Abstract
ELECTRE TRI-nB is a method designed to sort alternatives evaluated on several attributes into ordered categories. It is an extension of ELECTRE TRI-B that uses several limiting profiles, instead of one, to delimit each category. In a previous paper we have characterized the ordered partitions that can be obtained with ELECTRE TRI-nB, using a simple axiom called linearity. The simplicity of this characterization crucially rests on the possibility to use as many limiting profiles as we like to delimit a category. This is not completely realistic and there is a need to study models in which the number of limiting profiles delimiting each category is restricted. This note starts the investigation of such models. We specifically study the case of models using one or two profiles together with an outranking relation based on unanimity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In the rest of this text, we write “profile” instead of “limiting profile”. All the profiles used below are limiting profiles.
Throughout the paper, Remarks contain comments that can be skipped at first reading. Remarks may be useful later in the paper however.
Our conventions concerning binary relations are standard. They are exactly as in Bouyssou et al. (2021).
Although this case has little practical importance (see Fernández et al. 2017, p. 216, 2nd col., beginning of Sect. 2.1 or Roy 1996, p. 235), more powerful results can be obtained when \(n = 2\). It is easy to devise a condition equivalent to requiring, together with \(i\)-linear, that \({\succsim }_{i}\) has at most \(\ell \ge 2\) equivalence classes. Imposing that each relation \({\succsim }_{i}\) has at most \(\ell + 1\) equivalence classes is necessary if \(\langle {\mathcal {A}},{\mathcal {U}}\rangle \) has a representation in Model \((E^{c}_{\le \ell })\). When \(n=2\), this requirement is not only necessary but also sufficient to guarantee the existence of a representation in Model \((E^{c}_{\le \ell })\). Since this case is of little importance, we leave the easy proof of this fact to the interested reader. Simple examples show that the result does not generalize to the case \(n \ge 3\).
References
Bouyssou D, Marchant T (2007) An axiomatic approach to noncompensatory sorting methods in MCDM, I: the case of two categories. Eur J Oper Res 178(1):217–245
Bouyssou D, Marchant T (2007) An axiomatic approach to noncompensatory sorting methods in MCDM, II: More than two categories. Eur J Oper Res 178(1):246–276
Bouyssou D, Marchant T (2009) Ordered categories and additive conjoint measurement on connected sets. J Math Psychol 53(2):92–105
Bouyssou D, Marchant T (2010) Additive conjoint measurement with ordered categories. Eur J Oper Res 203(1):195–204. https://doi.org/10.1016/j.ejor.2009.07.032
Bouyssou D, Marchant T, Pirlot M (2020) The size of the largest antichains in products of linear orders. TOP. https://doi.org/10.1007/s11750-020-00587-6
Bouyssou D, Marchant T, Pirlot M (2021) A theoretical look at ELECTRE TRI-nB and related sorting models. arXiv:2008.09484
Fernández E, Figueira JR, Navarro J, Roy B (2017) ELECTRE TRI-nB: a new multiple criteria ordinal classification method. Eur J Oper Res 263(1):214–224. https://doi.org/10.1016/j.ejor.2017.04.048
Fernández E, Figueira JR, Navarro J (2019) An indirect elicitation method for the parameters of the ELECTRE TRI-nB model using genetic algorithms. Appl Soft Comput J 77:723–733. https://doi.org/10.1016/j.asoc.2019.01.050
Goldstein WM (1991) Decomposable threshold models. J Math Psychol 35(1):64–79
Greco S, Matarazzo B, Słowiński R (2001) Conjoint measurement and rough set approach for multicriteria sorting problems in presence of ordinal criteria. In: Colorni A, Paruccini M, Roy B (eds) A-MCD-A. Aide Mulcritère à la Décision / Multiple Criteria Decision Aid, European Commission, Joint Research Centre, Luxembourg, pp 117–144
Roy B (1996) Multicriteria methodology for decision aiding. Kluwer, Dordrecht, Original version in French: “Méthodologie multicritère d’aide à la décision”. Economica, Paris, 1985
Roy B, Bouyssou D (1993) Aide multicritère à la décision : méthodes et cas. Economica, Paris. 2-7178-2473-1
Słowiński R, Greco S, Matarazzo B (2002) Axiomatization of utility, outranking and decision-rule preference models for multiple-criteria classification problems under partial inconsistency with the dominance principle. Control Cybern 31(4):1005–1035
Vind K (1991) Independent preferences. J Math Econ 20(1):119–135
Vind K (2003) Independence, additivity, uncertainty. Springer, Berlin
Yu W (1992) Aide multicritère à la décision dans le cadre de la problématique du tri : Concepts, méthodes et applications. Université Paris Dauphine, Paris, France, Thèse de doctorat ((in French))
Acknowledgements
We are grateful to three anonymous referees and the area editor for their careful reading of the manuscript and their constructive comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bouyssou, D., Marchant, T. & Pirlot, M. A note on ELECTRE TRI-nB with few limiting profiles. 4OR-Q J Oper Res 20, 443–463 (2022). https://doi.org/10.1007/s10288-021-00485-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10288-021-00485-y