Abstract
We study the transitivity of fuzzy preference relations, often considered as a fundamental property providing coherence to a decision process. We consider the transitivity of fuzzy relations w.r.t. conjunctors, a general class of binary operations on the unit interval encompassing the class of triangular norms usually considered for this purpose. Having fixed the transitivity of a large preference relation w.r.t. such a conjunctor, we investigate the transitivity of the strict preference and indifference relations of any fuzzy preference structure generated from this large preference relation by means of an (indifference) generator. This study leads to the discovery of two families of conjunctors providing a full characterization of this transitivity. Although the expressions of these conjunctors appear to be quite cumbersome, they reduce to more readily used analytical expressions when we focus our attention on the particular case when the transitivity of the large preference relation is expressed w.r.t. one of the three basic triangular norms (the minimum, the product and the Łukasiewicz triangular norm) while at the same time the generator used for decomposing this large preference relation is also one of these triangular norms. During our discourse, we pay ample attention to the Frank family of triangular norms/copulas.
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References
Alsina C. (1985) On a family of connectives for fuzzy sets. Fuzzy Sets and Systems 16: 231–235
Alsina C., Frank M. J., Schweizer B. (2006) Associative functions: triangular norms and copulas. World Scientific Publishing Company, Singapore
Alsina C., Nelsen R. B., Schweizer B. (1993) On the characterization of a class of binary operations on distribution functions. Statistics & Probability Letters 17: 85–89
Arrow K. J. (1951) Social choice and individual values. Wiley, London
Bilgiç T. (1998) Interval-valued preference structures. European Journal of Operational Research 105: 162–183
Dasgupta M., Deb R. (1996) Transitivity and fuzzy preferences. Social Choice and Welfare 13: 305–318
Dasgupta M., Deb R. (2001) Factoring fuzzy transitivity. Fuzzy Sets and Systems 118: 489–502
De Baets B., De Meyer H., Díaz S (2009) On an idempotent transformation of aggregation functions and its application on absolutely continuous Archimedean copulas. Fuzzy Sets and Systems 160: 733–751
De Baets B., Fodor J. (1997) Twenty years of fuzzy preference structures (1978–1997). Belg. J. Oper. Res. Statist. Comput. Sci. 37: 61–82
De Baets, B., & Fodor, J. (2003). Additive fuzzy preference structures: The next generation. In B. De Baets & J. Fodor, Principles of fuzzy preference modelling and decision making (pp. 15–25). London: Academic Press.
De Baets B., Janssens S., De Meyer H. (2006) Meta-theorems on inequalities for scalar fuzzy set cardinalities. Fuzzy Sets and Systems 157: 1463–1476
De Baets B., Mesiar R. (1998) T-partitions. Fuzzy Sets and Systems 97: 211–223
De Baets, B., & Van de Walle, B. (1997). Minimal definitions of classical and fuzzy preference structures. In Proceedings of the Annual Meeting of the North American Fuzzy Information Processing Society (pp. 299-304). USA: Syracuse, New York.
De Baets B., Van De Walle B., Kerre E. (1995) Fuzzy preference structures without incomparability. Fuzzy Sets and Systems 76: 333–348
Díaz S., De Baets B., Montes S. (2003) On the transitivity of fuzzy indifference relations. Lecture Notes in Artificial Intelligence 2715: 87–94
Díaz S., De Baets B., Montes S. (2007) Additive decomposition of fuzzy pre-orders. Fuzzy Sets and Systems 158: 830–842
Díaz S., De Baets B., Montes S. (2008) On the compositional characterization of complete fuzzy pre-orders. Fuzzy Sets and Systems 159: 2221–2239
Díaz S., Montes S., De Baets B. (2004) Transitive decomposition of fuzzy preference relations: The case of nilpotent minimum. Kybernetika 40: 71–88
Díaz S., Montes S., De Baets B. (2007) Transitivity bounds in additive fuzzy preference structures. IEEE Transactions on Fuzzy Systems 15: 275–286
Fernández E., Navarro J., Duarte A. (2008) Multicriteria sorting using a valued preference closeness relation. European Journal of Operational Research 185: 673–686
Fodor J., Roubens M. (1994) Valued preference structures. European Journal of Operational Research 79: 277–286
Fodor J., Roubens M. (1994) Fuzzy preference modelling and multicriteria decision support. Kluwer Academic Publishers, Dordrecht
Fodor J., Roubens M. (1995) Structure of transitive valued binary relations. Mathematical Social Sciences 30: 71–94
Genest C., Quesada-Molina J. J., Rodríguez-Lallena J. A., Sempi C. (1999) A characterization of quasi-copulas. Journal of Multivariate Analysis 69: 193–205
Hájek P. (1998) Metamathematics of fuzzy logic. Kluwer Academic Publishers, Dordrecht
Haven E. (2002) Fuzzy interval and semi-orders. European Journal of Operational Research 139: 302–316
Herrera-Viedma E., Herrera F., Chiclana F., Luque M. (2004) Some issues on consistency of fuzzy preference relations. European Journal of Operational Research 154: 98–109
Janssens S., De Baets B., De Meyer H. (2004) Bell-type inequalities for quasi-copulas. Fuzzy Sets and Systems 148: 263–278
Klement E. P., Mesiar R., Pap E. (2000) Triangular norms. Kluwer Academic Publishers, Dordrecht
Nelsen, R. (1994). An Introduction to Copulas, Lecture Notes in Statistics, 139, Springer.
Ovchinnikov S. (1991) Similarity relations, fuzzy partitions, and fuzzy orderings. Fuzzy Sets and Systems 40: 107–126
Roubens, M., & Vincke, Ph. (1985). Preference Modelling. Lecture Notes in Economics and Mathematical Systems, 76. Springer.
Saminger-Platz S., De Baets B., De Meyer H. (2006) On the dominance relation between ordinal sums of conjunctors. Kybernetika 42: 337–350
Saminger-Platz S., De Baets B., De Meyer H. (2009) Differential inequality conditions for dominance between continuous Archimedean t-norms. Mathematical Inequalities and Applications 12: 191–208
Saminger S., Mesiar R., Bodenhofer U. (2002) Domination of aggregation operators and preservation of transitivity. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 10: 11–35
Sarkoci P. (2005) Domination in the families of Frank and Hamacher t-norms. Kybernetika 41: 349–360
Van de Walle B., De Baets B., Kerre E. (1998) A plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures Part 1: General argumentation. Fuzzy Sets and Systems 97: 349–359
Van de Walle B., De Baets B., Kerre E. (1998) Characterizable fuzzy preference structures. Annals of Operations Research 80: 105–136
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Díaz, S., De Baets, B. & Montes, S. General results on the decomposition of transitive fuzzy relations. Fuzzy Optim Decis Making 9, 1–29 (2010). https://doi.org/10.1007/s10700-010-9074-1
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DOI: https://doi.org/10.1007/s10700-010-9074-1