AUTHOR COPY Journal of Intelligent & Fuzzy Systems 26 (2014) 1601–1617 DOI:10.3233/IFS-130841 IOS Press 1601 Some new hybrid weighted aggregation operators under hesitant fuzzy multi-criteria decision making environment Huchang Liao a and Zeshui Xu a,b,∗ a Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai, China b College of Sciences, PLA University of Science and Technology, Nanjing, China Abstract. Hesitant fuzzy set, as a new generalized type of fuzzy set, is an efficient and powerful structure in expressing uncertainty and vagueness and has attracted more and more scholars’ attention. The aim of this paper is to develop some new aggregation operators to fuse hesitant fuzzy information. The hesitant fuzzy hybrid arithmetical averaging (HFHAA) operator, the hesitant fuzzy hybrid arithmetical geometric (HFHAG) operator, the quasi HFHAA operator and the quasi HFHAG operator are proposed and their properties are investigated. On the basis of these proposed operators, some algorithms are introduced to aid multi-criteria single person decision making and multi-criteria group decision making respectively. Some examples are provided to illustrate the practicality and validity of our proposed procedures. Keywords: Group decision making, hesitant fuzzy set, hybrid weighted aggregation operator, multi-criteria decision making 1. Introduction Since it was originally introduced by Zadeh [1], the fuzzy set has turned out to be one of the most efficient decision aid techniques providing the ability to deal with uncertainty and vagueness. In realistic decision making, imprecision may arise due to the unquantifiable information, incomplete information, unobtainable information, partial ignorance, and so forth [2]. To cope with imperfect and imprecise infor- mation whereby two or more sources of vagueness appear simultaneously, Zadeh’s traditional fuzzy set shows some limitations [3]. The traditional fuzzy set uses a crisp number in unit interval [0, 1] as a member- ship degree of an element to a set; however, very often, such a crisp number is difficult to be determined for the ∗ Corresponding author. Zeshui Xu, Tel./Fax: +86 25 84483382; E-mails: xuzeshui@263.net; liaohuchang@163.com (H. Liao). decision maker. On the other hand, if a group of decision makers are asked to evaluate the candidate alternatives, they often find some disagreements among themselves. Since the decision makers may have different opinions over the alternatives and they can’t persuade each other easily, a consensus result is hard to be obtained but a set of possible values. In such case, the traditional fuzzy set also can not be used to depict the group’s opinions. Hence, the classical fuzzy set has been extended into several different forms, such as the intuitionistic fuzzy set [4], the interval-valued intuitionistic fuzzy set [5], the type 2 fuzzy set [6], the type n fuzzy set [7], the fuzzy multisets (also named the fuzzy bags) [8], and so on [9]. All these extensions are based on the same rationale that it is not clear to assign the membership degree of an element to a fixed set [10]. Recently, on the basis of the above extensional forms of the fuzzy set, Torra and Narukawa [10, 11] proposed a new general- ized type of fuzzy set called hesitant fuzzy set (HFS), 1064-1246/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved