IJPAM: Volume 47, No. 4 (2008)

ITERATIVE REFINEMENT FOR INVARIANT SUBSPACES
OF MATRICES WITH APPLICATION TO
THE PROMETHEE-GAIA METHOD

Mario Ahues$^1$, David Duvivier$^2$, Alain Largillier$^3$, Nadine Meskens$^4$
$^{1,3}$Laboratoire de Mathématiques de l'Université de Saint-Etienne (LaMUSE)
Université Jean Monnet
23 Rue du Dr. Paul Michelon
Saint-Étienne, 42100, FRANCE
$^1$e-mail: mario.ahues@univ-st-etienne.fr
$^3$e-mail: larg@univ-st-etienne.fr
$^2$Laboratoire d'Informatique du Littoral
Université du Littoral Côte d'Opale
Maison de la Recherche Blaise Pascal
50, Rue Ferdinand Buisson - BP 719, Calais, 62228, FRANCE
e-mail: david.duvivier@lil.univ-littoral.fr
$^4$Louvain School of Management and Catholic
University of Mons (FUCaM)
151, Chaussée de Binche, Mons, 7000, BELGIUM
e-mail: nadine.meskens@fucam.ac.be


Abstract.Recent results in operational research have provided decision makers with an adaptable tool that is able to synthesize the performance measures of several solutions to be compared in a short time. In the context of multicriteria limited-time decision making problems, two of the authors have developed hybrid models composed of two mathematical models, a set of dedicated heuristics, a stochastic local search, meta-heuristic and a simulation model. According to the decision makers, the solutions are ranked on the basis of several criteria whose importance determines this ranking. A multicriteria method is incoporated into the hybrid one. In order to summarize the huge amount of resulting data/information, we have embeded the Promethee II multicriteria method and the Gaia plane. This extension requires to compute eigenelements on the output produced by Promethee what is done in two steps: 1) Get rough approximations to the desired part of the spectrum and its spectral subspace; 2) Refine them with an iterative scheme. A Newton-based scheme is proposed and applied to a matrix issued from the Promethee-Gaia method.

Received: August 18, 2008

AMS Subject Classification: 32P05

Key Words and Phrases: operational research, hybrid models, Promethee II multicriteria method, Gaia plane, Promethee-Gaia method

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 47
Issue: 4