Geometry and Analytics of the Multifacility Weber Problem

TL;DR

This paper presents a radical-based solution for the multifacility Weber problem, providing explicit formulas and conditions for network existence, and explores network dynamics and extensions to the general case.

Contribution

It introduces a novel radical-based method for solving the multifacility Weber problem with explicit formulas and conditions, extending to general cases.

Findings

Explicit formulas for facility coordinates are derived.

Conditions for network existence are established.

Network dynamics under parameter variation are analyzed.

Abstract

For the Weber problem of construction of the minimal cost planar weighted network connecting four terminals with two extra facilities, the solution by radicals is proposed. The conditions for the existence of the network in the assumed topology and the explicit formulae for coordinates of the facilities are presented. The obtained results are utilized for investigation of the network dynamics under variation of parameters. Extension of the results to the general Weber problem is also discussed.