# A Petri Net-Based Algorithm for Solving the One-Dimensional Cutting Stock Problem
## 7 Results  
For each tested instance of the datasets, we report the best results obtained with each algorithm for the number of stocks used and the time taken in seconds. Additionally, we provide the \(lb\) value for all the instances and the percentage above this value (\(\%\)\(>\)\(lb\)) obtained by each algorithm. The FBS-PN-CSP, G&S, and LLA algorithms were compared only with the _hinterding_ and _falkenauer_\(U\)_ datasets since the LLA algorithm ran out of memory for the other datasets. The minimum value of stocks used among the compared algorithms is highlighted in boldface. For the G&S algorithm, we provide the best solution found within the 10 executions, and likewise, for the FBS-PN-CSP algorithm, we include the best combinations of beam width and filter width where the best stock result was found. The results of the _hinterding_, _hard10_, and _wuegau1_ datasets are reported in this section. The results of the _hard28_, _binpack1_, _binpack2_, _binpack3_, and _binpack4_ datasets are included in Appendix A. For the _hinterding_ dataset, the three algorithms obtained the same optimal theoretical value for the 1a-5a instances and the 7a instance as shown in Table 2. The FBS-PN-CSP and the G&S algorithms also obtained the same best result for the 6a instance. Only the G&S had the optimal theoretical value for the 8a-10a instances. The FBS-PN-CSP algorithm had better results than the LLA algorithm for the 6a, 8a, and 9a instances, while for the 10a instance, both had the same result. Although the FBS-PN-CSP algorithm was not better than the G&S algorithm for instances 8a-10a, it obtained a percentage above the optimal theoretical value of less than 1%. The FBS-PN-CSP algorithm was more efficient than the G&S algorithm in six out of seven instances where they are equal, namely 2a-7a. However, the LLA algorithm had better times in all the instances where it had the same results as the other two algorithms. The beam width of the FBS-PN-CSP algorithm seems to increase as the number of items and the stock length increase. The highest beam width and filter width values were obtained in the 10a instance, which has the higher number of items and the longest stock. In 12 out of 17 instances of the _wae_\(gau1\)_ dataset, the FBS-PN-CSP algorithm matched the best results of the G&S algorithm (Table 3), and in 9 of these 12 instances, it had better times. In 4 out of the 17 instances, the FBS-PN-CSP had better results, and only in one instance, the G&S was better than the FBS-PN-CSP. In 7 out of the 17 instances, the FBS-PN-CSP obtained the optimal theoretical value, meanwhile, the G&S did so in only four instances. Regarding beam width, four instances required the higher tested values, namely 25 and 35, but with a small filter width. For the _hard10_ instances, the G&S algorithm was better than the FBS-PN-CSP algorithm as shown in Table 4. However, the G&S algorithm only had the optimal theoretical value in one instance, namely HARD9. Even with the smaller value of the beam width, the time required for the FBS-PN-CSP was far higher than the time of the G&S algorithm. For this