# Multi-Objective Optimisation in Multi-QoS Routing Strategy for Software-Defined Satellite Network
## 5 Proposed Methodology
If \(\widetilde{h}=\left\{\widetilde{A}_{i}\in\widetilde{h}|\widetilde{A}_{i}= \left(p_{i1}^{u},p_{i2}^{u},p_{i3}^{u},p_{i4}^{u};P_{1}\left(\widetilde{A}_{i}^ {u}\right),P_{2}\left(\widetilde{A}_{i}^{u}\right)\right)\left(p_{i1}^{l},p_{i2 }^{l},p_{i3}^{l},p_{i4}^{l};P_{1}\left(\widetilde{A}_{i}^{l}\right),P_{2}\left( \widetilde{A}_{i}^{l}\right)\right)\right\}\) is an interval type-2 fuzzy set, then score function \(\mathsf{s}\left(\widetilde{h}\right)=\frac{1}{|\widetilde{h}|}\sum_{\widetilde{A} \widetilde{e}\widetilde{h}}\mathsf{s}\left(\widetilde{A}\right)=\frac{1}{| \widetilde{h}|}\sum_{\widetilde{A}\widetilde{e}\widetilde{h}}\mathsf{s}\left[ \frac{p_{i1}^{u}+p_{i2}^{u}}{2}+\frac{P_{1}\left(\widetilde{A}^{u}\right)+P_ {2}\left(\widetilde{A}^{u}\right)+P_{1}\left(\widetilde{A}^{l}\right)+P_{2} \left(\widetilde{A}^{l}\right)}{4}\right]\times\left(\frac{p_{i1}^{u}+p_{i2}^{u }+p_{3}^{u}+p_{4}^{u}+p_{1}^{l}+p_{2}^{l}+p_{3}^{l}+p_{4}^{l}}{8}\right).\) Where \(|\widetilde{h}|\) is the number of elements of interval type-2 fuzzy set \(\widetilde{h}\). And the value of score function \(\mathsf{s}\left(\widetilde{h}\right)\) is in [0, 1]. For \(\mathsf{s}\left(\widetilde{h}\right)\), if \(\mathsf{s}\left(\widetilde{h}_{1}\right)\gg\mathsf{s}\left(\widetilde{h}_{2}\right)\), then \(\widetilde{h}_{1}\gg\widetilde{h}_{2}\). The calculation of Step5 and Step6 is based on the formulas in Section 3. Step7: The dominance matrix is established on the dominance relationship. Using the score function, the interval number of each target is calculated. The boundary of \begin{table}
\begin{tabular}{l l l} \hline
**Label** & **Linguistic Terms** & **Corresponding IT2 HFNs** \\ \hline AL & Absolutely low & (0.0, 0.0, 0.0, 0.0; 1,1) (0.0, 0.0, 0.0, 0.0; 1,1) \\ VL & Very Low & (0.0075, 0.0075, 0.015, 0.0525; 0.8,0.8), (0,0.02,0.07; 1,0.1) \\ ML & Slightly low & (0.2325, 0.255, 0.325, 0.3575; 0.8,0.8), (0.17,0.22,0.36,0.42; 1,0.1) \\ M & Middle & (0.4025, 0.4525, 0.5375, 0.5675; 0.8,0.8), (0.32,0.41,0.58,0.65; 1,0.1) \\ MH & Slightly high & (0.65, 0.6725, 0.7575, 0.79; 0.8,0.8), (0.58,0.63,0.80,8.6; 1,0.1) \\ H & High & (0.7825, 0.815, 0.885, 0.9075; 0.8,0.8), (0.72,0.78,0.92,0.97; 1,0.1) \\ VH & Very high & (0.9475, 0.985, 0.9925, 0.8,0.8), (0.93,0.98,1.0,1.0; 1,0.1) \\ AH & Absolutely high & (1.0, 1.0, 1.0, 1.0; 1.0,1.0), (1.0, 1.0, 1.0, 1.0) \\ \hline \end{tabular}
\end{table} 
Table 1: Linguistic term set and their corresponding values. the interval number is optimistic evaluation and pessimistic evaluation respectively. If \(I_{1}=[a_{1},b_{1}]\), \(I_{2}=[a_{2},b_{2}]\), then relationship of \(I_{1}\) and \(I_{2}(I_{1}>I_{2})\)can be calculated as:  
\[\mathrm{R}(I_{1}>I_{2})=\frac{\max(0,b_{1}-a_{2})-\max(0,a_{1}-b_{2})}{(b_{1}-a _{1})+(b_{2}-a_{2})}\]  
Similarly, \(\mathrm{R}(I_{2}>I_{1})\) can be calculated. \(\mathrm{R}(I_{1}>I_{2})+\mathrm{R}(I_{2}>I_{1})=1\). Specially, when \(a_{1}=a_{2}\) and \(b_{1}=b_{2}\), \(\mathrm{R}(I_{1}>I_{2})=\mathrm{R}(I_{2}>I_{1})\)=0.5. If we want to compare how much \(I_{1}\) is greater than \(I_{2}\), the following equation can be conducted:  
\[DM_{12}=\max(0,\mathrm{R}(I_{1}>I_{2})-\mathrm{R}(I_{2}>I_{1})).\]  
Step8: Use non dominance rule that [PERSON] et al. (2012) put forward:  
\[NDM_{i}=|\min((1-DM_{1}),(1-DM_{2}),\ldots,(1-DM_{n}))|(n\
eq i).\]  
Step9: Normalization. \[NDM_{i}=\frac{NDM_{i}}{\sum_{i=1}^{n}NDM_{i}}\]