## 3 Results
### Risk Determination Based on RUL Prediction
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & **Sensor** & **Sensor** & **Sensor** & **Sensor** & **Sensor** & **Sensor** & **Sensor** & **Sensor** & **Sensor** & **Sensor** \\  & **2** & **3** & **4** & **7** & **8** & **11** & **12** & **13** & **15** & **17** & **20** & **21** \\ \hline VC & 0.0567 & 0.2760 & 0.5110 & 0.1239 & 0.0020 & 0.4573 & 0.1117 & 0.0020 & 0.3408 & 0.2907 & 0.3519 & 0.3514 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Variation coefficients (VC) of the rest 12 variables. Figure 6: The functional modules and their connection. maximizing the value and minimize the risk of system portfolios. Therefore, a multi-objective algorithm is employed to solving the optimization problem. The non-dominated sorting generic algorithm (NSGA) is a kind of widely used multi-objective algorithm, which exhibits a good performance for retaining elites in offspring. On the other side, the differential evolution (DE) is a nice genetic operator, which plays really well on keeping population diversity. Thus, the paper uses the non-dominated differential evolution (NSDE) algorithm, which fuses the two advantages of NSGA and DE, to solve the system portfolio optimization problem. The corresponding parameters are set as follows. The population size is Pop = 100, the number of iterations is Gen = 1000, the mutation probability is 0.01 and the crossover probability is 0.2. Due to the certain randomization of all genetic algorithms, The NSDE also generates results with certain fluctuant. A typical method to guarantee the optimality of generated result is running the algorithm for multiple times, and then select the best individuals by comparing the corresponding multiple results. In the case, the program is iterated for 10 times to generate 10 Pareto results with each containing 200 individuals, shown in (a) of Figure 8. Then, the 10 sets of Pareto results are combined together to obtain the best 200 individuals among them, as shown in (b) of Figure 8.  
In detail, the 200 individuals of the best Pareto set are shown in the system option diagram in Figure 9. The rectangular area is divided into \(100\times 200\) rectangles according to the number of system candidates and the dimension of the non-dominated weapon system portfolios. Each rectangle represents whether a system candidate is selected in the Pareto set. If a weapon system \(i\) is selected by the \(j\) th non-dominated system portfolio, the \(i\) th row and \(j\) th column rectangular block will be colored black, otherwise, it is left blank. From Figure 10, it can be seen that some systems are frequently selected in the Pareto set. However, some systems are seldom selected or even never selected. To compare the importance degree of different systems, the frequencies for all systems of being selected in the Pareto set is counted. Systems of S6, S9, S15, S16, S22, S25, S32, S39, S50, S52, S65, S69, S70, S71, S72, S73, S75, S79, S83, S90, S98, and S99 are selected by at least one system portfolio in the Pareto set. In addition, the systems S9, S25, S50 are quite important according to the high selected numbers in the Pareto set. Further, the rank of systems according to selected numbers is: S25 > S9 > S50 > S39 > S69 > S75 > S83 > S22 > S71 > S32 > S73 > S16 > S98 > S72 > S15 > S6 > S65 > S99 > S70 > S79 > S52 > S90, which to some extend indicates the importance degrees of selected systems. As regarding to the rest systems, they can be directly neglected in the system portfolio selection process. Figure 8: Result exhibition: (**a**) Result of 10 times of running; and (**b**) result of 200 best individuals. By deeper analysis, it can be discovered that some systems tend to always be selected together. Therefore, a frequent item set mining algorithm of Apriori is applied to identify the association rules, shown in Table 6. The support parameter indicates the ratio between the simultaneously appearing frequency and all items, which means the probability of appearing simultaneously.