Fuzzy Relation equations Based on composition with Application

Abstract: Fuzzy relation equations (FREs) play a significant role in the modeling and analysis of systems under uncertainty, especially where imprecise information is prevalent. In this paper investigates the solution structures of fuzzy relation equations involving composition operations, particularly the max– min compositions. We explore conditions for the existence and uniqueness of solutions, present methods for determining the greatest and minimal solutions, and analyze their applications in decision-making, medical diagnosis, control systems, and pattern recognition. Furthermore, the study highlights algorithmicapproaches for solving FREs and demonstrates their applicability through illustrative examples. The results contribute to a deeper understanding of fuzzy relational modeling and offer a foundation for future work in fuzzy systems and soft computing.