## 3 Solution to the Problem of Comparing OFN/vOFNs  
This section presents a novel similarity measure, which is based on [PERSON]'s approach discussed previously. However, the proposed measure overcomes a key drawback of the original Canberra distance--its sensitivity to values approaching zero. The JC metric is tailored specifically for comparing ordered fuzzy numbers (OFNs), yet it can be generalized to other fuzzy number arithmetic systems by omitting the components that are specifically related to the nature of OFNs. As discussed previously, a Vectorial Ordered Fuzzy Number (vOFN) consists of an ordered pair of vectors representing its increasing edge \(f\) and decreasing edge \(g\). Figure 2 depicts two such numbers, with the starting and ending points of these vectors labeled as \(f_{A}(0),f_{A}(1)\), \(g_{A}(1)\), and \(g_{A}(0)\).  
**Lemma 1**.: _Consider two vOFNs, \(A\) and \(B\), each characterized by four boundary values corresponding to their rising and falling edge functions. Specifically, number \(A\) is represented by the set \(\{f_{A}(0),f_{A}(1),g_{A}(1),g_{A}(0)\}\), and number \(B\) by the set \(\{f_{B}(0),f_{B}(1),g_{B}(1),g_{B}(0)\}\). The comparison between \(A\) and \(B\) is carried out by evaluating the JC metric, defined as follows:_  
\[JC=\sum_{i=0}^{1}\frac{f_{A(i)}-f_{B(i)}}{|f_{A(i)}|+|f_{B(i)}|}+\sum_{i=0}^{1 }\frac{g_{A(i)}-g_{B(i)}}{|g_{A(i)}|+|g_{B(i)}|} \tag{11}\]  
_where the following hold: If the condition \(|f_{A(i)}|+|f_{B(i)}|=0\) or \(|g_{A(i)}|+|g_{B(i)}|=0\) is encountered, an affine transformation (translation) along the positive \(OX\) axis is required. This can be achieved by adding a constant value of \(1\) to the coordinates of both fuzzy numbers. The process should be repeated until no denominator becomes zero._  
**Definition 13**.: _When the JC metric equals \(JC=0\) and the fuzzy numbers \(A\) and \(B\) share the same order, the two numbers are considered identical._  
Figure 11: Power of a vOFN with a natural exponent.  
**Definition 14**.: _When the JC metric results in JC \(=0\) and the fuzzy numbers A and B have reversed orders, the number with a positive order is regarded as larger._  
**Definition 15**.: _If the value of the JC metric is JC \(<0\), fuzzy number A is deemed smaller than fuzzy number B, irrespective of their order._  
**Definition 16**.: _If the JC metric yields JC \(>0\), fuzzy number A is considered larger than fuzzy number B, regardless of the order of the numbers._  
Numerous examples, frequently used for testing comparison operators, are available in the existing literature [9; 29; 39; 40]. These examples, presented in vOFN notation, are displayed below alongside their corresponding visual representations (Figure 12). The selected examples include both disjoint fuzzy numbers as well as those that are nested within each other, thus posing a more complex interpretative challenge compared to standard operations on real numbers.  
Using the JC metric, as described in the preceding section, allows for the comparison and classification of fuzzy numbers. For each pair of numbers compared, a value is generated based on the JC metric. The similarity rules outlined in the definitions above can then be applied to determine the relationship between the individual pairs of fuzzy numbers.