# Nonlinear Optimization and Adaptive Heuristics for Solving Irregular Object Packing Problems
## 5 Further Scalable Object Configuration Problems  
### Scalable Models  
Conceptually, scalable object packings can be characterized by postulating that all packed objects have a similar or identical geometric structure--while their sizes and shapes still are or can be different to some extent, depending on the type of the objects. For example, one can think of collections of circles, spheres, ellipses, or ovals to pack: next, we will present models for the last three types of problems. To create scalable optimization models of interest, we have to guarantee by suitable model parameterization that the resulting models do not become \"too simple\" to handle. Although infinite tessellations created in finite regions can have wonderful aesthetic value (think of some historical mosaic designs or modern works by [PERSON] and followers), here we want to define finite packings, which become unbounded in size as the number of objects increases. For example, in the GCP model class, we can consider a collection of circles with radii \(r_{i}=i^{1/2}\), or \(r_{i}=i\) for \(i=1,\ \ldots,\ N\) to be packed into a corresponding \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No. of Circles** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline \hline \multicolumn{5}{l}{Deviation from best known} \\ \hline conopt & 1.0000000 & 1.0198816 & 1.0176073 & 1.041774 & 1.0166012 & 1.1035550 \\ ipopt & 1.0000000 & 1.0430155 & 1.0176073 & 1.0256125 & 1.0479264 & 1.0625176 \\ lgo & 1.0000000 & 1.0000000 & 1.0176073 & 1.0000000 & 1.0277057 & 1.0250992 \\ lgo & 1.0000000 & 1.0430155 & 1.0176073 & 1.0256125 & 1.0479264 & 1.0630476 \\ minos & 1.0000000 & 1.0430155 & 1.0176073 & 1.0134856 & 1.0542614 & 1.0648117 \\ snopt & 1.0000000 & 1.0430155 & 1.0176070 & 1.0256126 & 1.0479264 & 1.0648117 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 1: A solved GCP model instance. container circle. In both of these cases, the size of the corresponding optimal container will approach infinity, as \(N\to\infty\). Without going into further details, we remark that properly combined scalable OP models directly lead to new scalable OP models. As an example, think of packing finite collections of object types--each defined by iterative formulas--into a container. In scalable OP problems with objects having the same basic geometry (such as collections of circles, or spheres, ellipses, ovals, triangles, rectangles, polygons, etc. ), the number of decision variables that serve to jointly describe the object positions is typically a _linear_ function of the number of objects considered. The number of constraints is partially dictated by the pairwise non-overlapping criterion: the number of such constraints increases quadratically with the number of packed objects. Additional variables and constraints may be present, depending on the problem definition and structure. The GCP model class can serve to illustrate the above observations. For conceptual clarity, we refer to the typology of cutting and packing problems suggested by [PERSON] et al. [54], which facilitates the organization and categorization of the OP literature. In terms of their problem classification, the GCP problem class and each of the scalable model classes presented here are strongly heterogeneous assortment problems with non-identical packed items and non-orthogonal layouts with free rotations allowed, where one container is considered with a single objective (volume minimization) in the presence of additional no-overlap and strict containment constraints. To illustrate further scalable OP model classes, next, we discuss specific cases and examples based on our related studies [55; 56; 57; 58; 59; 60]. While [55] is a direct extension of the GCP model, the three other model-types discussed are more complicated to describe: therefore, for these models, only concise problem formulations are presented and illustrated here, with reference to our related works.