## 2 Materials and Methods
### Sensitivity Analysis  
To verify the reliability of the results obtained, a sensitivity analysis was carried out. During this sensitivity analysis, we retraced the procedure by which the results were obtained and identified the steps most affected by uncertainties and subjectivity,  
Figure 8: Map of the vulnerability parameter for the province of Ferrara computed as a function of the building age. considering their influence on the final ranking. Specifically, the choice of the preference function and the choice of weights appeared to be the most subjective. As for the choice of the preference function, a previous study [26] recommends assuming a linear preference function endowed with the definition of \(p\), \(q\) thresholds. Two approaches are adopted for the determination of \(p\) and \(q\): The so-called zero-max method, which imposes that the indifference threshold \(q\) is assigned the value of zero while the preference threshold \(p\) is set to be equal to the maximum difference between the evaluations of the criteria. The mean-std method requires the calculation of the average value and the standard deviation of a set of differences between the evaluations of the criteria. In the mean-std method, the indifference threshold is assigned the value of the difference between the average value and standard deviation, while the sum between the average value and standard deviation is assigned to the preference threshold. Following [26; 27; 28], we adopted the preference function of the linear type for the quantitative criteria, that is flood hazard, land use, the age of buildings, and population density. However, the algorithm was also run by choosing the usual preference function, which is the simplest possible one. The thresholds were computed as shown in Table 2. As for the sensitivity on the weights, the four scenarios described in Table 3 have been considered. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**Criteria**} \\ \cline{2-7}  & **Flood Hazard** & **PGA** & **Land Use** & \begin{tabular}{c} **Strategic** \\ **Buildings** \\ \end{tabular} & \
\begin{tabular}{c} **Age of** \\ **Buildings** \\ \end{tabular} &
\begin{tabular}{c} **Population** \\ **Density** \\ \end{tabular} \\ \hline Min/Max & max & Max & max & max & max & max \\ \hline Weight & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline Preference function & Usual & Linear & Linear & Usual & Linear & Linear \\ \hline Thresholds & absolute & Absolute & absolute & absolute & absolute & absolute \\ \hline q: Indifference, zero-max & n/a & 0.000 & 0.0000 & n/a & 0.000 & 0.000 \\ \hline p: Preference (zero-max) & n/a & 0.098 & 0.1896 & n/a & 0.158 & 523.00 \\ \hline s: Gaussian (zero-max) & n/a & n/a & n/a & n/a & n/a & n/a \\ \hline q: Indifference (mean-std) & n/a & 0.093 & 0.0261 & n/a & 0.0676 & 16.10 \\ \hline p: Preference (mean-std) & n/a & 0.155 & 0.1081 & n/a & 0.766 & 238.60 \\ \hline s: Gaussian (mean-std) & n/a & n/a & n/a & n/a & n/a & n/a \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Preference functions and the associated thresholds \(p\), \(q\), and \(s\). \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Sensitivity Analysis: Increase of Single Criteria Weights**} \\ \hline Scenario 0 & \multicolumn{2}{c}{All criteria have the same weight.} \\ \multicolumn{2}{c}{\(p=17\%\)} \\ \hline \multicolumn{2}{c}{Increase the weight of the \(i\)-th criterion by 50\% compared to its} \\ Scenario 1 & \multicolumn{2}{c}{initial value.} \\ \multicolumn{2}{c}{\(p_{i}=25.5\%\); \(p_{\text{other criteria}}=14.9\%\)} \\ \hline \multicolumn{2}{c}{Increase the weight of the \(i\)-th criterion by 50\% compared to its} \\ Scenario 2 & \multicolumn{2}{c}{previous value.} \\ \multicolumn{2}{c}{\(p_{i}=38.2\%\); \(p_{\text{other criteria}}=12.3\%\)} \\ \hline \multicolumn{2}{c}{Increase the weight of the \(i\)-th criterion by 50\% compared to its} \\ Scenario 3 & \multicolumn{2}{c}{previous value.} \\ \multicolumn{2}{c}{\(p_{i}=57.4\%\); \(p_{\text{other criteria}}=8.5\%\)} \\ \end{tabular}
\end{table}