## 3 Results
### Usual Preference Function  
When the usual preference function is used, the algorithm assumes equal weights. We recall that, here, thresholds \(p\) and \(q\) are not required. Basically, what is provided to the analyst is an order of priority where the municipalities in the province of Ferrara are ordered from the most sensitive to combined flood and seismic risk to the one that is least affected. Table 4 shows the final ranking of the alternatives.  
This is not the only way to visualize the results: The PROMETHEE rainbow plot, shown in Figure 9, allows one to highlight, for each alternative, the criteria that positively or negatively affect the results. In Figure 9, the colors are representative of the criterion: Yellow indicates the criteria relating to exposure, red is used for seismic hazard, green for vulnerability, and blue for flood hazard. For example, for the municipality of Ferrara (first in the ranking), it can be observed that the criterion that has a negative effect is the one relating to the flood hazard, whereas the other criteria have a positive effect on the Ferrara municipality. On the contrary, in the municipality of Jolanda di Savoia (last in the ranking), the only criterion that has a positive influence is the one relating to vulnerability, while all the others have a negative influence.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rank** & **Alternatives** & \(\mathbf{\Theta}\) & \(\mathbf{\Theta^{+}}\) & \(\mathbf{\Theta^{-}}\) \\ \hline
1 & Ferrara & 0.6111 & 0.7302 & 0.119 \\ \hline
2 & Cento & 0.5873 & 0.7222 & 0.1349 \\ \hline
3 & Tresigallo & 0.4127 & 0.6111 & 0.1984 \\ \hline
4 & Vigarano Mainarda & 0.2857 & 0.5873 & 0.3016 \\ \hline
5 & Mirabello + Sant\({}^{\prime}\)Agostino & 0.2698 & 0.5794 & 0.3095 \\ \hline
6 & Argenta + Portomaggiore & 0.2381 & 0.5238 & 0.2857 \\ \hline
7 & Bondeno & 0.1825 & 0.4921 & 0.3095 \\ \hline
8 & Copparo & 0.0238 & 0.4127 & 0.3889 \\ \hline
9 & Poggio Renatico & 0.0238 & 0.4524 & 0.4286 \\ \hline
10 & Comacchio & 0.0000 & 0.4048 & 0.4048 \\ \hline
10 & Formignana & 0.0000 & 0.381 & 0.381 \\ \hline
12 & Voghiera & \(-0.0238\) & 0.3651 & 0.3889 \\ \hline
13 & Lagosanto & \(-0.0317\) & 0.3889 & 0.4206 \\ \hline
14 & Berra & \(-0.1587\) & 0.3016 & 0.4603 \\ \hline
15 & Masi Torello & \(-0.1746\) & 0.2937 & 0.4683 \\ \hline
16 & Ro & \(-0.1905\) & 0.2857 & 0.4762 \\ \hline
17 & Fiscaglia & \(-0.2063\) & 0.2778 & 0.4841 \\ \hline
18 & Mesola & \(-0.2857\) & 0.2381 & 0.5238 \\ \hline
19 & Ostellato & \(-0.3571\) & 0.1984 & 0.5556 \\ \hline
20 & Goro & \(-0.3651\) & 0.1984 & 0.5635 \\ \hline
21 & Codigoro & \(-0.3651\) & 0.2222 & 0.5873 \\ \hline
22 & Jolanda di Savoia & \(-0.4762\) & 0.1429 & 0.619 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Ranking of alternatives for the usual preference function.  
Based on the ranking provided by PROMETHEE, it is possible to create a risk map of the municipalities of the province of Ferrara that highlights high-priority areas as those with a high level of combined flood and seismic risk, medium priority areas as the areas characterized by a medium combined-risk level, and, finally, low combined-risk areas.  
This map is shown in Figure 10. It can be seen that the three risk levels are identified by three different colors: Red is used for high risk, orange for medium risk, and yellow for low risk.  
Figure 10: Multiple-risk map for the Ferrara province obtained for the usual preference function (Type 1 in Table 1). 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 9: PROMETHEE rainbow plot for the usual preference function. On the vertical axis, the preference function \(\Theta\) is reported. The yellow bar indicates the criteria relating to exposure, red is used for seismic hazard, green for vulnerability, and blue for hydraulic hazard.