# Crossover Operator Inspired by the Selection Operator
## 3 Experiment and Results Discussion
### Optimization of Algorithm Metaparameters  
The proposed crossover method uses two metaparameters: the length of a transferred fragment and the selection strength coefficient \(a\) used in the proposed fragment selection method. To optimize these parameters, we conducted preliminary research before the comparative analysis. The criterion of the best individual (defined in Equation (7)) reaches its minimum (best) value at a high value (30) of the \(a\) coefficient (Figure 8). It strongly favors sequence fragments with the lowest (for the donor) and highest (for the recipient) internal cost. However, the randomness of the crossover operator's operation is still preserved, distinguishing it from the deterministic operator. Figure 9 shows an analysis of the influence of the fragment length on the obtained minimum value for the example of the sequencing problem of 100 tasks. As can be seen, the optimal fragment lengths for different cost functions (different sequencing problem definitions) are similar, although they differ slightly. The fragment being selected must be long enough to transfer significant, valuable information to the second individual. However, as the length of the fragments increases, the means (8) of the connections within them begin to converge. \[\bar{C}_{j}=\frac{1}{M-1}\sum_{i=1+j}^{M-1+j}c(\overline{\mathbf{P}}_{i}, \overline{\mathbf{P}}_{i+1}) \tag{8}\]  
where \(\bar{C}_{j}\) is the average value of connections for the \(j\)-th fragment. Figure 8: Preliminary impact analysis of metaparameter \(a\) on the effectiveness of the new crossover method (the criterion value of the best individual) for sequencing 100 tasks to determine its optimal value. The determined value (red dot) for the minimum was used in the main comparative analysis of the new operator and the existing operators. At this point, the probabilities (3) and (4) for selecting individual fragments also converge. Consequently, the selection pressure decreases, and the process effectively becomes a random drawing of fragments. As a result, the partially directed nature of the new crossover operator, its basic feature, disappears. In general, the standard deviation of the mean connections in fragments, \(\bar{C}_{j}\), is inversely proportional to the square root of the fragment length:  
\[\text{std}(\bar{C}_{j})\propto\frac{1}{\sqrt{M-1}} \tag{9}\]  
The given formula is only an estimation, assuming that the cost values between tasks are independent. In the presence of some correlation between these costs, the value will differ to some extent, but the general trend will be preserved. The exact rate at which the connections between the fragments analyzed in the selection process become too similar (a function of M in the denominator) depends on the specificity and complexity of the problem, expressed by functions of the costs of connections between tasks, and has no relation to the global number of tasks. Figure 9: Preliminary impact analysis of exchanged sequence fragment length on the effectiveness of the new crossover method (the criterion value of the best individual) for sequencing 100 tasks to determine its optimal value for different cost definitions: Euclidean (**A**), Euclidean2 (**B**), and Manhattan (**C**). The determined value (red dot) for the minimum was used in the main comparative analysis of the new operator and the existing operators. To confirm this theory, an analysis was performed on the influence of the length of the optimized sequence on the optimal length of the transmitted fragment (Figure 10) for different cost definitions (different problems). A stabilized value for the optimal fragment length is visible. Some fluctuations are due only to the limited number of samples (10 experiments) performed for a given fragment length. Based on these analyses, a method for selecting the fragment length metaparameter can be proposed. The procedure is to determine metaparameters for an arbitrary number of tasks, although preferably similar to the scale of the number of tasks that will occur in the target problem size.