Quintic fuzzy sets: A new class of fuzzy sets for solving multi-criteria decision-making problems under uncertainty

This study proposes a new class of fuzzy sets called Quintic Fuzzy Set (QuFS), where the sum of the fifth power of the membership degree and non-membership degree is less than or equal to one. We introduce the complement operator and basic operations on QuFS and further develop various theorems based on these fundamental set operations. We also present the standard modal operators on QuFS and discuss its elementary properties. We then define a novel score and accuracy function for ranking QuFS, as well as the Euclidean and Hamming distances between two QuFSs. A supplier selection example is presented in detail to illustrate the viability and utility of the proposed QuF-TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method. We also present a multi-criteria decision-making (MCDM) problem on COVID-19 lab selection to perform a comparative analysis of QuFS with Fermatean Fuzzy Set (FFS) and Pythagorean Fuzzy Set (PFS). Furthermore, we demonstrate an existing real case study on selection of Health care waste treatment (HCWT) technique in India and perform a comparative analysis with TOPSIS, SAW (Simple Additive Weighting) and WASPAS (Weighted Aggregated Sum Product Assessment) MCDM methods in QuF framework. With the advent of QuFS, a decision-maker can now employ the fuzzy notion to manage complicated and uncertain human judgements. The comparative results indicate that the suggested QuF measures and QuF MCDM technique outperform other fuzzy concepts regarding effectiveness and validity. Due to the restrictive nature of the current fuzzy set theories, the decision-makers in some complex, uncertain real-life problems are not free to choose the degree of membership and non-membership of their choice. Compared to the Intuitionistic Fuzzy Set (IFS), PFS, and FFS, the QuFS developed in this study has a bigger space grade space. This feature of QuFS enables decision-makers to structure complex, uncertain circumstances with greater flexibility in terms of membership and non-membership degrees.