# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 5 Conclusions  
In this paper, a method is proposed to handle the reliability problems under multiple failure models with an adaptive MRGP model. An RMQGS method is first proposed to optimize the uniformity and randomness of initial samples to improve the accuracy of the initial MRGP model. Two novel learning functions for describing the different iterative cycles are presented to build the adaptive MRGP model. The cumulative probability under multiple failure models is estimated by the MRGP-SS method with a few performance function calls. The main conclusions can be summarized as follows:  
(1) Compared with LHS and RGS methods, the sample points generated by the proposed RMQGS method can be evenly and randomly distributed in the entire sample space, which effectively avoids the problem of sample points falling on the design space boundary and duplication;  
(2) Two numerical examples illustrate the effectiveness of the proposed method in handling multiple failure modes in series and parallel relationships. For the series system, the proposed method improves the efficiency of MRGP + U, MRGP + H, and MRGP + EFF by 30%, 110%, and 44.4%, respectively, under the condition of similar calculation accuracy. For the series system, the proposed method can give both the accuracy and efficiency of calculation compared with MRGP + U, MRGP + H, and MRGP + EFF, and obtain better failure probability estimation with the least number of complex performance function calls;  
(3) The reliability analysis of the motor hanger and LEEX under multiple failure modes verifies the engineering application value of the proposed method. The results show that the reliability of the motor hanger under two series failure modes is 0.9969, and the error is only 3.1% compared with the MC simulation method, but the efficiency is significantly improved. The reliability of LEEX under three parallel failure modes is 0.9976, which is the same as the calculation result of MRGP + H and MRGP + EFF, but the efficiency is improved by 51.53% and 32.76%, respectively.  
Figure 10: Reliability analysis results of LEEX under multiple failure modes. (**a**) CDF curve of IEEX; (**b**) Iterative curve of CDF.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(N_{\text{calls}}\) & \(P\)/\(10^{-3}\) & \(\beta\) & \(\delta p\)/\% \\ \hline MC simulation & \(10^{6}\) & 2.5 & 2.8070 & — \\ Proposed & 78 & 2.4 & 2.8202 & 0.4681 \\ method & 99 & 2.3 & 2.8338 & 0.9548 \\ MRGP + H & 161 & 2.4 & 2.8202 & 0.4681 \\ MRGP + EFF & 116 & 2.4 & 2.8202 & 0.4681 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Results of example 4.5 with different methods.  
**Author Contributions:** Conceptualization, P.Z. and Z.W.; methodology, P.Z.; software, G.Y.; validation, Z.M., and P.S.; formal analysis, J.W.; investigation, P.Z.; resources, X.G.; data curation, X.G.; writing--original draft preparation, P.Z.; writing--review and editing, Z.W.; visualization, Z.W., and Z.M.; supervision, Z.W.; project administration, G.Y.; funding acquisition, P.Z. and Z.W. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by Sichuan Science and Technology Program (grant No. 2020 JDJQ0036), Guangdong Basic and Applied Basic Research Foundation (grant No. 2021A1515110308), Natural Science Foundation of Sichuan Province (grant No. 2022 NSFSC1941) and Sichuan Province Engineering Technology Research Center of General Aircraft Maintenance (grant No. GAMRC2021 YB03).  
**Institutional Review Board Statement:** Not applicable.  
**Infrared Consent Statement:** Not applicable.  
**Data Availability Statement:** The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.  
**Conflicts of Interest:** The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.