www.iaset.us editor@iaset.us INTERVAL-VALUED INTUITIONISTIC HESITANT FUZZY EINSTEIN GEOMETRIC AGGREGATION OPERATORS A. UMA MAHES WARI 1 & P. KUMARI 2 1 Associate Professor, Department of Mathematics, Quaid-E-Millath Government, College for Women, Chennai, Tamil Nadu, India 2 Assistant Professor, Department of Mathematics, D.G. Vaishnav College, Chennai, Tamil Nadu, India ABSTRACT Aggregation of fuzzy information in hesitant fuzzy environment is a new branch of hesitant fuzzy set (HFS) theory. HFS theory introduced by Torra and Narukowa has attracted significant interest from researchers in recent years. In this paper, we investigate the interval valued intuitionistic hesitant fuzzy (IVIHF) aggregation operators with the help of Einstein operations. First some new operations such as Einstein sum, Einstein product, and Einstein scalar multiplication on the interval valued intuitionistic hesitant fuzzy elements (IVIHFEs) are introduced. Then, some IVIHF aggregation operators such as interval valued intuitionistic hesitant fuzzy Einstein weighted geometric (IVIHFWG ∊ ) operators and the interval valued intuitionistic hesitant fuzzy Einstein ordered weighted geometric (IVIHFOWG ∊ ) operator are developed. Some of the properties of IVIHFEs are discussed in detail. KEYWORDS: Einstein Operations, Hesitant Fuzzy Set, Interval Valued Intuitionistic Hesitant Fuzzy Elements, Interval Valued Intuitionistic Hesitant Fuzzy Einstein Weighted Geometric (IVIHFWG ∊ ) Operators I. INTRODUCTION Fuzzy Set Theory by Zadeh [1] has been extended to several theories such as Atanassov's intuitionistic fuzzy set (AIFS) theory [2]. AIFSs is further generalized by Atanassov and Gargov [3] to accommodate the membership and non-membership functions to assume interval values, thereby introducing the concept of interval-valued intuitionistic fuzzy sets (IVIFSs). This extension mixes imprecision and hesitation. Recently, Torra and Narukawa [4] and Torra [5] proposed the hesitant fuzzy set (HFS), which is another generalization form of fuzzy set. The characteristic of HFS is that it allows membership degree to have a set of possible values. Therefore, HFS 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. Xia et al [6] developed hesitant fuzzy aggregation operators. Combining the heronian mean and hesitant fuzzy sets, some new hesitant fuzzy Heronian mean (HFHM) operators are explored in [7]. Aggregation operators are essential mathematical tool for fuzzy decision-making. This tool is extended to the interval valued intuitionistic hesitant fuzzy environment. All aggregation operators introduced previously are based on the algebraic product and algebraic sum of intuitionistic fuzzy values (IFVs) or hesitant fuzzy elements (HFEs) to carry out the combination process. The algebraic operations algebraic product and algebraic sum are not the unique operations that can be used to perform the intersection and union. Einstein product and Einstein sum are good alternatives for they typically give the same smooth approximation as algebraic product and algebraic sum. For intuitionistic fuzzy information, International Journal of Computer Science and Engineering (IJCSE) ISSN(P): 2278-9960; ISSN(E): 2278-9979 Vol. 3, Issue 3, May 2014, 125-140 © IASET