*4.6. Discussion*

Figure 3 shows the effect of the initialization method on the different optimizers applied over the selected datasets. The proposed bWOA-S and bWOA-V can reach the global optimal solution in almost half of the datasets, compared to the algorithms in all initialization methods. The limited search space in the case of binary algorithms explains the enhanced performance due to the balance between global and local searching. The balance between local and global searching assists the optimization algorithm to avoid early convergence and local optimal values. The small initialization keeps away the initial search agents from the optimal solution; however, in the large initialization, the search agents are closest to the optimal solution, although they have low diversity. While the mixed initialization method improves the performance of all compared algorithms, the two proposed algorithms are superior even in a high-dimensional dataset as in Table 9.

**Table 5.** Statistical mean fitness measure calculated on the different datasets for the compared algorithms using large initialization.


**Table 6.** Average classification accuracy on the different datasets for the compared algorithms using large initialization.


The standard deviation in the obtained fitness values on the different datasets for the compared algorithms averaged over the initialization methods is given in Table 10. As shown in this table, the proposed bWOA-V can reach the optimal solution better than compared algorithms, regardless of the initialization used.

With regard to the time consumption for optimization of these 11 test datasets, Table 11 presents the results of the average time obtained by the two proposed versions and other compared algorithms with 20 independent runs. As can be concluded from Table 11, bWOA-V ranks first among the algorithms. bWOA-S ranks fifth, but it is better than PSO and bALO, as it significantly outperforms the other compared algorithms with a little more time consumption.


**Table 7.** Statistical mean fitness measure calculated on the different datasets for the compared algorithms using mixed initialization.

**Table 8.** Average classification accuracy on the different datasets for the compared algorithms using mixed initialization.



**Table 9.** Results for high dimensional datasets.

**Table 10.** Standard deviation fitness function on the different datasets averaged for the compared algorithms over the three initialization methods.


**Figure 3.** Statistical mean fitness averaged on the different datasets for the different optimizers using the different initializers.



On the other hand, Tables 12 and 13 summarize the experimental results of the best and worst obtained fitness for the compared algorithms over 20 independent runs.

The mean selected features obtained from the compared algorithms are shown in Table 14.

Table 14 reports the ratio of mean selected features obtained from the compared algorithms. In Table 14, the performance of bWOA-V is superior in keeping its good classification accuracy by selecting a lower number of features.

This reveals the outstanding performance of bWOA-V in searching for both features' reduction and enhancing the optimization process.


**Table 12.** Best fitness function on the different datasets averaged for the compared algorithms over the three initialization methods.

**Table 13.** Worst fitness function on the different datasets averaged for the compared algorithms over the three initialization methods.


In order to compare each runs results, a non-parametric statistical called Wilcoxon's rank sum (WRS) test was carried out over the 11 UCI datasets at 5% significance level, and the *p*-values are given in Table 15. From this table, *p*-values for the bWOA-V are mostly less than 0.05, which proves that this algorithm's superiority is statistically significant. This means that bWOA-V exhibits a statistically superior performance compared to the other compared algorithms in the pair-wise Wilcoxon signed-ranks test.


**Table 14.** Average selection size on the different datasets averaged for the compared algorithms over the three initialization methods.

**Table 15.** The Wilcoxon test for the average fitness obtained by the compared algorithms.


Moreover, Figure 4 outlines the best and worst acquired fitness function value averaged over all the datasets, using small, mixed and large initialization. Figure 5 shows the classification accuracy average. From these figures, it can be proven that the bWOA-V performs better than other compared algorithms, such as PSO and bALO, which confirms bWOA-V's searching capability, especially in the large initialization.

In order to show the merits of bWOA-S and bWOA-V qualitatively, Figures 6–8, show the boxplots results for the three initialization methods obtained by all compared algorithms. According to these figures, bWOA-S and bWOA-V have superiority since the boxplot of bWOA-S and bWOA-V are extremely narrow and located under the minima of PSO, bALO, and the original WOA. In summary, the qualitative results prove that the two proposed algorithms are able to provide remarkable convergence and coverage ability in solving FS problems. Another fact worth mentioning here is that the boxplots show that bALO and PSO algorithms provide poor performance.

(**a**) Random initialization

(**b**) Small initialization

(**d**) Large initialization

initialization

(**c**) Mixed

(**b**) Small initialization

(**d**) Large initialization

**Figure 5.** Average classification accuracy and average selection size obtained on the different datasets averaged for the compared algorithms over the three initialization methods.

**Figure 6.** Small initialization boxplot for the compared algorithms on the different datasets.

**Figure 7.** Mixed initialization boxplot for the compared algorithms on the different datasets.

**Figure 8.** Large initialization boxplot for the compared algorithms on the different datasets.

#### **5. Conclusions and Future Work**

In this paper, two binary version of the original whale optimization algorithm (WOA), called bWOA-S and bWOA-V, have been proposed to solve the FS problem. To convert the original version of WOA to a binary version, S-shaped and V-shaped transfer functions are employed. In order to investigate the performance of the two proposed algorithms, the experiments employ 24 benchmark datasets from the UCI repository and eight evaluation criteria to assess different aspects of the compared algorithms.The experimental results revealed that the two proposed algorithms achieved superior results compared to the three well-known algorithms, namely PSO, bALO (three variants), and the original WOA. Furthermore, the results proved that bWOA-S and bWOA-V both achieved smallest number of selected features with best classification accuracy in a minimum time. In addition, the Wilcoxon's rank-sum nonparametric statistical test was carried out at 5% significance level to judge whether the results of the two proposed algorithms differ from the best results of the other compared algorithms in a statistically significant way. More specifically, the results proved that the bWOA-s and bWOA-V have merit among binary optimization algorithms. For future work, the two binary algorithms introduced here will be applied to high-dimensional real-world applications and will be used with more common classifiers such as SVM and ANN to verify the performance. The effects of different transfer functions on the performance of the two proposed algorithms are also worth investigating. This algorithm can be applied for many problems other than FS. We can also investigate a multi-objective version.

**Author Contributions:** A.G.H.: Software, Resources, Writing—original draft, editing. D.O.: Conceptualization, Data curation, Resources, Writing—review and editing. E.H.H.: Supervision, Methodology, Conceptualization, Formal analysis, Writing—review and editing. A.A.J.: Formal analysis, Writing—review and editing. X.Y.: Formal analysis, Writing—review and editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external fundin.

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