# 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
As each generation progresses, the population size (\(p\)) for each generation (\(k\)) diminishes by a factor of:

\[p^{k+1}=p^{k}*\gamma \tag{7}\]

The quantity of offspring is adjusted to determine the size of the new population, which is defined by Equation (7). This strategy reduces the possible solutions at each generation. Then, a restart procedure is executed when the population size falls below a threshold of \(p_{min}\) individuals. This restart procedure involves preserving only the restricted chromosome list (RCL) individuals, explained next, in the current population while generating \(p_{max}-|RCL|\) new individuals at random. Additionally, these parameters are utilized in the calculation of the stop criterion through Equation (8), which determines the maximum number of generations (\(max_{gen}\)) for the ABRKGA. Consequently, a minor reduction in population size (with \(\gamma\) values close to 1) results in fewer generations. Conversely, if the population size decreases faster, it necessitates more generations and restarts. \[max_{gen}=\gamma^{(}log_{\gamma}^{p_{min}}-p_{max}) \tag{8}\]

In order to generalize, the population chromosomes are arranged in ascending order, placing the best chromosomes at the top and the worst at the bottom. This arrangement applies to minimization problems, as is the case for LCPP. The population is divided into two groups, with the best individuals retained in a restricted chromosome list (RCL). The RCL consists of individuals whose fitness surpasses a threshold value (\(f_{e}\)) determined by the parameter \(\alpha\) within the range [0, 1]:

\[f_{e}=f_{min}+\alpha(f_{max}-f_{min}) \tag{9}\]

The fitness of the best individual in the current population is denoted as \(f_{min}\), while the fitness of the poorest individual is represented as \(f_{max}\). When \(\alpha=0\), only individuals with the highest fitness are included in the RCL, whereas \(\alpha=1\) includes all individuals in the RCL. The value of parameter \(\alpha\) is adjusted periodically based on the progress of the search. The maximum number of individuals that can be included in the RCL is limited by \(p_{e}\). Lastly, the individuals in the RCL are replicated and carried over to the next population. The parameter \(\alpha\) determines the number of elite individuals by influencing the size of the RCL (defined by Equation (9)). Initially, \(\alpha\) is set to 0.3 and decreases with each generation according to Equation (10). The minimum value for \(\alpha\) is 0.1. This range accounts for the suboptimal solutions commonly encountered in the early generations and the tendency for population stagnation after a certain number of iterations. This range was considered equality in the proposal of [PERSON] et al. (2018). \[\alpha=0.10+(1-\frac{gk}{max_{gen}})*(0.30-0.10) \tag{10}\]

The parameters \(k_{e}\) (ranging from 0.10 to 0.25) and \(k_{m}\) (ranging from 0.05 to 0.20) are employed to determine the elite (\(p_{e}\)) and mutant (\(p_{m}\)) partitions within the population. Once again, the objective of the search process is to promote diversification during the initial phase and gradually intensify the focus as the population evolves. The parameter control \(k_{e}\) is defined by Equation (11), while \(k_{m}\) is defined by Equation (12), with \(g_{k}\) representing the current generation. Consequently, the maximum size of the elite partition is smaller in the early generations when the average quality of individuals is lower. However, this number progressively increases in subsequent generations. Conversely, the number of mutants decreases as the search process unfolds. \[k_{e}=0.10+\frac{gk}{max_{gen}}*(0.25-0.10) \tag{11}\]

\[k_{m}=0.05+(1-\frac{gk}{max_{gen}})*(0.20-0.05) \tag{12}\]

In order to maintain diversification within the RCL, perturbations are employed to enable the ABRKGA to break free from local optima and overcome the limitations of solely introducing random individuals. Identical fitness values solely determine the similarity between individuals, for simplicity.