Author Contributions
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.; writing—original draft preparation, A.R. and D.E.; writing—review and editing, A.R. and F.C.; visualization, A.R.; supervision, F.C.; project administration, F.C.; funding acquisition, F.C. All authors have read and agreed to the published version of the manuscript.
Figure 1.
Flow diagram of the repair process described in the methodology section. The process is a sequential flow starting with the camera calibration method to obtain the correct coordinates. Subsequently, the items are codified to apply the repair method. The muéganos are created from non-overlapping and low-waste items. Finally, the greedy algorithm optimizes the results.
Figure 1.
Flow diagram of the repair process described in the methodology section. The process is a sequential flow starting with the camera calibration method to obtain the correct coordinates. Subsequently, the items are codified to apply the repair method. The muéganos are created from non-overlapping and low-waste items. Finally, the greedy algorithm optimizes the results.
Figure 2.
Binarized material and items. (a) Sample cowhide material to accommodate items. (b–d) Footwear items nesting in the sample material.
Figure 2.
Binarized material and items. (a) Sample cowhide material to accommodate items. (b–d) Footwear items nesting in the sample material.
Figure 3.
Image (a) shows a chessboard pattern over the material. The chessboard should cover most of the material for the best calibration. In image (b), the camera captures the material for the nesting process.
Figure 3.
Image (a) shows a chessboard pattern over the material. The chessboard should cover most of the material for the best calibration. In image (b), the camera captures the material for the nesting process.
Figure 4.
Item coding contains all the items of the nesting. Each item is represented by five alleles: identifier (), rotation angle (), position in rows and columns within the material (,), and packing item ().
Figure 4.
Item coding contains all the items of the nesting. Each item is represented by five alleles: identifier (), rotation angle (), position in rows and columns within the material (,), and packing item ().
Figure 5.
Sample of the item movements during the repair process. The black area represents the material, the white area represents the items, and the gray area represents the overlaps.
Figure 5.
Sample of the item movements during the repair process. The black area represents the material, the white area represents the items, and the gray area represents the overlaps.
Figure 6.
Muégano structure. Each muégano is a list of the contained items. The number of items depends on the largest number of adjacent pixels between edges of neighboring items.
Figure 6.
Muégano structure. Each muégano is a list of the contained items. The number of items depends on the largest number of adjacent pixels between edges of neighboring items.
Figure 7.
Sample nesting results. Inside of the red ellipse: example of muégano formed by two items. Each item has a different gray level.
Figure 7.
Sample nesting results. Inside of the red ellipse: example of muégano formed by two items. Each item has a different gray level.
Figure 8.
Flow diagram of the greedy sequential algorithm to fill the area of the unused material. The random item is selected, and its position is generated. If the item is non-overlapped with another one, it is placed. The process is looped while there is available space in the material.
Figure 8.
Flow diagram of the greedy sequential algorithm to fill the area of the unused material. The random item is selected, and its position is generated. If the item is non-overlapped with another one, it is placed. The process is looped while there is available space in the material.
Figure 9.
Comparison of waste between the results obtained with and without muéganos. The blue box-and-whisker plot shows a better improvement of the waste over the non-muéganos box-and-whisker plot. The mean of the results using muéganos is around .
Figure 9.
Comparison of waste between the results obtained with and without muéganos. The blue box-and-whisker plot shows a better improvement of the waste over the non-muéganos box-and-whisker plot. The mean of the results using muéganos is around .
Figure 10.
Comparison of time between the results obtained with and without muéganos. The blue box-and-whisker plot shows a greater spread of computational time cost, and the mean of the results is around 305 s due to the size of the muéganos structure. However, it is compensated by reduced waste.
Figure 10.
Comparison of time between the results obtained with and without muéganos. The blue box-and-whisker plot shows a greater spread of computational time cost, and the mean of the results is around 305 s due to the size of the muéganos structure. However, it is compensated by reduced waste.
Figure 11.
Nesting results obtained (a) without muéganos, and (b) with muéganos. The result obtained from muéganos and item nesting generates less waste than the non-muéganos solution due to the minimum separation between items inside the muéganos structure.
Figure 11.
Nesting results obtained (a) without muéganos, and (b) with muéganos. The result obtained from muéganos and item nesting generates less waste than the non-muéganos solution due to the minimum separation between items inside the muéganos structure.
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.
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.
Table 1.
Calibration variables resume.
Table 1.
Calibration variables resume.
Variable | Description |
---|
C | Number of chessboard images |
, | Focal length for axis x and y, respectively |
, | Optical center |
, , | Radial distortion |
, | Tangential distortion |
K | Calibration matrix |
d | Distortion vector |
| i-th 2D coordinate (pixel) |
u, v | Pixel coordinates for axis x and y, respectively |
M | Projection matrix |
| i-th values from projection matrix |
| i-th 3D coordinate (world) |
, , | World coordinates for axis x, y and z, respectively |
| Waste of the material in pixels |
| Waste of the material in m2 |
| Area of the rectangle around the material in pixels |
| Area of the rectangle around the material in m2 |
Table 2.
Variables for characterization of items involved in the constraints.
Table 2.
Variables for characterization of items involved in the constraints.
Variable | Description |
---|
| Material with items packed |
| Set of alleles by items |
| Fitness function value |
| Items that are overlapping |
| Pixels that are overlapping |
| Items that are outside of valid area |
| Pixels that are outside of valid area |
| Sum of pixels of the nested items. |
| Number of nested items |
Table 3.
Alleles identifying each item.
Table 3.
Alleles identifying each item.
Variable | Description |
---|
| Identifier item |
| Rotation angle |
| Row position within the material |
| Column position within the material |
| Packing item |
Table 4.
Results obtained without Muéganos.
Table 4.
Results obtained without Muéganos.
Runs | Waste (%) | Waste in m2 | Time | Admissible Orientations |
---|
1 | | | | |
2 | | | | |
3 | | | | |
4 | | | | |
5 | | | | |
6 | | | | |
7 | | | | |
8 | | | | |
9 | | | | |
10 | | | | |
11 | | | | |
12 | | | | |
13 | | | | |
14 | | | | |
15 | | | | |
16 | | | | |
17 | | | | |
18 | | | | |
19 | | | | |
20 | | | | |
Average | | | | |
Table 5.
Results obtained with Muéganos.
Table 5.
Results obtained with Muéganos.
Runs | Waste (%) | Waste in m2 | Time | Admissible Orientations |
---|
1 | 28.64 | 0.4513 | 339.52 | |
2 | 28.73 | 0.4528 | 380.47 | |
3 | 29.36 | 0.4628 | 290 | |
4 | 29.52 | 0.4654 | 275.48 | |
5 | 29.21 | 0.4605 | 264.16 | |
6 | 28.82 | 0.4593 | 315.52 | |
7 | 28.4 | 0.4480 | 321.24 | |
8 | 28.76 | 0.4536 | 306.85 | |
9 | 28.63 | 0.4515 | 282.33 | |
10 | 28.49 | 0.4494 | 359.65 | |
11 | 28.89 | 0.4556 | 290.64 | |
12 | 28.9 | 0.4554 | 303.83 | |
13 | 28.99 | 0.4567 | 268.73 | |
14 | 29.23 | 0.4607 | 274.65 | |
15 | 29.2 | 0.4602 | 318.56 | |
16 | 28.5 | 0.4492 | 291.57 | |
17 | 28.79 | 0.4537 | 282.16 | |
18 | 29.11 | 0.4588 | 305.44 | |
19 | 29.21 | 0.4605 | 308.56 | |
20 | 28.8 | 0.4539 | 324.28 | |
Average | | | | |
Table 6.
Comparative results with the state-of-the-art.
Table 6.
Comparative results with the state-of-the-art.
Method and Reference | Waste (%) | Efficiency (%) | Waste (%) Repair Method (Ours) | Efficiency (%) Repair Method (Ours) |
---|
MIP [4] | | | | |
BRKGA70 [27] | | | | |
FOMLS|RPP [30] | | | | |