Continuity of utility functions representing fuzzy preferences (original) (raw)
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Fono and Gwét [On strict lower and upper sections of fuzzy orderings, Fuzzy Sets and Systems 139 (2003) 583–599] introduced and studied strict lower section, strict upper section and strict order interval of a given weakly complete fuzzy pre-order R when the minimal regular fuzzy strict component of R is defined by the residual coimplicator of the ŁŁukasiewicz t-conorm. And they gave an example of application of fuzzy strict lower sections in economics. In this paper, we generalize their framework for the minimal regular fuzzy strict component of R defined by the residual complicator of a continuous t-conorm. We show that fuzzy strict sections and fuzzy strict order intervals, defined in this general case, generate fuzzy topologies.
Preference relation on fuzzy utilities based on fuzzy leftness relation on intervals
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We define a new preference relation #p(x, y) between two fuzzy numbers or utilities x and y, based on the fuzzy leftness relationship between intervals. A key property of #r(x, y) is that it satisfies the well-known min-transitivity property: #p(x, z )/> min{ (#p(x, y), p p(y, z)}; the previous definitions of preference relation for fuzzy utilities satisfy only certain weaker forms of transitivity. 1 A fuzzy number is a convex fuzzy set with a finite support, i.e., #n(x)~ 0 outside a finite interval. The membership function yA(x) of a convex fuzzy set consists of a non-decreasing part followed by a non-increasing part. Each a-cut A ~ = {x: /L4(x)/> 7} of such a fuzzy set A is an interval (open, closed, or semi-open).
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Complete pre-orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations. In this paper, we investigate this characterization in a fuzzy setting. We consider two types of completeness (weak completeness and strong completeness) and decompose a fuzzy pre-order by means of an indifference generator, in particular a Frank t-norm. In the weakly complete case, we identify the strongest type of transitivity of the indifference and strict preference relations in function of the generator used for constructing them. In the strongly complete case, we lay bare a stronger type of transitivity of the strict preference relation. We conclude the paper with a rather negative result: there is no hope to obtain a compositional characterization of weakly complete fuzzy pre-orders, and hence also not of fuzzy pre-orders in general.
On Interval-Valued Fuzzy Soft Preordered Sets and Associated Applications in Decision-Making
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Recently, using interval-valued fuzzy soft sets to rank alternatives has become an important research area in decision-making because it provides decision-makers with the best option in a vague and uncertain environment. The present study aims to give an extensive insight into decision-making processes relying on a preference relationship of interval-valued fuzzy soft sets. Firstly, interval-valued fuzzy soft preorderings and an interval-valued fuzzy soft equivalence are established based on the interval-valued fuzzy soft topology. Then, two crisp preordering sets, namely lower crisp and upper crisp preordering sets, are proposed. Next, a score function depending on comparison matrices is expressed in solving multi-group decision-making problems. Finally, a numerical example is given to illustrate the validity and efficacy of the proposed method.
A Functional Representation of Fuzzy Preferences
Theoretical Economics Letters, 2017
This paper defines a well-behaved fuzzy order and finds a simple functional representation for the fuzzy preferences. It includes the existing utility theory for exact preferences (no fuzziness) as a special case. It is a simple and intuitive extension of the utility theory under uncertainty, which can potentially be used to explain a few known paradoxes against the existing expected utility theory.
Conditional extensions of fuzzy preorders
Fuzzy Sets and Systems
The problem of embedding incomplete into complete rela- tions has been an important topic of research in the context of crisp relations and their applications. Several variations of the acclaimed Szpilrajn’s theorem have been provided, inclusive of the case when some order conditions between elements are imposed on the extension. We extend the analysis of that topic by Alcantud to the fuzzy case. By appealing to generators to decompose (fuzzy) preference relations into strict preference and indifference relations, we give general extension results for the corresponding concept of compatible extension of a fuzzy reflexive relation. Then we investigate the conditions under which compatible order extensions exist such that certain elements are connected by the asymmetric part, resp., and certain other elements by the symmetric part, to respective elements with degree 1.
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.
On strict lower and upper sections of fuzzy orderings
Fuzzy Sets and Systems, 2003
Strict lower sections, strict upper sections and open order intervals (strict order intervals) are classical notions associated to crisp orderings (crisp total preorders). In this paper, we extend these notions to fuzzy orderings. We determine all the fuzzy orderings (they contain all the crisp orderings) which make it possible to compare two fuzzy strict lower sections and two fuzzy strict upper sections. We then deduce the properties (equality, inclusion and intersection) of fuzzy strict order intervals. In this way, we obtain fuzzy extensions of well-known properties of crisp strict sections and crisp strict order intervals, and we display one example of their applications.