Using tropical optimization to solve constrained minimax single-facility location problems with rectilinear distance

TL;DR

This paper applies tropical optimization techniques to solve constrained minimax single-facility location problems with rectilinear distance, providing explicit closed-form solutions and demonstrating practical applications in surveillance system design.

Contribution

It introduces a novel tropical optimization approach for constrained location problems, offering new closed-form solutions and a transformation technique applicable to various optimization challenges.

Findings

Derived explicit solutions for constrained minimax location problems.

Extended tropical optimization methods to new application areas.

Demonstrated practical implementation with numerical examples and a surveillance system case.

Abstract

The aim of this paper is twofold: first, to extend the area of applications of tropical optimization by solving new constrained location problems, and second, to offer new closed-form solutions to general problems that are of interest to location analysis. We consider a constrained minimax single-facility location problem with addends on the plane with rectilinear distance. The solution commences with the representation of the problem in a standard form, and then in terms of tropical mathematics, as a constrained optimization problem. We use a transformation technique, which can act as a template to handle optimization problems in other application areas, and hence is of independent interest. To solve the constrained optimization problem, we apply methods and results of tropical optimization, which provide direct, explicit solutions. The results obtained serve to derive new solutions of the location problem, and of its special cases with reduced sets of constraints, in a closed form, ready for practical implementation and immediate computation. As illustrations, numerical solutions of example problems and their graphical representation are given. We conclude with an application of the results to optimal location of the central monitoring facility in an indoor video surveillance system in a multi-floor building environment.