## 2 Materials and Methods
### The PROMETHEE Method  
The proposed multi-hazard risk analysis procedure for the region under study is based on PROMETHEE [26; 27; 28; 29], a Multiple-Criteria Decision Analysis method. It belongs to the class of aggregation methods based on outranking relationships. It is known for its simplicity and the ability to analyze information from multiple sources. PROMETHEE allows one to jointly compare data originally expressed in different units and scales. A flux diagram explaining the various steps of the PROMETHEE-based analysis can be found in reference [29]. PROMETHEE deals with maximization or minimization problems with \(k\) different criteria of the kind  
\[\max(\textit{orm}\min)\{g_{1}(a),g_{2}(a),\ldots,g_{k}(a)|a\in A\}, \tag{1}\]  
Figure 2: (**a**) Seismic classification of municipalities in Emilia ([[https://ambiente.regione.emilia-romagna.it/en/geologia/seismic-risk/seismic-classification](https://ambiente.regione.emilia-romagna.it/en/geologia/seismic-risk/seismic-classification)]([https://ambiente.regione.emilia-romagna.it/en/geologia/seismic-risk/seismic-classification](https://ambiente.regione.emilia-romagna.it/en/geologia/seismic-risk/seismic-classification)), accessed on 26 January 2022). (**b**) Seismic Hazard Map of Italy (free source from INGV webpage [[http://zonesismiche.mi.ingv.it/](http://zonesismiche.mi.ingv.it/)]([http://zonesismiche.mi.ingv.it/](http://zonesismiche.mi.ingv.it/)), accessed on 26 January 2022). where \(A\) is a finite set of possible alternatives and function \(g_{j}(a)\) represents the performance of the \(j\)-th criterion. Let us consider two alternatives, \((a,b)\in A\). We have the following cases:  
\[\left\{ \begin{array}{l}\forall j:g_{j}(a)\geq g_{j}(b)\\ \exists k:g_{k}(a)>g_{k}(b)\end{array} \Leftrightarrow aPb,\right.\]  
\[\left. \begin{array}{l}\forall j:g_{j}(a)=g_{j}(b)\Leftrightarrow alb,\\ \end{array} \right. \tag{2}\]  
\[\left\{ \begin{array}{l}\exists s:g_{s}(a)>g_{s}(b)\\ \exists r:g_{r}(a)<g_{r}(b)\end{array} \Leftrightarrow aRb,\right.\]  
where \(P\), \(I\), and \(R\) denote preference (\(P\)), indifference (\(I\)), or incompatibility relations (\(R\)) of one alternative over the other, respectively. By comparing all the alternatives for each criterion, a hierarchy of alternatives belonging to the starting space A will be obtained. When comparing two actions,\((a,b)\in A\). the result of this comparison is expressed in terms of the preference function \(\wp:\ A\times A\rightarrow(0,1)\) that represents the intensity of the preference of alternative \(a\) towards alternative \(b\). Therefore, \(\wp(a,b)=0\) indicates no preference of \(a\) over \(b\) (or indifference), \(\wp(a,b)\simeq 0\) indicates a weak preference of \(a\) over \(b\), \(\wp(a,b)\simeq 1\) indicates a strong preference of \(a\) over \(b\), and \(\wp(a,b)=1\) indicates a strict preference of \(a\) over \(b\). In practice, the preference function will often be a function of the difference between the evaluations of the two alternatives considered:  
\[\wp(a,b)=P(g(a)-g(b))=P(d), \tag{3}\]  
where \(P\) is a non-decreasing function, equal to zero for negative values of \(d\). PROMETHEE offers six types of preference functions (see Table 1). Therefore, a preference index is defined as follows:  
\[\left\{ \begin{array}{l}\pi(a,b)=\sum\limits_{j=1}^{k}P_{j}(a,b)w_{j}\\ \pi(b,a)=\sum\limits_{j=1}^{k}P_{j}(b,a)w_{j}\end{array} \right., \tag{4}\]  
where \(\pi(a,b)\) expresses the degree to which \(a\) is preferred to \(b\) over all criteria and vice versa, and \(w_{j}\) is the weight of each criterion and expresses a measure of the importance of the relative criterion.