Xuzhu Wang

Taiyuan University of Technology · College of Mathematics

In this paper, first we give the definitions of various indicators of fuzzy relations and their basic properties. Then we investigate the relationships between these indicators, particularly between those of T-transitivity, negative S-transitivity, T–S-semitransitivity and T–S-Ferrers property. The investigation is divided into three parts: results...

This work aims at the discussion of reasonable properties for the ordering of fuzzy quantities. In the fuzzy literature more than 35 indices exist for the comparison of fuzzy quantities. To grasp this amalgam of indices we split them up into three classes (with linguistic approaches excluded). In this paper we briefly introduce the ordering indices...

Mathematics of Fuzziness—Basic Issues introduces a basic notion of ‘fuzziness’ and provides a conceptual mathematical framework to characterize such fuzzy phenomena in Studies in Fuzziness and Soft Computing. The book systematically presents a self-contained introduction to the essentials of mathematics of fuzziness ranging from fuzzy sets, fuzzy r...

In this paper, we focus on the characterizations of some properties of fuzzy relations by using the notions of traces. The investigation is two-fold. On one hand, we attempt to reformulate the results obtained by Fodor under weakened conditions. On the other hand, we present some new characterizations of properties, mainly T-asymmetry, T-S-semitran...

In this paper, we focus on the characterizations of various property indicators of fuzzy relations by means of left and right traces. The characterized indicators include those of reflexivity, T-asymmetry, S-completeness, T-transitivity, negative S-transitivity, T-S-semitransitivity and T-S-Ferrers property. The investigation can be regarded as an...

Da Ruan passed away suddenly, left us not only with the enormous sadness and shock but also the endless recollections of the past. I was sent to Belgium by Shanxi provincial government as a visiting scholar in 1988. Considering my research area is related to mathematics of fuzziness, in 1991 I contacted Professor Etienne Kerre from Ghent University...

In this paper, we firstly present some results on the relationships between the transitivity-related properties of fuzzy relations, including T-transitivity, negative S-transitivity, T-S-semitransitivity and T-S-Ferrers relation. Then the relationships between these transitivity-related indicators are further investigated. With our research, some c...

In this paper we introduce a definition of consistency of the judgement matrix in the fuzzy Analytic Hierarchy Process (AHP) and give a general expression of all fuzzy weights under the condition of consistency. Finally, based on our discussion, the geometric average method is suggested for fuzzy weights calculation in the practical decision-making...

The investigation of rationality plays a central role in the study of fuzzy choice functions. In this paper, we deal with some important rationality conditions of fuzzy choice functions, including Weak (Strong) Fuzzy Congruence Axiom, Weak (Strong) Axiom of Fuzzy Revealed Preference, fuzzy versions of famous crisp conditions α, β, γ and δ etc. in t...

The objective of the paper is to extend Bandyopadhyay's results on rationality conditions of crisp choice functions. By fuzzifying the rationality conditions in the crisp case, we present a necessary and sufficient condition for the acyclic rationality and a characterization theorem for the W_φ-transitive rationality under a strong De Mor...

In this paper we present an extension of the famous representation theorem of Negoita and Ralescu. In this theorem the authors proved that the structure of lattice-valued fuzzy sets is dual isomorphic to the structure of lattice-valued flou sets for a special kind of lattices satisfying an additional condition that resembles the characterization of...

In this chapter, we present an overview of choice functions. Firstly, we introduce some important research topics on classic choices which serve as a guideline for the fuzzification research of choice functions. Then we begin with the fuzzy choice function defined by Banerjee. After a brief introduction to Banerjee’s work, various preferences deriv...

In this paper, we mainly fuzzify the rationality conditions of choice functions employed by Suzumura and investigate their relationships systematically under the assumption that every involved choice set is normal. We prove that the rationality condition FRR is equivalent to WFCA, and furthermore, that both conditions are equivalent to WAFRP under...

In this paper, for fuzzy uncertainty of data in disaster risk assessment, we suggest a method to granule the information. Its characteristics lie in exploiting the tolerance for imprecision, uncertainty and partial truth to achieve tractability, robustness, low solution cost and better support with reality. Its operation is more simple and transpar...

As known to us, a relation is a subset of the Cartesian product of two sets. A relation is naturally fuzzified while a subset is fuzzified. In fact, whether two objects have a relation is not always easy to determine. For example, the relation “greater than” on the set of real numbers is a crisp one because we can determine the order relation of an...

As known to us, the theory of classical sets is the foundation on which modern mathematics rests. When sets are fuzzified, some traditional pure mathematical branches are accordingly generalized. In this chapter, we introduce three well-developed fuzzified mathematical areas briefly to have a glance at how a pure mathematical theory can be fuzzifie...

In the traditional multi-attribute decision analysis, there is a well-defined problem-solving model–the Simple Additive Weighting (SAW) method. This model can be formulated as follows. Let A1, A2, ..., An be n alternatives and C1,C2, ..., Cm m attributes with the corresponding weights w1, w2, ..., wm respectively. If the evaluation of the alternat...

This chapter aims to recall basic concepts of set theory and abstract algebra including set, relation, isomorphism, lattice, Boolean algebra and soft algebra, which will serve as the base of the remaining chapters in the book.

In this chapter, we focus on the introduction of fundamentals in fuzzy set theory, including some set-theoretic operations and their extensions, the decomposition of a fuzzy set, and mathematical representations of fuzzy sets in terms of a nest of sets. Towards the end of the chapter, fuzzy sets taking values in [0,1] are extended to those on a lat...

The major subject of this chapter is fuzzy control, one of the most successful application areas of fuzzy set theory. Nowadays, many fuzzy products are visible in the market. Almost every fuzzy product is related to a fuzzy control problem. It is no exaggeration to say that fuzzy set theory is highly accepted partly because of the great success of...

The rationality of fuzzy choice function is characterized based on the intuitive negation and the fuzzy Schwartz's conditions. Under the assumption that every choice set is normal, the conditions so defined are proved to be sufficient to guarantee the quasi-transitive, pseudo-transitive and semi-order rationality of fuzzy choice functions but no lo...

Firstly, this paper summarizes researches on the ranking of fuzzy quantities, which includes the description of ordering problems, application fields, review and classification of ordering indices, rationality of an ordering approach. Then, based on our research, we develop a software package. This paper gives a detail interpretation of the functio...

In order to reflect the fuzziness existing in a choice problem, Banerjee introduces the concept of fuzzy choice function. Meanwhile, Banerjee proposes three congruence conditions in assessing the rationality of a fuzzy choice function. In this paper, we prove some dependencies among these conditions although Banerjee asserts that they are independe...

In this paper, an axiomatization definition for the ranking of interval numbers is suggested from the mathematical and practical angle, then two families of ordering relations, θ- order and L-θ- order respectively, are proposed and some rationality properties related to them are investigated. Finally, the essence and rationality of θ- order and L-θ...

Majority order is defined to determine ranking orders between fuzzy numbers. Some reasonable properties of the derived order relations are investigated in detail. In particular, the approaches are mathematically interpreted by means of the Lebesgue measure

Given a fuzzy relation, S. Ovchinnikov and M. Roubens (1991) introduce a very general definition of fuzzy strict preference. The author investigates its transitivity-related properties including weak transitivity, consistency, acyclicity, etc

As a continuation of the first part related to the first and second class of ordering approaches this paper deals with the fulfilment of reasonable properties in the third class of ordering approaches. To do so we briefly introduce fuzzy relations on which the third class of approaches is based. Then we recall some transitivity-related concepts and...

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...

In this paper, we classify all the approaches to order fuzzy quantities into three classes. Then we focus our attention on the investigation of dependency among the elements of the first class of ordering approaches.

The purpose of this paper is twofold. Firstly, we would like to comment on the study of similarity measures carried out by Pappis and Karacapilidis (1993). Their definition of ‘approximate equality’ of fuzzy sets is modified and relevant properties related to this correlated definition are listed. Secondly, a new class of similarity measures, extra...

Several definitions of the transitivity property of fuzzy orderings that can be found in the literature, and the relations between them are investigated. Afterwards, we discuss Ovchinnikov’s transitivity condition in more detail.

We first carry out further investigations into relations between several existing transitivity-related notions in ranking alternatives or fuzzy numbers. Then we pay attention to the discussion of transitivity properties of some practical fuzzy preference relations introduced by Dubois and Prade. Finally, we develop an ordering procedure of alternat...