A New Class of Irregular Packing Problems Reducible to Sphere Packing in Arbitrary Norms

Abstract

Packing irregular objects composed by generalized spheres is considered. A generalized sphere is defined by an arbitrary norm. For three classes of packing problems, balance, homothetic and sparse packing, the corresponding new (generalized) models are formulated. Non-overlapping and containment conditions for irregular objects composed by generalized spheres are presented. It is demonstrated that these formulations can be stated for any norm. Different geometrical shapes can be treated in the same way by simply selecting a suitable norm. The approach is applied to generalized spheres defined by Lp norms and their compositions. Numerical solutions of small problem instances obtained by the global solver BARON are provided for two-dimensional objects composed by spheres defined in Lp norms to demonstrate the potential of the approach for a wide range of engineering optimization problems.

Packing Spheres into a Minimum-Height Parabolic Container

“…The proposed approach was numerically tested for eleven instances of the problem ( 9), (10) with different (a) dimensions, n = 2, 3, 4, 5; (b) numbers of spheres, m = 50, 100, 200; b) radii, r j , j = 1, . .…”

“…Since we cannot t j explicitly, then we need to solve the optimization problems of the form (7) for each sphere S j (y j ) on each iteration of the solution process of the problem (10), (11). We do not use the solvers BARON [28] or IPOPT [29] for the original problem because of the dynamic nature of the Φ-function.…”

“…Apply the modification of the FDM to solve the problem (10), (11) for a set of feasible starting points. 4.…”

“…Taking into account the problem ( 15), ( 16), the following step-by-step algorithm is employed to solve problem (10), (11).…”

“…Rich theoretical and empirical results are presented in these papers, where various NLP models and exact/heuristic/metaheuristic solution techniques are accompanied by extensive computational experiments to demonstrate the efficiency of the proposed approaches. Different NLP models and solution algorithms for packing 3D objects into regular 3D containers (cuboids, spheres, and cylinders) can be found, e.g., in [9,10] for spherical and in [11] for ellipsoidal shapes, together with corresponding empirical results obtained for different numbers of objects and various container shapes. In all these works, geometric tools for modeling non-overlapping and containment conditions in Euclidean and non-Euclidean [4,5,9] metrics are provided.…”