Listing the families of Sufficient Coalitions of criteria involved in Sorting procedures

Certain sorting procedures derived from ELECTRE TRI such as MR-Sort or the Non-Compensatory Sorting (NCS model) model rely on a rule of the type: if an object is better than a profile on a “sufficient coalition” of criteria, this object is assigned to a category above this profile. In some cases the strength a coalition can be numerically represented by the sum of weights attached to the criteria and a coalition is sufficient if its strength passes some threshold. This is the type of rule used in the MR-Sort method. In more general models such as Capacitive-MR-Sort or NCS model, criteria are allowed to interact and a capacity is needed to model the strength of a coalition. In this contribution, we want to investigate the gap of expressivity between the two models. In this view, we explicitly generate a list of all possible families of sufficient coalitions for a number of criteria up to 6. We also categorize them according to the degree of additivity of a capacity that can model their strength. Our goal is twofold: being able to draw a sorting rule at random and having at disposal examples in view of supporting a theoretical investigation of the families of sufficient coalitions.

[1]  Thierry Marchant,et al.  An axiomatic approach to noncompensatory sorting methods in MCDM I: The case of two categories (juin 2005) , 2005 .

[2]  Bernard Roy,et al.  Aide multicritère à la décision : méthodes et cas , 1993 .

[3]  Thierry Marchant,et al.  An axiomatic approach to noncompensatory sorting methods in MCDM, II: More than two categories , 2007, Eur. J. Oper. Res..

[4]  Vincent Mousseau,et al.  Learning the parameters of a majority rule sorting model taking attribute interactions into account , 2014 .

[5]  William Emmanuel S. Yu,et al.  Aide multicritere a la decision dans le cadre de la problematique du tri , 1992 .

[6]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[7]  Tamon Stephen,et al.  Counting inequivalent monotone Boolean functions , 2012, Discret. Appl. Math..

[8]  Marc Pirlot,et al.  Learning the Parameters of a Multiple Criteria Sorting Method , 2011, ADT.

[9]  Sascha Kurz,et al.  On Dedekind’s problem for complete simple games , 2010, Int. J. Game Theory.

[10]  Marc Pirlot,et al.  Learning a Majority Rule Model from Large Sets of Assignment Examples , 2013, ADT.

[11]  B. Leclerc,et al.  Finite Ordered Sets: Concepts, Results and Uses , 2012 .

[12]  Peter L. Hammer,et al.  Boolean Functions - Theory, Algorithms, and Applications , 2011, Encyclopedia of mathematics and its applications.

[13]  Liu Jinyan,et al.  Preference Elicitation from Inconsistent Pairwise Comparisons for Multi-criteria Ranking with Multiple Reference Points. , 2013, ICISO 2013.

[14]  Thierry Marchant,et al.  Multiattribute preference models with reference points , 2013, Eur. J. Oper. Res..

[15]  Antoine Rolland,et al.  Reference-based preferences aggregation procedures in multi-criteria decision making , 2013, Eur. J. Oper. Res..