# 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
The \(e\)-BRKGA strategy outperforms the \(e\)-ABRKGA variant, striking a commendable balance between solution optimality and computational efficiency. Therefore, for practical applications with faster results, the \(e\)-BRKGA approach emerges as the more favorable choice over the \(e\)-ABRKGA approach. ## 6 Conclusions

This research paper presents the LCPP (laser cutting path planning) problem, which effectively captures real-world conditions by evaluating distinct cutting and air-moving speeds. The consequence of cutting and sliding rates in actual machinery is often overlooked in the existing literature, making this investigation a valuable contribution to the field. By presenting a self-adaptive parameters evolutionary approach, namely, the adaptive biased random-key genetic algorithm (ABRKGA), and a heuristic to try to construct more significant fitness of individuals, the authors demonstrate their effectiveness in tackling practical instances of the problem. These methods hold promise in addressing functional challenges related to the LCPP in real-world systems. Computational tests were conducted to conclude our investigation, comprising four distinct scenarios and considering 50 available instances in the literature [9]. In the first scenario, tests are conducted using the conceptual model in mathematical programming proposed for the LCPP (Section 2.2). Our findings revealed that the optimal solution was obtained only for instances containing a maximum of 20 nodes and 20 edges within a runtime limit of 1 h.

The other three experiments (Sections 5.3-5.5) were designed to facilitate a comparative analysis between solution quality and execution time. By conducting these experiments, our objective was to assess how different approaches or algorithms perform in terms of finding high-quality solutions within a limited time frame. This comparison allows us to understand the trade-offs between solution optimality and computational efficiency, providing valuable insights into the strengths and weaknesses of each method under consideration. Such an analysis is crucial in selecting the most suitable approach for practical applications, where finding reasonably good solutions is often paramount. In this context, we conclude, through the analysis of the test tables, that applying the heuristic in conjunction with BRKGA provided the best trade-off between solution quality and speed. Additionally, ABRKGA can be an excellent alternative when there is no prior study on layout patterns for the LCPP. Accurate thermal evaluation methods can be seamlessly integrated into cutting path algorithms without substantially increasing computation time. As a result, there is a compelling motivation for CAM software developers to integrate these strategies into their software solutions. This enhanced flexibility empowers businesses to tailor their manufacturing processes in response to evolving objectives, ensuring better adaptability in the dynamic market landscape. For future work, the objective is to leverage this versatile framework (\(e\)-BRKGA and ABRKGA) to tackle more complex multi-objective optimizations of laser cutting paths, thereby further enhancing the already promising results obtained through an essential linear combination of diverse objective functions. This continuous pursuit of optimization will enable industries to unlock greater efficiencies and precision in their manufacturing operations. **Author Contributions:** Conceptualization, B.A.J. ; data curation, M.C.S. and G.N.d.C. ; methodology, J.W.L.C., B.A.J. and M.C.S. ; software, G.N.d.C. ; validation, B.A.J. ; writing--original draft, M.C.S., B.A.J. and P.R.P. ; writing--review and editing, B.A.J. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Publicly available datasets were analyzed in this study.