## 2 Materials and Methods
### Dynamic Selection Adaptive Genetic Algorithm (DSAGA)
#### 2.3.4 Crossover Operation  
The crossover operation, a fundamental element of genetic algorithms, mirrors the genetic exchange found in biological evolution [40]. Its principal function is to utilize gene selection from multiple parent individuals to produce offspring with unique combinations of traits, potentially enhancing their characteristics by inheriting the best attributes of their parents.  
Among the crossover techniques available in genetic algorithms, single-point crossover, multi-point crossover, uniform crossover [41], and PMX (partially mapped crossover) [42] operation have gained prominence. Employing a single-point crossover technique requires modifications to the algorithm due to its potential effects [43]. In single-point crossover, a randomly selected position divides two parent chromosomes into two segments, and offspring are generated by exchanging fragments from the parents. This approach could disrupt advantageous gene combinations on the chromosomes. Within the scheduling problems, this could lead the algorithm to local optima, resulting in premature convergence [44]. On the other hand, multi-point crossover is a genetic procedure that divides the chromosome into several segments by randomly choosing multiple sites, thereby facilitating information exchange between these sections. This process also helps leave some initial chromosome information intact without altering it [45].  
Considering uniform crossover may lead to a disregard for the local structure of the parent chromosome, ensuing potential disruption to the overall structure, which could make it challenging to retain specific universal features of the parent chromosome. This study identified the PMX operation as the preferred crossover method, owing to its broad adoption in genetic algorithms. The PMX procedure functions by choosing a crossover point between two parent individuals and then initiating gene segment exchanges, ensuring the preservation of certain parent information while also enabling new combinations in the offspring.  
The crossover operation entails a random selection of the crossover point, followed by dividing the gene fragments of both parents into two sections. Subsequently, gene portions located beyond the crossover point from one parent are transferred to an offspring. The procedure also includes identifying unique genes from the second parent that retain their original sequence and incorporating them into the offspring's genome. Finally, gene segments anterior to the crossover point are completed, ensuring that the offspring's genetic sequence is of uniform length.  
In solving FJSP with the highest degree of efficacy, setting the crossover rate as close to 1 as feasible is crucial [46], ensuring comprehensive population evolution. Set the crossover rate, denoted as [47]  
\[p_{c}=0.8 \tag{20}\]  
This rate, commonly applied in genetic algorithm implementations, facilitates hastening convergence to the optimum solution while preserving enough genetic diversity to prevent premature convergence, thereby contributing to the algorithm's enhanced performance. This aspect is particularly relevant in applications like FJSP, where the diffusion of advantageous traits from parent chromosomes throughout the population is used to identify the best possible solution.