# Hierarchical Fuzzy MCDA Multi-Risk Model for Detecting Critical Urban Areas in Climate Scenarios
## 2 Materials and Methods
### The GHF-MCDA Model  
In [6], a fuzzy-based multicriteria decision-making model was implemented on a GIS platform to find, among the subzones in which the area of study is partitioned, the ones best suited to meeting a set of criteria expressed as fuzzy sets. Formally, A = {a1, a2,..., am} is the set of alternatives correspondent to the set of homogeneous subzones, and r1, r2,..., rn is the set of n criteria. Each criterion rk is expressed as a linguistic label for the fuzzy set:  
\[\mathrm{R_{k}}=\left\{\mu_{R_{k}}(a_{1})/a_{1},...,\mu_{R_{k}}(a_{i})/a_{i},...,\mu_{R_{k}}(a_{m})/a_{m}\right\}\,\mathrm{k}=1,\;2,\;\ldots,\;\mathrm{k} \tag{1}\]  
where \(\mu_{R_{k}}(a_{i})\) is the membership degree assigned to the i_th_ alternative in meeting the k_th_ criterion. The relevance of the k_th_ criterion in selecting the best alternative is modeled in [27] with a weight \(\alpha_{k}\) called the _Coefficient of Relative Significance_ (for short, CRS). The CRS are assigned to the n criteria, respecting the following constraint:  
\[\frac{1}{n}\sum_{k=1}^{n}\alpha_{k}=1 \tag{2}\]  
The Saaty's AHP pairwise comparison matrix is assigned by the expert to obtain the CRS's. When **B** is the pairwise comparison nxn matrix where bkj, k, j = 1,...,n; it gives the relevance of the criterion rk with respect to the criterion rj. The values assigned to the pairwise comparison matrix element bkj are shown in Table 1.  
The conjugate elements bjk are set to 1/bkj and the diagonal elements bkk are set to 1. The CRS values \(\alpha_{1}\), \(\alpha_{2}\),..., \(\alpha_{n}\) are given by the components of the eigenvector **w** = (w1, w2,..., wk,... wn) correspondent to the greatest real eigenvalue solution of the characteristic equation  
\[|\textbf{B}-\lambda\textbf{I}|=0 \tag{3}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Relevance of rk on rj** & **bkj** & **Relevance of rk on rj** & **bkj** \\ \hline Equally important & 1 & Equally important & 1 \\ Equally or slightly more important & 2 & Equally or slightly less important & 1/2 \\ Slightly more important & 3 & Slightly less important & 1/3 \\ Slightly to much more important & 4 & Slightly to way less important & 1/4 \\ Much more important & 5 & Way less important & 1/5 \\ Much to far more important & 6 & Way to far less important & 1/6 \\ Far more important & 7 & Far less important & 1/7 \\ Far more important to extremely more & 8 & Far less important to extremely less important & 1/8 \\ important & 8 & important & 1/8 \\ Extremely more important & 9 & Extremely less important & 1/9 \\ \hline \end{tabular}
\end{table} 
Table 1: Values of the Saaty’s pairwise comparison matrix. where **I** is the identity nxn matrix. The CRS value \(\alpha_{k}\) of the kth criterion is given by:  
\[\alpha_{k}=n\cdot w_{k} \tag{4}\]  
To find the best alternative, the fuzzy set intersection of the n fuzzy sets R1, R2,..., Rn is built, given by  
\[R=\bigcap_{k=1}^{n}R_{k}^{\alpha_{k}} \tag{5}\]  
with membership function  
\[\mu_{R}(a_{i})=\min_{k=1,...,n}(\mu_{R_{k}}^{\alpha_{k}}(a_{i})) \tag{6}\]  
The alternative that best meets all the criteria is the one with the greatest membership degree in the fuzzy set R.  
Moreover, in [20], a suitability thematic map of the subzones is constructed by classifying them based on the membership values \(\mu_{R}(a_{i})\) i = 1,..., m.  
In order to model complex criteria that characterize urban analysis problems, in [6], a hierarchical tree extension of this fuzzy MCDA model, called GHF-MCDA, was tested. In GHF-MCDA, a criterion is broken down into sub-criteria, and the fuzzy set corresponding to a criterion is built starting from the fuzzy sets associated with its sub-criteria. The n criteria at the top level are denoted with: \(r_{1}(^{1})\), \(r_{2}(^{1})\),..., \(r_{n}(^{1})\), where the subscript in round brackets (1) denotes the level in the hierarchy. Let us suppose that the kth criterion \(r_{k}(^{1})\) is decomposed in the next level in \(n_{k}\) sub-criteria \(r_{k1}(^{2})\), \(r_{k2}(^{2})\),..., \(r_{\text{knk}}(^{2})\).