# 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} \hline \hline  & \multicolumn{3}{c}{**FBS-PN-CSP**} & \multicolumn{5}{c}{**G\&S**} \\ \cline{3-10}
**Instances** & _lb_ & **Stock** & \(\beta\) & \(\alpha\) & \% \(>\)_lb_ & **Time (s)** & **Stock** & \% \(>\)_lb_ & **Time (s)** & **Avg. Stock** & **Avg. Time** \\ \hline HARD0 & 55 & 59 & 5 & 2 & 7.27 & 5673.86 & **56** & 1.82 & 79.64 & 64.40 & 129.34 \\ HARD1 & 56 & 60 & 5 & 2 & 7.14 & 5669.64 & **57** & 1.79 & 327.97 & 59.20 & 394.00 \\ HARD2 & 56 & 60 & 5 & 2 & 7.14 & 5707.35 & **57** & 1.79 & 132.01 & 57.00 & 276.31 \\ HARD3 & 55 & 59 & 5 & 2 & 7.27 & 5663.16 & **56** & 1.82 & 96.01 & 57.90 & 296.15 \\ HARD4 & 56 & 60 & 5 & 2 & 7.14 & 5628.17 & **57** & 1.79 & 139.28 & 57.00 & 331.95 \\ HARD5 & 55 & 59 & 5 & 2 & 7.27 & 5454.04 & **56** & 1.82 & 241.52 & 60.20 & 325.61 \\ HARD6 & 56 & 60 & 5 & 2 & 7.14 & 5462.07 & **57** & 1.79 & 123.19 & 57.00 & 370.79 \\ HARD7 & 54 & 59 & 5 & 2 & 9.26 & 5645.90 & **56** & 3.70 & 273.06 & 61.10 & 273.82 \\ HARD8 & 56 & 59 & 5 & 2 & 5.36 & 5447.00 & **57** & 1.79 & 112.20 & 59.30 & 314.82 \\ HARD9 & 56 & 60 & 5 & 2 & 7.14 & 5529.23 & **56** & 0.00 & 265.34 & 64.40 & 266.41 \\ \hline \hline \end{tabular} From the algorithms’ stock results for each instance, the minimum is in boldface.  
\end{table} 
Table 4: Best stock results for _hard10_ instances. \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**FBS-PN-CSP**} & \multicolumn{5}{c}{**G\&S**} \\ \cline{3-10}
**Instances** & _lb_ & **Stock** & \(\beta\) & \(\alpha\) & \% \(>\)_lb_ & **Time (s)** & **Stock** & \% \(>\)_lb_ & **Time (s)** & **Avg. Stock** & **Avg. Time** \\ \hline TEST0005 & 28 & **29** & 5 & 2 & 3.57 & 56.71 & **29** & 3.57 & 139.58 & 29.00 & 352.83 \\ TEST0014 & 23 & **24** & 5 & 2 & 4.35 & 39.37 & **24** & 4.35 & 64.83 & 24.00 & 243.59 \\ TEST0022 & 14 & **15** & 5 & 2 & 7.14 & 19.90 & **15** & 7.14 & 48.72 & 15.00 & 322.90 \\ TEST0030 & 27 & **28** & 5 & 2 & 3.70 & 52.06 & **28** & 3.70 & 61.94 & 28.00 & 312.79 \\ TEST0044 & 14 & **14** & 25 & 4 & 0.00 & 367.03 & 15 & 7.14 & 380.16 & 42.00 & 38.02 \\ TEST0049 & 11 & **11** & 25 & 2 & 0.00 & 276.48 & 12 & 9.09 & 241.32 & 15.50 & 304.08 \\ TEST0054 & 14 & **14** & 35 & 3 & 0.00 & 484.86 & **14** & 0.00 & 356.10 & 47.80 & 146.63 \\ TEST0055 & 15 & **16** & 5 & 2 & 6.67 & 62.91 & **16** & 6.67 & 427.36 & 34.90 & 154.52 \\ TEST0055 & 20 & **20** & 15 & 14 & 0.00 & 257.22 & 96 & 380.00 & 0.00 & 96.00 & 0.01 \\ TEST0058 & 20 & 21 & 5 & 2 & 5.00 & 34.71 & **20** & 0.00 & 350.58 & 20.60 & 351.83 \\ TEST0065 & 15 & **16** & 5 & 2 & 6.67 & 22.52 & **16** & 6.67 & 18.94 & 16.00 & 309.42 \\ TEST0068 & 12 & **13** & 5 & 2 & 8.33 & 61.89 & **13** & 8.33 & 240.98 & 49.00 & 143.78 \\ TEST0075 & 13 & **13** & 25 & 6 & 0.00 & 418.74 & 14 & 7.69 & 294.42 & 47.30 & 320.35 \\ TEST0082 & 24 & **25** & 5 & 2 & 4.17 & 36.54 & **25** & 4.17 & 136.45 & 25.00 & 426.39 \\ TEST0084 & 16 & **16** & 15 & 15 & 0.00 & 116.07 & **16** & 0.00 & 76.77 & 17.80 & 223.36 \\ TEST0095 & 16 & **17** & 5 & 2 & 6.25 & 77.83 & **17** & 6.25 & 88.63 & 17.00 & 354.11 \\ TEST0097 & 12 & **12** & 15 & 15 & 0.00 & 110.32 & **12** & 0.00 & 397.77 & 25.20 & 197.42 \\ \hline \hline \end{tabular} From the algorithms’ stock results for each instance, the minimum is in boldface.  
\end{table} 
Table 3: Best stock results for _wave_gau1_ instances. instances, the value was 5, which is the smallest value tested. For the _binpack2_ instances, the FBS-PN-CSP had the same results as the G&S algorithm in 19 instances (Table A3), and these were equal to the corresponding optimal theoretical values. All the results of the G&S algorithm were equal to the optimal theoretical value. The LLA algorithm had the optimal theoretical values in half of the instances and the best times among the three algorithms. The FBS-PN-CSP algorithm had better times than the G&S in ten of the instances where they had the same stock result. The beam width was mostly 15 throughout the _binpack2_ instances.