Abstract: This research investigates the reliability and availability of a multi-state k-out-of-n system composed of three interconnected subsystems: main power supply units, backup generators, and uninterruptible power supply units. Each subsystem operates under specific failure and repair protocols, including warm and cold standby arrangements. State transitions are modeled using stochastic processes, and the Sumudu transform technique is employed to solve the complex differential equations governing the system's behavior. Fermatean fuzzy sets are incorporated to effectively manage uncertainty. Numerical illustrations evaluate availability and reliability under various scenarios, demonstrating the influence of failure rates, maintenance delays, and repair personnel availability on system performance. The fuzzy approach provides a broader range of reliability and availability estimates compared to traditional Markov methods. The integration of advanced mathematical tools and Fermatean fuzzy logic enhances decision-making, maintenance planning, and cost optimization, enabling the design of robust, efficient, and cost-effective systems in complex engineering environments with prevalent uncertainties.

Key words: reliability, availability, multi-state k-out-of-n system, sumudu transform, fermatean fuzzy set