In this paper an optimized multidimensional hyperspheres packing problem (HPP) is considered for a bounded container. Additional constraints, such as prohibited zones in the container or minimal allowable distances between spheres can also be taken into account. Containers bounded by hyper- (spheres, cylinders, planes) are considered. Placement constraints (non-intersection, containment and distant conditions) are formulated using the phi-function technique. A mathematical model of HPP is constructed and analyzed. In terms of the general typology for cutting & packing problems, two classes of HPP are considered: open dimension problem (ODP) and knapsack problem (KP). Various solution strategies for HPP are considered depending on: a) objective function type, b) problem dimension, c) metric characteristics of hyperspheres (congruence, radii distribution and values), d) container’s shape; e) prohibited zones in the container and/or minimal allowable distances. A solution approach is proposed based on multistart strategies, nonlinear programming techniques, greedy and branch-and-bound algorithms, statistical optimization and homothetic transformations, as well as decomposition techniques. A general methodology to solve HPP is suggested. Computational results for benchmark and new instances are presented.

中文翻译：

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优化包装多维超球体：统一方法

在本文中，针对有界容器考虑了优化的多维超球体包装问题（HPP）。还可以考虑其他约束条件，例如容器中的禁止区域或球体之间的最小允许距离。考虑以超球为边界的容器（球体，圆柱体，平面）。放置约束（不相交，围堵和远距离条件）是使用phi函数技术制定的。建立并分析了HPP的数学模型。就切割和包装问题的一般类型而言，考虑了两类HPP：开放尺寸问题（ODP）和背包问题（KP）。考虑HPP的各种解决方案策略取决于：a）目标函数类型，b）问题维度，c）超球体的度量特征（一致性，半径分布和值），d）容器的形状；e）容器中的禁区和/或最小允许距离。提出了一种基于多启动策略，非线性编程技术，贪婪和分支定界算法，统计优化和同构变换以及分解技术的解决方案。建议了解决HPP的通用方法。给出了基准测试和新实例的计算结果。建议了解决HPP的通用方法。给出了基准测试和新实例的计算结果。建议了解决HPP的通用方法。给出了基准测试和新实例的计算结果。