## 2 Vectorial Ordered Fuzzy Numbers (VOFNs)
The calculation presented graphically in Figure 4 proceeds as follows:  
_Example calculation:_  
A = [-2, -4, -4, -7] B = [8,3,3,1] C = A + B  
C = [6,1,1,-6]  
**Definition 6** (Difference between vOFNs [12; 13; 22]). : _Given three ordered fuzzy numbers \(A=(\overrightarrow{F_{A}},\overrightarrow{G_{A}})\), \(B=(\overrightarrow{F_{B}},\overrightarrow{G_{B}})\), and \(C=(\overrightarrow{F_{C}},\overrightarrow{G_{C}})\), it is said that C is the result of subtracting B from \(A\), denoted as \(C=A-B\), if the following holds:_  
\[\forall_{\mu\in[0,1]}[\overrightarrow{F_{A}}-\overrightarrow{F_{B}}= \overrightarrow{F_{C}}\wedge\overrightarrow{G_{A}}-\overrightarrow{G_{B}}= \overrightarrow{G_{C}}] \tag{4}\]  
Hence, subtraction in the context of OFNs corresponds to the addition of an opposite value to the subtrahend, i.e., \(A-B=A+(-B)\). The order of the OFN indicates the position of the increasing part of the membership curve \(\overrightarrow{F_{A}}\) in relation to the decreasing part of the fuzzy number \(\overrightarrow{G_{A}}\). An illustration of this arithmetic operation is shown in Figure 5.  
_A calculation example is as follows:_  
A = [-1,0,1,4] B = [5,3,2,0] C = A - B = A + (B) C = [-1,0,1,4]+[-5,-3,-2,0] C = [-6, -3, -1, 4]  
Figure 4: Sum of vOFNs. **Definition 7** (Element neutral to the sum and the difference of vOFNs [17,27]). : _The result of A-A subtraction is the real number \(r=0\). The real number \(r\in R\) is represented by the pair \((\overrightarrow{F_{r}},\overrightarrow{G_{r}})\), whereby the following hold:_  
\[\forall_{\mu\in[0,1]}[C_{A}=(l_{A},p_{A})\times\{0\}\cup[1^{-}_{A},1^{+}_{A}] \times\{1\}\cup\overrightarrow{F_{A}}\cup\overrightarrow{F_{B}}] \tag{1}\]0  
_A computational example is provided below, and its graphical interpretation is shown in \[\forall_{\mu\in[0,1]}[C_{A}=(l_{A},p_{A})\times\{0\}\cup[1^{-}_{A},1^{+}_{A}] \times\{1\}\cup\overrightarrow{F_{A}}\cup\overrightarrow{F_{B}}] \tag{1}\]1_  
A = [5,4,3,2]  
-A = [-5, -4, -3, -2] A-A = A+(-A)=[0,0,0,0]  
\[\forall_{\mu\in[0,1]}[C_{A}=(l_{A},p_{A})\times\{0\}\cup[1^{-}_{A},1^{+}_{A}] \times\{1\}\cup\overrightarrow{F_{A}}\cup\overrightarrow{F_{B}}] \tag{1}\]2 Difference between vOFNs. \[\forall_{\mu\in[0,1]}[C_{A}=(l_{A},p_{A})\times\{0\}\cup[1^{-}_{A},1^{+}_{A}] \times\{1\}\cup\overrightarrow{F_{A}}\cup\overrightarrow{F_{B}}] \tag{1}\]3 Element neutral to the sum. **Definition 8** (Multiplication of a vOFN by a scalar [12; 13; 27]). : _Let two ordered fuzzy numbers \(A=(\overrightarrow{F_{A}},\overrightarrow{G_{A}})\) and \(C=(\overrightarrow{F_{C}},\overrightarrow{G_{C}})\) be given, along with a scalar \(r\in\mathbb{R}\). It is said that \(C\) is the product of \(A\) multiplied by the scalar \(r\), if the following holds:_  
\[\forall_{\mu\in[0,1]}[C_{A}=(l_{A},p_{A})\times\{0\}\cup[1^{-}_{A},1^{+}_{A}] \times\{1\}\cup\overrightarrow{F_{A}}\cup\overrightarrow{F_{B}}] \tag{1}\]4  
_A computational example is provided below, and its graphical interpretation is shown in \[\forall_{\mu\in[0,1]}[C_{A}=(l_{A},p_{A})\times\{0\}\cup[1^{-}_{A},1^{+}_{A}] \times\{1\}\cup\overrightarrow{F_{A}}\cup\overrightarrow{F_{B}}] \tag{1}\]5_  
There is an ordered fuzzy number \(A=[0,1,3,4]\). Determine \(C=r*A\) for \(r=2\). \(C=2*A=2*[0,1,3,4]=[0,2,6,8]\). **Definition 9** (Multiplication of vOFNs [12; 13]). : _Consider three ordered fuzzy numbers \(A=(\overrightarrow{F_{A}},\overrightarrow{G_{A}})\), \(B=(\overrightarrow{F_{B}},\overrightarrow{G_{B}})\), and \(C=(\overrightarrow{F_{C}},\overrightarrow{G_{C}})\).