# A Petri Net-Based Algorithm for Solving the One-Dimensional Cutting Stock Problem
## 6 Experiments
For the second phase of the G&S algorithm, we used the Scipy package with the _linprog_ library. We also set the same parameter values as in [50], that is, specifically: a time limit of 600 s for the G&S algorithm, a mutation rate of 10%, a maximum age of 2, and a maximum number of cutting patterns for the tree construction of 1,000,000. Additionally, we also executed the G&S algorithm 10 times for each instance. However, we did not set a time limit in the second phase of the G&S algorithm, as the solutions from the ILP solver were obtained in an acceptable time. \[lb=\left\lceil\frac{\sum_{i}^{m}l_{i}d_{i}}{L}\right\rceil \tag{13}\]  
One of the issues with beam search algorithms is the selection of the beam width \(\beta\). The smaller the \(\beta\) value, the faster the search, but this comes with the loss of optimality due to aggressive pruning. We conducted preliminary experiments to determine suitable \(\beta\) and \(\alpha\) values, considering the condition \(\alpha\leq\beta\). These experiments were carried out with the _hinterding_ dataset and we tested for \(\beta\) values chosen from 5 to 45. In all ten instances of this dataset, we observed that the best solutions were found with a beam width between 5 and 25. For small instances, the best solutions were found with lower \(\beta\) values while for large instances the values were higher. However, with higher values (namely from 30 to 45) there was no improvement in the solutions among the tested instances. Thus, for all the experiments with the FBS-PN-CSP algorithm, we used the interval \(5\leq\beta\leq 35\) for the beam width, at steps 5, 15, 25, and 35. We chose these \(\beta\) values since from the preliminary experiments, at these values we observed improvements in the solutions of the tested instances. Furthermore, apart from finding no solution improvements in the preliminary experiments, higher values of \(\beta\) impact the efficiency of the FBS algorithm. However, for the _hinterding_ dataset, we also implemented the \(\beta\) value of 45 as in the preliminary experiments. For the _hard10_ dataset, we only could implement a beam width of \(\beta=5\) due to the length of the stock, as it will be discussed in Section 8. The filter width was set in the interval \(2\leq\alpha\leq\beta\) for each value of \(\beta\). With \(\alpha=1\), no good solution was found in any instance. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Dataset** & \begin{tabular}{l} **Number of** \\ **Instances** \\ \end{tabular} & \begin{tabular}{l} **Min-Max Number** \\ **of Item Types** \\ \end{tabular} & \begin{tabular}{l} **Number of** \\ **Different Item** \\ **Types (Std. Dev.)** \\ \end{tabular} & \
\begin{tabular}{l} **Number of** \\ **Orders 1** \\ \end{tabular} &
\begin{tabular}{l} **Stock Length 2** \\ \end{tabular} \\ \hline _hinterding_ & 10 & 8–36 & 4 (10.55) & 20, 50, 60, 126, 200, 400, 600 & 14, 15, 25, 25, 4300, 86, 120, 120, 120, 120 \\ _hard28_ & 28 & 136–189 & 21 (14.43) & 160, 180, 200 & 1000 \\ _hard10_ & 10 & 197–200 & 4 (0.89) & 200 & 100,000 \\  & & & & 114, 96, 57, 111, & \\ _wae\_gau1_ & 17 & 33–64 & 14 (9.39) & 164, 141, 144, 142, 239, 91, 60, 163, 228, 86, 92, 153, 119 \\ _falkenauer\_Li_: & & & & 22, 86, 92, 153, 119 \\ _binpack1_ & 20 & 58–68 & 11 (2.71) & 120 & 150 \\ _binpack2_ & 20 & 71–81 & 8 (2.14) & 250 & 150 \\ _binpack3_ & 20 & 80–81 & 2 (0.40) & 500 & 150 \\ _binpack4_ & 20 & 81 & 1 (0) & 1000 & 150 \\ \hline \hline \end{tabular}  
* Among all instances. 2 For each instance.  
\end{table} 
Table 1: Data sets details used in this study.