# Hierarchical Fuzzy MCDA Multi-Risk Model for Detecting Critical Urban Areas in Climate Scenarios
## 2 Materials and Methods
### The Proposed Model  
We apply the hierarchical method [4] to evaluate which subzones of an urban area are the most critical in the presence of multi-risk scenarios. In particular, we investigate which subzones are most vulnerable in the presence of multiple risk scenarios. The GHF-MCDA model is implemented in a GIS-based framework with the aim of determining which subzones are most vulnerable in the presence of multiple risks. Following the hierarchical decomposition model of an urban settlement into sub-systems proposed in [5], the urban settlement is partitioned into subsystems, and the vulnerability of each subsystem is assessed. In our study, the GHF-MCDA model was tested to detect which subzones of an urban settlement were most vulnerable to the two climatic phenomena of heatwaves and pluvial flooding, taking into consideration the vulnerabilities of the physical system, broken down into the subsystems of buildings and open spaces. The hierarchy of the criteria is built on two hierarchical levels. The first level is made up of four criteria: residential building heatwave vulnerability, open space heatwave vulnerability, residential building pluvial flooding vulnerability, and open space pluvial flooding vulnerability. Each criterion is partitioned into a set of sub-criteria related to the set of indicators used in [5] to evaluate the vulnerability of the sub-systems of buildings and open spaces to heatwave and pluvial flooding phenomena. Figure 2 shows the hierarchy of the criteria and the characteristics needed to calculate the set of indicators. The radix node, called _Subzone criticality_, is a value in the normalized interval [0, 1] corresponding to the suitability of the subzone to be vulnerable to both phenomena. The fuzzy sets of each sub-criterion are assigned to the domain [0, 5] in which the values of the intermediate indicators fall (Figure 3). The fuzzy sets of the four criteria are assigned to the normalized domain [0, 1]. The [PERSON]'s pairwise comparison matrix of a criterion is built taking into account the weights of the intermediate indicators assigned in [5]. The four pairwise comparison matrices are shown, respectively, in Tables 3-6. Figure 2: Hierarchical structure of the criteria. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Open Space Area** & **Soil Permeability** & \
\begin{tabular}{c} **Flow Capacity of Storm** \\ **Drainage System** \\ \end{tabular} &
\begin{tabular}{c} **Maintenance of Storm** \\ **Drainage System** \\ \end{tabular} \\ \hline Open space area & 1 & 1/4 & 1/6 & 1/3 \\ \hline Soil permeability & 4 & 1 & 1/3 & 3 \\ \hline Flow capacity of storm & & & & \\ drainage system & 6 & 3 & 1 & 4 \\ \hline Maintenance of storm & & & & \\ drainage system & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Pairwise comparison matrix of the criterion open space pluvial flooding vulnerability. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **SVF** & **Albedo** & **Solar Exposure** & **NDVI** \\ \hline SVF & 1 & 6 & 3 & 4 \\ \hline Albedo & 1/6 & 1 & 1/4 & 1/3 \\ \hline Solar exposure & 1/3 & 4 & 1 & 3 \\ \hline NDVI & 1/4 & 3 & 1/3 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Pairwise comparison matrix of the criterion open space heatwave vulnerability. Figure 3: Fuzzy sets assigned to criteria (**a**) and sub-criteria (**b**). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Building Coverage** & **Percentage of Building** & **Ground Floor** \\  & **Ratio** & **on the Sidewalk** & **Activities** & **Roof Typology** \\ \hline Building coverage ratio & 1 & 1/6 & 1/9 & 1/3 \\ \hline Percentage of building & & & & \\ on the sidewalk & 6 & 1 & 1/3 & 4 \\ \hline Ground floor activities & 9 & 3 & 1 & 6 \\ \hline Roof typology & 3 & 1/4 & 1/6 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Pairwise comparison matrix of the criterion residential building pluvial flooding vulnerability.