## 2 Methodology
### A Multi-Criteria Decision-Making (MCDM) Using Fuzzy AHP and Fuzzy TOPSIS
#### 2.2.2 Fuzzy TOPSIS  
Step 1. Determine the decision makers' (DMs') ratings for each criterion  
Set \(x_{ijt}=\left(e_{ijt},f_{jt},g_{jt}\right),\ i=1,\ldots,n,j=1,\ldots,m,\ t=1, \ldots,l\), as the appropriate rating defined for alternatives \(A_{i}\), by decision makers \(D_{t}\), for each criterion \(C_{j}\). The average value of the rating, \(x_{ij}=\left(e_{ijt},f_{ij},g_{ij}\right)\), can be calculated as:  
\[x_{ij}=\frac{1}{l}\otimes\left(x_{ij1}\oplus x_{ij2}\oplus\ldots\oplus x_{ijt }\oplus\ldots\oplus x_{ijl}\right) \tag{10}\]  
where:  
\[ \begin{array}{l}e_{ij}=\frac{1}{l}\sum\limits_{t=1}^{l}e_{ijt}\\ f_{ij}=\frac{1}{l}\sum\limits_{t=1}^{l}f_{jt}\\ g_{ij}=\frac{1}{l}\sum\limits_{t=1}^{l}g_{jt}\end{array}  \tag{11}\]  
Step 2. Normalize the average value of the decision makers for each criterion  
To ensure compatibility between values and units of ratings and weights, these values need to be normalized into comparable ranges. Suppose \(r_{ij}=\left(a_{ij},b_{ij},c_{ij}\right)\) is the average value of alternative \(A_{i}\) for criterion \(C_{j}\). The normalized value \(x_{ij}\) can be calculated as:  
\[x_{ij}=\left(\frac{a_{ij}}{c_{j}^{*}},\frac{b_{ij}}{c_{j}^{*}},\frac{c_{ij}}{ c_{j}^{*}}\right)j\in B \tag{12}\]  
\[x_{ij}=\left(\frac{a_{j}^{-}}{c_{ij}},\frac{a_{j}^{-}}{b_{ij}},\frac{a_{j}^{- }}{a_{ij}}\right)j\in C \tag{13}\]  
where:  
\[a_{j}^{-}=\min_{i}a_{ij},c_{j}^{*}=\max_{i}c_{ij},i=1,\ldots,n;j=1,\ldots,m. \tag{14}\]  
Step 3. Determine the normalized rating-weight value  
The normalized rating-weight value \(G_{i}\) is calculated by multiplying the weight values of the criteria by the rating value of the solutions as:  
\[ \begin{array}{l}G_{i}=x_{ij}\otimes w_{j}\\ i=1,\ldots,n,j=1,\ldots,m\end{array}  \tag{15}\]  
Step 4. Calculate values \(A^{+},A^{-},d_{i}^{+}\) and \(d_{i}^{-}\)  
The fuzzy positive ideal solution (FPIS,\(A^{+}\)) and fuzzy negative ideal solution (FNIS,\(A^{-}\)) are calculated as:  
\[A^{+}=\left(1.0,\ 1.0,\ 1.0\right) \tag{16}\]  
\[A^{-}=\left(0.0,\ 0.0,\ 0.0\right) \tag{17}\]Distance of each alternative \(A_{i},i=1,\ldots,n\) from \(A^{+}\) and \(A^{-}\) is calculated as:  
\[d_{i}^{+} = \sqrt{\sum_{i=1}^{n}\left(G_{i}-A^{+}\right)^{2}} \tag{18}\] \[d_{i}^{-} = \sqrt{\sum_{i=1}^{n}\left(G_{i}-A^{-}\right)^{2}} \tag{19}\]  
where: \(d_{i}^{+}\) is the shortest distance of the alternative \(A_{i}\) and \(d_{i}^{-}\) is the longest distance of the alternative \(A_{i}\)  
Step 5. Calculate the closeness coefficient  
The closeness coefficients (\(CC_{i}\)) used to determine the ranking order of the alternatives are calculated as:  
\[CC_{i}=\frac{d_{i}^{-}}{d_{i}^{+}+d_{i}^{-}} \tag{20}\]  
The higher a coefficient, the closer that alternative is to the positive ideal solution (PIS) and the further that alternative is from the negative ideal solution (NIS). From these closeness coefficients, the best alternative among the given ones is determined. Following the fuzzy TOPSIS, the sensitivity analysis is conducted to determine the impact of different conditions of selected criteria (\(C_{j}\)) and decision makers (\(DM_{t}\)) and on ranking alternatives (\(A_{i}\)). Figure 2 shows the relationships between a general goal, 10 selected criteria (\(C_{i}\), \(i\) = 1...10), and 8 studied alternatives (\(A_{j}\), \(j\) = 1... 8). The general goal is determining factors affecting the likelihood of flood hazards. Criteria consist of slope (C\({}_{1}\)), terrain elevation (C\({}_{2}\)), Creek-bottom terrain (C\({}_{3}\)), soil type (C\({}_{4}\)), stream density (C\({}_{5}\)), drainage channel density (C\({}_{6}\)), distance to water body (C\({}_{7}\)), high tide-induced flooding area (C\({}_{8}\)), land use and land cover (C\({}_{9}\)), and water storage by transportation corridors (C\({}_{10}\)). These criteria define \"vulnerability\" in flood studies. The selection of the criteria is based on the quality and quantity of the available information.