# A Petri Net-Based Algorithm for Solving the One-Dimensional Cutting Stock Problem
## 7 Results
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Instance**} & \multirow{3}{*}{\(lb\)} & \multicolumn{3}{c}{**FBS-PN-CSP**} & \multicolumn{3}{c}{**G\&S**} & \multicolumn{3}{c}{**LLA**} \\ \cline{3-14}  & & **Stock** & \(\beta\) & \(\alpha\) & \% & **Time** & **Stock** & **\%** & **Time** & **Avg.** & **Avg.** & **Stock** & **\%** & **Time** \\  & & & & & \(>\)\(lb\)** & **(s)** & & & \(>\)\(lb\)** & **(s)** & **Stock** & **Time** & & \(>\)\(lb\)** & **(s)** \\ \hline
1a & 9 & 9 & 5 & 5 & 0.00 & 0.08 & 9 & 0.00 & 0.01 & 9.00 & 9.18 & 9 & 0.00 & 0.00 \\
2a & 23 & **23** & 5 & 2 & 0.00 & 0.19 & **23** & 0.00 & 34.15 & 23.00 & 304.45 & **23** & 0.00 & 0.00 \\
3a & 15 & **15** & 5 & 5 & 0.00 & 0.23 & **15** & 0.00 & 2.09 & 15.00 & 269.23 & **15** & 0.00 & 0.05 \\
4a & 19 & **19** & 15 & 2 & 0.00 & 1.24 & **19** & 0.00 & 27.16 & 19.00 & 266.60 & **19** & 0.00 & 0.03 \\
5a & 51 & **53** & 5 & 5 & 3.92 & 25.71 & **53** & 3.92 & 35.83 & 53.00 & 276.56 & **53** & 3.92 & 5.06 \\
6a & 78 & **79** & 15 & 4 & 1.28 & 9.87 & **79** & 1.28 & 21.91 & 79.00 & 243.05 & 80 & 2.56 & 1.13 \\
7a & 68 & **68** & 15 & 15 & 0.00 & 9.89 & **68** & 0.00 & 110.58 & 68.00 & 373.81 & **68** & 0.00 & 0.53 \\
8a & 143 & 144 & 15 & 3 & 0.70 & 23.18 & **143** & 0.00 & 11.25 & 143.00 & 273.35 & 145 & 1.40 & 29.14 \\
9a & 149 & 150 & 15 & 15 & 0.67 & 31.43 & **149** & 0.00 & 35.82 & 149.00 & 339.76 & 152 & 2.01 & 10.86 \\
10a & 215 & 216 & 25 & 22 & 0.47 & 117.90 & **215** & 0.00 & 12.10 & 215.00 & 294.33 & 216 & 0.47 & 13.39 \\ \hline \hline \end{tabular} From the algorithms’ stock results for each instance, the minimum is boldface.  
\end{table} 
Table 2: Best stock results for _hinterding_ instances. reason, we could not implement larger values for the beam width since the computational cost was prohibited, although it did not run out of memory. Despite the results for stock used by the FBS-PN-CSP algorithm, it was not far from those of the G&S and could be at least equal if higher values of the beam width can be implemented. The results of the FBS-PN-CSP and G&S algorithms are shown in Table 11 for the _hard28_ dataset. Both algorithms obtained the same best results except for the BPP900 and BPP781 instances, where the FBS-PN-CSP had the better results. Nevertheless, neither obtained the optimal theoretical value. The FBS-PN-CSP was more efficient in 15 out of 27 instances where both algorithms obtained the same result. As for the beam width of the FBS-PN-CSP, this was the minimum value of the tested values, that is, \(\beta=5\) for all the instances. Table 12 shows the best stock results for the _binpack1_ instances of the _falkenauer_U_ dataset, where the FBS-PN-CSP, G&S, and LLA algorithms are compared. The FBS-PN-CSP and the G&S algorithms obtained the optimal theoretical values of each instance, but the FBS-PN-CSP had the best times. The LLA algorithm achieved the optimal theoretical values in 12 out of 20 instances and had better times than the FBS-PN-CSP algorithm. Only in two instances did the FBS-PN-CSP use the higher beam width values (25 and 35), but in most