# Crossover Operator Inspired by the Selection Operator
## 3 Experiment and Results Discussion
### Comparative Analysis

Table 5: F-test and \(t\)-test confirming a statistically significant reduction in the mean value of the optimization criterion (total cost) achieved after 1000 generations for the 500-task problem and the percentage value of the criterion reduction for the new crossover compared with the previous methods with confidence intervals for a confidence level of 0.99. The crossover operators compared in this work also differ in computational complexity. Completely random operators like PMX, OX, and Single-Point Crossover are the fastest, as shown in Figure 18, because they do not take into account the cost of connections between tasks during selecting crossover points and they do not have to analyze all connections. In contrast, the SCX operator and the new probabilistic method presented in this work are slower than completely random operators. This is because they require calculating the distance between all tasks in individuals before deciding on the location of the intersection points of the chain. The SCX method is slower than the probabilistic operator presented in this work because it accurately compares the distance from the current task to the following task in both parents when generating an offspring. The analysis of the optimization progress presented in Figures 19-21 turns out to be interesting. The proposed crossover method gains an advantage over the compared crossover methods already at the beginning of the optimization. This compensates for the longer duration of the new crossover. In the studies discussed above, a constant population size of 200 individuals was assumed. To eliminate the possibility that the population size influenced the advantage of the new crossover operator, a supplementary analysis was conducted to assess the impact of population size on the results obtained for the various operators in the example variant. As shown in Figure 22, the new operator maintains its advantage across the entire population size range examined. Notably, only the new method demonstrates some improvement in efficiency with an increase in population size. Figure 17: Effectiveness of the crossover operator for the 250-task problem with discrete parameters (five levels of values). The mean criterion value (three definitions of cost: Euclidean, Euclidean\({}^{2}\), and Manhattan) of the best individual achieved after 1000 generations. Figure 18: Comparison of execution time of 200 generations for the 500-task problem. Figure 19: Change in the criterion value over time for the 500-task problem (cost based on Euclidean distance) over 1000 generations. Figure 20: Change in the criterion value over time for the 500-task problem (cost based on Euclidean distance) over 1000 generations. For the other methods, the number of generations used ensures that they reach their respective local minima, and increasing the population size does not improve their solutions. In contrast, the new crossover method is able to benefit from a larger population size, enabling it to improve its solution by sampling more of the solution space within a single generation. However, this improvement comes at the cost of longer computation times per generation.