Picture Fuzzy Directed Hypergraphs With Applications Toward Decision-Making and Managing Hazardous Chemicals

Many real-life problems present multiple options necessitating modern researchers to identify the most suitable choice or decision. In this context, fuzzy sets have proven to be valuable tools for addressing such problems. Picture fuzzy sets (PFSs), due to their increased flexibility and broader domain as compared to other extensions of fuzzy sets (FSs) have been extensively used in solving various decision-making problems. To achieve optimal decisions, various fuzzy graphs and fuzzy hypergraph structures are employed. Picture fuzzy directed hypergraphs (PFDHGs) offer greater utility in addressing problems with multiple options or uncertainties. In this study, we introduce novel concepts of PFDHGs, along with their applications in decision-making theory and hazardous chemicals management. Initially, we present the foundational concepts of PFDHGs, including essential terminologies such as hyper-edge and height. Subsequently, we explore various types of PFDHGs, such as simple PFDHGs and support simple PFDHGs. Additionally, we introduce the concept of
$({\widetilde {\alpha }}, {\widetilde {\beta }}, {\widetilde {\gamma }})$
-level hyperarcs and elucidate the fundamental sequence in PFDHGs based on these arcs. The study further examines elementary PFDHGs, partial PFDHGs, and ordered PFDHGs, investigating the relationships among them. Structural properties of PFDHGs are analyzed through different types of homomorphisms of PFDHGs. We also define the order and size of PFDHGs. Finally, we present the applications of PFDHGs in decision-making and managing hazardous chemicals, emphasizing the innovative contributions and practical implications of our work.