A parametric model for determining consensus priority vectors from fuzzy comparison matrices

In fuzzy clustering, we always use fuzzy equivalence matrix to find the clustering results. Although fuzzy equivalence matrix is one of the most important tools in fuzzy clustering, it’s hard to judge whether a fuzzy matrix is a fuzzy equivalence matrix directly. In this paper, some simple and useful properties of the fuzzy equivalence matrices have been proposed based on the basic properties proposed in the classi cal articles. Using these properties, the judgment whether a fuzzy matrix is a fuzzy equivalence matrix may become an easy work. Keywords: Fuzzy clustering, fuzzy equivalence matrix 1. INTRODUCTION Given a finite set of objects X, and a measure of similarity for an arbitrary pair of objects, the problem of clustering is to find a partition of X such that the degree of similarity is high for objects within each block (cluster) and low for objects in different blocks. The purpose of clustering is to find a natural classification of given objects on the basis of a conceived concept of similarity. The classification may help to uncover a structure in given data, improve our understanding of a complex system, or suggest a meaningful simplification of a given system [1]. The requirement that given objects be classified into pairwise disjoint subsets (crisp clusters) is overly strong in many practical applications. Fuzzy set theory allows us to weaken it and, hence, make clustering more realistic. So, fuzzy clustering becomes one of the hottest research branches of data mining. One of the approaches to fuzzy clustering is based on fuzzy equivalence relations defined on X. Every fuzzy equivalence relation induces a crisp partition in each of its .-cuts. Using this approach, the fuzzy clustering problem becomes the problem of identifying a fuzzy equivalence relation on X in terms of a conceived concept of similarity. Although this may not be possible to do directly, it is usually easy to determine a fuzzy compatibility relation in terms of an appropriate distance function defined on X. A distance function is always reflexive and symmetric, but it is not transitive. However, once we have a compatibility relation, we can readily obtain a meaningful fuzzy equivalence relation by calculating its transitive closure. The resulting fuzzy equivalence relation offers the user a sequence of crisp partitions on X, which are linearly ordered by refinement ordering and graded by values of . [0, 1]. The user is given a flexibility to choose one of these partitions on the basis of additional criteria [2-4]. But, to calculate the transitive closure of a N×N fuzzy matrix , we need to calculate the 2kth power of the matrix, where k = 1, 2, , N. In the processing, after we calculate the 2k+1th power of the matrix and find that the 2k+1th power of the matrix is equal to the 2kth power of the matrix, the 2kth power of the matrix is the transitive closure of the matrix. In fact, if we can find the 2kth power of the matrix is a fuzzy equivalence matrix directly, we do not need to calculate the 2k+1th power of the matrix. If we can judge a fuzzy matrix is a fuzzy equivalence matrix by means of some simple properties of the equivalence matrix, it may be helpful. In this paper, some properties of the fuzzy equivalence matrix have been proposed and the judgment whether a fuzzy *white70121@163.com

Preference modelling is a fundamental part of several applied fields but at the same time it has its own interesting theoretical problems. There exists a well-known axiomatic approach to fuzzy preference structures. In this axiomatic framework, general constructions of strict preference, indifference and incomparability relations associated with a fuzzy large preference relation are established via solving a system of functional equations. In the present paper we overview the solutions of two key functional equations (one of Frank, and one of Alsina) under different conditions on the underlying logic operations. Consequences of these general results to fuzzy preference modelling are also summarized

In this paper we first consider the problem of extending fuzzy (weak and strict) preference relations, represented by fuzzy preorders on a set to a fuzzy preferences on subsets, and we characterise different possibilities. Based on their properties, we then semantically define and axiomatize several two-tiered graded modal logics to reason about the corresponding different notions of fuzzy preferences.

Consistency and consensus reaching process for group decision making based on complete interval distributed preference relations under social network analysis

Recently, many methods based on fuzzy rough sets are proposed to reduce fuzzy attributes. The common characteristic of these methods is that all of them are based on fuzzy equivalence relation. In other words, the underlying concept of rough sets, indispensability relation, is generalized to fuzzy equivalence relation. Here fuzzy equivalence relation is the binary relation, which is reflexive, symmetric and transitive. This paper tries to generalize the fuzzy equivalence relation to fuzzy similarity relation, which is more helpful to keeping the fuzzy information of initial data than fuzzy equivalence relation. Based on the fuzzy similarity relation, fuzzy matrix computation for information system is proposed which can be used to reduce fuzzy attributes. Firstly, fuzzy similarity relation who is isomorphic with the fuzzy similarity matrix is given as fuzzy indispensability relation. Then all the information of initial data, such as the similarity among objects and fuzzy inconsistence degree between two objects, can be represented by fuzzy similarity matrix. Secondly, by considering that the small perturbation of the fuzzy similarity matrix can be ignorable, we propose some basic concepts of knowledge reduction such as fuzzy attributes reduct, core and fuzzy significance of attributes etc in this paper. Thirdly, a heuristic algorithm based on the fuzzy significance of attributes is proposed to find close-to-minimal fuzzy attributes reduct. Finally, experimental comparisons with other methods of attributes reduction are given. The experimental results show that our method is feasible and effective

Fuzzy preference relations (Orlovsky 1978) (also known as reciprocal preference relation (Baets et al. 2006; Xu 2007b,f; Xu and Chen 2008b)) and multiplicative preference relations (Saaty 1980) are the most common tools to express the DMs’ preferences over alternatives in decision making. However, sometimes, to get a more reasonable decision result, a decision organization, which contains a lot of DMs (or experts), is authorized to provide the preferences by comparing each pair of alternatives using 0-1 scale, and when providing the degrees to which an alternative is superior to another, it is not very sure about a value but has hesitancy between several possible values. In such cases, these several possible values can be considered as a HFE, and a hesitant fuzzy preference relation is constructed when all the preferences over a set of alternatives are provided (Xia and Xu 2013).

Group decision making with preference information on alternatives is an interesting and important research topic which has been receiving more and more attention in recent years. The purpose of this paper is to investigate multiple-attribute group decision-making (MAGDM) problems with distinct uncertain preference structures. We develop some linear-programming models for dealing with the MAGDM problems, where the information about attribute weights is incomplete, and the decision makers have their preferences on alternatives. The provided preference information can be represented in the following three distinct uncertain preference structures: 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first establish some linear-programming models based on decision matrix and each of the distinct uncertain preference structures and, then, develop some linear-programming models to integrate all three structures of subjective uncertain preference information provided by the decision makers and the objective information depicted in the decision matrix. Furthermore, we propose a simple and straightforward approach in ranking and selecting the given alternatives. It is worth pointing out that the developed models can also be used to deal with the situations where the three distinct uncertain preference structures are reduced to the traditional ones, i.e., utility values, fuzzy preference relations, and multiplicative preference relations. Finally, we use a practical example to illustrate in detail the calculation process of the developed approach.

Compatibility measures and consensus models for group decision making with intuitionistic multiplicative preference relations

Normalizing rank aggregation method for priority of a fuzzy preference relation and its effectiveness

The ordinal consistency of a fuzzy preference relation

There are alternative ways of decomposing a given fuzzy weak preference relation into its antisymmetric and symmetric components. In this paper I have provided support to one among these alternative specifications. It is shown that on this specification the fuzzy analogue of the General Possibility Theorem is valid even when the transitivity restrictions on the individual and the social preference relations are relatively weak. In the special case where the individual preference relations are exact but the social preference relation is permitted to be fuzzy it is possible to distinguish between different degrees of power of the dictator. This power increases with the strength of the transitivity requirement.

A fuzzy multi-criteria decision-making model by associating technique for order preference by similarity to ideal solution with relative preference relation

This is a first approach to the study of the connection between fuzzy preference relations and fuzzy choice functions. In particular we depart from a fuzzy preference relation and we study the conditions it must satisfy in order to get a fuzzy choice function from it. We are particulary interested in one function: G-rationalization. We discuss the relevance of the completeness condition on the departing preference relation. We prove that not every non-complete fuzzy preference relation leads to a choice function.Keywordsfuzzy preference relation G-rational choice functioncomplete fuzzy relation

Some issues on consistency of fuzzy preference relations

On the basis of geometric consistency (GCI) of multiplicative preference relations and the transformation relation between multiplicative preference relations and fuzzy preference relation, inconsistency measure of fuzzy preference relation and the thresholds associated with it are studied. A new satisfactory consistency concept of fuzzy preference relation is defined. For a fuzzy preference relation, which does not possess satisfactory consistency, an improving consistency approach is provided. Then, programming models for estimating interval priority weights from fuzzy preference relations is established. Finally, an example is given to illustrate the validity and practicality of the proposed methods.