Abstract
We discuss some new approaches to preference aggregation, keeping the natural property of transitivity of strict preferences in mind. In a previous paper, we discussed various ways in which to construct and process strict partial order relations in the context of ranking objects on the basis of multiple criteria. We now broaden the scope to include more general expressions of preferences as inputs and introduce the concept of a NIP-triple, composed of a relation of necessary couples, a relation of impossible couples and a relation of possible couples. The use of NIP-triples allows for a more straightforward characterization of the consistent and prioritized consistent union as well as a smooth formulation of algorithmic implementations. We also introduce a NIP-triple closing operation, which can be combined with the consistent union operations for increased flexibility. Some properties of the proposed operations are examined. The consistent union operation is commutative, as is its composition with the closing operation. Both the consistent and prioritized consistent union are associative, but not when they are composed with the closing operation. Nevertheless, the composed operations surely have their use, which is also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andréka, H., Ryan, M., & Schobbens, P.-Y. (2002). Operators and laws for combining preference relations. Journal of Logic and Computation, 12, 13–53.
Bouyssou, D. (1990). Building criteria: A prerequisite for MCDA. In C. A. Bana e Costa (Ed.), Readings in multiple criteria decision aid (pp. 58–80). Berlin: Springer.
Brüggemann, R., & Bartel, H.-G. (1999). A theoretical concept to rank environmentally significant chemicals. Journal of Chemical Information and Computer Sciences, 39, 211–217.
Brüggemann, R., Voigt, K., Kaune, A., Pudenz, S., Komossa, D., & Friedrich, J. (1988). Vergleichende Ökologische Bewertung von Regionen in Baden-Württemberg (Technical report). GSF—Forschungszentrum für Umwelt und Gesundheit. GSF-Bericht 20/98.
Chomicki, J. (2003). Preference formulas in relational queries. ACM Transactions on Database Systems, 28, 1–40.
Chomicki, J. (2006). Database querying under changing preferences. Annals of Mathematics and Artificial Intelligence, 50, 79–109.
De Baets, B., & De Meyer, H. (2003). On the existence and construction of T-transitive closures. Information Sciences, 152, 167–179.
De Baets, B., & Fodor, J. (2003). Additive fuzzy preference structures: the next generation. In B. De Baets & J. Fodor (Eds.), Principles of fuzzy preference modelling and decision making (pp. 15–25). Gent: Academia Press.
De Baets, B., Van de Walle, B., & Kerre, E. (1998). Characterizable fuzzy preference structures. Annals of Operation Research, 80, 105–136.
Denœux, Th., & Masson, M. (2011). Evidential reasoning in large partially ordered sets. Application to multi-label classification, ensemble clustering and preference elicitation. Annals of Operations Research, this issue.
Dubois, D., & Prade, H. (1988). Possibility theory. New York: Plenum Press.
Fishburn, P. C. (1974a). Paradoxes of voting. The American Political Science Review, 68, 537–546.
Fishburn, P. C. (1974b). Social choice functions. SIAM Review, 16(1), 63–90.
Fishburn, P. C. (1977). Condorcet social choice functions. SIAM Journal on Applied Mathematics, 33, 469–489.
Floyd, R. W. (1962). Algorithm 97: Shortest path. Communications of the ACM, 5, 345.
Fürnkranz, J., & Hüllermeier, E. (2003). Pairwise preference learning and ranking. Lecture Notes in Computer Science, 2837, 145–156.
Hüllermeier, E., Fürnkranz, J., Cheng, W., & Brinker, K. (2008). Label ranking by learning pairwise preferences. Artificial Intelligence, 172(16–17), 1897–1916.
Naessens, H., De Meyer, H., & De Baets, B. (2002). Algorithms for the computation of T-transitive closures. IEEE Transactions on Fuzzy Systems, 10, 541–551.
Nedbal, R. (2005). Relational databases with ordered relations. Logic Journal of the IGPL, 13, 587–597.
Pini, M. S., Rossi, F., Venable, K. B., & Walsh, T. (2009). Aggregating partially ordered preferences. Journal of Logic and Computation, 19(3), 475.
Pudenz, S., Brüggemann, R., Komossa, D., & Kreimes, K. (1998). An algebraic/graphical tool to compare ecosystems with respect to their pollution by Pb, Cd III: Comparative regional analysis by applying a similarity index. Chemosphere, 36, 441–450.
Rademaker, M., De Baets, B., & De Meyer, H. (2008). New operations for informative combination of two partial order relations with illustrations on pollution data. Computational Chemistry and High Throughput Screening, 11, 745–755.
Roubens, M., & Vincke, Ph. (1985). Lecture notes in economics and mathematical systems: Vol. 250. Preference modeling. Berlin: Springer.
Van de Walle, B., De Baets, B., & Kerre, E. (1995). Fuzzy preference structures without incomparability. Fuzzy Sets and Systems, 76, 333–348.
Vincke, Ph. (1992). Exploitation of a crisp relation in a ranking problem. Theory and Decision, 32, 221–240.
Warshall, S. (1962). A theorem on boolean matrices. Journal of the ACM, 9, 11–12.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rademaker, M., De Baets, B. Consistent union and prioritized consistent union: new operations for preference aggregation. Ann Oper Res 195, 237–259 (2012). https://doi.org/10.1007/s10479-011-0852-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-011-0852-0