Quasi-Kernels of Outranking Relations

  • Pierre Hansen
  • Martine Anciaux-Mundeleer
  • Philippe Vincke
Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 130)

Abstract

In several decision-aid methods for discrete multiple criteria problems an outranking relation is used to express the preferences which are sufficiently strongly established on a given set of actions. The actions corresponding to a kernel of the graph G induced by the outranking relation can be selected for further study — provided such a kernel exists. As this is not always the case, ROY recently proposed to select the actions corresponding to a quasi-kernel of G. In this paper, we study how this can be done. The weakness of a quasi-kernel of G is defined as the number of vertices of G not in Q and having no successor in Q. The problem of determining a quasi-kernel of minimum weakness is expressed as a mixed-integer program and a specialized branch-and-bound algorithm is proposed to solve it. Computational experience on random graphs is discussed. A classification of the vertices of G according to whether they belong or not to one or all quasi-kernels of G or to one or all quasi-kernels of minimum weakness of G is presented. Finally, it is shown how the class of any vertex of G can be determined by applying a few times the algorithm after fixing some of the variables at 0 or at 1.

Keywords

Metra 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • Pierre Hansen
    • 1
  • Martine Anciaux-Mundeleer
    • 2
  • Philippe Vincke
    • 2
  1. 1.Institut d’Economie Scientifique et de GestionLilleFrance
  2. 2.Université Libre de BruxellesBruxellesBelgium

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