# Repair Method and Mueganos Structure Applied to the Nesting Problem in Finite Materials
## 2 Methodology
### Mueganos Detection and Structure
The result obtained from _mueganos_ and item nesting generates less waste than the non-_mueganos_ solution due to the minimum separation between items inside the _mueganos_ structure. Table 6 displays the comparative results of our work against other methods. The three initial columns present references and values of efficiency and waste obtained from the reported methods. The final two columns exhibit outcomes from the proposed method in this study, derived through item utilization from the compared methods. ## 4 Discussion and Future Work

This paper proposes a strategy to repair solutions obtained with overlaps using metaheuristics, which provides an excellent alternative to solve the nesting problem and allows an improved matching process. The _mueganos_ structure aims to compact the items to reduce waste, and this work demonstrates that feasible solutions are obtained to compete with the results achieved in previous research. It is important to mention that the repair process and the _mueganos_ can be applied to any solution to solve the nesting problem, as long as there is an overlap in any solution. The method proposed in this research is applied to real problems in industries such as the leather-footwear industry, where waste reduction is crucial for optimal material utilization. Many companies still manually conduct nesting processes, relying on operator \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method and Reference** & **Waste (\%)** & **Efficiency (\%)** & **Waste (\%) Repair Method (Ours)** & **Efficiency (\%) Repair Method (Ours)** \\ \hline MIP [4] & 18.46 & 81.54 & 17.89 & **82.11** \\ BRKGA70 [27] & 26.68 & 73.32 & 25.35 & **74.65** \\ FOMLS1 RPP [30] & 31.68 & 68.31 & 30.91 & **69.09** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparative results with the state-of-the-art. Figure 12: Results obtained by our method using state-of-the-art patterns. We used (**a**) the items from [4], (**b**) from [30], and (**c**) from [27] to compare the waste and efficiency. expertise, impacting time and material waste. Automation of the nesting process is vital for efficiency. Specialized companies have provided us with the required materials (including leather material and footwear patterns) to conduct real-size testing. Additionally, we established an arrangement within the factory to capture images of the materials, facilitating the subsequent calibration and image digitization processes. The proposed method offers a significant advantage in automating the nesting process within these industries, leading to decreased material waste and increased efficiency. Although this method has advantages over other reported ones, it has certain limitations, but working on these can lead to even better results. For example, time production is a important factor in the industry when working in real-time, and this improvement could make the algorithm more efficient. In future work, the time factor will be improved by using parallel programming with threads in the implementation of the algorithm. Additionally, it is convenient to keep a count of all the items needed to complete the manufacturing process depending on the type of industry in which the algorithm is applied. To further improve results, it is necessary to work on determining the quality of the skin to give an adequate positioning to each item. The skin segmentation is by type of cut, since each item must have a certain elasticity, and therefore must be placed in a suitable orientation angle. Another future work is to calculate the efficiency of wastage by _mueganos_ to provide solutions with the least possible waste. Finally, a further improvement will be to achieve the formation of _mueganos_ per production unit; for example, if a shoe needs five items for its manufacture, then the _mueganos_ should contain these five items. Conceptualization, A.R. and F.C. ; methodology, A.R. and F.C. ; software, A.R. ; validation, A.R. and F.C. ; formal analysis, A.R. and F.C. ; investigation, A.R. ; resources, A.R., F.C. and D.E. ; data curation, A.R.