Multiple Attribute Decision Making Based on Hesitant Fuzzy Einstein Geometric Aggregation Operators

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. 1. Introduction 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 [3] 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 [4]. Research on aggregation operators with intuitionistic fuzzy information has received increasing attention as shown in the literature. Xu [5] 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 [6] 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 [6] and [5], many generalized intuitionistic fuzzy aggregation operators have been investigated [5–30]. Recently, Torra and Narukawa [31] and Torra [32] 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