# Evolutionary Algorithms for Optimization Sequence of Cut in the Laser Cutting Path Problem
## 4 Overview of Application of ABRKGA to LCPP
### An Eulerian Heuristic to Generate Improved Individuals for LCPP
This perturbation strategy involves randomly swapping values in a vector of random keys, with the strength of perturbation defined by the self-adaptive parameter \(\beta\), which ranges from 0.001 to 0.1. A random variable is generated for each chromosome gene for each similar individual. This random variable determines whether a specific random key will undergo modification through gene swaps at random positions. This probability is set to a low value, otherwise, the search process may devolve into randomness. The parameter \(\beta\) is included in position \(2n+2\) of the chromosome and evolves alongside the other genes associated with the problem. Within this framework, individuals with identical fitness values are subjected to perturbations, with the intensity of perturbation determined by \(\beta\), a self-adaptive parameter. The value of \(\beta\), calculated using Equation (13), is incorporated into chromosome \(c\) at position \(2n+2\) and undergoes evolutionary changes. \[\beta=0.001+c_{2n+2}*(0.1-0.001) \tag{13}\]

The mutation stage produces a set of \(p_{m}\) mutant individuals through a random generation process. The identical module for generating the initial population creates these mutants. The crossover process combines \(p-|RCL|-p_{m}\) pairs of parents through mating, with one parent chosen from the RCL and the other randomly selected from the remaining population. The parameterized uniform crossover technique ([PERSON], 2018) is applied, which involves generating uniform random numbers \(r_{j}\) within the range of \([0,1]\) for each gene of the chromosome. For each gene position \(j\), if \(r_{j}\) is less than or equal to \(\rho_{e}\), then the offspring inherits the \(j\)-th allele from the RCL parent; otherwise, it inherits the allele from the other parent. The value of the self-adaptive parameter \(\rho_{e}\), located at position \(2n+1\) of the chromosome, is always selected from the non-RCL parent. The parameterized uniform crossover is designed to prioritize the inheritance of alleles from the RCL parent chromosomes. This method is achieved through the self-adaptive parameter \(\rho_{e}\), which determines the probability of an offspring inheriting the vector component from the RCL parent. The calculation of \(\rho_{e}\), as defined by Equation (14), involves the utilization of the random-key \(2n+1\) from the non-RCL chromosome \(c\). \[\rho_{e}=0.65+c_{2n+1}*(0.80-0.65) \tag{14}\]

It is essential to point out that the values for defining the calculations of the ABRKGA control parameters equations are based on the work of [39; 41]. Table 1 presents each suggested value range for the parameters applied. Furthermore, the parameter \(\gamma\) is a configuration parameter, but it is relatively straightforward to adjust. Like in [8], we have defined three specific values for \(\gamma\in 0.999,0.998,0.997\) based on the available running time. All of these values lead to a gradual decrease in population size. The primary distinction lies in the maximum number of generations, affecting the running time and solution space. Table 2 provides an overview of the \(\gamma\) and \(max_{gen}\) values and their respective characteristics. \(p1\) : The search is conducted within a reasonable computational time, ensuring that the minimum population size remains around 70% of the initial population size. \(p2\): The search process is executed within a reasonable time, necessitating the restart of the population to its maximum size once. \(p3\): The search is carried out over an extended duration, requiring four population restarts to be performed. The objective of the ABRKGA is to utilize parameter tuning to enhance exploration during the initial phase of the search and increase exploitation throughout the evolutionary process. The evolutionary process of the ABRKGA is depicted in Figure 3.0 The decoder represents the problem-dependent component, responsible for converting random-key vectors into solutions and evaluating their fitness.