## 4 Methods
### Fuzzy TOPSIS  
The TOPSIS method is one of the common MCDM approaches [18]. The option chosen in this method should be the minimum distance from the Positive Ideal Solution and the greatest distance from the Negative Ideal Solution [19].  
Fuzzy TOPSIS is a frequently used multi-criteria decision-making system that can rank many alternatives with the advantages of quickly identifying the best alternative among the leading methods [20].  
This method includes 6 main steps:  
**Step 1:** The decision matrix is created.  
**Step 2:** The decision matrix is subjected to normalization process. Normalization is a mathematical process that allows the comparison of each criterion to [0, 1], and to compare the results. The mathematical representation of a fuzzy normalized decision matrix is given in Equations (4)-(7).  
\[r_{i,j}=\frac{X_{i,j}}{u_{j}^{+}}=\left(\frac{l_{X_{i,j}}}{u_{j}^{+}};\frac{m_ {X_{i,j}}}{u_{j}^{+}};\frac{u_{X_{i,j}}}{u_{j}^{+}}\right)\ j\ \epsilon B \tag{4}\]  
\[r_{i,j}=\frac{l_{j}^{-}}{X_{i,j}}\!\!=\!\left(\frac{l_{j}^{-}}{u_{ij}^{+}}; \frac{l_{j}^{-}}{m_{ij}^{+}};\frac{l_{j}^{-}}{l_{ij}^{+}}\right)\ j\ \epsilon\ C \tag{5}\]  
\[u_{j}^{+}=max\big{(}u_{i,j}\ \big{)}\ \forall i=1,\ldots,m\ \ j\ \ \epsilon\ B \tag{6}\]  
\[l_{j}^{-}=min\big{(}l_{i,j}\ \big{)}\ \ \ \forall i=1,\ldots,m\ \ j\ \ \epsilon\ C \tag{7}\]  
\(r_{i,j}\): normalized fuzzy decision matrix.  
Equations for the normalization process are handled in two cases: benefit and cost. If the criterion is a criterion of benefit or cost criterion, the above procedures are performed.  
**Step 3:**\(w_{j}\) is the weight of each criterion and the decision matrix is multiplied by \(w_{j}\) values to obtain a weighted normalized fuzzy decision matrix.  
\[\widetilde{v}_{i,j}=\widetilde{r}_{i,j}*w_{j}\]  
\(\widetilde{v}_{i,j}\): weighted normalized fuzzy decision matrix.  
**Step 4:** In this step, fuzzy positive ideal solution \(\mathbf{A^{+}}\), and fuzzy negative ideal solution \(\mathbf{A^{-}}\) are obtained. Mathematical equations for obtaining ideal solutions are given in Equations (8)-(11):  
\[\mathbf{A^{+}}=(V_{1}^{+};\ V_{J}^{+},\ldots\ldots,\ V_{n}^{+}) \tag{8}\]  
\[\mathbf{A^{-}}=(V_{1}^{-};\ V_{J}^{-},\ldots\ldots,\ V_{n}^{-}) \tag{9}\]  
\[V_{J}^{+}=\left\{\begin{array}{ll}(1,1,1)j\ \epsilon B\\ (0,0,0)j\ \epsilon C\end{array}\right. \tag{10}\]  
\[V_{J}^{-}=\left\{\begin{array}{ll}(0,0,0)j\ \epsilon B\\ (1,1,1)j\ \epsilon C\end{array}\right. \tag{11}\]  
\(A^{+}\): fuzzy positive ideal solution;  
\(A^{-}\): fuzzy negative ideal solution.  
For the positive ideal solution and for the negative ideal solution, it can be seen that the fuzzy expression of the structures determined in terms of the benefit and cost criteria differ from each other.  
**Step 5:** After obtaining fuzzy positive and negative ideal solutions, n-dimensional separation distances are obtained.  
The fuzzy optimal ideal solution distance and the worst ideal solution distance are calculated via Equations (12) and (13), respectively. The distance measurement between two fuzzy numbers is represented as d (.. ).  
\[\mathbf{D}_{I}^{*}=\sum_{j=1}^{n}\mathbf{d}(\widetilde{\mathrm{v}}_{\mathrm{ij}}, \ \widetilde{\mathrm{v}}_{\mathrm{i}}^{*})\ \ \ \ \ \ \mathrm{j}=1,2,...\mathrm{J} \tag{12}\]  
\[\mathbf{D}_{I}^{-}=\sum_{j=1}^{n}\mathbf{d}(\widetilde{\mathrm{v}}_{\mathrm{ij}},\ \widetilde{\mathrm{v}}_{\mathrm{i}}^{-})\ \ \ \ \ \mathrm{j}=1,2,...\mathrm{J} \tag{13}\]  
In the current study, the distance between two fuzzy numbers was measured via the Vertex Method in Equation (14):  
\[\mathrm{d}\Big{(}\widetilde{\mathrm{A}}_{1},\widetilde{\mathrm{A}}_{2}\Big{)} =\sqrt{1/3[(l_{1}-l_{2})^{2}+(m_{1}-m_{2})^{2}+(u_{1}-u_{2})^{2}} \tag{14}\]  
**Step 6:** The closeness index is calculated for each criterion in the last step. The calculation of this index is in Equation (15):  
\[\mathbf{CC}_{I}\ =\ \frac{\mathbf{D}_{I}^{-}}{\mathbf{D}_{I}^{*}+\mathbf{D}_{I}^{-}}\ \ \ \ \ \mathrm{j}=1,2,.. \mathrm{J} \tag{15}\]  
_CC\({}_{I}\)_ :** closeness coefficient