## 2 Methodology
### Uncertainty Reduction by Interval-Valued Pythagorean Fuzzy Set  
Fuzzy system theory (FST) has undergone significant developments and diversification, becoming a cornerstone in computational intelligence and decision sciences [32]. The Fuzzy Set (FS) extends the classical set theory by allowing elements to have degrees of membership rather than binary inclusion or exclusion. Building on this foundational concept, Fuzzy Logic (FL) employs the principles of FST to develop a system of approximate reasoning. Unlike classical binary logic, which restricts truth values to zero or one, FL enables the handling of intermediate truth values, thus allowing for more flexible and human-like reasoning processes [33; 34; 35]. This capability is particularly advantageous in fields where precise mathematical models are infeasible or human reasoning needs to be simulated. Fuzzy Inference Systems (FISs), as implementations of fuzzy logic, provide a framework for decision-making through approximate reasoning and linguistic terms [36]. FISs can process both crisp and fuzzy inputs, generating fuzzy outputs through rule-based inference. A successful implementation of FISs heavily relies on well-designed fuzzy rules. FISs have demonstrated effectiveness in various applications, particularly where human judgment plays a crucial role [37]. In the field of enhancing traffic safety, FISs were proposed to assess collision risks and guide avoidance maneuvers for maritime autonomous surface ships (MASS) [38]. The near-collision decision-making mechanism using FISs could further improve local path planning [39]. However, FISs encounter significant limitations when dealing with massive and hierarchically structured variables. Specifically, the rule-based design of FISs becomes exponentially complex, leading to potential rule conflicts and inconsistencies. Moreover, while a FIS can evaluate factors based on individual characteristics, it struggles to effectively capture the relative importance relationships between the factors. Therefore, our work combines FST variants with the MCDM method to structure and solve complex decision-making problems involving extensive and hierarchically organized criteria. FST variants are widely used to overcome the inaccuracies that arise from the reliance on expert scoring in MCDM methods. Over the past decades, FST variances have enhanced the capability to tackle various complex systems. The Intuitionistic Fuzzy Set (IFS) represents a significant extension [40]. IFS incorporates the membership degree and non-membership degree of elements, which can provide a more thorough description of uncertainty [40]. The Interval-Valued Fuzzy  
Figure 1: The proposed framework. Set (IVFS) is another important extension [41]. The IVFS uses intervals instead of single values to represent the membership degree of elements [42]. Though IVFS improves uncertainty modeling by capturing data variability, computational complexity is introduced due to interval arithmetic. The Pythagorean Fuzzy Set (PFS) offers a more recent and innovative extension [43]. Unlike the IFS, where the sum of the membership and non-membership degrees must be lower than or equal to one, the PFS relaxes this constraint to the sum of the squares of these degrees. Moreover, the mathematical structure of the PFS allows for using Pythagorean theorem-based calculations, simplifying the aggregation and comparison of fuzzy values [43]. This structure is particularly beneficial in MCDM contexts. Building on the strengths of both the IVFS and PFS, the Interval-Valued Pythagorean Fuzzy Set (IVPFS) represents a further evolution by combining interval representation with the Pythagorean structure [44]. IVPFS further enhances the PFS framework by representing membership and non-membership degrees as intervals. The interval representation captures the range of and variability in uncertainty more explicitly, providing a more robust modeling approach [44]. The hybrid approach offers a more robust mechanism for modeling uncertainty than either IVFS or PFS alone.