# Multicriterial and restricted location problems with polyhedral gauges

• Stefan Nickel
Conference paper
Part of the Operations Research Proceedings book series (ORP, volume 1994)

## Summary

Given a finite set $$Ex = \{ E{x_1},E{x_2}...,E{x_M}\}$$ in the plane, a forbidden region R, we want to find a point $$X \notin \operatorname{int} \,\left( R \right),such\,that \sum\nolimits_{i = 1}^M {{w_i}\gamma E{x_i}\left( {X - E{x_i}} \right)}$$ is minimized.

This is a variant of the well-known Weber Problem, where we measure the distance with polyhedral gauges. The unit ball of a polyhedral gauge may be any convex polyhedron containing the origin. This large class of distance functions allows very general (practical) settings — such as asymmetry — to be modeled. Each Ex i is allowed to have its own gauge γ E x i and the forbidden region R enables us to include negative information in the model.

Efficient algorithms and structural properties for that non-convex optimization problem based on combinatorial methods are presented. Also the connection to point-objective location problems is discussed.

## Keywords

Location Problem Convex Polyhedron Level Curve Combinatorial Algorithm Multiobjective Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Zusammenfassung

Gegeben sei eine endliche Menge $$Ex = \{ E{x_1},E{x_2}...,E{x_M}\}$$ in der Ebene und ein verbotenes Gebiet R. Wir suchen ein $$X \notin \operatorname{int} \,\left( R \right),so\,da{\ss} \sum\nolimits_{i = 1}^M {{w_i}\gamma E{x_i}\left( {X - E{x_i}} \right)}$$ minimiert wird.

Dies ist eine Variante des klassischen Weber Problems, wobei die Entfernungsfunktion durch polyedrische Gauges gegeben ist. Der Einheitskreis eines polyedrischen Gauges kann ein beliebiges konvexes Polyeder sein, welches den Ursprung enthält. Diese Klasse von Entfernungsfunktionen erlaubt die Modellierung sehr allgemeiner (praxisnaher) Gegebenheiten, wie z.B. Asymmetrie. Jedes Ex i kann seinen eigenen Gauge γ E x i besitzen und das verbotene Gebiet R erlaubt es uns, negative Informationen in das Modell einzuarbeiten.

Effiziente Algorithmen und strukturelle Eigenschaften für dieses nichtkonvexe Optimierungsproblem, basierend auf kombinatorischen Methoden, werden dargestellt. Ferner wird der Zusammenhang mit point-objective Lokationsproblemen diskutiert.

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