Abstract This paper addresses a mixed-integer linear programming model for solving the Multidimensional Multi-Way Number Partitioning Problem (MDMWNPP), the most general version of the family of Number Partitioning Problems. First, a contextualization concerning the Two-Way Number Partitioning Problem (TWNPP), the Multi-Way Number Partitioning Problem (MWNPP), the Multidimensional Two-Way Number Partitioning Problem (MDTWNPP), and the approached problem are presented. MDMWNPP is a generalization of MDTWNPP for the number k of subsets greater than two. After the proposed model is introduced, some properties are shown, and the main constraints are proved. To validate the consistency of the proposed mathematical formulation, a comparison, using instances from the literature, between the proposed model for k=2 and a model from the literature for MDTWNPP, is performed. The proposed model showed competitive results against this specialized model and, besides, solved instances of the problem for values of k equals to 3 and 4.

中文翻译：

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求解多维多维数划分问题的混合整数线性规划模型

摘要 本文介绍了一种混合整数线性规划模型，用于解决多维多路数划分问题 (MDMWNPP)，这是数划分问题族的最一般版本。首先，介绍了关于双向数划分问题 (TWNPP)、多向数划分问题 (MWNPP)、多维二维数划分问题 (MDTWNPP) 和解决问题的上下文。MDMWNPP 是 MDTWNPP 对大于 2 的子集数量 k 的推广。引入所提出的模型后，展示了一些性质，并证明了主要约束条件。为了验证所提出的数学公式的一致性，使用文献中的实例进行比较，在 k=2 的建议模型和 MDTWNPP 文献中的模型之间执行。所提出的模型显示出与此专用模型相比具有竞争力的结果，此外，还解决了 k 值等于 3 和 4 的问题实例。