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Deriving a minimum distance-based collective preorder: a binary mathematical programming approach

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Abstract

Deriving the “closest” (minimal distance) collective judgment to all individual opinions is a complex aggregation problem that has been widely studied in group decision-making literature. However, most of the existing literature does not consider individual opinions expressed as partial preorders (i.e., a preference system which includes the incomparability relation). In this paper, we propose a method based on binary linear programming to derive a minimum distance-based collective preorder from individual preferences relational systems (p.r.s.). This method is threefold. First, each member determines a preorder (partial or total) over the set of alternatives. Second, an aggregation algorithm is proposed to derive at least one collective and not necessary transitive p.r.s. at minimum distance from all individual preorders. Third, a binary linear programming optimization will transform each non-transitive collective p.r.s. into a collective preorder (i.e. a transitive p.r.s.). The proposed method has three main advantages: (1) it deals with incomparability (partial preorders), (2) the relative importance of the members is explicitly considered and (3) the collective p.r.s. obtained after the aggregation step might be “exploited” according to different decision-making problematics (i.e. ranking, choice and sorting).

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References

  • Arrow KJ (1951) Social Choice and individual values. Wiley, New York

    Google Scholar 

  • Ben Abdelaziz F, Martel JM, Mselmi A (2004) IMGD: an interactive method for multi-objective group decision aid. J Oper Res Soc 55(5): 464–474

    Article  Google Scholar 

  • Ben Khélifa S, Martel JM (2001) A distance-based collective weak ordering. Group Decis Negotiat 10: 317–329

    Article  Google Scholar 

  • Banzhaf JF (1965) Weighted voting Doesn’t Work: a mathematical analysis. Rutgers Law Rev 19: 317–343

    Google Scholar 

  • Borda JC (1781) Mémoire sur les Élections au Scrutin. Histoire de l’Académie Royale de Science, Paris

    Google Scholar 

  • Brans JP, Mareschal B, Vincke Ph (1984) PROMETHEE: a new family of outranking methods in multicriteria analysis. In: Brans JP (eds) Operational research’84. North-Holland, Amsterdam, pp 408–421

    Google Scholar 

  • Condorcet (Marquis de) MJANC (1785) Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voies. Imprimante Royale, Paris

  • Cook WD, Seiford LM (1978) Priority ranking and consensus formation. Manag Sci 24: 1721–1733

    Article  Google Scholar 

  • Cook WD, Kress M (1985) Ordinal ranking with intensity of preference. Manag Sci 31: 26–32

    Article  Google Scholar 

  • Cook WD, Kress M, Seiford LM (1986) An axiomatic approach to distance on partial orderings. RAIRO Recherche Opérationnelle 20: 115–122

    Google Scholar 

  • Cook WD (2006) Distance-based and ad hoc consensus models in ordinal preference ranking. Eur J Oper Res 172: 369–385

    Article  Google Scholar 

  • Copeland H (1951) A reasonable social choice welfare function. Seminar on application of mathematics to social sciences. University of Michigan

  • Deegan J, Packel EW (1979) A new index of power for simple n-person games. Int J Game Theory 7: 113–123

    Article  Google Scholar 

  • DeGroot M (1974) Reaching a consensus. J Am Stat Assoc 69: 118–121

    Article  Google Scholar 

  • Figueira J, Roy B (2002) Determining the weights of criteria in the ELECTRE type methods with a revised Simos’ procedure. Eur J Oper Res 139: 317–326

    Article  Google Scholar 

  • French JRP (1956) A formal theory of social power. Psychol Rev 63(3): 181–194

    Article  Google Scholar 

  • Hudry O, Leclerc B, Monjardet B, Barthélemy J-P (2005) Médianes métriques et laticielles. In: Dubois D, Pirlot M, Bouyssou D, Prade H (eds) Concepts et méthodes pour l’aide à la décision 2005

  • Jabeur K, Martel JM (2002a) Détermination d’un (ou plusieurs) systèmes(s) relationnel(s) de préférence (s.r.p) collectif(s) à partir des s.r.p. individuels. Document de travail # 011-2002, Faculté des Sciences de l’Administration (FSA), Université Laval

  • Jabeur K, Martel JM (2002b) Quantification de l’importance relative des membres d’un groupe en vue d’établir un préordre collectif. Info Syst Oper Res 40: 81–198

    Google Scholar 

  • Jabeur K, Martel JM, Ben Khélifa S (2004) A distance-based collective preorder integrating the relative importance of the group’s members. Group Decis Negotiat 13: 327–349

    Article  Google Scholar 

  • Jabeur K (2004) Une démarche générale d’aide aux membres d’un groupe à la recherche d’un résultat de consensus. Ph.D. thesis, Université Laval, Canada

  • Jabeur K, Martel J-M (2007a) A collective choice method based on individual preferences relational systems (p.r.s). Eur J Oper Res 177: 1549–1565

    Article  Google Scholar 

  • Jabeur K, Martel J-M (2007b) An ordinal sorting method for group decision-making. Eur J Oper Res 180: 1272–1289

    Article  Google Scholar 

  • Keeney RL, Kirkwood CW (1975) Group decision making using cardinal social welfare functions. Manag Sci 22: 430–437

    Article  Google Scholar 

  • Keeney RL, Raiffa H (1976) Decision with multiple objectives: preferences and value tradeoffs. Wiley, New York

    Google Scholar 

  • Kemeny JG, Snell JL (1962) Preference ranking: an axiomatic approach. Mathematical models in the social sciences, pp 9–23

  • Martel JM, Ben Khélifa S (2000) Deux propositions d’aide multicritère à la décision de groupe. In: Ben Abdelaziz, Haouari et Mellouli (eds) Optimisation et Décision. Centre de publication Universitaire, Tunis, pp 213–228

    Google Scholar 

  • Pomerol J-C, Barbara-Romero S (1993) Choix multicritère dans l’entreprise: principe et pratique. Collection informatique, Éditions Hermes

    Google Scholar 

  • Ramanathan R, Ganesh LS (1994) Group preference aggregation methods employed in AHP: an evaluation and an intrinsic process for deriving members’ weightages. Eur J Oper Res 79: 249–265

    Article  Google Scholar 

  • Roy B (1978) ELECTRE III: Algorithme de Classement Basé sur une Représentation des Préférence en Présence de Critères Multiples. Cahiers du CERO 21: 3–24

    Google Scholar 

  • Roy B, Bouyssou D (1993) Aide multicritère à la décision: méthodes et cas. Economica, Paris

    Google Scholar 

  • Roy B, Slowinski R (1993) Criterion of distance between technical programming and socio-economic priority. RAIRO Recherche Opérationnelle 27(1): 45–60

    Google Scholar 

  • Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Political Sci Rev 48(3): 787–792

    Article  Google Scholar 

  • Schärlig A (1996) Pratiquer Électre et Prométhée. Collection Diriger l’entreprise, Presses polytechniques et universitaires romandes, Lausanne

  • Vincke P (1989) L’aide multicritère à la décision. Éditions de l’Université de Bruxelles, Belgique

    Google Scholar 

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Authors and Affiliations

  1. Institut Supérieur de Commerce et de Comptabilité de Bizerte (ISCCB), 7021 Zarzouna, Bizerte, Tunisie

    Khaled Jabeur

  2. Faculté des Sciences de l’Administration, Université Laval, Ste-Foy, QC, G1K 7P4, Canada

    Jean-Marc Martel

  3. Defence R&D Canada Valcartier, 2459, Pie-XI Blvd North, Val-Bélair, QC, G3J 1X5, Canada

    Adel Guitouni

Authors
  1. Khaled Jabeur
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  2. Jean-Marc Martel
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  3. Adel Guitouni
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Correspondence to Khaled Jabeur.

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Jabeur, K., Martel, JM. & Guitouni, A. Deriving a minimum distance-based collective preorder: a binary mathematical programming approach. OR Spectrum 34, 23–42 (2012). https://doi.org/10.1007/s00291-009-0192-5

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