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Quasi-Kernels of Outranking Relations

  • Pierre Hansen
  • Martine Anciaux-Mundeleer
  • Philippe Vincke
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

Random Graph Multiple Criterion Decision Initial Vertex Terminal Vertex Regression Step 
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.

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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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