# Multi-Objective Optimisation in Multi-QoS Routing Strategy for Software-Defined Satellite Network
## 5 Proposed Methodology  
In this section, we will introduce the proposed multi-QoS routing algorithm that uses the Interval type-2 fuzzy set to make fuzzy decisions. In general, the steps used to make fuzzy decisions that uses Interval type-2 fuzzy set are the following:  
Step1: Formulate the multi criteria decision (Multi-QoS) solving problem by determining criteria set as \(\text{B}=\{m_{1},m_{2},m_{3},\ldots,m_{n}\}\) with the criteria weight vector \(\text{W}=\{w_{1},w_{2},\)\(\ldots,w_{n}\}\) and \(\sum_{j=1}^{n}w_{j}=1\). Figure 3: Queue model. Figure 2: Multi-QoS routing model. Step2: Determine the linguistic term set, semantic and linguistic expressions. Define \(V^{g}\) is the context free expression of linguistic term set and S is a linguistic term set and S = { \(S_{0},S_{1},S_{2},S_{3},\cdots,S_{d}\)}. There is a rule: if \(i\ll j\), then \(S_{i}\ll S_{j}\). And then define the operation: \(\max\bigl{(}S_{i},S_{j}\bigr{)}=S_{j},\min\bigl{(}S_{i},S_{j}\bigr{)}=S_{i}\), \(V^{g}=\{at least,\ at\ most,\ between,\ is\ and\ S_{0},S_{1},S_{2},\ S_{3},\cdots,S_{d}\}\). The rule is: R = {_at least_\(\gg\) \"; at most := \(\ll\) ; between \(<\cdots<\) ; is\" = \"}. Step3: Assume that there are k experts and collect the negative and positive comments of all \(k\) experts on all criteria. In order to describe interval type-2 fuzzy set, define the evaluations are in \(\left[r_{ij}^{-},r_{ij}^{+}\right],r_{ij}^{-},r_{ij}^{+}\) respectively represent the minimum of negative evaluation value and maximum of positive evaluation value of the \(i\) expert on the \(j\) goal. Step4: The data given by experts is de-fuzzed. The corresponding value is given in Table 1.  
From Table 1, the data given by experts can be expressed by interval type-2 fuzzy set:  
\[\widetilde{h}=\left\{\widetilde{A}_{i}\in\widetilde{h}|\widetilde{A}_{i}= \left(p_{i1}^{u},p_{i2}^{u},p_{i3}^{u},p_{i4}^{u};P_{1}\left(\widetilde{A}_{i }^{u}\right),P_{2}\left(\widetilde{A}_{i}^{u}\right)\right),\left(p_{i1}^{l},p _{i2}^{l},p_{i3^{\prime}}^{l},p_{i4}^{l};P_{1}\left(\widetilde{A}_{i}^{l} \right),P_{2}\left(\widetilde{A}_{i}^{l}\right)\right)\right\}\]  
where \(\widetilde{h}_{ij}\) is interval type-2 fuzzy set, \(i\) denotes scheme and \(j\) denotes criterion. Step5: According to all target demand data, evaluate all schemes based on (\(w_{j}1\leq j\leq n\)):  
\[\widetilde{h}_{i}=\sum_{j=1}^{n}w_{j}\widetilde{h}_{ij}(\text{where }i=1,2,3,\ldots,m)\]  
\[=\cup_{\widetilde{A}_{i}\in\widetilde{h}_{i1},\widetilde{A}_{i2}\in \widetilde{h}_{i2},\ldots,\widetilde{A}_{i\alpha}\in\widetilde{h}_{i\alpha}} \left\{\left( \begin{array}{c}k^{-1}(\sum_{j=1}^{n}w_{j}k(p_{ij1}^{u})),k^{- 1}(\sum_{j=1}^{n}w_{j}k(p_{ij2}^{u})),\\ k^{-1}(\sum_{j=1}^{n}w_{j}k(p_{ij4}^{u})),min_{j}\left(P_{1}\left(\widetilde{A}_{ ij}^{u}\right)\right),min_{j}\left(P_{2}\left(\widetilde{A}_{ij}^{u}\right) \right)\end{array} \right),\right.\]  
\[\left( \begin{array}{c}k^{-1}(\sum_{j=1}^{n}w_{j}k(p_{ij1}^{l})),k^{-1}(\sum _{j=1}^{n}w_{j}k(p_{ij2}^{l})),k^{-1}(\sum_{j=1}^{n}w_{j}k(p_{ij3}^{l})),\\ k^{-1}(\sum_{j=1}^{n}w_{j}k(p_{ij4}^{l})),min_{j}\left(P_{1}\left(\widetilde{A}_{ ij}^{l}\right)\right),min_{j}\left(P_{2}\left(\widetilde{A}_{ij}^{l}\right) \right)\end{array} \right)\right\}\]  
Step6: Calculate the score \(\mathsf{s}\left(\widetilde{h}_{ij}\right)\)(\(i=1,2,3,\ldots,m\)) for each scheme.