## 3 Solution to the Problem of Comparing OFN/vOFNs
### Discussion and Comparison with Alternative Methods  
The results derived in the previous section are straightforward and unambiguous, and they remain consistent across a wide selection of test cases gathered from both classical and recent benchmark studies. By incorporating examples drawn from the existing literature, such as [9; 29; 39; 40], we ensured that the evaluation of the JC metric would not be limited to author-generated scenarios. This step was necessary to demonstrate the metric's robustness in standard benchmark situations and to support a meaningful comparison with traditional Ordered Fuzzy Number methods. Moreover, the performance of the JC metric has been benchmarked against a comprehensive set of comparison operators (Table 1) described in the literature. These include distance-based methods, defuzzification-based approaches, and operators specifically designed for handling Ordered Fuzzy Numbers (OFNs). To help the reader better understand these references without requiring an in-depth study of external sources, a brief description of each representative method included in the comparison is provided below. * **GR (General Ranking Method)**--A classical ranking method based on centroid calculations and distance metrics, originally proposed to compare triangular and trapezoidal fuzzy numbers in a scalarized manner [41]. * **ML (Modified Linguistic Ranking)**--Introduced to align numerical results with linguistic preferences by using weighting functions and ranking indices [41]. * **[PERSON]'s 0.5-cut, 0.9-cut methods**--Utilize alpha-cuts to reduce fuzzy sets to intervals and then compare representative crisp values [31]. * **[PERSON] and [PERSON]** proposed several comparative measures based on the concept of distance between fuzzy sets [39]. * **F1, F2, F3 (Yager's Methods)--A series of fuzzy comparison techniques using score functions and area-based evaluations to handle non-normal and overlapping fuzzy numbers [29]. * **CH (Chang's index)--Uses a distance-based approach derived from the expected values of fuzzy numbers and their support intervals [9]. * **PCM (Partial Comparison Method)--It was developed as a response to ambiguity in comparing fuzzy numbers, especially when their membership functions overlap significantly [33]. * **K (Kerre's Method)--Based on t-norms and fuzzy implications to determine dominance relationships [42]. * **BK (Baas-Kwakkernaak Method)--Uses centroid-based distance and ranking functions [43]. * **PSD (Possibility-Based Dominance)--**Proposed by [PERSON] and [PERSON], this approach compares fuzzy numbers based on possibility theory, evaluating dominance through \(\alpha\)-cut analysis [6]. Compared to these methods, the JC metric presents several advantages. First, it is invariant under uniform translation and scaling, which makes it suitable for applications where relative differences are more meaningful than absolute magnitudes. Second, it correctly reflects symmetry and antisymmetry in fuzzy number relationships, including reversed orientation--a property not consistently preserved in traditional methods. Furthermore, the JC metric integrates naturally into the vectorial representation of Ordered Fuzzy Numbers, allowing a unified approach for operations and comparisons. Unlike some earlier methods that require a case-by-case analysis or rely on defuzzification (which may result in information loss), the JC metric operates entirely within the fuzzy framework while preserving interpretability. The comparative analysis confirms that the JC metric produces results compatible with the well-established methods in most benchmark scenarios, while also correctly resolving certain edge cases that cause ambiguity or inconsistency in other approaches.