## 3 Results
### Linear Preference Function  
The ranking of alternatives for the linear preference function and zero-max method is shown in Table 5, while the corresponding multi-risk map is shown in Figure 11. These maps were obtained by associating the quantitative criteria, i.e., flood hazard, land use, age of buildings, and population density, with a linear preference function, while thresholds \(q\) and \(p\) were determined with the zero-max method.  
For a linear preference function of the aforementioned quantitative criteria, and thresholds \(q\) and \(p\) determined by the mean-std method, we obtained the results shown in Table 6 and Figure 12.  
By comparing the results obtained from the usual and the linear preference functions, it can be understood that changes of the preference function do not reflect large changes of the final risk maps. The only difference is that the risk levels of the municipality of Vigarano Mainarda swap with Bondeno, and Fiscaglia swaps with Lagosanto.  
By comparing the maps in Figures 11 and 12, obtained with the thresholds chosen with the zero-max and mean-std methods, respectively, we observe that the risk levels of Lagosanto, Vigarano Mainarda, and Codigoro increase. Particularly, we observe that the choice of the preference function affects the final ranking of the alternatives especially when the thresholds are chosen according to the mean-std method.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rank** & **Alternatives** & \(\mathbf{\Theta}\) & \(\mathbf{\Theta^{+}}\) & \(\mathbf{\Theta^{-}}\) \\ \hline
1 & Cento & 0.459 & 0.5086 & 0.0496 \\ \hline
2 & Ferrara & 0.3545 & 0.393 & 0.0385 \\ \hline
3 & Tresigallo & 0.1821 & 0.2642 & 0.0821 \\ \hline
4 & Mirabello + Sant’Agostino & 0.1444 & 0.2622 & 0.1179 \\ \hline
5 & Argenta + Portomaggione & 0.1352 & 0.2182 & 0.0829 \\ \hline
6 & Bondeno & 0.1257 & 0.2069 & 0.0812 \\ \hline
7 & Vigarano Mainarda & 0.112 & 0.255 & 0.143 \\ \hline
8 & Copparo & 0.0505 & 0.1684 & 0.1179 \\ \hline
9 & Poggio Renatico & 0.031 & 0.2216 & 0.1906 \\ \hline
10 & Comacchio & 0.0224 & 0.1631 & 0.1406 \\ \hline
10 & Voghiera & \(-\)0.0422 & 0.0984 & 0.1406 \\ \hline
12 & Formignana & \(-\)0.0721 & 0.0937 & 0.1657 \\ \hline
13 & Fiscaglia & \(-\)0.0761 & 0.0868 & 0.1628 \\ \hline
14 & Lagosanto & \(-\)0.0898 & 0.1385 & 0.2283 \\ \hline
15 & Codigoro & \(-\)0.101 & 0.1155 & 0.2164 \\ \hline
16 & Ostellato & \(-\)0.1092 & 0.0736 & 0.1828 \\ \hline
17 & Ro & \(-\)0.1362 & 0.0589 & 0.195 \\ \hline
18 & Masi Torello & \(-\)0.1371 & 0.0557 & 0.1928 \\ \hline
19 & Berra & \(-\)0.1551 & 0.0688 & 0.2239 \\ \hline
20 & Jolanda di Savoia & \(-\)0.1947 & 0.0408 & 0.2355 \\ \hline
21 & Mesola & \(-\)0.2203 & 0.0353 & 0.2556 \\ \hline
22 & Goro & \(-\)0.283 & 0.0137 & 0.2968 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Ranking of alternatives for the linear preference function and zero-max method.  
Figure 11: Multiple-risk map for the Ferrara province for the linear preference function with the thresholds chosen with the zero-max method. The risk levels strictly indicate the relative priority ranking for decision-makers and do not indicate the effective safety level of the various municipalities.  
Figure 12: Multiple-risk map for the Ferrara province for the linear preference function with the thresholds chosen with the mean-std method. The risk levels strictly indicate the relative priority ranking for decision-makers and do not indicate the effective safety level of the various municipalities.  
Regardless of the preference function chosen, the maps obtained present a similar risk trend, i.e., the territory is divided into two parts: The municipalities of the western part of the territory of the province of Ferrara, plus Ferrara and Tresigallo, are characterized by a medium-high risk level; the upper-eastern part of the province is characterized by a medium-low risk level.