Fuzzy Hamacher Aggregation Functions and Their Applications to Multiple Attribute Decision Making

Abstract

Hamacher t-norm and t-conorm have been widely applied in fuzzy multiple attribute decision making (MADM) to combine assessments on each attribute, which are generally expressed by Atanassov's intuitionistic fuzzy (AIF) numbers, interval-valued intuitionistic fuzzy (IVIF) numbers, hesitant fuzzy (HF) elements, and dual hesitant fuzzy (DHF) elements. Due to the fact that AIF numbers and HF elements are special cases of IVIF numbers and DHF elements, respectively, two propositions can be established from analyzing numerical examples and real cases concerning MADM with IVIF and DHF assessments in the literature: (1) the monotonicity of alternative scores derived from Hamacher arithmetic and geometric aggregation operators with respect to the parameter r in Hamacher t-norm and t-conorm and (2) the relationship between alternative scores generated by Hamacher arithmetic and geometric aggregation operators, given the same r. The authors provide the theoretical proof of the two propositions and propose a new method to compare alternatives in MADM with consideration of all possible values of r.