# A Cellular Automata-Based Crossover Operator for Binary Chromosome Population Genetic Algorithms
## 4 Genetic Algorithms
### Crossover Methods Used by GA  
Crossover operators are used to generate offspring chromosomes by recombining selected individuals from the current population to form the new generation. Among the most used crossover operators are single-point, two-point, k-point, uniform, partially matched, order, precedence preserving crossover, shuffle, reduced surrogate, and cycle. Single-point crossing determines the offspring by taking the values of the genes from one parent up to the crossing point and the genes from the other parent after the crossing point position. Application examples for one-point crossover are shown in Figure 9, where the gene values are binary values, and in Figure 10, where the gene values are real values. Figure 8: The general structure of a GA, which includes the stages of selecting chromosomes from the current population, recombining the selected chromosomes, carrying out some mutations on the resulting chromosomes, and evaluating the quality of the new chromosomes through the fitness function that also determines the component of the new generation of chromosomes of the algorithm. Figure 9: Single-point crossover for two chromosome parents with genes with binary values and having the crossover point in position 3. In a two-point and k-point crossing, two or more random crossing points are selected, and the offspring are obtained by taking the genetic information from each parent by swiping in the crossing points. Application examples for k-point crossover (\(k\in\mathbb{N}^{*}\), \(k=3\)) are shown in Figure 11, where the gene values are binary values, and in Figure 12, where the gene values are real values. Uniform crossover is achieved by taking genes from the first or second parent evenly based on a uniform random parameter, for each gene, that decides which parent the gene is taken. Application examples for uniform crossover (the uniform parameter establishes the modification of the genes on positions 4 and 7) are shown in Figure 13, where the gene values are binary values, and in Figure 14, where the gene values are real values. Figure 11: k-point crossover for two chromosome parents with genes with binary values and having the crossover points in position 3, 5, and 7; k = 3. Figure 12: k-point crossover for two chromosome parents with genes with real values and having the crossover points in position 3, 5, and 7; k = 3. Partially Mapped Crossover (PMX) was proposed by [PERSON] and [PERSON] and is one of the most used operators in genetic algorithm applications [43]. The working method for this operator includes several stages: two parents are selected for recombination, the values of the two cutting positions are determined that determine the mapping of the values from one parent to the other, and analysis of the mapping rules and their application to the gene areas outside the mapping area at the level of the two parents is conducted. In the end, two descendants are obtained that in the cutting area keep the genes of one of the parents, and outside the cutting area, certain values are updated based on the mapping between the two parents. An example regarding the steps mentioned for the PMX crossover is shown in Figure 15 by considering the cut positions equal to 3 and 6 for two parent chromosomes with nine genes. Figure 14: The uniform crossover for two chromosome parents with genes with real values and having the uniform parameter updating the genes at positions 4 and 7. Figure 13: The uniform crossover for two chromosome parents with genes with binary values and having the uniform parameter updating the genes at positions 4 and 7. Order Crossover (OX) is another crossover method that is based on the establishment of cutoff points that induce the preservation of the values of the genes from one parent in the respective intervals, and the rest of the genes are completed with the values from the other parent, from which the values taken are already excluded from the first parent [44].