Abstract

The work considers a minimax two-facility location problem in a multidimensional space with Chebyshev distance under interval constraints on the feasible location area. The problem involves two groups of facilities with known coordinates and the objective to select optimal location coordinates for two new facilities under given constraints. The location of the new facilities is considered optimal if it minimizes the maximum of the following values: the distance between the first new facility and the farthest facility in the first group, the distance between the second new facility and the farthest facility in the second group, and the distance between the first and second new facilities. The location problem is formulated as a multidimensional optimization problem in terms of tropical mathematics, a field focused on the theory and applications of algebraic systems with idempotent operations. A direct analytical solution to the problem is derived using methods and results of tropical optimization. The obtained result describes the optimal location area for the new facilities in a parametric form that enables the formal analysis of solutions and direct calculations.

中文翻译：

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具有切比雪夫距离的空间中双设施选址问题的求解

摘要

该工作考虑了在可行位置区域的区间约束下具有 Chebyshev 距离的多维空间中的极小极大双设施位置问题。该问题涉及两组坐标已知的设施，目标是在给定约束下为两个新设施选择最佳位置坐标。如果新设施的位置最小化以下值中的最大值，则认为新设施的位置是最佳的：第一个新设施与第一组中最远设施之间的距离，第二个新设施与第二组中最远设施之间的距离，以及第一个和第二个新设施之间的距离。定位问题在热带数学方面被表述为多维优化问题，一个专注于具有幂等运算的代数系统的理论和应用的领域。使用热带优化的方法和结果推导出该问题的直接解析解。获得的结果以参数形式描述了新设施的最佳位置区域，可以对解决方案进行正式分析和直接计算。