# Crossover Operator Inspired by the Selection Operator
## 3 Experiment and Results Discussion
### Comparative Analysis  
After determining the optimal parameter values for the example task, a comparison of the obtained total sequence costs was carried out for the algorithm using the new crossover method and algorithms using previously developed crossover methods (Single-Point Crossover [14; 15], PMX [22], OX [23], and SCX [40]). In order to facilitate the analysis of the results, an identical stopping condition was adopted for all comparable crossover operators--reaching 1000 generations, and after such a number of iterations, the obtained values of the minimized criteria (Equation (7)) were compared. A constant population size was used--200 individuals. In a single crossover operation, two individuals participate and the way genes are exchanged between them determines the efficiency of the crossover operator. For this reason, it seems that comparative studies can be based on the constant population size. However, the chromosome length (the number of optimized tasks) can have an influence on the crossover efficiency. Therefore, the crossover operators were compared for three sequence lengths (tasks 250, 500, and 1000). These sizes are significantly larger compared with the size for which the metaparameter values were determined to confirm their scalability. Various degrees of complexity in the shape of solutions hypersurface are provided by using three different cost definitions (based on the Euclidean distance, the square of this distance, and the Manhattan distance). As can be seen in the results presented in Figures 11-13 and Tables 1-3 for all tested sequence lengths (250, 500 and 1000 tasks) and for the three definitions of the transition cost between tasks, the new crossover method provides a significant advantage over previously developed sequence crossover methods. Interestingly and worth emphasizing, these four reference crossover operators achieve a very similar level of the minimized criterion in their group. Against this background, the new crossover operator distinguishes itself very clearly for each cost definition and seems to introduce a significant improvement. In our opinion, the experimental conditions did not favor the new operator in any way. For different cost definitions and problem sizes, the new crossover operator maintains its advantage. This suggests that the developed operator represents a new and more effective class of crossover operators. In order to confirm the statistical significance of the reduction in the mean criterion value obtained for the new crossover in comparison to the previous crossover methods (shown in Tables 1-3), the \(t\)-test was performed. For an independent two-sample \(t\)-test, the variances of the two groups should be approximately equal. Therefore, this test was preceded by an F-test. For both tests, the populations from which the samples are drawn should be normally distributed. This condition was confirmed for all samples by the Shapiro-Wilk test. The results of which are shown in Tables 4-6. The analysis was carried out for the four literature crossover methods compared with the new proposed method. A significance level of \(\alpha=0.01\) was adopted for both tests. If the \(p\)-value is less than the accepted value of \(\alpha\), the null hypothesis (\(H_{0}\)) should be rejected in favor of the alternative hypothesis (\(H_{1}\)). In the case of the F-test, the null hypothesis assumes equality of variances, and in the case of the \(t\)-test, equality of means. For each variant, the equality of variances and a statistically significant change in the mean were confirmed.