# An Improved Adaptive Genetic Algorithm for Two-Dimensional Rectangular Packing Problem
## 4 Computational Experiments
### Computational Experiment for Zero-Waste Instances
#### 4.1.3 Computational Experiment for \(N\) Instances  
The \(N\) benchmark instances were generated by [PERSON] et al., and optimal solutions were also known beforehand. It includes 13 groups of test instances, and the number of items to be packed increases from 10 in the first group to 3152 in the thirteenth group. For detailed information on \(N\) instances, refer to [PERSON] et al. [19]. The comparison results of our experiments are shown in Table 3. It should be noted that the '-' in N13 means that no experimental calculation has been performed. We can recognize from the results shown in Table 3 that SGA can only obtain a 100% filling rate for N1-N3 instances with a small scale. Both AGA [26] and AGA [29] fail to obtain a 100% filling rate in five instances, and HAGA fails to obtain a 100% filling rate in three instances N10-N12. Comparing the N5 and C41-C43 instances of almost the \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instances**} & \multirow{2}{*}{\(n\)} & \multirow{2}{*}{**Sheet**} & \multicolumn{2}{c}{**SGA**} & \multicolumn{2}{c}{**AGA [26]**} & \multicolumn{2}{c}{**AGA [29]**} & \multicolumn{2}{c}{**HAGA**} \\ \cline{3-11}  & & & **F/\%** & **Time/s** & **F/\%** & **Time/s** & **F/\%** & **Time/s** & **F/\%** & **Time/s** \\ \hline C11 & 16 & \(20\times 20\) & 100 & 2.41 & 100 & 2.14 & 100 & 2.16 & 100 & 1.64 \\ C12 & 17 & \(20\times 20\) & 100 & 2.49 & 100 & 2.33 & 100 & 2.43 & 100 & 1.7 \\ C13 & 16 & \(20\times 20\) & 100 & 2.28 & 99.65 & 2.14 & 100 & 2.1 & 100 & 1.62 \\ C21 & 25 & \(40\times 15\) & 100 & 10.83 & 100 & 8.55 & 100 & 7.79 & 100 & 5.54 \\ C22 & 25 & \(40\times 15\) & 100 & 9.03 & 100 & 7.87 & 100 & 7.41 & 100 & 5.22 \\ C23 & 25 & \(40\times 15\) & 100 & 9.19 & 100 & 7.58 & 100 & 7.23 & 100 & 5.37 \\ C31 & 28 & \(60\times 30\) & 100 & 13.41 & 100 & 11.82 & 100 & 10.72 & 100 & 8.92 \\ C32 & 29 & \(60\times 30\) & 100 & 13.73 & 100 & 11.54 & 100 & 10.94 & 100 & 9.12 \\ C33 & 28 & \(60\times 30\) & 100 & 12.81 & 100 & 10.81 & 100 & 10.66 & 100 & 8.86 \\ C41 & 49 & \(60\times 60\) & 100 & 22.52 & 100 & 21.94 & 100 & 21.07 & 100 & 19.8 \\ C42 & 49 & \(60\times 60\) & 100 & 23.28 & 100 & 21.55 & 100 & 20.9 & 100 & 19.17 \\ C43 & 49 & \(60\times 60\) & 100 & 22.37 & 100 & 21.87 & 100 & 21.32 & 100 & 20.14 \\ C51 & 73 & \(60\times 90\) & 99 & 338.13 & 100 & 46.23 & 100 & 70.11 & 100 & 41.82 \\ C52 & 73 & \(60\times 60\) & 100 & 57.46 & 100 & 42.53 & 100 & 40.27 & 100 & 38.31 \\ C53 & 73 & \(60\times 60\) & 99.62 & 327.17 & 100 & 44.07 & 100 & 68.44 & 100 & 42.19 \\ C61 & 97 & \(80\times 120\) & 98.69 & 483.98 & 100 & 127.85 & 100 & 106.97 & 100 & 100.85 \\ C62 & 97 & \(80\times 120\) & 100 & 243.48 & 100 & 53.43 & 100 & 132.54 & 100 & 54.78 \\ C63 & 97 & \(80\times 120\) & 100 & 386.6 & 100 & 111.63 & 100 & 187.37 & 100 & 105.42 \\ C71 & 196 & \(160\times 240\) & 98.43 & 1686.13 & 98.94 & 1643.48 & 99.08 & 1603.96 & 99.52 & 1551.02 \\ C72 & 197 & \(160\times 240\) & 98.67 & 1825.58 & 98.97 & 1779.65 & 99.23 & 1748.54 & 100 & 1501.37 \\ C73 & 196 & \(160\times 240\) & 98.34 & 1672.37 & 98.82 & 1619.88 & 98.94 & 1572.38 & 100 & 1335.34 \\  & Average & & 99.65 & 341.2 & 99.83 & 266.61 & 99.87 & 269.3 & 99.98 & 232.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the filling rate (%) and calculation time (s) on the \(C\) instances. same scale, you will find that the N5 instance consumes a lot more time than the C41-C43 instances. For example, the calculation time of AGA [26] is 22.37 s on the C43 instance, but it takes 33.98 s on the N5 instance. These indicate that solving \(N\) instances is more difficult than solving the \(C\) instances, but the HAGA is still better than the two types of adaptive methods AGA [26] and AGA [29] to solve the 2 DRP problem. Then, by comparing SGA and the adaptive genetic algorithm, we figure out that AGA [26], AGA [29], and HAGA reduce the time by 90.46, 114.08, and 154.49 s, respectively compared with SGA, and the average filling rate of the adaptive genetic algorithm is generally higher than that of SGA.