## 5 Illustrative Example
### Performance of Mitigation Strategies  
Under a given \(m\), BC-based, RE-based, and MI-based strategies select the bridges one by one according to their importance ranking, and the JMI-based strategy is identified as the bridge set with the highest joint importance value. The most direct way to measure the performance of different mitigation strategies is to compare their effects on reducing the failure probability of the network. The failure probabilities based on the four importance ranking approaches as well as the possible failure probability interval calculated by enumeration are given in Figure 5 under different conditions. It can be seen from the figure that the results of BC-based and RE-based strategies have an obvious gap with the minimum failure probabilities (i.e., optimal solutions), especially in Case 3 and Case 4 with different bridge vulnerabilities. In contrast, both MI-based and JMI-based strategies show good risk mitigation performance in all cases, and the JMI-based strategy performs even better in some situations as enlarged in Figure 5. Indeed, the JMI-based approach can be taken as a revision of the MI-based ranking by considering the second-order interactive effect between the selected bridges and the candidate bridges through Equation (9). For example, under the problem of \(m=2\) in Case 2, the MI-based approach selects B9 and B1 according to Figure 4b, however, the JMI-based approach selects B9 and B5 because they have better complementary effect, which is reflected in \(MI(\lambda_{5};\varphi|\lambda_{9})=0.055>MI(\lambda_{1};\varphi|\lambda_{9}) =0.053\). Similarly, under the problem of \(m=6\) in Case 3, the JMI-based approach selects [B1, B3, B5, B7, B8, B9] instead of [B1, B2, B3, B5, B8, B9] because \(MI(\lambda_{7};\varphi|\{\lambda_{1},\lambda_{3},\ \lambda_{5},\lambda_{8},\lambda_{9}\})=0.098>MI( \lambda_{2};\varphi|\{\lambda_{1},\lambda_{3},\ \lambda_{5},\lambda_{8},\lambda_{9}\})=0.081\). Table 4 lists the sum of the failure probability gaps between each strategy and the optimal solution, where the total deviation of the MI-based strategy is less than 2% and that of the JMI-based approach is less than 1% in each case.  
Figure 5: Network failure probability under different approaches and conditions. (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 4.  
In addition to the selection of retrofitted bridges, a reasonable budget is also a key concern to decision makers. However, the decision making on budget is often a more complex problem and requires more computation. Taking the problem in this paper as an example, investigating the optimal budget needs to estimate the minimum network failure probability for each value of \(m\). This computational burden is terrible for the resource-constrained decision models that are widely proposed. However, the method presented in this paper could provide an easy access to the problem. Specifically, the JMI values of all \(m\) values can be calculated based on the same sample set, and thus generate little extra computation. Considering that the failure probability gap between the JMI-based strategy and the optimal solution is quite small, the investment benefit of a budget can be approximately estimated by JMI analysis combined with MCS within a low computation cost, because only one MCS is needed for the JMI-based strategy per \(m\) value, instead of solving the optimization problem at each \(m\) value. With the JMI-based curve as shown in Figure 5, the decision makers could know the relationship between the value of \(m\) and the mitigation benefit in advance, so as to make a better investment decision.