Algebraic solution to a constrained rectilinear minimax location problem on the plane

TL;DR

This paper presents an algebraic method for solving a constrained rectilinear minimax location problem on the plane, utilizing eigenvalues and eigenvectors of matrices in idempotent algebra to find optimal solutions.

Contribution

It introduces a novel algebraic approach based on eigenvalue properties in idempotent algebra to solve a specific constrained location problem.

Findings

Provides a new algebraic solution reducing the problem to eigenvalue computation

Demonstrates the approach on a rectilinear minimax location problem with rectangular constraints

Shows the effectiveness of eigenvalue methods in constrained location optimization

Abstract

We consider a constrained minimax single facility location problem on the plane with rectilinear distance. The feasible set of location points is restricted to rectangles with sides oriented at a 45 degrees angle to the axes of Cartesian coordinates. To solve the problem, an algebraic approach based on an extremal property of eigenvalues of irreducible matrices in idempotent algebra is applied. A new algebraic solution is given that reduces the problem to finding eigenvalues and eigenvectors of appropriately defined matrices.