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)


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.


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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  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

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