On the Dynamic Problem of Optimal Set Partitioning with Determination of Subset Center Coordinates
Abstract
Relevance. Optimal set partitioning is one of the key problems in modern applied mathematics and optimization theory, with wide-ranging applications in logistics, computer science, bioengineering, complex systems modeling, and artificial intelligence. Of particular interest are dynamic variants of set partitioning problems, where the conditions of the problem change over time, and the partitioning must adapt to the evolving system dynamics. In the vast majority of applied problems, optimal set partitioning is directly linked to the minimization of an objective functional, which inherently depends not only on the shapes or contours of the subsets but also on other defining parameters that are crucial for determining the desired subsets. In classical formulations, such parameters often include the centers of the subsets. Practical applications of problems in this form arise in economics, logistics, medicine, architecture, and other areas of human activity.
Objective. The main goal of this study is to formulate a single-product dynamic optimal set partitioning problem with the determination of the coordinates of the centers of the resulting subsets, to develop an algorithm for solving the dynamic problem, to conduct a numerical experiment, and to analyze the obtained results in order to confirm their reliability.
Methods. The primary research methods used in this work include optimization theory techniques, qualitative theory of differential equations, and numerical methods for solving optimization problems.
Results. The main results of the study include the formulation of a single-product dynamic optimal set partitioning problem with determination of subset center coordinates, the development of a solution algorithm, the outcomes of the numerical experiment, and the analysis of the results obtained.
Conclusions. This article presents a novel dynamic optimal set partitioning problem with determination of subset center coordinates. An algorithm for solving the problem is proposed, and a numerical experiment is conducted. The results confirm the validity of the proposed approach and demonstrate its potential applicability to solving real-world problems.
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References
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