# Algorithmic Advances for 1.5-Dimensional Two-Stage Cutting Stock Problem
## 5 Computational Results
### Discussion  
The performance of the algorithms was evaluated based on several metrics, including the number of patterns generated at the initial and second stages (\(n_{1}\) and \(n_{2}\)), the average number of intermediate orders (\(m_{1}\)), the average percentage difference in the objective value (\(\Delta\%\) Int. Obj. Value), and the average difference in the total computation time (\(\Delta\) Tot. time). Table 2 presents the computational results for instances II and I2. For smaller instances of I1, which feature single length ratios and single lengths for finished rolls, both algorithms performed adequately. The G&S algorithm, however, tended to generate a significantly larger number of second-stage patterns (\(n_{2}\)) compared to the R&CG algorithm, leading to increased computational effort. Despite this, the G&S algorithm achieved comparable integer objective values, as indicated by the small average differences in the objective value (\(\Delta\%\) Int. Obj. Value). The R&CG algorithm, with its fewer generated patterns, demonstrated greater computational efficiency, requiring less total computation time in most cases. In bigger instances of II and in the more complex instances of I2, which incorporate multiple length ratios, multiple lengths for finished rolls, and enforce a minimum trim loss at the first stage, the performance gap between the two algorithms became more pronounced. The G&S algorithm continued to generate a substantially higher number of second-stage patterns, which significantly increased the computational time. For example, in instances I2 with \(m_{2}=50\), the G&S algorithm generated an average of 815,093 s-stage \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **II** & **I2** \\ \hline Stock size & 10,000 & 10,000 \\ Intermediate order width range & (1250; 2500) & (2000; 3000) \\ Length ratios & 20 & \(\{20;30\}\) \\ Order width & UR(300; 500) & UR(100; 500) \\ Order length & 1 & US\(\{1;2;3\}\) \\ Order amount & UR(10; 100) & UR(10; 100) \\ First stage min trim & 0 & 100 \\ Second stage min trim & 50 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of instance generator. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline \multirow{3}{*}{**Instance**} & \multirow{3}{*}{\(m_{2}\)} & \multicolumn{3}{c}{**G& \&S**} & \multicolumn{3}{c}{**R&CG**} \\ \cline{3-10}  & & & **Avg. \(n_{1}\)** & **Avg. \(n_{2}\)** & **Avg. \(m_{1}\)** & **Avg. \(n_{1}\)** & **Avg. \(n_{2}\)** & **Avg. \(m_{1}\)** & **Avg. \(m_{1}\)** & **Avg. \(m_{1}\)** \\ \hline I1 & 10 & 27.0 & 21,857.4 & 1.0 & 2.0 & 36.1 & 1.0 & \(+0.7\) & \(-5.6\) \\ I1 & 20 & 26.0 & 42,098.3 & 1.0 & 2.0 & 82.5 & 1.0 & \(-0.2\) & \(-45.5\) \\ I1 & 30 & 34.0 & 51,636.1 & 1.0 & 2.0 & 113.9 & 1.0 & \(-43.3\) & \(-232.0\) \\ I1 & 40 & 26.5 & 39,165.6 & 1.0 & 2.0 & 155.1 & 1.0 & \(-86.1\) & \(+3.1\) \\ I1 & 50 & 25.6 & 39,207.0 & 1.0 & 2.0 & 192.2 & 1.0 & \(-86.1\) & \(-64.7\) \\ \hline I2 & 10 & 25.3 & 39,604.3 & 5.0 & 17.9 & 98.5 & 17.9 & \(-59.9\) & \(-362.8\) \\ I2 & 20 & 19.9 & 165,498.6 & 5.0 & 18.0 & 236.4 & 18.0 & \(-75.8\) & \(-169.0\) \\ I2 & 30 & 19.0 & 248,692.0 & 5.0 & 18.0 & 351.1 & 18.0 & \(-85.8\) & \(-332.0\) \\ I2 & 40 & 23.9 & 343,933.2 & 5.0 & 18.0 & 485.3 & 18.0 & \(-87.6\) & \(-443.5\) \\ I2 & 50 & 14.0 & 815,093.0 & 10.0 & 18.0 & 644.0 & 18.0 & \(-82.2\) & \(-595.6\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational results for instances II and I2. patterns, whereas the R&CG algorithm generated only 644 patterns. This difference resulted in the G&S algorithm having a much longer computation time, with an average decrease in the total time of -595.6 s compared to the R&CG algorithm. Moreover, the G&S algorithm exhibited a worse performance in terms of the solution quality for these complex instances. The average difference in the integer objective value (\(\Delta\%\) Int. Obj. Value) was negative, indicating that the R&CG algorithm found better solutions with lower trim losses.