# Repair Method and Mueganos Structure Applied to the Nesting Problem in Finite Materials
### Related Work
[PERSON] and [PERSON] [26]proposed a genetic algorithm to generate a sequence of patterns and use an allocation algorithm to place the sequence using a sliding method integrated with a neural network. In the nesting problem, it is important to address the issue of overlaps between items. To avoid overlaps, it is required to apply techniques such as the repair of unfeasible solutions or the no-fit raster concept to check overlapping between any two-dimensional generic-shaped items. [PERSON] et al. [27] used the no-fit raster concept, which is used to check overlaps between any two-dimensional items in a generic way. They also implement a biased random-key genetic algorithm (BRKGA) to determine the sequence in which these items are packed. Once the sequence is obtained, they propose two heuristics based on the bottom-left movements and the no-fit raster concept, which are used to sort these items in the given container by observing objective criteria. [PERSON] and [PERSON] [28] proposed a one-pass heuristic, placing items at different positions in the material to avoid the overlaps using the no-fit polygon (NFP) technique, thus generating different partial solutions, until the best one was found. Details of the NFP algorithm are found in [29]. [PERSON] et al. [4] used a greedy strategy by applying a Mixed-Integer Programming (MIP) model for the nesting problem. [PERSON] et al. [30] proposed an adaptation of the no-fit polygon to consider the penetration depth of the items. It is based on an iterative compaction scheme using a separation algorithm based on a fast obstruction map local search (FOMLS). Repair algorithms are essential to improve and avoid overlapping items in the nesting problem. These algorithms can be developed using a greedy or random repair approach or a metaheuristic strategy to find feasible solutions. However, eliminating overlapping items is not the most effective solution as it can lead to more waste material. There are several possible methods to repair an individual that is not a feasible solution. [PERSON] et al. [31] explained that one of these methods is to create a repaired version that is a local solution for that particular individual. This repaired version can replace the original individual in the population under some probability. The repair of an unfeasible individual is related to learning and evolution, which interact with each other to improve and find a repaired and feasible solution called the Baldwin effect [32]. The contributions and innovations of this paper are summarized as follows:  
1. A proposed repair method can be applied in a wide range of heuristic and metaheuristic techniques for solution approximation to solve the nesting problem. 2. A calibration method to enhance the measurement and scaling of items and material for the nesting process. 3. Enhanced efficiency and waste reduction through the use of _mueganos_ as an innovative concept. 4. The _mueganos_ structure facilitates nesting of elements in a more compact form and minimizing waste. 5. Local optimization derived from the formation and optimal position of _mueganos_. 6. The sequential greedy algorithm (SGA) allows filling the available areas in the material with random pattern selection. 7. Improved application of metaheuristic algorithms when working with chromosomes or repaired solutions (e.g., genetic algorithms). 8. Experimental results underscore that the proposed method, particularly incorporating _mueganos_, leads to improved waste reduction and solution quality compared to previous methodologies. The paper is structured as follows:  
* Methodology: The procedure used to solve the nesting problem is explained, including the repair method proposed. * Experimental Results: The results obtained with the proposed methodology are presented. * Conclusions and future work: The conclusions drawn from the results are discussed, and suggestions for future work are provided.