## 4 Case Studies
### Reliability Uncertainty Analysis on an AEPS
#### 4.2.2 Calculation and Analysis
For example, compared to the other four solutions, solution 5, which improved the most components, had the poorest effect in system reliability improvement. That is, because in solution 5, though seven components were improved, the sum of the importance measure indicators of these components was only 0.23. In contrast, the importance measure index of both the component LG BUS and BTB was greater than 0.35, which definitely led to the improvement of solution 1 and 2 being better than solution 5, though only one component was improved in the former two solutions. To be more specific, if the designers want to improve the system with only one component, there are 11 solutions; if the designers want to improve the system with two components, there are \(C_{11}^{2}=55\) solutions; if the designers want to improve the system with three components, there are \(C_{11}^{3}=165\) solutions, and so on. In the face of such a huge number of optional solutions, if there is no importance measure index providing guidance, the formulation of improvement solutions in the system reliability design stage will inevitably become blind, and difficult to work. Obviously, as the number of components in the system increases, the problem becomes more difficult. In contrast, since the proposed importance measure index can correctly reflect the contribution of components to the uncertainty of system reliability, then according to the components' ranking of the index from high to low, the effect of components on the improvement in system reliability will decrease in turn. Based on this, the designer can quickly propose an effective optimization solution from the huge number of options. (2) Based on the ranking of the proposed importance measure indicators from high to low and the analysis in point (1), this paper summarizes the \"quantity and degree optimization criteria\" for system reliability design. Figure 6: Kernel density function of system reliability after different improvement solutions are applied. (**a**) Kernel density function of system reliability after solution 1 is applied; (**b**) Kernel density function of system reliability after solution 2 is applied; (**c**) Kernel density function of system reliability after solution 3 is applied; (**d**) Kernel density function of system reliability after solution 4 is applied; (**e**) Kernel density function of system reliability after solution 5 is applied. (**f**) The comparison of the kernel density function of system reliability of five solutions. The criterion is described as follows. (i) When the number of components that can be optimized/reselected is required to be \(m\) at most, the sum value \(w\%\) (\(\eta_{1}+\eta_{2}+\ldots+\eta_{m}=w\%\)) of the importance measure indicators of the first \(m\) components in the index ranking determines that the degree by which the system reliability uncertainty can be reduced. (ii) When the degree of system reliability uncertainty is required to be reduced by \(w\%\) at least, and if the sum of the importance measure indicators of the first \(m\) components in the ranking satisfies the inequality \(\eta_{1}+\eta_{2}+\ldots+\eta_{m}\geq w\%\) and \(\eta_{1}+\eta_{2}+\ldots+\eta_{m-1}\leq w\%\), then to meet the requirement above, at least \(m\) components are required to be optimized/reselected. This criterion is illustrated with the results obtained in this system. For criterion (i), if a maximum of three components are required to optimize the system, and the top three components in the index ranking are LG BUS, BTB1, and LGB, the sum value 0.8745 of the importance indicators of the three components determines that the uncertainty of system reliability can be reduced by 87.45% at most. For criterion (ii), if the uncertainty degree of system reliability is required to be reduced by at least 90%, as the importance index sum value of the first three components is 0.8745 and the importance index sum value of the first four components is 0.9715, four components at least should be improved to meet the design requirements.