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Interval-valued preference structures. (English) Zbl 0960.91508
Summary: Different languages that are offered to model vague preferences are reviewed and an interval-valued language is proposed to resolve a particular difficulty encountered with other languages. It is shown that interval-valued languages are well defined for De Morgan triples constructed by continuous triangular norms, conorms and a strong negation function. A new transitivity condition for vague preferences is suggested and its relationships to known transitivity conditions is established. A complete characterization of interval-valued preference structures is also provided.

MSC:
91B08 Individual preferences
91B14 Social choice
03E72 Theory of fuzzy sets, etc.
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[1] Alsina, C., On a family of connectives for fuzzy sets, Fuzzy sets and systems, 16, 231-235, (1985) · Zbl 0603.39005
[2] Alsina, C.; Trillas, E.; Valverde, L., On some logical connectives for fuzzy sets theory, Journal of mathematical analysis and applications, 93, 15-26, (1983) · Zbl 0522.03012
[3] Anand, P., Are the preference axioms really rational?, Theory and decision, 23, 189-214, (1987)
[4] Arrow, K., Social choice and individual values, (1963), Wiley New York
[5] Barrett, C.R.; Pattanaik, P.K., On vague preferences, (), 69-84
[6] Barrett, C.R.; Pattanaik, P.K.; Salles, M., On choosing rationally when preferences are fuzzy, Fuzzy sets and systems, 34, 197-212, (1990) · Zbl 0688.90003
[7] Bilgiç, T., Measurement-theoretic frameworks for fuzzy set theory with applications to preference modelling, ()
[8] Bilgiç, T.; Türkşen, I., Measurement-theoretic justification of fuzzy set connectives, Fuzzy sets and systems, 76, 3, 289-308, (1995) · Zbl 0852.04006
[9] Bilgiç, T.; Türkşen, I., Model-based localization for an autonomous mobile robot, ()
[10] Doignon, J.-P.; Monjardet, B.; Roubens, M.; Vincke, P., Biorder families, valued relations, and preference modelling, Journal of mathematical psychology, 30, 435-480, (1986) · Zbl 0612.92020
[11] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049
[12] Dubois, D.; Prade, H., A class of fuzzy measured based on triangular norms: a general framework for the combination of information, International journal of general systems, 8, 43-61, (1982) · Zbl 0473.94023
[13] Dyer, J.; Fishburn, P.; Steuer, R.; Wallenius, J.; Zionts, S., Multiple criteria decision making, multiattribute utility theory: the next ten years, Management science, 38, 5, 645-654, (1992) · Zbl 0825.90620
[14] Ellsberg, D., Risk, ambiguity and the savage axioms, Quarterly journal of economics, 75, 643-669, (1961) · Zbl 1280.91045
[15] Fishburn, P.C., Binary choice probabilities: on the varieties of stochastic transitivity, Journal of mathematical psychology, 10, 327-352, (1973) · Zbl 0277.92008
[16] Fishburn, P.C., On the theory of ambiguity, International journal of information and management sciences, 2, 2, 1-16, (1991) · Zbl 0755.90003
[17] Fishburn, P.C., The axioms and algebra of ambiguity, Theory and decision, 34, 119-137, (1993) · Zbl 0780.90004
[18] Fodor, J.C.; Roubens, M., Fuzzy strict preference relations in decision making, (), 1145-1149
[19] Fodor, J.C.; Roubens, M., Fuzzy preference modelling and multicriteria decision support, vol. 14 of theory and decision library, () · Zbl 0827.90002
[20] Fodor, J.C.; Roubens, M., Valued preference structures, European journal of operational research, 79, 2, 277-286, (1994) · Zbl 0812.90005
[21] Gisin, V.B., On transitivity of strict preference relations, Fuzzy sets and systems, 67, 293-301, (1994) · Zbl 0845.90011
[22] Gottwald, S., Fuzzy sets and fuzzy logic: foundations of application from a mathematical point of view, (1993), Vieweg Wiesbaden · Zbl 0782.94025
[23] Machina, M.J.; Schmeidler, D., A more robust definition of subjective probability, Econometrica, 60, 4, 745-780, (1992) · Zbl 0763.90012
[24] Mostert, P.S.; Shields, A.L., On the structure of semi groups on a compact manifold with boundary, Annals of mathematics, 65, 117-143, (1957) · Zbl 0096.01203
[25] Ovchinnikov, S., Similarity relations, fuzzy partitions, and fuzzy orderings, Fuzzy sets and systems, 40, 107-126, (1991) · Zbl 0725.04003
[26] Ovchinnikov, S., Social choice and łukasiewicz logic, Fuzzy sets and systems, 43, 3, 275-290, (1991) · Zbl 0742.90010
[27] Ovchinnikov, S.; Roubens, M., On strict preference relations, Fuzzy sets and systems, 43, 319-326, (1991) · Zbl 0747.90006
[28] Ovchinnikov, S.; Roubens, M., On fuzzy strict preference, indifference and incomparability relations, Fuzzy sets and systems, 47, 313-318, (1992) · Zbl 0765.90002
[29] Ovchinnikov, S.; Roubens, M., On fuzzy strict preference, indifference and incomparability relations, Fuzzy sets and systems, 49, 15-20, (1992) · Zbl 0768.90005
[30] Piaget, J., Traité de logique: essai de logistique opératoire, (1949), A. Colin Paris
[31] Roberts, F., Measurement theory, (1979), Addison-Wesley Reading, MA
[32] Roy, B., The outranking approach and the foundations of electre methods, Theory and decision, 31, 1, 49-73, (1991)
[33] Savage, L.J., The foundations of statistics, (1972), Dover Publications New York · Zbl 0121.13603
[34] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific journal of mathematics, 10, 313-334, (1960) · Zbl 0091.29801
[35] Schweizer, B.; Sklar, A., Associative functions and statistical triangle inequalities, Universitatis debreceniensis. institutum mathematicum. publicationes mathematicae, 8, 169-186, (1961) · Zbl 0107.12203
[36] Schweizer, B.; Sklar, A., Associative functions and abstract semigroups, Universitatis debreceniensis. institutum mathematicum. publicationes mathematicae, 10, 69-81, (1963) · Zbl 0119.14001
[37] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), North-Holland Amsterdam · Zbl 0546.60010
[38] Skala, H.J., On many-valued logics, fuzzy sets, fuzzy logics and their applications, Fuzzy sets and systems, 1, 129-149, (1978) · Zbl 0396.03024
[39] Suppes, P.; Krantz, D.; Luce, R.; Tversky, A., ()
[40] Türkşen, I.B., Klein groups in fuzzy inference, (), 556-560
[41] Türkşen, I.B., Interval valued fuzzy sets based on normal forms, Fuzzy sets and systems, 20, 191-210, (1986) · Zbl 0618.94020
[42] Türkşen, I.B.; Bilgiç, T., Interval valued strict preference, (), 593-599 · Zbl 0868.90004
[43] Türkşen, I.B.; Bilgiç, T., Interval-valued strict preference with zadeh triples, Fuzzy sets and systems, 78, 2, 183-195, (1996), (Special Issue on fuzzy MCDM) · Zbl 0868.90004
[44] Weber, S., A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms, Fuzzy sets and systems, 11, 115-134, (1983) · Zbl 0543.03013
[45] Yuan, B.; Pan, Y.; Wu, W., On normal form based interval-valued fuzzy sets and their applications to approximate reasoning, International journal of general systems, 23, 3, 241-254, (1995) · Zbl 0850.04004
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