**4. Conclusion**

Inspired by SORA's method, a novel sequential RBDO method is proposed which integrates the shifting constraint's strategy with quantile-based probability of failure estimation instead of MPP. Searching for global MPP is problematic when being performed in the random design space. Hence, in the proposed method, the RBDO structure is disintegrated in the probability space instead. This quantile-based algorithm uses a combination of different techniques to ensure a better result is obtained in each iteration so that wherever the problem is paused, it ensures us that the current design is more reliable than the previously generated ones. Once the deterministic optimization problem is solved, the failure probability estimation process in the probability space begins which is based on the quantile calculation with sampling. To enhance the computational efficiency in sampling, in the first level, the samples are generated adaptively. Then, the second-level sampling is done focusing on the failure region with the knowledge obtained from first-level sampling. Furthermore, the samples of previous iterations are reused to build a mixture distribution for which most of the samples are distributed in the failure region to provide an accurate estimate of the quantile. After the quantile is calculated, with an efficient adaptive step size for the optimal shifting value, the algorithm is able to quickly converge to the target failure probability. We explored the efficiency of the proposed method through three benchmark problems. The results show that the proposed method will decrease the computational cost to less than half of what is reported about other existing methods in very small target probabilities (*P<sup>t</sup> <sup>f</sup>* ¼ 0*:*001) and down to 9% in mild target probabilities (*Pt <sup>f</sup>* <sup>¼</sup> <sup>0</sup>*:*1 and *<sup>P</sup><sup>t</sup> <sup>f</sup>* ¼ 0*:*01)). It is also able to provide promising results both for highly nonlinear problems and problems with mixture of normal–nonnormal variables as well as random-deterministic variables. As an extension to this work, the same framework will be combined with the IS concept to reach a potentially more efficient algorithm for smaller failure probabilities, and we will explore it in the future works.
