# Multi-Objective Optimisation in Multi-QoS Routing Strategy for Software-Defined Satellite Network
## 3 Fuzzy Sets Theory
### Interval Type-2 Fuzzy Set  
The T1 FSs (type-1 fuzzy sets) is a method proposed in 1965. The value of elements in Type-1 is represented by a real value in [0, 1]. Type-2 fuzzy sets have the characteristics of primary and secondary membership. In this part, we introduce the definition of Type-2 fuzzy sets and the definition of Interval Type-2 fuzzy sets. Trapezoidal type-1 fuzzy set can be represented by \(\widetilde{A}=\{p_{1},p_{2},p_{3},p_{4};P_{1}(A),P_{2}(A)\}\), where \(p_{1},p_{2},p_{3},p_{4}\) are members of the set \(X\). \(P_{1}(A)\), \(P_{2}(A)\) are the non-zero values of the mapping function. **Definition 3**. : _Type-2 fuzzy set is an extension of the T1 FSs. The membership degree of Type-2 fuzzy set is a Type-1 fuzzy set. Assuming that \(\widetilde{\widetilde{A}}\)is the Type-2 fuzzy set of set \(X\), then the member relationship functions\(\widetilde{\widetilde{A}}\)is satisfied with the following conditions:_  
\[\widetilde{\widetilde{A}}=\left\{\left((x,u),s_{\widetilde{\widetilde{A}}}(x,u )\right)|\forall x\in X,\forall u\in J_{x}\subseteq[0,1],0\leq s_{\widetilde{ \widetilde{A}}}(x,u)\leq 1\right\}\]  
_where \(J_{x}\) represents the interval of [0, 1], so the Type-2 fuzzy set \(\widetilde{\widetilde{A}}\) also can be represented as follows:_  
\[\widetilde{\widetilde{A}}=\int\limits_{x\in X}\int\limits_{u\in J_{x}}u_{ \widetilde{\widetilde{A}}}(x,u)|(x,u)\]  
**Definition 4**. : _Assuming that \(\widetilde{A}\) is the Type-2 fuzzy set of set \(X\) and the member relationship function is \(s_{\widetilde{A}}\), if all \(s_{\widetilde{A}}(x,u)=1\), then \(\widetilde{A}\) is called the Interval Type-2 fuzzy set of set \(X\). Interval Type-2 fuzzy set is a special fuzzy set and can be represented by:_  
\[\widetilde{A}=\int\limits_{x\in X}\int\limits_{u\in J_{x}}1(x,u)\]  
_where \(J_{x}\) represents the interval of [0, 1]._  
**Definition 5**. : _The membership function of Type-1 fuzzy set is composed of the upper and the lower membership functions of Interval Type-2 fuzzy set. We use interval Type-2 fuzzy set to solve multi-QoS routing problem. The upper and the lower membership functions of Interval Type-2 fuzzy set are defined as:_  
\[\widetilde{A}_{i}=\left(\widetilde{A}_{i}^{u},\widetilde{A}_{i}^{l}\right)= \left(p_{i1}^{u},p_{i2}^{u},p_{i3}^{u},p_{i4}^{u};P_{1}\left(\widetilde{A}_{i} ^{u}\right),P_{2}\left(\widetilde{A}_{i}^{u}\right)\right),\left(p_{i1}^{l},p _{i2}^{l},p_{i3}^{l},p_{i4}^{l};P_{1}\left(\widetilde{A}_{i}^{l}\right),P_{2} \left(\widetilde{A}_{i}^{l}\right)\right)\]  
_where \(\widetilde{A}_{i}^{u},\widetilde{A}_{i}^{l}\) denote the upper and the lower membership functions and they are TIFSs; \(p_{11}^{u},p_{12}^{u},p_{i3}^{u}\), \(p_{i4}^{u}\) and \(p_{11}^{l},p_{12}^{l},p_{13}^{l},p_{14}^{l}\) denote the reference points of interval Type-2 fuzzy set \(\widetilde{A}\). \(P_{1}\left(\widetilde{A}_{i}^{u}\right),P_{2}\left(\widetilde{A}_{i}^{u}\right)\) denote the value of the upper membership function of \(p_{12}^{u}\) and \(p_{i3}^{u}\). \(P_{1}\left(\widetilde{A}_{i}^{l}\right)\), \(P_{2}\left(\widetilde{A}_{i}^{l}\right)\) denote the value of the lower membership function of \(p_{12}^{l},p_{13}^{l}\). Where \(1\ll i\ll n\), \(n\) is the number of elements of the \(x\) set. \(P_{1}\left(\widetilde{A}_{i}^{u}\right),P_{2}\left(\widetilde{A}_{i}^{u} \right),P_{1}\left(\widetilde{A}_{i}^{l}\right)\), \(P_{2}\left(\widetilde{A}_{i}^{l}\right)\) both are the value of [0, 1]._  
**Definition 6**.