Dependability plays a crucial role in determining the performance success of a system, focusing on understanding the factors contributing to system failures. The Failure Mode and Effects Analysis (FMEA) method, a traditional safety technique widely utilized across various safety-critical sectors, employs the Risk Priority Number (RPN) to gauge criticality and prioritize failure modes. However, it faces limitations, particularly in scenarios with ambiguous or uncertain information. Therefore, this study introduces a fuzzy criticality assessment approach to evaluate system failure modes, offering an alternative prioritization to the conventional method. Furthermore, a novel hybrid method is proposed, merging the Grey Relational Analysis (GRA) and Fuzzy Analytic Hierarchy Process. This hybrid approach addresses the shortcomings related to the absence of established inference rules, relying heavily on experience, and assigns weights to three equally significant parameters: severity, detection, and frequency, a departure from the traditional method. Through a real gas turbine system case study, this approach demonstrates promising outcomes in risk assessment and prioritizing failure modes, effectively handling various forms of ambiguity, uncertainty, and diverse expert judgments.