Some Hesitant Fuzzy Einstein Aggregation Operators and Their Application to Multiple Attribute Group Decision Making

Abstract

Hesitant fuzzy sets, as a new generalized type of fuzzy set, has attracted scholars’ attention due to their powerfulness in expressing uncertainty and vagueness. In this paper, motivated by the idea of Einstein operation, we develop a family of hesitant fuzzy Einstein aggregation operators, such as the hesitant fuzzy Einstein Choquet ordered averaging operator, hesitant fuzzy Einstein Choquet ordered geometric operator, hesitant fuzzy Einstein prioritized weighted average operator, hesitant fuzzy Einstein prioritized weighted geometric operator, hesitant fuzzy Einstein power weighted average operator, and hesitant fuzzy Einstein power weighted geometric operator. And we also study some desirable properties and generalized forms of these operators. Then, we apply these operators to deal with multiple attribute group decision making under hesitant fuzzy environments. Finally, a numerical example is provided to illustrate the practicality and validity of the proposed method.

An explainable multi-attribute decision model based on argumentation

“…Recently several researchers proposed fuzzy MADM models to handle criteria's evaluations with various complex fuzzinesses. As an example, to deal with degrees to which an alternative decision satisfies criteria, Park et al (2017) develop two approaches based on the concepts of entropy, cross-entropy, and similarity measure; Zhang (2016) propose one based on prioritised aggregation operator (more operators of this sort can be found in Luo et al (2003bLuo et al ( , 2015), and Yu et al (2016) also do so by developing several operators to aggregate all the criteria's evaluations. However, to the best of our knowledge, none of these approaches have any functionality for explaining the decisions they recommend.…”

Some Hesitant Fuzzy Hamacher Power-Aggregation Operators for Multiple-Attribute Decision-Making

“…The HFS has the advantage of representing the membership degree of one element to a set by a set of possible values between 0 and 1, so it is an effective tool to represent a decision-maker's hesitation in expressing his/her preferences for objects than the FS or its classical extensions. In this regard, the HFS theory has been applied to many practical applications such as decision-making [9][10][11][12][13][14][15][16][17][18].…”

“…In real life, we often face situations where input arguments are expressed not as exact numerical values but as interval numbers [27], intuitionistic fuzzy numbers [28][29][30], interval-valued intuitionistic fuzzy numbers [31], linguistic variables [32][33][34], uncertain linguistic variables [35][36][37], or hesitant fuzzy elements (HFEs) [10,11]. Many extensions of power-aggregation operators have been proposed to address these situations: the uncertain power-aggregation operators [25,38,39], intuitionistic fuzzy power-aggregation operators [26,40], interval-valued intuitionistic fuzzy power-aggregation operators [40], linguistic power-aggregation operators [41][42][43][44] and hesitant fuzzy power-aggregation operators [15,18]. In particular, with respect to HFEs, Zhang [15] proposed a family of hesitant fuzzy power-aggregation operators, including the hesitant fuzzy power-weighted average/geometric (HFPWA or HFPWG), generalized hesitant fuzzy power-weighted average/geometric (GHFPWA or GHFPWG), hesitant fuzzy power-ordered weighted average/geometric (HFPOWA or HFPOWG), and generalized hesitant fuzzy power-ordered weighted average/geometric (GHFPOWA or GHFPOWG) operators, and applied them to solve multiple criteria group decision-making problems under hesitant fuzzy environment.…”

“…Yu [16] extended the Einstein t-norm and t-conorm to HFEs, and developed some hesitant fuzzy Einstein aggregation operators based on the Einstein product and Einstein sum operational rules on HFEs. By mean of these operational rules on HFEs, Yu et al [18] proposed a wide range of hesitant fuzzy power-aggregation operators, such as the hesitant fuzzy Einstein power-weighted average/geometric (HFEPWA or HFEPWG), generalized hesitant fuzzy Einstein power-weighted average/geometric (GHFEPWA or GHFEPWG), hesitant fuzzy Einstein power-ordered weighted average/geometric (HFEPOWA or HFEPOWG), and generalized hesitant fuzzy Einstein power-ordered weighted average/geometric (GHFEPOWA or GHFEPOWG) operators, and applied them to deal with MADM with hesitant fuzzy information. Hamacher [46] proposed a more generalized t-norm and t-conorm, called the Hamacher t-norm and t-conorm.…”

Probabilistic Linguistic Aggregation Operators Based on Einstein t-Norm and t-Conorm and Their Application in Multi-Criteria Group Decision Making

“…However, the algebraic operational laws are not the only operational laws for information fusion. The Einstein operations are equally useful tools to substitute the algebraic operations [9]. Zhao et al [10], in their research introduced Einstein product as a t-norm and Einstein sum as t-conorm.…”