*5.3. Comparisons*

To evaluate our algorithm in a different scenario than that of a random operator, in this Section we compare the results obtained by the algorithm that uses discretization by k-means, with the results published in [34,67]. In these works, the harmony search algorithm was used to optimize the buttress retaining wall. To evaluate the comparison, the best value will be analyzed for each of the different heights of the wall, in addition to comparing the distribution of the total solutions obtained for the different heights through violin plots. To determine that the difference is statistically significant, we will use the non-parametric Wilcoxon signed-rank test.

In Figures 11 and 12, the best values obtained for the k-means and HS algorithms are compared. In the comparison, all the parameters were kept fixed except for the height, which like the previous experiments, varied from 6 to 14 m. When analyzing the heights of 6 to 8 m in Figure 11, it is observed

that the cost of the solutions obtained by both algorithms is similar. As the wall grows in height, the curves have a greater separation, with the height of 14 (m) obtaining the greatest difference, this being 4.87% in favor of k-means. In the case of the comparison of the algorithms when optimizing the emissions of CO2, the curves behave similarly to that of the cost optimization case. For small values of wall height, very similar values are obtained. As the height of the wall increases, the quality of the k-means solutions improves compared to the HS. this is seen in Figure 12.

**Figure 11.** Comparison between the best solutions obtained by the k-means and HS algorithms in cost optimization.

**Figure 12.** Comparison between the best solutions obtained by the k-means and HS algorithms in emission optimization.

Figures 13 and 14 show the comparison of the total solutions obtained by both algorithms. In the case of the cost optimization shown in Figure 13, we see that the interquartile range of the solutions obtained by k-means has better performance than those of HS. Up to height 12, the dispersion of the solutions is kept quite small in both algorithms. From height 12 onwards, this dispersion increases considerably in both k-means and HS. The case of the optimization of emissions presents a similar behavior, increasing its dispersion considerably from height 13. The above is observed in Figure 14.

**Figure 13.** Violin plot comparison between k-means and HS results for cost in euros.

**Figure 14.** Violin plot comparison between k-means and HS results for CO2 emissions.

## **6. Conclusions**

In this article, a hybrid algorithm was proposed which uses the unsupervised k-means learning technique to construct discrete versions of optimization algorithms that work in continuous spaces. The optimization problem of a counterfort retaining wall was addressed, considering the cost and emission of CO2 as objective functions. The cuckoo search optimization algorithm was used to be discretized. Additionally, a random operator was constructed to determine the contribution of the k-mean operator in the optimization process. It was concluded that k-means produces better results than the random operator and in many cases this does it systematically, thus reducing the dispersion of the solutions. In addition, when we compare k-means with HS, we observe that as we increase the height, where the optimization problem becomes more difficult because it is more difficult to obtain stability of the wall with respect to overturning and sliding, k-means is more robust than HS reaching the height of 14 (m) at a difference of 4.76% in favor of k-means in optimizing emissions and 4.87% in minimizing costs. On the other hand, when we analyze the dispersion of the set of solutions, we see that k-means once again perform better than HS, especially for heights greater than 12 (m).

There are several possible directions for further extensions and improvements of the present work. The first line arises from observing the configuration parameters presented in Table 4. The configuration procedure can be simplified and improved by incorporating adaptive mechanisms that allow the parameters to be modified in accordance with the feedback obtained from the candidate solutions. A second line considers the incorporation of an intelligent agen<sup>t</sup> that uses value gradient policies or action methods frequently used in reinforcement learning, in order to have information on the performance of the optimization algorithms with which we could modify the parameters dynamically. Finally, another possible line of research is to explore the population managemen<sup>t</sup> of solutions dynamically. Through analyzing the history of exploration and exploitation of the search space, one can identify regions where it is necessary to increase the population and others where it is appropriate to decrease it.

**Author Contributions:** Conceptualization, V.Y., J.V.M. and J.G.; methodology, V.Y., J.V.M. and J.G.; software, J.V.M. and J.G.; validation, V.Y., J.V.M. and J.G.; formal analysis, J.G.; investigation, J.V.M. and J.G.; ; data curation, J.V.M.; writing—original draft preparation, J.G.; writing—review and editing, V.Y., J.V.M. and J.G.; funding acquisition, V.Y. and J.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** The first author was supported by the Grant CONICYT/FONDECYT/INICIACION/11180056, the other two authors were supported by the Spanish Ministry of Economy and Competitiveness, along with FEDER funding (Project: BIA2017-85098-R)

**Conflicts of Interest:** The authors declare no conflict of interest.
