Solution of a Two-Facility Location Problem in a Space with Chebyshev Distance

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.