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Group decision making and consensus under fuzzy preferences and fuzzy majority. (English) Zbl 0768.90003
The paper develops fuzzy set-based models for fundamental relations of strict preference, indifference, and incomparability. This generalization is aimed at preserving all classical properties found in preference modelling. Recall that in this theory the above binary relations are defined in a given family $A$ of alternatives as follows: Strict preference: $aPb$ iff $aRb$ and not $bRa$; Indifference: $aIb$ iff $aRb$ and $bRa$; Incomparability: $aJb$ iff not $aRb$ and not $bRa$, where $R$ denotes a binary relation of weak preference, say $aRb$ iff $a$ is at least as good as $b$. The main results pertain to an extension of the classical results by proposing fuzzy models for the above relations. It is proved that a “reasonable” generalization (preserving the properties found in the Boolean case) should be based upon Lukasiewicz-like De Morgan triples.

##### MSC:
 91B08 Individual preferences 03E72 Fuzzy set theory
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##### References:
 [1] Aizerman, M. A.: New problems in the general choice theory. Soc. choice welf. 2, 235-282 (1985) · Zbl 0583.90004 [2] Blin, J. M.; Whinston, A. P.: Fuzzy sets and social choice. J. cybernet. 4, 17-22 (1973) · Zbl 0303.90009 [3] Fedrizzi, M.: Group decisions and consensus: A model using fuzzy sets theory (in italian). Rivista scienze econ. Soc. A 9, No. 1, 12-20 (1986) [4] Fedrizzi, M.; Kacprzyk, J.: On measuring consensus in the setting of fuzzy preference relations. Non-conventional preference relations in decision making, 129-141 (1988) · Zbl 0652.90004 [5] Fedrizzi, M.; Kacprzyk, J.; Zadron\dot{}, S.: An interactive multi-user decision support system for consensus reaching processes using fuzzy logic with linguistic quantifiers. Decision support systems 4, 313-327 (1988) [6] Kacprzyk, J.: Collective decision making with a fuzzy majority rule. Proc. WOGSC congress, 153-159 (1984) [7] Kacprzyk, J.: Zadeh’s commonsense knowledge and its use in multicriteria, multistage and multiperson decision making. Approximate reasoning in expert systems, 105-121 (1985) [8] Kacprzyk, J.: Some ’commonsense’ solution concepts in group decision making via fuzzy linguistic quantifiers. Management decision support systems using fuzzy sets and possibility theory, 125-135 (1985) [9] Kacprzyk, J.: Group decision-making with a fuzzy majority via linguistic quantifiers, part I: A consensory-like pooling; part II: A competitive-like pooling. Cybernet. and systems 16, 131-144 (1985) · Zbl 0602.90006 [10] Kacprzyk, J.: Group decision making with a fuzzy linguistic majority. Fuzzy sets and systems 18, 105-118 (1986) · Zbl 0604.90012 [11] Kacprzyk, J.: Towards an algorithmic/procedural ’human consistency’ of decision support systems: A fuzzy logic approach. Applications of fuzzy sets in human factors, 101-116 (1986) [12] Kacprzyk, J.: On some fuzzy cores and ’soft’ consensus measures in group decision making. The analysis of fuzzy information 2, 119-130 (1987) · Zbl 0652.90011 [13] Kacprzyk, J.: Towards ’human consistent’ decision support systems through commonsense-knowledge-based decision making and control models: A fuzzy logic approach. Comput. artificial intelligence 6, 97-122 (1987) · Zbl 0631.68076 [14] Kacprzyk, J.; Fedrizzi, M.: ’Soft’ consensus measures for monitoring real consensus reaching processes under fuzzy preferences. Control cybernet. 15, 309-323 (1986) · Zbl 0636.90001 [15] Kacprzyk, J.; Fedrizzi, M.: A ’soft’ measure of consensus in the setting of partial (fuzzy) preferences. European J. Oper. res. 34, 315-325 (1988) · Zbl 0652.90004 [16] Kacprzyk, J.; Fedrizzi, M.: A ’human-consistent’ degree of consensus based on fuzzy logic with linguistic quantifiers. Math. social sci. 18, 275-290 (1989) · Zbl 0685.90008 [17] Kacprzyk, J.; Fedrizzi, M.: Multiperson decision making models using fuzzy sets and possibility theory. (1990) · Zbl 0724.00034 [18] Kacprzyk, J.; Fedrizzi, M.; Nurmi, H.: Group decision making with fuzzy majorities represented by linguistic quantifiers. Approximate reasoning tools for artificial intelligence, 126-145 (1990) [19] Kacprzyk, J.; Fedrizzi, M.; Nurmi, H.: Fuzzy logic with linguistic quantifiers in group decision making and consensus formation. An introduction to fuzzy logic applications in intelligent systems, 263-280 (1992) [20] Kacprzyk, J.; Nurmi, H.: Linguistic quantifiers and fuzzy majorities for more realistic and human-consistent group decision making. Fuzzy methodologies for industrial and systems engineering, 267-281 (1989) [21] Kacprzyk, J.; Roubens, M.: Non-conventional preference relations in decision making. (1988) · Zbl 0642.00025 [22] Kacprzyk, J.; Yager, R. R.: Linguistic quantifiers and belief qualification in fuzzy multicriteria and multistage decision making. Control cybernet. 13, 155-173 (1984) · Zbl 0551.90091 [23] Kacprzyk, J.; Yager, R. R.: ’Softer’ optimization and control models via fuzzy linguistic quantifiers. Inform. sci. 34, 157-178 (1984) · Zbl 0562.90098 [24] Kacprzyk, J.; Yager, R. R.: Using fuzzy logic with linguistic quantifiers in multiobjective decision making and optimization: A step towards more human-consistent models. Stochastic versus fuzzy approaches in multiobjective mathematical programming under uncertainty, 331-350 (1990) · Zbl 0728.90051 [25] Kacprzyk, J.; Zadron\dot{}ny, S.; Fedrizzi, M.: An interactive user-friendly decision support system for consensus reaching based on fuzzy logic with linguistic quantifiers. Fuzzy computing, 307-322 (1988) [26] Kahneman, D.; Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47, 263-291 (1979) · Zbl 0411.90012 [27] Loewer, B.; Laddaga, R.: Destroying the consensus. Synthese 62, 79-96 (1985) [28] Machina, M.: Expected utility analysis without the independence axiom. Econometrica 50, 277-323 (1982) · Zbl 0475.90015 [29] Nurmi, H.: Approaches to collective decision making with fuzzy preference relations. Fuzzy sets and systems 6, 249-259 (1981) · Zbl 0465.90006 [30] Nurmi, H.: Imprecise notions in individual and group decision theory: resolution of allais’ paradox and related problems. Stochastica 6, 283-303 (1982) · Zbl 0519.90006 [31] Nurmi, H.: Voting procedures: A summary analysis. British J. Political sci. 13, 181-208 (1983) [32] Nurmi, H.: Comparing voting systems. (1987) [33] Nurmi, H.: Assumptions on individual preferences in the theory of voting procedures. Non-conventional preference relations in decision making, 142-155 (1988) [34] Nurmi, H.; Fedrizzi, M.; Kacprzyk, J.: Vague notion in the theory of voting. Multiperson decision making models using fuzzy sets and possibility theory, 43-52 (1990) [35] Nurmi, H.; Kacprzyk, J.: On fuzzy tournaments and their solution concepts in group decision making. European J. Oper. res. 51, 223-232 (1991) · Zbl 0742.90009 [36] Ponsard, C.: An application of fuzzy subsets theory to the analysis of the consumer’s spatial preferences. Fuzzy sets and systems 5, 235-244 (1981) · Zbl 0454.90004 [37] Ponsard, C.: L’équilibre du consomateur dans un context imprécis. Sistemi urbani 3, 107-133 (1981) [38] Ponsard, C.: Producer’s spatial equilibria with a fuzzy constraint. European J. Oper. res. 10, 302-313 (1982) · Zbl 0483.90017 [39] Ponsard, C.: Partial spatial equilibria with fuzzy constraints. J. regional sci. 22, 159-175 (1982) · Zbl 0483.90017 [40] Ponsard, C.: Fuzzy sets in economics: foundations of soft decision theory. Management decision support systems using fuzzy sets and possibility theory, 25-37 (1985) [41] Ponsard, C.: Fuzzy mathematical models in economics. Fuzzy sets and systems 28, 273-283 (1988) · Zbl 0657.90017 [42] Tversky, A.; Kahneman, D.: Rational choice and the framing of decisions. J. of business 59, S251-S278 (1986) · Zbl 1225.91017 [43] Yager, R. R.: Quantifiers in the formulation of multiple objective decision functions. Inform. sci. 31, 107-139 (1983) · Zbl 0551.90084 [44] Yager, R. R.: Quantified propositions in linguistic logic. Internat. J. Man-machine stud. 19, 195-227 (1983) · Zbl 0522.03013 [45] Yager, R. R.: Aggregating evidence using quantified statements. Inform. sci. 36, 179-206 (1985) · Zbl 0584.94030 [46] Zadeh, L. A.: A computational approach to fuzzy quantifiers in natural languages. Comput. math. Appl. 9, 149-184 (1983) · Zbl 0517.94028 [47] Zadeh, L. A.: Syllogistic reasoning in fuzzy logic and its application to usuality and reasoning with dispositions. IEEE trans. Systems man cybernet. 175, 754-763 (1985) · Zbl 0593.03033