Optimized packing multidimensional hyperspheres: a unified approach.

Area of Science:

• Operations Research
• Computational Geometry
• Optimization

Background:

• The hyperspheres packing problem (HPP) is a complex optimization challenge with applications in various fields.
• Existing methods often struggle with multidimensional spaces and complex constraints.

Purpose of the Study:

• To develop an optimized mathematical model for the multidimensional hyperspheres packing problem (HPP) in bounded containers.
• To incorporate additional constraints such as prohibited zones and minimum distances between hyperspheres.
• To propose a general methodology for solving HPP considering diverse parameters.

Main Methods:

• Formulation of placement constraints (non-intersection, containment, distant conditions) using the phi-function technique.
• Mathematical modeling and analysis of HPP, including open dimension problems (ODP) and knapsack problems (KP).
• Application of a hybrid solution approach combining multistart strategies, nonlinear programming, greedy and branch-and-bound algorithms, statistical optimization, homothetic transformations, and decomposition techniques.

Main Results:

• A comprehensive mathematical model for multidimensional HPP with various constraints is presented.
• The study explores diverse solution strategies tailored to objective functions, problem dimensions, hypersphere characteristics, container shapes, and specific constraints.
• Computational results for benchmark and new instances demonstrate the effectiveness of the proposed approach.

Conclusions:

• A general and adaptable methodology for solving the multidimensional hyperspheres packing problem is proposed.
• The developed approach effectively handles complex constraints and diverse problem configurations.
• The findings contribute to advancing optimization techniques for packing problems in bounded containers.