Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP

The lack of consistency in decision making can lead to inconsistent conclusions. In fuzzy analytic hierarchy process (fuzzy AHP) method, it is difficult to ensure a consistent pairwise comparison. Furthermore, establishing a pairwise comparison matrix requires judgments for a level with n criteria (alternatives). The number of comparisons increases as the number of criteria increases. Therefore, the decision makers judgments will most likely be inconsistent. To alleviate inconsistencies, this study applies fuzzy linguistic preference relations (Fuzzy LinPreRa) to construct a pairwise comparison matrix with additive reciprocal property and consistency. In this study, the fuzzy AHP method is reviewed, and then the Fuzzy LinPreRa method is proposed. Finally, the presented method is applied to the example addressed by Kahraman et al. [C. Kahraman, D. Ruan, I. Do?an, Fuzzy group decision making for facility location selection, Information Sciences 157 (2003) 135-153]. This study reveals that the proposed method yields consistent decision rankings from only n − 1 pairwise comparisons, which is the same result as in Kahraman et al. research. The presented fuzzy linguistic preference relations method is an easy and practical way to provide a mechanism for improving consistency in fuzzy AHP method.     

On mixed fuzzy topological spaces and countability

In this paper, we introduce a new type of mixed fuzzy topological space. We define countability on mixed fuzzy topological spaces. We investigate its different quasi type properties.     

结合方向信息和模糊聚类的指纹图像二值化方法

尽管指纹图像具有局部方向一致性,但在一些局部区域,尤其是包含指纹奇异点的区域点方向变化仍然较大.鉴于此在求出指纹各个像素的点方向以后对指纹图像进行分块,引入了方向信息可靠块和方向信息不可靠块的概念.对方向信息可靠块提出一种用方向信息分割指纹图像的方法.对方向信息不可靠块,采用两种方法区分出指纹决和背景块:一是利用该块的均值、方差;二是利用邻域信息.在此基础上对不可靠块中的指纹块采用加权模糊C.均值聚类的方法对其进行二值化.实验表明:用本文提出的方法对指纹图像进行二值化处理后得到的二值图像脊线光滑无孔洞.对断线起到了一定的连接作用,而且对于方向变化较大的分块避免了二值化后纹线的偏移.     

On the structuring of systems with fuzzy relations

Interpretive structural modeling for binary relations and fuzzy structural modeling for fuzzy relations restrict the relation defined on the system being modeled to be either transitive or fuzzy semitransitive. Also, no cycles are allowed in the formal sense. In this paper, we present a new generalized fuzzy structural modeling (GFSM) methodology which does not impose any such restriction on the relation defined on the system while allowing the inclusion of cycles of any order. Algorithms are presented for deriving a minimum-edge connection matrix, automatic detection of hierarchies and cycle condensation. Examples are presented to demonstrate the working of the GFSM algorithms.     

On intuitionistic fuzzy rough sets and their topological structures

In this paper, lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of intuitionistic fuzzy approximation operators are examined. Relationships between intuitionistic fuzzy rough set approximations and intuitionistic fuzzy topologies are then discussed. It is proved that the set of all lower approximation sets based on an intuitionistic fuzzy reflexive and transitive approximation space forms an intuitionistic fuzzy topology; and conversely, for an intuitionistic fuzzy rough topological space, there exists an intuitionistic fuzzy reflexive and transitive approximation space such that the topology in the intuitionistic fuzzy rough topological space is just the set of all lower approximation sets in the intuitionistic fuzzy reflexive and transitive approximation space. That is to say, there exists an one-to-one correspondence between the set of all intuitionistic fuzzy reflexive and transitive approximation spaces and the set of all intuitionistic fuzzy rough topological spaces. Finally, intuitionistic fuzzy pseudo-closure operators in the framework of intuitionistic fuzzy rough approximations are investigated.     

On the primitivity of the division of fuzzy relations

 Allowing for flexible queries enables database users to express preferences inside elementary conditions and priorities between conditions. The division is one of the algebraic operators defined in order to query regular databases. This operation aims at the selection of A-elements which are connected with (at least) a given subset of B-elements, e.g., the stores which ordered all the items supplied by a given manufacturer. It is mainly used in the framework of the relational model of data, although it makes sense in object-oriented databases as well. In the relational context, the division is a non-primitive operation which may be expressed in terms of other operations, namely projection, Cartesian product and set difference. When fuzzy predicates appear, this operator needs to be extended to fuzzy relations and this requires the replacement of the usual implication by a fuzzy one. This paper proposes two types of meaning of the extended division and it investigates the issue of the primitivity of the extended operation (i.e., if the division of fuzzy relations is expressible in terms of other operations). The final objective is to decide whether this operator is necessary or not for the purpose of flexible querying and to help the design of a query language supporting flexible queries, among which those conveying a division of fuzzy relations.     

On measuring μ-T-inconditionality of fuzzy relations

In this paper two methods are proposed to measure the μ-T-inconditionality character of any fuzzy relation for any continuous t-norm T, and it is studied when both methods result to be equivalent.     

On the algebraic structure of binary lattice-valued fuzzy relations

From a general algebraic point of view, this paper aims at providing an algebraic analysis for binary lattice-valued relations based on lattice implication algebras—a kind of lattice-valued propositional logical algebra. By abstracting away from the concrete lattice-valued relations and the operations on them, such as composition and converse, the notion of lattice-valued relation algebra is introduced, LRA for short. The reduct of an LRA is a lattice implication algebra. Such an algebra generalizes Boolean relation algebras by general distributive lattices and can provide a fundamental algebraic theory for establishing lattice-valued first-order logic. Some important results are generalized from the classical case. The notion of cylindric filter is introduced and the generated cylindric filters are characterized.     

On intuitionistic fuzzy topologies based on intuitionistic fuzzy reflexive and transitive relations

Topologies and rough set theory are widely used in the research field of machine learning and cybernetics. An intuitionistic fuzzy rough set, which is the result of approximation of an intuitionistic fuzzy set with respect to an intuitionistic fuzzy approximation space, is an extension of fuzzy rough sets. For further studying the theories and applications of intuitionistic fuzzy rough sets, in this paper, we investigate the topological structures of intuitionistic fuzzy rough sets. We show that an intuitionistic fuzzy rough approximation space can induce an intuitionistic fuzzy topological space in the sense of Lowen if and only if the intuitionistic fuzzy relation in the approximation space is reflexive and transitive. We also examine the sufficient and necessary conditions that an intuitionistic fuzzy topological space can be associated with an intuitionistic fuzzy reflexive and transitive relation such that the induced lower and upper intuitionistic fuzzy rough approximation operators are, respectively, the intuitionistic fuzzy interior and closure operators of the given topology.     

基于模糊偏序关系评估决策的智能信息融合

针对信息融合系统应综合考虑获取信息所要求的资源、计算复杂度和时间所需要的最小成本的问题,提出了基于模糊偏序关系评估决策的智能信息融合系统,利用模糊偏序关系的排序方法,来对传感器进行评估决策和智能选择,并与证据理论相组合,应用于目标识别,仿真结果表明了该方法的有效性。     

On information energy of the image fuzzy sets

Information energy is a measure of useful information obtained from a fuzzy set. This paper investigates the information energy change between the original fuzzy set and the image fuzzy set, through the extension principle. Some methods, by which the information energy of the image fuzzy set can be directly obtained without calculating its membership function, are studied. Useful formulas calculating the information energy of the image of the same shape fuzzy numbers are specifically introduced.     

基于优势关系下模糊目标信息系统的属性约简

在现实生活中,许多信息系统不仅是模糊的还是基于优势关系的。在基于优势关系下模糊目标信息系统中引入了广义决策的概念,提出了分别保持下广义决策、上广义决策、广义决策不变的三种不同约简,进一步给出了各属性约简的判定定理和辨识矩阵,提供了在优势关系下模糊目标信息系统属性约简的具体方法。通过实例验证了该方法的有效性,进一步丰富了粗糙集理论。     

On fuzzy rough sets based on tolerance relations

Dubois and Prade (1990) [1] introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets, which was originally proposed by Pawlak (1982) [8]. Later, Radzikowska and Kerre introduced the so-called (I,T)-fuzzy rough sets, where I is an implication and T is a triangular norm. In the present paper, by using a pair of implications (I,J), we define the so-called (I,J)-fuzzy rough sets, which generalize the concept of fuzzy rough sets in the sense of Radzikowska and Kerre, and that of Mi and Zhang. Basic properties of (I,J)-fuzzy rough sets are investigated in detail.     

On distance and correlation measures of hesitant fuzzy information

A hesitant fuzzy set, allowing the membership of an element to be a set of several possible values, is very useful to express people's hesitancy in daily life. In this paper, we define the distance and correlation measures for hesitant fuzzy information and then discuss their properties in detail. These measures are all defined under the assumption that the values in all hesitant fuzzy elements (the fundamental units of hesitant fuzzy sets) are arranged in an increasing order and two hesitant fuzzy elements have the same length when we compare them. We can find that the results, by using the developed distance measures, are the smallest ones among those when the values in two hesitant fuzzy elements are arranged in any permutations. In addition, the derived correlation coefficients are based on different linear relationships and may have different results. © 2011 Wiley Periodicals, Inc.     

Fuzzy relations and fuzzy groups

Given a group S, we consider fuzzy relations on S, that is, maps from S × S into [0,1]. Of particular interest is to investigate conditions under which the fuzzy relation becomes a fuzzy subgroup on S × S. We prove that if σ is a fuzzy subset of S and μ_σ is the strongest fuzzy relation on S that is a fuzzy relation on σ, then μ_σ is a fuzzy subgroup if and only if σ is a fuzzy subgroup. A number of other results are obtained about the interrelationships between fuzzy relations on S (including the weakest fuzzy relation) and fuzzy subgroups on S × S.     

Preference relations based on hesitant-intuitionistic fuzzy information and their application in group decision making

Preference relations are a powerful quantitative decision approach that assists decision makers in expressing their preferences over alternatives. In real-life applications, decision makers may not be able to provide exact preference information with crisp numbers. To solve this problem, a hesitant-intuitionistic fuzzy number (Hesitant-IFN) is proposed in this paper, and a proposal for the hesitant-intuitionistic fuzzy preference relation (Hesitant-IFPR) and its complementary form (Hesitant-IFCPR) for uncertain preference information are presented. Compared with other preference relations, the proposed relations use hesitant fuzzy elements (HFEs) to express the priority intensities of decision makers and produce the corresponding non-priority intensities by a conversion formula. In addition, we have deduced the operational laws and comparative methods of Hesitant-IFNs and used such information to investigate the corresponding aggregation operators and the approximate consistency tests. Next, we have constructed a group decision-making approach under a hesitant-intuitionistic fuzzy environment. Finally, two case studies are presented to illustrate the preference relations, the approximate consistency tests and the group decision method.     

Confluence and termination of fuzzy relations

Confluence and termination are essential properties connected to the idea of rewriting and substituting which appear in abstract rewriting systems. The aim of the present paper is to investigate confluence, termination, and related properties from the point of view of fuzzy logic leaving the ordinary notions a particular case when the underlying structure of truth degrees is two-valued Boolean algebra. The main motivation of this study is the fact that in several natural situations, the notion of substitutability is inherently fuzzy rather than crisp.     

A decomposition of fuzzy relations

This study is concerned with a decomposition of fuzzy relations, that is their representation with the aid of a certain number of fuzzy sets. We say that some fuzzy sets decompose an original fuzzy refraction if the sum of their Cartesian products approximate the given fuzzy relation. The theoretical underpinnings of the problem are presented along with some linkages with Boolean matrices (such as a Schein rank). Subsequently, we reformulate the decomposition of fuzzy relations as a problem of numeric optimizing and propose a detailed learning scheme leading to a collection of decomposing fuzzy sets. The role of the decomposition in a general class of data compression problems (including those of image compression and rule-based system condensation) is formulated and discussed in detail.     

On improvement of stability conditions for continuous Mamdani-like fuzzy systems

This paper is concerned with continuous fuzzy systems with singleton consequents, called type II fuzzy systems. It first introduces the canonical form of an unforced type II fuzzy system and its stability theorem presented in the previous study. Then, improving the stability theorem, it gives necessary and sufficient conditions such that a type II fuzzy system is stable with respect to a global quadratic Lyapunov function.