An investigation into relations between some transitivity-related concepts.

*(English)*Zbl 0917.90025Summary: There exists a variety of transitivity notions in the literature to eliminate the possible inconsistency adherent to a fuzzy preference relation in ranking fuzzy quantities or alternatives. The relationships among max-min transitivity, restricted max-min transitivity, quasitransitivity, weak transitivity, consistency and acyclicity are investigated. We point out that max-min transitivity and \(\omega\)-transitivity are very strong restrictions on a fuzzy preference relation, and acyclicity is the weakest one. With a certain transitivity condition, an ordering procedure is suggested to obtain a total order relation among fuzzy quantities or alternatives based on a fuzzy preference relation.

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##### References:

[1] | Banerjee, A, Rational choice under fuzzy preferences: the orlovsky choice function, Fuzzy sets and systems, 53, 295-299, (1993) · Zbl 0801.90001 |

[2] | Basile, L, Ranking alternatives by weak transitivity relations, (), 105-112 |

[3] | Bortolan, G; Degani, R, A review of some methods for ranking fuzzy subsets, Fuzzy sets and systems, 15, 1-19, (1985) · Zbl 0567.90056 |

[4] | Buckley, J.J, Ranking alternatives using fuzzy numbers, Fuzzy sets and systems, 15, 21-31, (1985) · Zbl 0567.90057 |

[5] | Chen, S; Hwang, C, Fuzzy multiple attribute decision making, (), 101-486 |

[6] | Dubois, D; Prade, H, Ranking fuzzy numbers in the setting of possibility theory, Inform. sci., 30, 183-224, (1983) · Zbl 0569.94031 |

[7] | Kolodziejczyk, W, Orlovsky’s concept of decision-making with fuzzy preference relation - further results, Fuzzy sets and systems, 19, 11-20, (1986) · Zbl 0597.90004 |

[8] | Montero, F.J; Tejada, J, A necessary and sufficient condition for the existence of Orlovsky’s choice set, Fuzzy sets and systems, 26, 121-125, (1988) · Zbl 0641.90007 |

[9] | Nakamura, K, Preference relations on a set of fuzzy utilities as a basis for decision making, Fuzzy sets and systems, 20, 147-162, (1986) · Zbl 0618.90001 |

[10] | Ok, E.A, On the approximation of preferences by exact relations, Fuzzy sets and systems, 67, 173-179, (1994) · Zbl 0845.90014 |

[11] | Orlovsky, S.A, Decision-making with a fuzzy preference relation, Fuzzy sets and systems, 1, 155-167, (1978) · Zbl 0396.90004 |

[12] | Saade, J.J; Schwarzlander, H, Ordering fuzzy sets over the real line: an approach based on decision making under uncertainty, Fuzzy sets and systems, 50, 237-246, (1992) |

[13] | Wang, X; Kerre, E.E; Cappelle, B; Ruan, D, Transitivity of fuzzy orderings based on pairwise comparisons, J. fuzzy math., 3, 455-463, (1995) · Zbl 0839.90005 |

[14] | Wang, X; Ruan, D; Kerre, E.E, The use of weak transitivity in ranking fuzzy numbers, (), 322-332 |

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