## 2 Methodology
### A Combination of Fuzzy AHP and Fuzzy TOPSIS  
#### 2.2.1 Fuzzy AHP  
The AHP is one of the most common Multi-Criteria Decision Making (MCDM) instruments to deal with quantifiable and intangible criteria, which reflect the relative importance of the alternatives based on constructing a pair-wise comparison matrix [38; 39; 40]. Fuzzy AHP was developed to determine the weights of multiple criteria [41]. While the traditional AHP faces limitations on information imprecision and vagueness for decision making, the fuzzy AHP solves these problems of imprecision using linguistic variables, which are used to represent the relative importance between each pair of criteria [42]. Five linguistic variables corresponding with their fuzzy numbers are presented in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Linguistic Variables** & **Fuzzy Numbers** \\ \hline Extreme importance (EXI) & (7; 9; 9) \\ Very strong importance (VSI) & (5; 7; 9) \\ Strong importance (SI) & (3; 5; 7) \\ Moderate importance (MI) & (1; 3; 5) \\ Equal importance (EI) & (1; 1; 1) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Linguistic variables and fuzzy numbers [42].  
Figure 1: Location of Ky Khang and Vung Ang EZ in the Ky Anh district (Ha Tinh, Vietnam).  
The process of Fuzzy AHP is structured in four steps [41]. Let \(X=\{x_{1},x_{2}\dots\ x_{n}\}\) be an object set and \(G=\{g_{1},g_{2}\dots\ g_{m}\}\) be a goal set. According to the extent analysis method [42], each object \(\mathrm{x_{i}}\) is evaluated by performing an extent analysis for each goal, \(g_{i}\). Therefore, \(m\) extent analysis values for each object can be obtained by using following notation: \(M_{g^{1}}\), \(M_{g^{2}}\)...\(M_{g^{m}}\).  
Step 1. The fuzzy synthetic extent value (\(S_{i}\)) with respect to the ith criterion is calculated in the following way:  
\[S_{i}=\sum\limits_{j=1}^{m}M_{g_{i}}^{j}\otimes\left[\sum\limits_{i=1}^{n}\sum \limits_{j=1}^{m}M_{g_{i}}^{j}\right]^{-1} \tag{1}\]  
With:  
\[\sum\limits_{j=1}^{m}M_{g_{i}}^{j}=\left(\sum\limits_{j=1}^{m}l_{j},\sum \limits_{j=1}^{m}m_{j},\sum\limits_{j=1}^{m}u_{j}\right)\sum\limits_{i=1}^{n} \sum\limits_{j=1}^{m}M_{g_{i}}^{j}=\left(\sum\limits_{i=1}^{n}l_{i},\sum \limits_{i=1}^{n}m_{i},\sum\limits_{i=1}^{n}u_{i}\right)\] \[\left\{\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}M_{g_{i}}^{j} \right\}^{-1}=\left(\frac{1}{\sum\limits_{i=1}^{n}u_{i}},\frac{1}{\sum\limits_ {i=1}^{n}u_{i}},\frac{1}{\sum\limits_{i=1}^{n}l_{i}}\right)\]  
where: \(l\) is the lower limit value; \(m\) is the most promising value; \(u\) is the upper limit value.  
Step 2. The degree of possibility of \(S_{2}(l_{2},m_{2},u_{2})\geq S_{1}(l_{1},m_{1},u_{1})\) is defined as:  
\[V(S_{1}\geq S_{2})=\sup\limits_{y\geq x}\left[\min\left(\mu_{M_{1}}(x),\mu_{M _{2}}(y)\right)\right] \tag{2}\]  
where: \(x\) and \(y\) are the values on the axis of membership function of each criterion.  
\[V(S_{1}\geq S_{2})=\left\{\begin{array}{c}1\ if\ m_{1}>m_{2}\\ 0\ if\ l_{2}>u_{1}\\ (l_{2}-u_{1})/(l_{2}-u_{1}+m_{1}-m_{2})\ otherwise\end{array}\right\} \tag{3}\]  
Step 3. The possibility that a convex fuzzy number \(S\) is greater than \(k\) convex fuzzy numbers \(S_{i}\left(i=\overline{1,k}\right)\) is defined as:  
\[V(S\geq S_{1},S_{2},\dots S_{n})=V[(S\geq S_{1}),(S\geq S_{2}),\dots,(S\geq S _{n})]=\min V(S\geq S_{i}); \tag{4}\]  
Step 4. Calculate the normalized weight vectors W\({}^{\prime}\)  
\[W\prime=\left(d\
u(A_{1}),d\
u(A_{2}),\dots,d\
u(A_{n})\right)^{T} \tag{5}\]  
where: \(d\
u(A_{i})=\min V(S_{i}\geq S_{t})\); \(i=\overline{1,n}\); \(t=\overline{1,n}\); and \(i\
eq t\)  
The normalized weight vectors \(W^{\prime}\) is generated according to the pairwise comparisons of the criteria of the involved respondents. As far as the important the corresponding criterion is concerned, the higher the weight, the more important the corresponding criterion.