Pythagorean fuzzy $ N $-Soft PROMETHEE approach: A new framework for group decision making

The use of Pythagorean fuzzy $ N $-soft sets (PFNSs) enables the examination of belongingness and non-belongingness of membership degrees, as well as their combinations with $ N $-grading, in the unpredictable nature of individuals. This research aims to enhance our understanding of a popular multi-criteria group decision making (MCGDM) technique, Preference Ranking Organization Method for Enrichment of Evaluations, under the PFNS environment, aiding in making effective decisions for real-life problems, as fuzzy set theory is directly relevant to real-life applications. The PROMETHEE technique's main principle is to calculate the inflow and outflow streams of alternatives based on the deviation of their score degrees, ultimately providing partial and complete rankings of the given options. To capture the uncertainty of human nature, which demands both the association and disassociation of the considered criteria and provision of $ N $-grading, the PFNS PROMETHEE technique is introduced in this research article. First, an Analytic Hierarchy Process AHP is used to check the feasibility of the standard weights of the criteria. The article then explains the detailed method of the fuzzy $ N $-soft PROMETHEE technique to rank alternatives, with all the steps presented in an extensive flowchart for better understanding of the methodology. Furthermore, the practicality and viability of the proposed technique are demonstrated through an example of selecting the best chemical element in cloud seeding, where the most suitable choice is identified using an outranking directed graph. The credibility of the PFNS PROMETHEE technique is assessed by comparison with an existing method. Finally, the proposed technique's strengths and weaknesses are discussed to demonstrate its efficiency and drawbacks.