# Metaheuristics for the Minimum Time Cut Path Problem with Different Cutting and Sliding Speeds
## 5 Conclusions  
This study introduced a case of the MTCP problem, which more closely resembles real-world scenarios by distinguishing cutting and moving speeds. The cutting and the sliding rates for the equipment are essential parameters and, despite their relevance in real-world machinery, such features are often ignored in the literature. This study presented the minimum time cut path problem, including two presented evolutionary approaches, i.e., the genetic algorithm (GA) and biased random-key genetic algorithm (BRKGA) methods, which are suitable for addressing larger instances. Computational tests presented extensive hyper-parameter tuning using a large set of instances extracted from the literature, which can be leveraged in future research. For all instances analyzed, the proposed methodology achieved a gain in time of 47 s and an improvement in 94% of the tests performed. However, we noticed a decrease in performance when the input layouts had a relatively large number of vertices. This fact indicates a correlation between this parameter and the population size. Moreover, a suitable approach would consist of splitting the layout into parts, considering cutoff distances, so that the individual's representation can also be segmented. Hence, we would build \(N\) subgroups of edges, and the evaluation would be limited in finding the best sequence for each \(N\). Lastly, we would add the endpoint  
Figure 15: Evolution process defined for time limit reached instances in C-group. Configuration of BRKGA: [P = 5000, \(P_{e}=0.30\), and \(P_{m}=0.1\)]. times to the beginning of all predefined \(N\) by finding a feasible solution and enabling parallel processing. Future work will investigate how clustering methods such as K-Means can be integrated into the current approach to find feasible and quality solutions when addressing instances with a more significant number of parts positioned in a layout. This strategy will be integrated into the mathematical model. Another research direction is the investigation of an alternative representation for candidate solutions in the BRKGA algorithm. In the recent literature, there have been examples of methodology that allow the use of a purely discrete representation and the possibility of combining the discrete solutions using vector-like operations, similar to what is usually done for continuous solutions as in the random-key encoding [48; 49; 50]. Conceptualization, B.A.J. ; data curation, M.C.S. and G.N.d.C. ; methodology, L.J.P.d.A., B.A.J. and M.C.S. ; software, G.N.d.C. ; validation, B.A.J. ; writing--original draft, L.J.P.d.A., B.A.J. and P.R.P. ; writing--review & editing, L.J.P.d.A. and B.A.J. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. **Data Availability Statement:** Publicly available datasets were analyzed in this study. This data can be found here: [[https://github.com/GUIKAR741/metaheuristics-minimum-path](https://github.com/GUIKAR741/metaheuristics-minimum-path)]([https://github.com/GUIKAR741/metaheuristics-minimum-path](https://github.com/GUIKAR741/metaheuristics-minimum-path)) (accessed on 8 October 2021). The folder \"algorithms\" includes the codes of the two approaches developed for this study. The folder \"instances\" holds all input SVG layouts. The folder \"results\" presents the numerical experimental data. The folder \"result-evolution process\" contains the evolution graph of the best solution (\(\Upsilon\)) with the number of generations (X), and the folder \"results-final draw\" shows the ten images of the path sequence found (each instance \(\times\) execution), just for the BRKGA approach with hyper-parameters [\(P=5000\), \(P_{e}=0.30\), and \(P_{m}=0.1\)]. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: