## 5 Sensitivity Analysis in Capital Budgeting for Territorial Seismic Risk Mitigation  
The budget allocation for the retrofitting of portfolio RC buildings shown above was performed by assuming a hypothetical budget value. The choice of intervention for each single building is a function of the budget limit when the minimum risk of optimum solution is chosen according to the constraint. Thus, changing the budget value may change the decision of intervention. Actually, for this type of decision-making problem, the budget is assumed to be known. However, in some projects spending a partial amount of budget might be more appropriate than the whole. Hence, that would be a complementary analysis of the budget allocation to see if spending a different amount than the budget might change the final decision remarkably or not. The sensitivity analysis reported here was not made to find out which parameters had more impact on the final decision but to figure out how just one parameter (i.e., budget) could affect the final decision. The electronic platform can be of great use to achieve this goal. The analysis was conducted by changing the budget level in different discretized values from the minimum total cost to maximum total cost value. The former corresponds to the needed cost to achieve a minimum global seismic risk mitigation. Conversely, the maximum total cost is the investment allowing to upgrade seismic capacity of all buildings up to the full satisfaction, that in this framework was set to be the achievement of a vulnerability index \(\chi_{i}\geq 0.8\). Indeed, in this case, no further interventions were suggested to furtherly reduce seismic risk, matching the Italian national requirements [21]. Two decision parameters were considered to observe the effect of budget variation on the final decision: (1) number of buildings involved in a decision strategy by spending  
Figure 10: Optimal intervention strategy for a budget of \(2.5\times 10^{7}\)\(\epsilon\). Figure 9: Optimal budget allocation for an investment of \(2.5\times 10^{7}\)\(\epsilon\). any amount of budget rate and (2) total risk reduction achieved in comparison to the initial total risk level. The effect of budget variation on the two selected decision parameters for the case study presented in the previous section is shown in Figure 11 starting from zero budget to the maximum possible investment for the project. The variation in the number of buildings involved in each of the four decision strategies and the corresponding seismic risk reduction due to the amount of investment are depicted. If the budget is lower than the minimum total cost, all the buildings are in the condition of no intervention. In the case of a very limited budget, at least equal to the minimum total cost, it is not possible to intervene in all the buildings. Then, retrofit actions would be the first preferred option for the most vulnerable buildings (\(\chi\leq 0.2\)), to be sorted according to the lowest vulnerability indices corresponding to the most urgent buildings for intervention. In this case study, there was only one building needing to be demolished and rebuilt with a minimum cost of EUR 2.4 million, while the other 45 buildings were left in the condition of status quo. In this scenario a seismic risk reduction of 9.56% was achieved. In correspondence with the maximum total cost, the risk reduction could achieve the maximum value of 47.2%. This limit value is influenced by the choice, made in the framework, to stop the retrofit interventions on the buildings when their vulnerability indexes achieve the value of 0.8. This means that further increases in the investment would not contribute to furtherly reduce the seismic risk, provided that the decision would always result in \"no intervention\". Spending any budget amount in between the minimum and the maximum total cost allows intervention on a specific number of buildings with any of the presumed intervention strategies.