## 4 Research Methodology
### Phase III Calculation IVNNs Performance Score  
ELECTRE is a family of methods used for choosing, sorting and ranking, multi-criteria problems. ELECTRE III was developed by [PERSON] in 1978m, which is valued outranking relation.  
\(A=\{a_{1}a_{2},\ldots,a_{n}\}\) is the finite set of alternatives, \(Y=\{y_{1},y_{2},\ldots,y_{m}\}\) is the finite set of criteria, \(y_{j}(a)\) represents the performance of alternative a on criterion \(y_{j}\in Y\). Assume that all the criteria are of the gain type, which means the greater the value, the better. \(q_{j}\) is the indifference threshold, which represents two alternatives in terms of their evaluations on criterion \(y_{j}\). In general, \(q_{j}\) is a function of attribute value \(q_{j}(a_{i})\), which can be denoted as \(q_{j}(y_{i}(a_{i}))\); \(p_{j}(y_{j}(a_{i}))\) is preference threshold, which indicates that there is a clear strict preference of one alternative over the other in terms of their evaluations on criterion \(y_{j}\). In addition, \(v_{j}(y_{j}(a_{i}))\) is a veto threshold that indicates that the attribute value \(y_{j}(a_{i})\) of scheme \(a_{i}\) is lower than the attribute value \(y_{j}(a_{k})\) of scheme \(a_{k}\), and when it reaches or exceeds \(v_{j}(y_{i}(a_{i}))\), it is not recognized that the \(a_{i}\) is preferred to the \(a_{k}\). \(y_{j}(a_{i})-y_{j}(a_{k})\) which indicates the situation of preference of \(a_{i}\) over \(a_{k}\) for criterion \(C_{j}\). A weight \(w_{j}\) expresses the relative importance of criterion \(y_{j}\), as it can be interpreted as the voting power of each criterion to the outranking relation.  
Where \(y_{j}(a_{i})\) and \(y_{j}(a_{k})\) are expressed in the form of IVNNs in the paper, that is, \(y_{j}(a_{i})=\left\langle\left[T_{i}^{L},T_{i}^{U}\right],\left[I_{i}^{L},I_{ i}^{U}\right],\left[F_{i}^{L},F_{i}^{U}\right]\right\rangle\), \(y_{j}(a_{k})=\left\langle\left[T_{k}^{L},T_{k}^{U}\right],\left[I_{k}^{L},I_{ k}^{U}\right],\left[F_{k}^{L},F_{k}^{U}\right]\right\rangle\). In the calculation of Equation (8), let  
\[y_{j}(a_{i})=\left(T_{i}^{L}+T_{i}^{U}\right)-\left(I_{i}^{L}+I_{i}^{U}\right) -\left(F_{i}^{L}+F_{i}^{U}\right) \tag{22}\]  
\[y_{j}(a_{k})=\left(T_{k}^{L}+T_{k}^{U}\right)-\left(I_{k}^{L}+I_{k}^{U}\right) -\left(F_{k}^{L}+F_{k}^{U}\right) \tag{23}\]  
and so  
\[y_{j}(a_{i})-y_{j}(a_{k})=\left(T_{i}^{L}+T_{i}^{U}-T_{k}^{L}\right)+\left(I_{i }^{L}+I_{i}^{U}-I_{k}^{L}-I_{k}^{U}\right)+\left(F_{i}^{L}+F_{i}^{U}-F_{k}^{U} \right) \tag{24}\]  
\[T_{ij}=\left\{\begin{array}{l}\delta_{i}\frac{a_{ij}}{a_{ij}}(i\in\theta_{B} )\\ \lambda_{i}\frac{a_{ij}^{\min}}{a_{ij}}(i\in\theta_{C},a_{i}^{\min}\
eq 0)\\ \lambda_{i}(1-\frac{a_{ij}}{a_{i}^{\max}})(i\in\theta_{C},a_{i}^{\min}\
eq 0) \end{array}\right.\]  
\[I_{ij}=\left\{\begin{array}{l}\varepsilon_{i}\frac{a_{ij}}{a_{i}^{\min}}(i \in\theta_{B})\\ \mu_{i}\frac{a_{ij}^{\min}}{a_{ij}}(i\in\theta_{C},a_{i}^{\min}\
eq 0)and\\ \mu_{i}(1-\frac{a_{ij}}{a_{i}^{\max}})(i\in\theta_{C},a_{i}^{\min}\
eq 0) \end{array}\right. \tag{25}\]  
\[F_{ij}\left\{\begin{array}{l}F_{ij}^{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\The \(\delta,\lambda\) refer to the certainty parameters of the benefit criteria and cost criteria, respectively [23], while the \(\varepsilon,\mu\) stand for the uncertainty parameters of the benefit criteria and cost criteria respectively, and they obey rule \(0<\delta_{i}+\varepsilon_{i}\leq 1,0<\lambda_{i}+\mu_{i}\leq 1\).