## 5 Sensitivity Analysis in Capital Budgeting for Territorial Seismic Risk Mitigation
The number of buildings with no interventions evidently tends to decrease as the budget increases, in favor of the number of buildings with partial or full retrofitting programs. In particular, for lower values of the budget, the number of partial retrofitted buildings is higher than the number of fully retrofitted ones, while this condition is inverted from a certain threshold (around \(2.5\times 10^{7}\)\(\ell\)). Indeed, the increase in budget constantly makes for an increase in full retrofitting instead of partial retrofitting solutions. In addition, there is an indented trend on both curves that is explained by the philosophy within the framework, according to which is the priority to intervene first on those buildings with the highest vulnerability. For instance, in Figure 12a it may be observed that at a budget level of EUR 1.60 \(\times\)\(10^{7}\) eight buildings are partially retrofitted (yellow tag) and two buildings fully retrofitted (orange tag), while in Figure 12b, at the  
Figure 11: Sensitivity of number of buildings at each decision strategy and total risk reduction to the budget variation. next budget level (budget = EUR 1.64 \(\times\) 10\({}^{7}\)), nine buildings are partially retrofitted and one building is chosen for full retrofitting. Considering that an \"increase in the investment should make more risk reduction\" justifies the observation of a reduction in the number of fully retrofitted buildings and an increase in the number of partial retrofittings. Actually, buildings have different vulnerability indexes and decision making is made through different decision cases according to Table 1. Given a certain budget level, it may be more convenient for risk mitigation to invest in the partial retrofit of a very vulnerable building (i.e., belonging to case 2) instead of fully retrofitting a building in a less severe condition (i.e., of case 3). Furthermore, the former intervention may be more expensive than the latter considering the lower value of the corresponding vulnerability index. Therefore, in such a scenario, the budget increase may result in a higher number of partially retrofitted buildings with respect to the fully retrofitted ones. It is interesting to observe that the fully retrofitting solution in Figure 12b does not correspond to any of the two buildings tagged with the same retrofit intervention in Figure 12a. This is because buildings belonging to case 2 may equally be tagged as \"no intervention\", \"full retrofit\" or \"partial retrofit\". The final type of intervention indicated by the platform on the map results is the one that minimizes the territorial seismic risk within the budget limit. In addition, the cumulative number of buildings decided for each decision category is provided in Figure 13, where each colored area denotes the portion of budget range allocated for those specified number of buildings. The general trend is that first the budget goes for partial retrofitting. For instance, for a budget value of EUR 0.5 \(\times\) 10\({}^{7}\) there are three partial retrofitting solutions, one demolition and reconstruction, forty-two buildings with no interventions and no full retrofit solutions (Figure 14a). For a budget level of EUR 1.5 \(\times\) 10\({}^{7}\), the number of partial retrofit solutions rises to eight, while the budget is used for a full retrofit intervention for no buildings (Figure 14b). In a different way, when the budget is further increased to the value of EUR 2. 5 \(\times\) 10\({}^{7}\) (Figure 14c), the number of partial retrofit interventions decreases in favor of the number of full retrofit interventions, which in this case assume the same value (i.e., six buildings in partial retrofit and six buildings in full retrofit). This trend is confirmed when the maximum budget cost is achieved, which in this case corresponds to EUR 3.2 \(\times\) 10\({}^{7}\) (Figure 14d). Indeed, at this level, no partial retrofit solutions are planned, while a total of 14 buildings are to be fully retrofitted.