## 5 Conclusions  
Two reliability methods based on simulations are revisited. One method fits normality polynomials to the simulated data with fractile constraints, and the other approximates the linearized limit state surface at the design point using multi-linear regression; for the latter, a slight modification is proposed. Three limit state functions, a very simple one, other for a structural engineering application and another for a coastal engineering application, are employed. The most relevant findings of this study are that for the normality polynomial approach, a power law was found to adequately represent the mean and standard deviation of the error in the estimated reliability index as a function of the number of simulations. It could be used as an aid for decision makers to select a precision level (quantitatively) associated to a selected reliability index, thus reducing the number of required simulations by expressively accepting an error level. Additionally, it is found that the multi-linear regression approach is an excellent option to obtained accurate reliability levels, although a sufficiently large number of simulations is required (not prohibitive though). It also has the advantage that the design point and sensitivity factors are determined. Other findings in this study are:  
1. When the normality polynomial approach is used, the reliability index is dependent on the generated random numbers of each run, but it becomes stable for a large number of simulations. 2. The reliability index cannot always be determined with the Monte Carlo simulations, while the opposite occurs when normality polynomials are used, although significant deviations from the exact value are observed for small numbers of simulations. In general, for an intermediate number of simulations (e.g., \(1\times 10^{3}\)), the fitted normality polynomials lead to a better estimate of the reliability index than the Monte Carlo simulations. 3. When the mean relative error and its standard deviation are computed for the reliability index (compared to the exact value), for decreasing number of simulations the Monte Carlo simulation approach tend to larger mean errors and standard deviations than the normality polynomial approach. 4. 3 rd-order normality polynomials were mostly used; when 4 th order ones are used, the fitting leads to comparable results. 5. When the multi-linear regression approach is considered, a minimum number of simulations is required for successfully performing the regression (in the order of \(10^{4}\) to \(10^{5}\)simulations), but once this is ensured, a very precise reliability index is obtained (more precise than by using the first order reliability method (FORM)), and the design point and sensitivity factors are also determined and in good agreement with those determined with the FORM. 6. If the tolerance for the multi-linear regression approach is increased (i.e., if a wider range in the nearest of the failure surface is stipulated to gather the vectors of simulated data), the number of simulations can be reduced. **Author Contributions:** Conceptualization, A.-D.G.-S., F.C.-V., C.M., [PERSON]-G.V.-V. and A.H.-M.; Formal analysis, A.-D.G.-S., F.C.-V., C.M., J.-G.V.-V. and A.H.-M.; Funding acquisition, C.M., A.-D.G.-S.; Investigation, A.-D.G.-S., F.C.-V., C.M., [PERSON]-G.V.-V. and A.H.-M.; Methodology, A.-D.G.-S., F.C.-V., C.M., J.-G.V.-V. and A.H.-M.; Project administration, C.M. ; Supervision, A.-D.G.-S. and C.M. All authors have read and agreed to the published version of the manuscript. **Funding:** The financial support from the Erasmus Mundus Coastal and Marine Engineering and Management (CoMEM) programme for one of the authors of this study and from Universidad de Guanajuato (Division de Ingenierias and Campus Guanajuato) is gratefully acknowledged. **Acknowledgments:** We thank Laboratori d'Enginyeria Maritima, Universitat Politecnica de Catalunya. We are also very thankful to [PERSON] and \(\mathcal{O}\).