, and for all ; then does not hold necessarily in general. To illustrate that, an example is given as follows.
Example 23. Let , , , , , , and ; then and . By Definition 3, we have and . It follows that . Clearly, for , but .
4.2. Hesitant Fuzzy Einstein Ordered Weighted Averaging Operator
Definition 24. For a collection of the , a hesitant fuzzy Einstein ordered weighted averaging operator is a mapping such that where is a permutation of , such that for all and is aggregation-associated vector with and . In particular, if , then the operator is reduced to the operator of dimension (i.e., (17)).
Similar to the operator, the operator has the following properties.
Theorem 25. Let be a collection of HFEs and the weight vector of with and . Then where the equality holds if only if all are equal and the number of values in is only one.
From Theorem 25, we can conclude that the values obtained by the operator are not less than the ones obtained by the operator proposed by Xia and Xu . To illustrate that, let us consider the following example.
Example 26. Let , , and be three and suppose that is the associated vector of the aggregation operator.
By Definitions 3 and 4, we calculate the score values and the accuracy values of , , and as follows, respectively:
, , , .
According to Definition 5, we have . Then , , .
By the definition of , we have If we use the