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)


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.




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Agin,“Optimum Seeking with Branch and Bound; Manag. Sci., 13, (1966) 176–185.CrossRefGoogle Scholar
  2. [2]
    M. Anciaux-Mundeleer and P. Hansen,“On Kernels in Strongly Connected Graphs; Networks, to appearGoogle Scholar
  3. [3]
    C. Berge, Graphes et hypergraphes, Paris (Dunod, 1970 )Google Scholar
  4. [4]
    C. Berge,’Nouvelles extensions du noyau d’un graphe et ses applications en théorie des jeux:’Publi. Econ., 6 (1973) 5–11Google Scholar
  5. [5]
    P. Bertier et J. de Montgolfier,’bn Multicriteria Analysis: An Application to a Forest Management Problem;’Metra, 13, (1974) 33–45.Google Scholar
  6. [6]
    P. Buffet, J.P. Gremy, M. Marc et B. Sussman,“Peut-on choisir en tenant compte de critères multiples? Une méthode (ELECTRE) et trois applicationA’, Metra, 6, (1967), 283–316.Google Scholar
  7. [7]
    J.L. Guigou,’Un French Location Models for Production Units’ Regional and Urban Economics, 1 (1971).Google Scholar
  8. [8]
    P. Hansen, Programmes mathématiques en variables 0–1, Thèse, Université de Bruxelles (1974).Google Scholar
  9. [9]
    P. Hansen,“Les procédures d’exploration et d’optimisation par séparation et évaluation,”29–65, in B. Roy (ed.). Combinatorial Frog. (Reide1,1975).Google Scholar
  10. [10]
    E. Jacquet–Lagrèze,“How we can use the Notion of Semi-orders to Build Outranking Relations in Multiple Criteria Decision Making, Mctra, 13, (1974) 59–86Google Scholar
  11. [11]
    E. Lawler and D. Wood,“Branch and Bound Methods, A Survey, Oper. Res., 14, (1966), 699–719Google Scholar
  12. [12]
    L. Lovasz and V. Chvatal,“Every Directed Graph has a Semi-Kernel;’175, in C. Berge and D.K. Ray-Chauduri (eds) Hypergraph Seminar, Lecture Notes in Mathematics, No. 411, Berlin, Heidelberg, New York (Springer, 1974 )Google Scholar
  13. [13]
    M. Richardson,“On Weakly Ordered Systems,’ Bull. Amer. Math. Soc „ 52 (1946), 113–116.CrossRefGoogle Scholar
  14. [14]
    E. Roba, B. Sussman et M. Theys,“Les méthodes de choix multicritères appliquées à la sélection du personnel;’ in Models of Manpower, (English Universities Press, 1970 ).Google Scholar
  15. [15]
    B. Roy,’Classement et choix en présence de points de vue multiples (la méthode ELECTREj’, Revue, Fr. d’Inf. Rech. Oper., 2 (1968) 57–75.Google Scholar
  16. [16]
    B. Roy,“Problems and Methods with Multiple Objective Functions, Matlt. Prog., 1 (1971), 239–266Google Scholar
  17. [17]
    B. Roy,’tritères multiples et modélisation des préférences. L’apport des relations de surclassement,“ Revue d’Econ. Pol., 84 (1974)Google Scholar
  18. [18]
    B. Roy,’how Outranking Relation helps Multicriteria Decision Making’, Actes du Séminaire de Beaulieu-Sainte Assise, 6–7 décembre 1973 (CESMAP, 1975 )Google Scholar
  19. [19]
    B. Roy,“Management Scientifique et Aide â la Décision’, Rapport de synthèse No. 86; Direction Scientifique, SEMA (1974).Google Scholar
  20. [20]
    B. Roy et P. Bertier,“La méthode ELECTRE 2, une application au média-planning’ 291–302 in M. Ross (ed), Operational Research 72, (North-Holland, American Elsevier, 19731.Google Scholar
  21. [21]
    J. Von Neumann and O. Morgenstern, Theory of Games and Economic Be-haviour, Princeton (Princeton University Press’? 1953 ).Google Scholar

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

Personalised recommendations