## 4 Case Studies
### Rotatable Objects  
Objects with large aspect ratios may require alteration of their orientation to be fit into the available space. New sets of objects were generated to test the developed methodologies on the packing problems with rotational degrees of freedom. The orientation angle was restricted to be a multiple of 45\({}^{\circ}\) to cover diagonal placement possibilities while keeping the problem size minimal. In the first investigation, a (45 \(\times\) 45) square and four smaller circles of 39 units diameter were packed into the square domain. Figure 7 shows the optimal packing configuration for these objects. Despite the relatively simple characteristics of the problem, the optimal solution cannot be attained with the ordered placement based on decreasing size. The reason is that all three performance metrics (\(f_{DC}\), \(f_{DBL}\), and \(f_{A1}\)) tend to position the large square on a corner leaving room for only three of the circles. Only using \(f_{DC}\) with the small-to-big ordering leads to the optimal solution indicating that packing smaller objects first can be advantageous in certain problems. While the sequential placement performed ineffectively for this problem, concurrent placement with active constraints  
Figure 6: (**a**) One optimal solution for the objects used in the second study with restrained rotations. The objects perfectly fill the domain for optimal packing. (**b**) Demonstration of two possible positions for the placement of the second object during a sample run. For \(f_{DC}\) metric both positions are equally preferable whereas the bottom location is more preferable for \(f_{A1}\). Figure 7: The optimal solution for the objects used in the first study involving rotational degrees of freedom. provided success rates of 5% and 7% for \(f_{T1}\) and \(f_{A1}\) metrics, respectively. Therefore, if the problem size permits its use, the concurrent optimization strategy can offer versatility in tackling different problems. As another investigation, the concurrent optimization cases were repeated by removing collision and containment constraints. In the absence of constraints, \(f_{T1}\) and \(f_{A1}\) still yielded 5% and 4% success rates, respectively. These results demonstrate that these metrics intrinsically facilitate the containment of rotatable objects in addition to collision avoidance. The second case study concerned packing three rectangles and six circles into a square domain. The quantities and dimensions of the objects used in this study are given in Table 4. With the optimal packing, all objects in the set can be placed into the domain providing a filling ratio of 74.4% as shown in Figure 8.  
The concurrent optimization strategy was not effective for packing this set due to the increased number of objects and extra rotational degrees of freedom. For this problem, the filling ratios were compared for \(f_{DC}\), \(f_{DBL}\), and \(f_{A1}\) metrics while collision and containment constraints were held active. When the objects were ordered in decreasing length (diagonal length for the rectangles and diameter for the circles) the average filling ratios over 100 trials were found as 56.7% and 67.7% using \(f_{DC}\) and \(f_{DBL}\), respectively. For this ordering, the best results were achieved with \(f_{A1}\), which yielded an average filling ratio of 70.8%. In this case, \(f_{A1}\) outperformed the other metrics since it prefers positioning the smaller rectangles right next to the large rectangle, which is diagonally packed in the first step. Such placement leaves sufficient room at the corners where the large circles can be packed. When area-wise sorting was used, \(f_{DC}\), \(f_{DBL}\), and \(f_{A1}\) provided 66.3%, 60.4%, and 56.5% mean filling ratios, respectively. Hence, the relative effectiveness of the utilized metrics can be significantly influenced by the ordering strategy.