## 2 Regional Seismic Risk: Assessment and Mitigation through a Simplified Procedure
### Cost-Benefit Seismic Risk Mitigation Analysis  
Successively, the vulnerability index of the i-th building before making any intervention can be determined as follows  
\[\chi^{\text{i}}_{\text{pre-int}}=\frac{\lambda_{\text{SL}}}{\lambda^{\text{i}}_{ \text{pre-int}}} \tag{3}\]  
where \(\lambda_{\text{SL}}\) is the maximum allowable frequency of failure [26]. According to the obtained value of vulnerability indices, a decision measure has to be made for all the possible cases according to Table 1.  
The definition \"No intervention\" (N) indicates that the building is left as it is, so no retrofitting cost is needed. Any possible retrofitting option which makes the capacity of building to be at least equal to the demand (\(\chi=1.0\)) would satisfy the condition of full retrofitting (F). Alternatively, a lower level of target design (\(\chi=0.7\)) could be obtained by means of a partial retrofit (P), so that less of the retrofitting budget goes to any building in the portfolio. When the vulnerability index is very low (\(\chi_{\text{i}}\leq 0.2\)), buildings might be demolished and rebuilt (D) because they are considered as extremely risky (case 1). The new building would be designed and constructed according to the new building code, thus satisfying the minimum requirements of the seismic design criteria. Contrarily, when the vulnerability index is high enough (\(\chi_{\text{i}}\geq 0.8\)), no interventions (N) are programmed (case 4) [21]. Case 2 includes a vast range of vulnerability scenarios: in this case, three possible interventions are admitted, i.e., P, F or N. This allows decision makers to consider the circumstance in which very limited budget is available, thus requiring them to leave some buildings as they are, or to perform just a partial retrofit. When the vulnerability index is slightly lower than the admitted level (\(0.7\leq\chi_{\text{i}}<0.8\)), only full retrofit or no-intervention are possible decision strategies. Partial and full retrofitting cost is estimated as a partial amount of reconstruction cost, adopting specific cost models [27] as a function of the vulnerability index (the more vulnerable a building is, the larger the amount of money that needs to be paid to repair) and year of construction [19]. Finally, the post-retrofit seismic risk of each building is calculated assuming the probability of exceeding LS damage state of i-th building when it is decided to be repaired, either partially or fully, determined from the pre-intervention probability of failure multiplied by a scale factor [19]. The regional seismic risk in the post-retrofit scenario is the sum of the seismic risk associated to each building. The total cost due to seismic risk mitigation of the portfolio corresponds to the summation of the single intervention cost for each building. On the basis of the possible decision strategies to be assumed for each building belonging to the portfolio, several combinations are obtained depending on the number of buildings falling within case 1 to 4 according to Table 1. If \(n\) is the total number of buildings and letters \(a\) to \(d\) represent the number of buildings characterized by a vulnerability index belonging to the range of cases 1 to 4, respectively, the total number of combinations of the portfolio retrofit decisions would be Q = 1\({}^{\text{a}}\)\(\times\) 3\({}^{\text{b}}\)\(\times\) 2\({}^{\text{c}}\)\(\times\) 1\({}^{\text{d}}\). Each combination corresponds to a total retrofitting cost accounting for all the portfolio interventions and a total post-intervention seismic risk. This allows decision makers to construct a cost-benefit curve similar to the illustrative curve provided in Figure 3, where each dot indicator represents