Complex nth power root fuzzy sets: Theory, and applications for multi-attribute decision making in uncertain environments.

The newly introduced nth power root fuzzy set is a useful tool for expressing ambiguity and vagueness. It has an improved ability to manage uncertain situations compared to intuitionistic fuzzy set and Pythagorean fuzzy set theories, making nth power root fuzzy sets applicable in various everyday decision-making contexts. The notions of nth power root fuzzy sets and complex fuzzy sets are integrated in this study to offer complex nth power root fuzzy sets (CnPR-FSs), explaining its fundamental ideas and useful applications. The proposed CnPR-FS integrates the advantages of nth power root fuzzy set and captures both quantitative and qualitative analyses of decision-makers. It is shown that CnPR-FSs are a crucial tool that can describe uncertain data better than complex intuitionistic fuzzy sets and complex Pythagorean fuzzy sets. A key characteristic of CnPR-FSs is a constraint that guarantees the summation of the nth power of the real (and imaginary) part of the complex-valued membership degree and the 1/n power of the real (and imaginary) part of the complex-valued non-membership degree to be equal to or less than one. This allows for a broader representation of uncertain information. The study also explores the creation of customized comparison techniques, accuracy functions, and scoring functions for two complex nth power root fuzzy numbers. Furthermore, it investigates novel aggregation operators by providing in-depth descriptions of their characteristics, such as complex nth power root fuzzy weighted averaging (CnPR-FWA) as well as complex nth power root fuzzy weighted geometric (CnPR-FWG) operators based on CnPR-FSs. Through an in-depth analysis, this paper aims to determine the selection of the most suitable caterer and optimal venue for corporate events. The study's outcomes highlight the suggested method's effectiveness and practical application as compared to other approaches, providing insight into its practical applicability and efficacy.