Some properties of extended fuzzy preference relations using modalities (original) (raw)
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In decision making, in order to avoid misleading solutions, the study of consistency when the decision makers express their opinions by means of preference relations becomes a very important aspect in order to avoid misleading solutions. In decision making problems based on fuzzy preference relations the study of consistency is associated with the study of the transitivity property. In this paper, a new characterization of the consistency property defined by the additive transitivity property of the fuzzy preference relations is presented.
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Summary. Preference relations are the most common representation structures of information used in decision making problems because they are useful tool in modelling decision processes, above all when we want to aggregate experts' preferences into group preferences. Therefore, to establish rationality properties to be verified by preference relations is very important in the design of good decision making models.
Fuzzy Sets and Systems, 1998
The generalization of the concept of a classical (or crisp) preference structure to that of a fuzzy preference structure, expressing degrees of strict preference, indifference and incomparability among a set of alternatives, requires the choice of a de Morgan triplet, i.e., of a triangular norm and an involutive negator. The resulting concept is only meaningful provided that this choice allows the representation of truly fuzzy preferences. More specifically, one of the degrees of strict preference, indifference or incomparability should always be unconstrained to the preference modeller. This intuitive requirement is violated when choosing a triangular norm without zero divisors, since in that case fuzzy preference structures reduce to classical preference structures, and hence none of the degrees can be freely assigned. Furthermore, it is shown that the choice of a continuous non-Archimedean triangular norm having zero divisors is not compatible with our basic requirement: the sets of degrees of strict preference, indifference and incomparability in [0, 1 [ are always bounded from above by a value strictly smaller than 1. These fundamental results imply that when working with a continuous triangular norm, only Archimedean ones having zero divisors are suitable candidates. These arguments sufficiently support our plea for the use of Lukasiewicz triplets in the definition of fuzzy preference structures.
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In order to model the preferences of a decision-maker (DM) by means of fuzzy preference relations, a DM can utilize different preference formats (such as ordering of the alternatives, utility values, multiplicative preference relations, fuzzy estimates, and reciprocal as well as nonreciprocal fuzzy preference relations) to express his/her judgments. Afterward, the obtained information is utilized to construct fuzzy preference relations. Here we introduce a procedure that allows the use of so-called preference functions (which is a preference format utilized in the methods of PROMETHEE family) to construct nonreciprocal fuzzy preference relations. With diverse preference formats being offered, a DM can select the one that is the most convenient to articulate his/her preferences. In order to demonstrate the applicability of the proposed procedure a multicriteria decision-making problem related to the site selection for constructing a new hospital is considered here.
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Analysis of X, R models is considered as part of a general approach to solving optimization problems with fuzzy coefficients. This approach involves a modification of traditional optimization methods and is associated with formulating and solving one and the same problem within the framework of mutually interrelated models by constructing equivalent analogs with fuzzy coefficients in objective functions alone. The use of the approach allows one to maximally cut off dominated alternatives. The subsequent contraction of the decision uncertainty region is based on reducing the problem to multiobjective choice of alternatives in a fuzzy environment. Three techniques for processing of fuzzy preference relations reflecting criteria of quantitative as well as qualitative character are discussed. The results of the paper are universally applicable and are already being used to solve power engineering problems.
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We consider properties of intuitionistic fuzzy preference relations. We study preservation of a preference relation by lattice operations, composition and some Atanassov's operators like F α,β , P α,β , Q α,β , where α, β ∈ [0, 1]. We also define semi-properties of intuitionistic fuzzy relations, namely reflexivity, irreflexivity, connectedness, asymmetry, transitivity. Moreover, we study under which assumptions intuitionistic fuzzy preference relations fulfil these properties. In all these cases, if possible, we try to give characterizations of adequate properties.