Abstract
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•New score and accuracy functions of dual hesitant fuzzy elements are designed.•Some Frank operational rules of dual hesitant fuzzy sets are developed.•Frank hybrid weighted arithmetic and geometric aggregation operators are proposed.•Monotonicity of the two types of aggregation operators regarding the parameter in Frank operations and their relationships are investigated.•A flexible MADM approach with dual hesitant fuzzy information is proposed and applied to an investment evaluation problem.
Real-world decision-making problems are often complex and indeterminate. Thus, uncertainty and hesitancy are usually unavoidable issues being experienced by decision makers. Dual hesitant fuzzy sets (DHFSs) which are described in terms of the two functions, namely the membership hesitancy function and the non-membership hesitancy function, have been developed. In light of their properties, they are considered as a powerful vehicle to express uncertain information in the process of multi-attribute decision-making (MADM). In accordance with the practical demand, this study proposes a new MADM approach with dual hesitant fuzzy (DHF) assessments based on Frank aggregation operators. First, original score and accuracy functions of DHFS are developed to construct a new comparison method of DHFSs. The properties of the developed score and accuracy functions are analyzed. Second, we investigate the generalized operations of DHFS based on Frank t-norm and t-conorm. The generalized operations are then used to build the generalized arithmetic and geometric aggregation operators of DHF assessments in the context of fuzzy MADM. The monotonicity of arithmetic and geometric aggregated assessments with respect to a parameter in Frank t-norm and t-conorm and their relationship are also demonstrated. In particular, the monotonicity is employed to associate the parameter with the risk attitude of a decision maker, by which a method is designed to determine the parameter. A procedure of the proposed MADM method is presented. Finally, an investment evaluation problem is discussed by the proposed approach to demonstrate its applicability and validity. A detailed sensitivity analysis and a comparative study are also conducted to highlight the validity and advantages of the approach proposed in this paper. More importantly, we discuss the situations where Frank aggregation operators are replaced by Hamacher aggregation operators at the second step of the proposed approach, through re-considering the investment evaluation problem.