## 2 Methodology
### A Combination of Fuzzy AHP and Fuzzy TOPSIS
#### 2.2.2 Fuzzy TOPSIS  
The fuzzy TOPSIS was developed based on the attribute of the shortest and the longest distance from the positive ideal solution and the negative ideal solution [43]. The best alternative has the shortest distance to the positive ideal solution, and the longest distance from the negative ideal solution. The classical TOPSIS assumes that individual preferences are assigned with crisp values. However, one should consider uncertainty and imprecision in the environmental practices. Fuzzy TOPSIS is more feasible because it incorporates the fuzzy environment uncertainty in decision making.  
The process of Fuzzy TOPSIS used in this study follows six steps [43].  
Step 1. Calculate the aggregate fuzzy ratings for the solutions.  
If \(x_{ijt}=\left(e_{ijt},f_{jt},g_{ijt}\right)\), \(i=\overline{1,n}\), \(j=\overline{1,m}\), \(t=\overline{1,l}\) is the fuzzy aggregated rating of solution \(A_{i}\), by decision maker \(D_{t}\), with respect to each criteria \(C_{j}\). The fuzzy aggregated rating \(x_{ij}=\left(e_{ij},f_{ij},g_{ij}\right)\), is given by:  
\[x_{ij}=\frac{1}{l}\otimes\left(x_{ij1}\oplus x_{ij2}\oplus\ldots\oplus x_{ijt }\oplus\ldots\oplus x_{ijl}\right) \tag{6}\]  
where: \(e_{ij}=\frac{1}{l}\sum\limits_{t=1}^{l}e_{ijt}\), \(f_{ij}=\frac{1}{l}\sum\limits_{t=1}^{l}f_{ijt}\), and \(g_{ij}=\frac{1}{l}\sum\limits_{t=1}^{l}g_{ijt}\)  
Step 2. Calculate normalized fuzzy decision matrix.  
Data is normalized to obtain a comparable scale using linear scale transformation. Suppose \(r_{ij}=\left(a_{ij},b_{ij},c_{ij}\right)\) is the mean of alternative solutions \(i\) for criterion \(j\). The normalized value \(x_{ij}\) can be calculated as:  
\[x_{ij}=\left(\frac{a_{ij}}{c_{j}^{*}},\frac{b_{ij}}{c_{j}^{*}},\frac{c_{ij}}{ c_{j}^{*}}\right)j\in B \tag{7}\]  
\[x_{ij}=\left(\frac{a_{j}}{c_{ij}},\frac{a_{j}}{b_{ij}},\frac{a_{j}}{a_{ij}} \right)j\in C \tag{8}\]  
where: \(a_{j}=\min_{i}a_{ij}\), \(c_{j}^{*}=\max_{i}c_{ij}\), \(i=\overline{1,n}\), \(j=\overline{1,m}\)  
Step 3. Construct the weighted normalized matrix.  
The weighted-normalized value \(G_{i}\) is given by multiplying the normalized value \(x_{ij}\) of the decision matrix by the weight assigned to the criterion \(j\).  
\[G_{i}=x_{ij}\otimes w_{j},\ i=\overline{1,n},\ j=\overline{1,m} \tag{9}\]  
Step 4. Determine the fuzzy positive ideal and negative ideal solutions.  
The positive ideal solution maximizes the benefit criteria and minimizes the cost criteria; whereas the negative ideal solution maximizes the cost criteria and minimizes the benefit criteria. The positive and negative ideal solutions are found out in the following way:  
\[\text{A}^{+}=\left[\text{A}_{j}^{+}\right]_{\text{ij}}=\left[\text{v}_{j\max }\right]_{\text{ij}} \tag{10}\]  
\[\text{A}^{-}=\left[\text{A}_{j}^{-}\right]_{\text{ij}}=\left[\text{v}_{j\min }\right]_{\text{ij}^{\prime}}j=\overline{1,m} \tag{11}\]  
Calculate the distance/separation from:  
* Positive Ideal Separation (\(d^{+}\)):  
\[d_{i}^{+}=\sqrt{\sum\limits_{i=1}^{n}(G_{i}-A^{+})^{2}} \tag{12}\]  
* Negative Ideal Separation (\(d^{-}\)):  
\[d_{i}^{-}=\sqrt{\sum\limits_{i=1}^{n}(G_{i}-A^{-})^{2}} \tag{13}\]  
where: \(d_{i}^{+}\) and \(d_{j}^{-}\) are the distances of each alternative \(A_{i}\) from positive and negative ideal solutions.  
Step 5. Calculate the Relative closeness coefficient (CC) to the ideal solution.  
\[CC_{i}=\frac{d_{i}^{-}}{d_{i}^{+}+d_{i}^{-}};\ 0<CC_{i}<1;\ i=\overline{1,n} \tag{14}\]  
\[CC_{i}=1\ \text{if}\ A_{i}=A^{+};CC_{i}=0\ \text{if}\ A_{i}=A^{-}\]  
Step 6. Rank alternatives  
The rank of considered alternatives can be decided, according to the descending order of closeness coefficient (CC). When the closeness coefficient is closer to 1 the corresponding solution is the optimal one.