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A review of fuzzy set aggregation connectives. (English) Zbl 0582.03040

This paper provides an extensive survey on fuzzy-set-theoretic operations, and emphasizes the relevance of the theory of functional equations in the axiomatical construction of classes of such operations and the derivation of functional representations. The second part is devoted to the application of fuzzy set theory to multifactorial evaluation. Some links between this approach and multiattribute utility theory are explored. Problems of modeling the importance of criteria, as well as of choosing a proper aggregation connective in a given situation, are also discussed.
Reviewer: Wang Peizhuang

MSC:

03E72 Theory of fuzzy sets, etc.
91B16 Utility theory
39B99 Functional equations and inequalities

Software:

ELECTRE
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[1] J. Aczel, Lectures on Functional Equations and Applications; J. Aczel, Lectures on Functional Equations and Applications · Zbl 0139.09301
[2] Albert, P., The algebra of fuzzy logic, Fuzzy Sets and Systems, 1, 203-230 (1978) · Zbl 0407.03031
[3] Alsina, C.; Trillas, E.; Valverde, L., Do we need max, min, l — \(j\) in fuzzy set theory, (Yager, R., Fuzzy Sets and Possibility Theory: Recent Developments (1981), Pergamon: Pergamon Oxford), 275-297
[4] Alsina, C.; Trillas, E.; Valverde, L., On some logical connectives for fuzzy set theory, J. Math. Anal. Appl., 93, 15-26 (1983) · Zbl 0522.03012
[5] Arora, P. N., On characterizing some generalizations of Shannon’s entropy, Inform. Sci., 21, 13-22 (1980) · Zbl 0469.94007
[6] Bellman, R. E.; Giertz, M., On the analytic formalism of the theory of fuzzy sets, Inform. Sci., 5, 149-156 (1973) · Zbl 0251.02059
[7] Bellman, R. E.; Zadeh, L. A., Decision-making in a fuzzy environment, Management Sci., 17, B141-B164 (1970) · Zbl 0224.90032
[8] Chatalic, P., Mémoire de fin d’études et D.E.A. (1984), ENSEEIHT: ENSEEIHT Toulouse, France
[9] Czogala, E.; Drewniak, J., Associative monotonic operations in fuzzy set theory, Fuzzy Sets and Systems, 12, 249-270 (1984) · Zbl 0555.94027
[10] Dhombres, J. G., Recent applications of functional equations, (Proceedings of the 2d International Conference on Mathematics at the Service of Man. Proceedings of the 2d International Conference on Mathematics at the Service of Man, Las Palmas, Spain (1982)), 14-19 · Zbl 0512.39003
[11] Dombi, J., A general class of fuzzy operations, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators, Fuzzy Sets and Systems, 8, 149-163 (1982) · Zbl 0494.04005
[12] Dombi, J., Basic concepts for a theory of evaluation: The aggregative operator, European J. Oper. Res., 10, 282-293 (1981) · Zbl 0488.90003
[13] Dombi, J.; Vas, Z., Basic theoretical treatment of fuzzy connectives, Acta Cybernet., 6, 191-201 (1983) · Zbl 0517.94026
[14] Dubois, D., Ensembles flous et conception assistée par ordinateur, (Research Report No. 199 (1980), IMAG: IMAG Grenoble)
[15] Dubois, D., Triangular norms for fuzzy sets, (P. Klement, E., Proceedings of the 2d International Seminar on Fuzzy Set Theory (15-20 Sept. 1980), J. Kepler Universität: J. Kepler Universität Linz, Austria), 39-68
[16] Dubois, D., Modèles mathématiques de l’imprécis et de l’incertain, en vue d’applications aux techniques d’aide à la décision, (Thèse d’Etat (1983), Univ. de Grenoble) · Zbl 0546.94036
[17] Dubois, D.; Prade, H., New results about properties and semantics of fuzzy set-theoretic operators, (Wang, P. P.; Chang, S. K., Fuzzy sets: Theory and Applications to Policy Analysis and Information Systems (1980), Plenum: Plenum New York), 59-75
[18] Dubois, D.; Prade, H., Fuzzy Sets and Systems. Theory and Applications (1980), Academic: Academic New York · Zbl 0444.94049
[19] Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE. Trans. Automat. Control, 26, 926-936 (1981) · Zbl 1457.68262
[20] Dubois, D.; Prade, H., A class of fuzzy measures based on triangular norms, Internat. J. Gen. Systems, 8, 43-61 (1982) · Zbl 0473.94023
[21] Dubois, D.; Prade, H., Fuzzy set-theoretic differences, and inclusions and their use in fuzzy arithmetics and analysis, (Proceedings of the 5th International Seminar on Fuzzy Set Theory. Proceedings of the 5th International Seminar on Fuzzy Set Theory, Linz, Austria (1983)) · Zbl 0549.03020
[22] Dubois, D.; Prade, H., Criteria aggregation and ranking of alternatives in the framework of fuzzy set theory, (Zimmermann, H.-J.; Zadeh, L. A.; Gaines, B., Fuzzy Sets and Decision Analysis. Fuzzy Sets and Decision Analysis, TIMS Studies in the Management Sciences, Vol. 20 (1984)), 209-240
[23] D. Dubois and H. Prade, Fuzzy set-theoretic differences and inclusions and their use in solving fuzzy equations, Control Cybernet.; D. Dubois and H. Prade, Fuzzy set-theoretic differences and inclusions and their use in solving fuzzy equations, Control Cybernet. · Zbl 0549.03020
[24] Dubois, D.; Prade, H., Théorie des Possibilités: Applications à la Représentation des Connaissances en Informatique (1985), Masson: Masson Paris · Zbl 0674.68059
[25] Dujmovic, J. J., Evaluation, comparison and optimization of hybrid computers using the theory of complex criteria, (Dekker, L., Simulation of Systems (1976), North-Holland: North-Holland Amsterdam), 553-566
[26] Dummett, M., Elements of Intuitionism (1974), Oxford U.P: Oxford U.P Oxford
[27] H. Dyckhoff, Basic concepts for a theory of evaluation: Hierarchical aggregation via autodistributive connectives in fuzzy set theory, European J. Oper. Res.; H. Dyckhoff, Basic concepts for a theory of evaluation: Hierarchical aggregation via autodistributive connectives in fuzzy set theory, European J. Oper. Res. · Zbl 0559.90047
[28] Dyckhoff, H.; Pedrycz, W., Generalized means as model of compensatory connectives, Fuzzy Sets and Systems, 14, 143-154 (1984) · Zbl 0551.03035
[29] Esteva, F.; Trillas, E.; Domingo, X., Weak and strong negation functions for fuzzy set theory, (Proceedings of the 11th IEEE International Symposium on Multiple-Valued Logic. Proceedings of the 11th IEEE International Symposium on Multiple-Valued Logic, Oklahoma City (1981)), 23-26 · Zbl 0548.03036
[30] Farquhar, P. H., A survey of multiattribute utility theory and its applications, TIMS Stud. Management Sci., 6, 59-89 (1977)
[31] Frank, M. J., On the simultaneous associativity of \(F(x, y)\) and \(x + y − F(x, y)\), Aequationes Math., 19, 194-226 (1979) · Zbl 0444.39003
[32] Fuchs, L., Partially Ordered Algebraic Systems (1963), Pergamon: Pergamon Oxford · Zbl 0137.02001
[33] Fung, L. W.; Fu, K. S., An axiomatic approach to rational decision-making in a fuzzy environment, (Zadeh, L. A.; Fu, K. S.; Tanaka, K.; Shimura, M., Fuzzy Sets and Their Applications to Cognitive and Decision Processes (1975), Academic: Academic New York), 227-256 · Zbl 0366.90003
[34] Gaines, B. R., Foundations of fuzzy reasoning, Int. J. Man-Machine Stud., 8, 623-668 (1976) · Zbl 0342.68056
[35] Giles, R., Lukasiewicz logic and fuzzy set theory, Int. J. Man-Machine Stud., 8, 313-327 (1976) · Zbl 0335.02037
[36] Hamacher, H., (Trappl, R.; Klir, G. J.; Ricciardi, L., Über logische Verknupfungen unscharfer Aussagen und deren Zugehörige Bewertungs-funktionen. Über logische Verknupfungen unscharfer Aussagen und deren Zugehörige Bewertungs-funktionen, Progress in Cybernetics and Systems Research, Vol. 3 (1978), Hemisphere: Hemisphere New York), 276-287 · Zbl 0435.03018
[37] Hersh, H. M.; Caramazza, A., A fuzzy set approach to modifiers and vagueness in natural languages, J. Exp. Psychol. Gen., 105, 254-276 (1976)
[38] Hersh, H. M.; Caramazza, A.; Brownell, H. H., Effects of context on fuzzy membership functions, (Gupta, M. M.; Ragade, R. K.; Yager, R. R., Advances in Fuzzy Set Theory and Applications (1979), North-Holland: North-Holland Amsterdam), 389-408
[39] Higashi, M.; Klir, G. J., On measures of fuzziness and fuzzy complements, Int. J. Gen. Systems, 8, No. 3 (1982) · Zbl 0484.94047
[40] Kayser, D., Vers une modélisation du raisonnement “approximatif”, (Borillo, M., Représentation des Connaissances et Raisonnement dans les Sciences de l’Homme (Sept. 1979), Pub. INRIA: Pub. INRIA Le Chesnay, France), 440-457, St Maximin
[41] Keeney, R. L.; Raiffa, H., Decisions with Multiple Objectives: Preferences and Value Trade-offs (1976), Wiley: Wiley New York · Zbl 0488.90001
[42] Klement, E. P., Construction of fuzzy σ-algebras using triangular norms, J. Math. Anal. Appl., 85, 543-566 (1982) · Zbl 0491.28003
[43] Klement, E. P., Operations on fuzzy sets: An axiomatic approach, Inform. Sci., 27, 221-232 (1984) · Zbl 0515.03036
[44] Kulka, J.; Novak, V., Have fuzzy operations a psychological correspondence, Studia Psych., 26, 131-140 (1984)
[45] Lakoff, G., Hedges: A study in meaning criteria and the logic of fuzzy concepts, J. Philos. Logic, 2, 458-508 (1973) · Zbl 0272.02047
[46] Ling, C. H., Representation of associative functions, Publ. Math. Debrecen, 12, 189-212 (1965) · Zbl 0137.26401
[47] Lowen, R., On fuzzy complements, Inform. Sci., 14, 107-113 (1978) · Zbl 0416.03047
[48] Menger, K., Statistical metrics, (Proc. Nat. Acad. Sci. U.S.A., 28 (1942)), 535-537 · Zbl 0063.03886
[49] Mizumoto, M., Fuzzy sets and their operations, Inform. and Control, 50, 160-174 (1982), Part II · Zbl 0488.04004
[50] Norwich, A. M.; Turksen, I. B., The fundamental measurement of fuzziness, (Yager, R. R., Fuzzy Set and Possibility Theory: Recent Developments (1982), Pergamon: Pergamon Oxford), 49-60 · Zbl 0538.94026
[51] Oden, G. C., Integration of fuzzy logical information, J. Exp. Psych. Human Perception and Performance, 3, 565-575 (1977)
[52] Ovchinnikov, S., General negations in fuzzy set theory, J. Math. Anal. Appl., 92, 234-239 (1983) · Zbl 0518.04003
[53] Ovchinnikov, S., Representations of synonymy and antonymy by automorphisms in fuzzy set theory, Stochastica, 5, 95-107 (1981) · Zbl 0476.04006
[54] Prade, H., Unions et intersections d’ensembles flous, BUSEFAL (Univ. P. Sabatier, Toulouse), 3, 58-62 (1980)
[55] Prade, H., Modal semantics and fuzzy set theory, (Yager, R., Fuzzy Set and Possibility Theory: Recent Developments (1982), Pergamon), 232-246
[56] Rescher, N., Many-Valued Logic (1969), McGraw-Hill: McGraw-Hill New York · Zbl 0248.02023
[57] Roy, B., Electre III: Un algorithme de classement fondé sur une représentation floue des préférences, en présence de critères multiples, Cahiers CERO, 20, 3-24 (1978) · Zbl 0377.90003
[58] Saaty, T. L., Exploring the interfaces between hierarchies, multiple objectives and luzzy sets, Fuzzy Sets and Systems, 1, 57-68 (1978) · Zbl 0378.94001
[59] Savage, L. J., The Foundations of Statistics (1972), Dover: Dover New York · Zbl 0121.13603
[60] Schweizer, B., Multiplications on the space of probability distribution functions, Aequationes Math., 12, 156-183 (1975) · Zbl 0305.22004
[61] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific J. Math., 10, 313-334 (1960) · Zbl 0091.29801
[62] Schweizer, B.; Sklar, A., Associative functions and statistical triangle inequalities, Publ. Math. Debrecen, 8, 169-186 (1961) · Zbl 0107.12203
[63] Schweizer, B.; Sklar, A., Associative functions and abstract semi-groups, Publ. Math. Debrecen, 10, 69-81 (1963) · Zbl 0119.14001
[64] Silvert, W., Symmetrie summation: A class of operations on fuzzy sets, IEEE Trans. Systems, Man Cybernet., 9, 659-667 (1979) · Zbl 0424.04003
[65] Smets, P., Elementary semantic operators, (Yager, R. R., Fuzzy Set and Possibility Theory: Recent Developments (1982), Pergamon), 247-257
[66] Sugeno, M., Fuzzy measures and fuzzy integrals: A survey, (Gupta, M. M.; Saridis, G. N.; Gaines, B. R., Fuzzy Automata and Decision Processes (1977), North-Holland: North-Holland Amsterdam), 89-102
[67] Thöle, U.; Zimmermann, H.-J.; Zysno, P., On the suitability of minimum and product operators for the intersection of fuzzy sets, Fuzzy Sets and Systems, 2, 167-180 (1979) · Zbl 0408.94030
[68] Trillas, E., Sobre funciones de negacion en la teoria de conjunctos diffuses, Stochastica, 3, 47-59 (1979)
[69] Trillas, E.; Riera, T., Towards a representation of “synonyms” and “antonyms” by fuzzy sets, BUSEFAL, No. 5, 42-68 (1981)
[70] Voxman, W.; Goetschel, R., A note on the characterization of max and min operators, Inform. Sci., 30, 5-10 (1983) · Zbl 0597.04002
[71] Yager, R. R., Multiple objectives decision-making using fuzzy sets, Internat. J. Man-Machine Stud., 9, 375-382 (1977) · Zbl 0371.90005
[72] Yager, R. R., On a general class of fuzzy connectives, Fuzzy Sets and Systems, 4, 235-242 (1980) · Zbl 0443.04008
[73] Yager, R. R., Comments on textured sets, IEEE Trans. Systems Man Cybernet., 11, 730-731 (1981)
[74] Yager, R. R., Some procedures for selecting fuzzy set-theoretic operators, Internat. J. Gen. Systems, 8, 115-124 (1982) · Zbl 0488.04005
[75] Zadeh, L. A., Fuzzy sets, Inform. and Control, 8, 338-353 (1965) · Zbl 0139.24606
[76] Zadeh, L. A., Probability measures of fuzzy events, J. Math. Anal. Appl., 23, 421-427 (1968) · Zbl 0174.49002
[77] Zadeh, L. A., A fuzzy set-theoretic interpretation of linguistic hedges, J. Cybernet., 2, 3, 4-34 (1972)
[78] Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3-28 (1978) · Zbl 0377.04002
[79] Zadeh, L. A., Fuzzy sets and information granularity, (Gupta, M. M.; Ragade, R. K.; Yager, R. R., Advances in Fuzzy Set Theory and Applications (1979), North-Holland: North-Holland Amsterdam), 3-18 · Zbl 0377.04002
[80] Zimmermann, H.-J., Results of empirical studies in fuzzy set theory, (Klir, G., Applied General System Research (1978), Plenum: Plenum New York), 303-312
[81] Zimmermann, H.-J.; Zysno, P., Latent connectives in human decision-making, Fuzzy Sets and Systems, 4, 37-51 (1980) · Zbl 0435.90009
[82] Zimmermann, H.-J.; Zysno, P., Decisions and evaluations by hierarchical aggregation of information, Fuzzy Sets and Systems, 10, 243-260 (1983) · Zbl 0519.90049
[83] Zysno, P., The integration of concepts within judgmental and evaluative processes, (Trappl, R.; Klir, G.; Pichler, F., Progress in Cybernetics and Systems Research, Vol. 8 (1982), Hemisphere: Hemisphere New York), 509-517
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