## 2 Vectorial Ordered Fuzzy Numbers (VOFNs)  
Building upon the definition of Ordered Fuzzy Numbers introduced by [PERSON] et al. [14; 21; 22; 27], examples of such numbers can be found in Figures 11-13. Additionally, we now introduce the concept of Vectorial Ordered Fuzzy Numbers as follows. **Definition 1** (Vectorial Ordered Fuzzy Number). : _A Vectorial Ordered Fuzzy Number (VOFN) is generally trapezoidal in shape, denoted by the coordinates \(A[a,b,c,d]\). This trapezoid is constructed by an ordered pair of vectors, \(\overrightarrow{F_{A}}\) and \(\overrightarrow{G_{A}}\). The vector \(\overrightarrow{F_{A}}\) starts at point \(f_{0}(a,0)\) and ends at \(f_{1}(b,1)\), while \(\overrightarrow{G_{A}}\) starts at \(g_{1}(c,1)\) and ends at \(g_{0}(d,0)\)._  
The orientation, direction, and magnitude of the two vectors are determined by the points between which they are drawn (Figure 1). **Definition 2** (Positive and Negative Order [27]). : _A Vectorial Ordered Fuzzy Number (VOFN) can also be represented as a triple \(A=(R,\mu_{A},S_{A})\), where \(\mu_{A}:R\rightarrow[0,1]\) is the membership function of the number, and \(S_{A}\in\{-1,0,1\}\) denotes its order. Based on their order, VOFNs can be categorized into two types:_  
* **VOFNs of a positive order--the ordering direction conforms to the increasing values along the OX axis. **
* **VOFNs of a negative order--the ordering direction is opposite to the increasing values along the OX axis. **  
Figure 1: vOFN formed by an ordered pair of vectors. A visual depiction (Figure 2) of an ordered fuzzy number \(A\) is represented by the pair \(A=(C_{A},S_{A})\), where \(C_{A}\subseteq\mathbb{R}^{2}\) is a curve defined as follows:  
\[\forall_{\mu\in[0,1]}[C_{A}=(l_{A},p_{A})\times\{0\}\cup[1^{-}_{A},1^{+}_{A}] \times\{1\}\cup\overrightarrow{F_{A}}\cup\overrightarrow{F_{B}}] \tag{1}\]  
where:  
\(\overrightarrow{F_{A}}\) and \(\overrightarrow{G_{A}}\subseteq\mathbb{R}^{2}\) are the graphs of the vectors;  
\(S_{A}\) represents the order;  
\(C_{A}\) is the membership curve. Definition 3 (Opposite vOFN): The opposite of a number \(A\), denoted as \(-A\), is defined as its mirror image with respect to the OY axis. Consequently, opposite vOFNs have opposite orders (Figure 3). Definition 4 (Reversal of the Order [27]): The reversal of the order of a fuzzy number \(A\) consists of swapping the upper part (vector \(\overrightarrow{F_{A}}\)) with the lower part (vector \(\overrightarrow{G_{A}}\)). This operation is denoted as follows:  
\[\forall_{\mu\in[0,1]}\Big{[}B=A|^{-}\Leftrightarrow\overrightarrow{G_{B}}= \overrightarrow{F_{A}}\wedge\overrightarrow{F_{B}}=\overrightarrow{G_{A}} \Big{]} \tag{2}\]  
where  
\(A\) is the ordered fuzzy number represented by the vector pair \((\overrightarrow{F_{A}},\overrightarrow{G_{A}})\);  
\(B\) is the outcome of the vOFN order reversal operation;  
\(|^{-}\) represents the reversal of the vOFN order. The number obtained through this operation is referred to as the reversed vOFN or a vOFN with reverse orientation (opposite order). Definition 5 (Sum of vOFNs [17; 21; 22; 27]): Let \(A=(\overrightarrow{F_{A}},\overrightarrow{G_{A}})\), \(B=(\overrightarrow{F_{B}},\overrightarrow{G_{B}})\), and \(C=(\overrightarrow{F_{C}},\overrightarrow{G_{C}})\) be three ordered fuzzy numbers. Then, \(C\) is said to be the sum, denoted as \(C=A+B\), if the following holds:  
\[\forall_{\mu\in[0,1]}\Big{[}\overrightarrow{F_{C}}=\overrightarrow{F_{A}}+ \overrightarrow{F_{B}}\wedge\overrightarrow{G_{C}}=\overrightarrow{G_{A}}+ \overrightarrow{G_{B}}\Big{]} \tag{3}\]  
An example illustrating the addition of two vOFNs \(A\) and \(B\) with the same order is shown in Figure 4.  
Figure 3: Opposite numbers: \(A\) and \(-A\). Figure 2: Fuzzy numbers with (**a**) positive order and (**b**) negative order. The operation of adding vOFNs with the same order yields the same result as the addition of classical convex fuzzy numbers.