# Relation \"Greater than or Equal to\" between Ordered Fuzzy Numbers
## 4 Relation \"Greater than or Equal to\" for Ordered Fuzzy Numbers
\begin{array}{l}\
u_{GE}(\overset{\leftrightarrow}{\mathcal{K}},\overset{ \leftrightarrow}{\mathcal{L}})=\
u_{GE}(\overset{\leftrightarrow}{\mathcal{K }}\boxplus\overset{\leftrightarrow}{\mathcal{L}},\llbracket 0\rrbracket)=\
u_{GE}(\llbracket 0 \rrbracket,\overset{\leftrightarrow}{\mathcal{L}}\boxplus\overset{\leftrightarrow }{\mathcal{K}})=\
u_{[GE]}(\llbracket 0\rrbracket,\Psi(\overset{\leftrightarrow}{\mathcal{L}}\boxplus \overset{\leftrightarrow}{\mathcal{K}}))=\\ =\
u_{[GE]}(\llbracket 0\rrbracket,\Psi(\overset{\leftrightarrow}{\mathcal{L}}) \odot(\odot\Psi(\boxplus\overset{\leftrightarrow}{\mathcal{K}})))=\
u_{[GE]}( \odot\Psi(\boxplus\overset{\leftrightarrow}{\mathcal{K}})\odot\Psi(\overset {\leftrightarrow}{\mathcal{L}}),\llbracket 0\rrbracket)=\\ \
u_{[GE]}(\Theta\,\Psi(\boxplus\overset{\leftrightarrow}{\mathcal{K}}),\Psi( \overset{\leftrightarrow}{\mathcal{L}}))=\
u_{[GE]}(\overset{\leftarrow}{ \Psi(\overset{\leftrightarrow}{\mathcal{K}})},\overset{\leftrightarrow}{ \Psi(\overset{\leftrightarrow}{\mathcal{L}})}).\text{ QED}\end{array}  \tag{96}\]  
**Example 3. **_Let us compare the OFN \(\overset{\leftrightarrow}{\mathcal{X}}=\overset{\leftrightarrow}{\mathcal{L} }(12,14,18,20,S_{X},E_{X})\) determined by (59) and the OFN \(\overset{\leftrightarrow}{\mathcal{Y}}=\overset{\leftrightarrow}{\mathcal{L }}(13,11,6,5,S_{Y},E_{Y})\) determined by (60). Using (93), (62), (64), and (41), we get_  
\[ \begin{array}{l}\
u_{GE}(\overset{\leftrightarrow}{\mathcal{Y}},\overset{ \leftrightarrow}{\mathcal{X}})=\
u_{[GE]}(\overset{\leftarrow}{\Psi( \overset{\leftrightarrow}{\mathcal{Y}})},\overset{\leftrightarrow}{\Psi( \overset{\leftrightarrow}{\mathcal{X}})})=\
u_{[GE]}(\mathcal{L}(5,6,11,13,E_{Y },S_{Y}),\ \mathcal{L}(12,14,18,20,S_{X},E_{X}))=\\ =\
u_{[GE]}(\mathcal{L}(5,6,11,13,L_{V},R_{Y}),\mathcal{L}(12,14,18,20,L_{U},R _{U}))=\
u_{[GE]}(\mathcal{V},\mathcal{U})=\frac{2}{5}.\end{array}  \tag{97}\]  
The simplicity of the calculations in the above example is apparent. In fact, Example 3 together with Theorem 5 shows that:  
* if compared OFNs are both positively oriented then the fuzzy preorder \(\widetilde{GE}\in\mathcal{F}(\mathbb{K}^{2})\) depends only on the interaction between the ending function of the first compared OFN and the starting function of the second compared OFN;
* if the first compared OFN is positively oriented and the second compared OFN is negatively oriented then the fuzzy preorder \(\widetilde{GE}\in\mathcal{F}(\mathbb{K}^{2})\) depends only on the interaction between the ending functions of compared OFN;
* if the first compared OFN is negatively oriented and the second compared OFN is positively oriented then the fuzzy preorder \(\widetilde{GE}\in\mathcal{F}(\mathbb{K}^{2})\) depends only on the interaction between the starting function of the first compared OFN;
* if compared OFNs are both negatively oriented, then the fuzzy preorder \(\widetilde{GE}\in\mathcal{F}(\mathbb{K}^{2})\) depends only on the interaction between the starting function of the first compared OFN and the ending function of the second compared OFN.