## 4 Case Studies
### Non-Rotatable Objects  
The developed methodology was initially utilized to obtain packing configurations for non-rotatable objects. The first study involved sets of rectangular objects whose heights and widths are given in Table 1. These sets were obtained by splitting one rectangle into two new ones at each step as demonstrated in Figure 5. Hence, the number of rectangles was regularly increased from 4 to 9 while going from the first case (a) to the last one (f). \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Number of Objects** & \multicolumn{8}{c}{**(Height, Width)**} \\ \hline
4 & \((50,60)\) & \((50,40)\) & \((30,100)\) & \((20,100)\) & & & & \\
5 & \((50,60)\) & \((50,40)\) & \((30,70)\) & \((30,30)\) & \((20,100)\) & & & \\
6 & \((50,60)\) & \((50,40)\) & \((20,70)\) & \((10,70)\) & \((30,30)\) & \((20,100)\) & & & \\
7 & \((50,60)\) & \((50,40)\) & \((20,70)\) & \((10,70)\) & \((30,20)\) & \((30,10)\) & \((20,100)\) & & \\
8 & \((50,60)\) & \((50,40)\) & \((20,70)\) & \((10,70)\) & \((30,20)\) & \((30,10)\) & \((15,100)\) & \((5,100)\) & \\
9 & \((50,60)\) & \((50,40)\) & \((15,70)\) & \((5,70)\) & \((10,70)\) & \((30,20)\) & \((30,10)\) & \((15,100)\) & \((5,100)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Heights and widths of the rectangles used in the sets used in the first study that involves no rotations. Figure 5: Sample optimal solutions for different sets of rectangles used in the first study with restrained rotations. When packed optimally the objects perfectly fill the domain and their number in the sets increases from (**a–f**) as one rectangle is split into two new ones at each step. Table 2 shows the success rates in packing all objects of each set for different optimization strategies and objective functions. For each case, the average results for 100 computations are presented. In the concurrent optimization, the effectiveness of maximizing total ones decreased with the increasing number of objects. Maximizing adjacent ones showed a similar trend but gave slightly better results for the higher number of objects. No collision avoidance constraints were considered in these studies since the higher values of the objectives also favor reducing the overlaps. For the sequential optimization, the optimal solutions were found for all trials for the sets with four and five objects. For the analyzed sets, sorting by diagonal length generally performed better compared to area sorting, which was more slightly more effective for only intermediate numbers (\(6,7\)) of objects. For the length sorting, using the distance to the bottom-left corner as the objective function provided the best results. However, maximizing adjacent ones also worked fairly well providing a 75% success rate for 9 objects. For the second study, another set of objects with the quantities and dimensions specified in Table 3 was used. This set perfectly occupies the domain for optimal packing, which requires placement of the objects according to a regular pattern. For concurrent optimization, the optimal packing could not be found using \(f_{T1}\) as the performance metric, where \(f_{A1}\) still succeeded 6 times out of 100 trials. For sequential placement, \(f_{A1}\) delivered the best results by far with its 40% success rate, where \(f_{DC}\) and \(f_{DBL}\) succeeded in 7% and none of the trials, respectively. The effectiveness of \(f_{A1}\) originates from its inclination towards maximizing the line of contact, which in turn aids the formation of the required pattern. In Figure 6, (a) one optimal solution for the second study and (b) positioning preferences of \(f_{DC}\) and \(f_{A1}\) during a sample run are demonstrated.