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 (α,β,γ)-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.

In this manuscript, we first initiate several types of effective arcs of intuitionistic fuzzy directed graphs, followed by discussions on different types of dominations in intuitionistic fuzzy directed graphs and their application in decision-making. The notion of dominations in fuzzy graphs, fuzzy directed graphs, intuitionistic fuzzy graphs and picture fuzzy graphs have been extensively discussed in the literature. Thus, the work presented in our study is two-fold: on one side, it extends the notion of domination in fuzzy directed graphs, while on the other side, it fills the gap existing in the literature. Initially, we introduce several types of effective arcs like semi-λ effective arc, semi-ν effective arcs in intuitionistic fuzzy directed graphs. Subsequently, we initiate the notions of dominations and domination numbers for several types of intuitionistic fuzzy directed graphs based on these effective arcs. We present numerous significant characterizations of dominations in intuitionistic fuzzy directed graphs by utilizing minimal dominating sets. Furthermore, we investigate the minimal dominating sets and domination numbers of intuitionistic fuzzy-dipaths and dicycles, and provide various fascinating results. Finally, by utilizing the concepts presented in this study, we provide the algorithm for the solution of the problem related to decision-making regarding the opening of restaurants in various areas of the different cities.