# An Improved Adaptive Genetic Algorithm for Two-Dimensional Rectangular Packing Problem
## 4 Computational Experiments
### Compare the Results with Other Algorithms  
We compare the proposed algorithm with the algorithms in Ref. [43; 44; 45]. A hybrid simulated annealing algorithm (HSA) proposed by [PERSON] et al. [43] uses a greedy strategy and simulated annealing to obtain an improved solution. A least-waste-first heuristic algorithm (LWF) proposed by [PERSON] et al. [44] presents a least wasted first heuristic algorithm to find the desirable solution. An iterative bidirectional heuristic placement (IBHP) proposed by [PERSON] et al. [45] combines the constructive heuristic algorithms of bidirectional heuristic placement (BHP) and iterated local search (ILS) with a shift strategy to improve the solution. Considering that Ref. [44] lacks some test instances, the results of LWF and HSA are taken directly from Ref. [43]. The HSA and LWF in C++ programming language were run on a Dell GX270 with 2.60 GHz CPU and 512 MB memory. The IBHP in C programming language was performed on an Intel Core i5 CPU at 2.90 GHz with 4 GB of RAM running on the Ubuntu 12.04 operating system. The comparisons of the average filling rate and the average calculation time between LWF, HSA, IBHP, and HAGA are shown in Table 5. The '-' in Table 5 means that the results are not reported by the corresponding algorithm. From Table 5, it can be observed that the average filling rate of HAGA on the C11-C73, cgcut1-cgcut3, and ngcut12 instances is 99.8%, 97.5755%, and 86.5549%, respectively, which outperforms LWF, HSA, and IBHP. Compared with LWF for these instances, the average filling rate of HAGA can be increased by 0.5356%, 1.6604%, and 0.999%, respectively. The maximum average filling rate can be improved by 1.6604% on cgcut1-cgcut3 for LWF. The average filling rate of HAGA is better than HSA except for N1-N12. Compared with IBHP for C11-C73, cgcut1-cgcut3, and ngcut1-ngcut12 instances, the average filling rate of HAGA can be increased by 0.0527%, 0.486%, and 1.1835%, respectively. Therefore, the HAGA achieved better quality for C11-C73, cgcut1-cgcut3, and ngcut1-ngcut12 instances within reasonable time, with the elimination of equipment and language differences. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instances**} & \multirow{2}{*}{\(\mathbf{n}\)} & \multirow{2}{*}{\(\mathbf{W}\)} & \multirow{2}{*}{\(\mathbf{H}\)} & \multirow{2}{*}{**Opt.**} & \multicolumn{2}{c}{**AGA [26]**} & \multicolumn{2}{c}{**AGA [29]**} & \multicolumn{2}{c}{**HAGA \\ \cline{5-10}  & & & & & **F/\%** & **Time/s** & **F/\%** & **Time/s** & **F/\%** & **Time/s** \\ \hline beng8 & 120 & 40 & 101 & 99.6782 & 98.9697 & 184.26 & 98.8132 & 193.07 & 99.6782 & 123.85 \\ beng9 & 160 & 40 & 126 & 99.3651 & 98.8396 & 1063.11 & 99.3651 & 880.52 & 99.3651 & 735.11 \\ beng10 & 200 & 40 & 156 & 99.6314 & 98.6238 & 1197.35 & 98.7903 & 1206.47 & 98.7342 & 1178.83 \\ \hline \multicolumn{10}{c}{Average} & \multicolumn{1}{c}{91.6365} & \multicolumn{1}{c}{130.29} & \multicolumn{1}{c}{91.7819} & \multicolumn{1}{c}{126.6} & \multicolumn{1}{c}{91.8369} & \multicolumn{1}{c}{110.76} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instances**} & \multicolumn{2}{c}{**LWF**} & \multicolumn{2}{c}{**HSA**} & \multicolumn{2}{c}{**IBHP**} & \multicolumn{2}{c}{**HAGA**} \\ \cline{2-10}  & **F/\%** & **Time/s** & **F/\%** & **Time/s** & **F/\%** & **Time/s** & **F/\%** & **Time/s** \\ \hline J1-J2 & - & - & - & - & - & - & - & 99.96 & 14.45 \\ C11-C73 & 99.4444 & 36.79 & 99.8078 & 41.27 & 99.9273 & - & 99.98 & 232.29 \\ N1-N12 & 99.748 & 29.83 & 99.9113 & 35.23 & 99.97 & - & 99.74 & 913.76 \\ cgcut1- & 95.9151 & 33.42 & 96.5341 & 40.00 & 97.0895 & - & 97.5755 & 99.55 \\ gcut1- & 90.6318 & 42.89 & 89.5653 & 60.02 & 89.6397 & - & 89.6145 & 108.36 \\ gcut1- & 85.5559 & 2.76 & 85.1418 & 25.01 & 85.3714 & - & 86.5549 & 18.38 \\ ngcut12 & 99.3778 & 6.07 & 99.3312 & 6.05 & 99.4326 & - & 99.3429 & 228.12 \\ beng10 & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}