On the representations of L-equivalence relations on L-fuzzy sets with applications to locally vague environments

ABSTRACT

L-equivalence relations on L-fuzzy sets, equivalent to L-valued equalities on ordinary sets, are modelling fuzzy similarity of two objects that are partial members of a fuzzily defined set. This paper deals with construction and representation properties of L-equivalence relations on L-fuzzy sets. To this end, we set up a method for constructing them by means of quasi-pseudo-metrics, and then point out how L-equivalence relations on L-fuzzy sets are represented by families of L-fuzzy subsets of L-fuzzy sets. The present results are applied to locally vague environments to elucidate conceptual structures of locally vague environments, and to show how the points in a locally vague environment are described.     

Vague集向Fuzzy集转化的一种新方法

结合Vague集理论及投票模型下的解释,对Vague集向Fuzzy集转化的方法进行了分析,指出已有方法存在的不足之处,提出了一种新的有效转化方法,并通过实例说明其有效性。     

Vague集向Fuzzy集转化的方法模型

结合Vague集理论及投票模型下的解释,对Vague集向Fuzzy集转化的方法进行了分析,提出了一种新的转化方法模型,并从理论上解释了已有方法的合理性与不足。     

一种Vague集转化为Fuzzy集的方法

已存在的Vague集向Fuzzy集转化的多种模型都存在着一定缺陷。论文从另一种角度探讨了Vague集到Fuzzy集的转化,它把模糊隶属度看成是参数k的函数,这样在计算模糊隶属度时,只需给定满足条件的参数k即可;另一方面,在进行模糊信息处理时,会遇到一些不确定因素,如果所得到的模糊隶属度与实际结果有些偏差,可根据参数k的数学性质有目的地调整k,而不必去重新建立方法模型,因此,此方法比已存在方法灵活,适用;文章也对已有的转化方法与所提模型进行了比较,说明了所提模型的正确性及有效性。     

A novel method for ranking of vague sets for handling the risk analysis of compressor system

Vague sets were first proposed by Gau and Buehrer [11] as an extension of fuzzy sets which encompass fuzzy sets, inter-valued fuzzy sets, as special cases. Vague sets consist of two parts, that is, the membership function and nonmembership function. Therefore, in accordance with practical demand these sets are more flexible than the existing fuzzy sets and provide much more information about the situation. In this paper, a new approach for the ranking of trapezoidal vague sets is introduced. Shortcomings in some existing ranking approaches have been pointed out. Validation of the proposed ranking method has been established through the reasonable properties of the fuzzy quantities. Further, the proposed ranking approach is applied to develop a new method for dealing with vague risk analysis problems to find the probability of failure, of each component of compressor system, which could be used for managerial decision making and future system maintenance strategy. Also, the proposed method provides a useful way for handling vague risk analysis problems.     

An enhanced method and its application for fuzzy multi-criteria decision making based on vague sets

Ye [Ye Jun. Improved method of multicriteria fuzzy decision making based on vague sets. Computer-Aid Design 2007;39:164–9] presented an improved method to handle multi-criteria fuzzy decision-making problems based on vague set theory. He/She provided some functions to measure the degree of suitability of each alternative with respect to a set of criteria presented by vague values. However, in some cases, these functions do not give sufficient information about alternatives. Therefore, in this paper, an enhanced method is provided to measure the accuracy membership of each alternative so as to give additional information for the decision maker. In addition, to making computing and ranking results easier and to increase the recruiting productivity, a computer-based decision-support system is also developed, which may help to make a decision more efficiently.     

Relations between complex vague soft sets

Complex vague soft sets are essentially vague soft sets characterized by an additional parameter called the phase term which is defined over the set of complex numbers. In this study, we introduce and discuss the relations between complex vague soft sets. We present the definitions of the Cartesian product of complex vague soft sets and subsequently that of complex vague soft relations. The definition of the composition of complex vague soft sets is also provided. The notions of symmetric, transitive, reflexive and equivalence complex vague soft relations are then proposed and the algebraic properties of these concepts are verified. The relation between complex vague soft sets is then discussed in the context of a real-life problem: the relation between the financial indicators of the Chinese economy which are characterized by their degrees of influence on the financial indicators of the Malaysian economy, and the time required for the former to affect the latter. Interpretations of the results obtained from this example are then proposed by relating them to recent significant real-life events in the Chinese and Malaysian economies.     

Vague集的模糊熵*

首先,在给出了反例证实现有方法在某些情况得出的结果与实际结论不一致情况下,提出了一个真正考虑了两种因素的改进的vague集模糊熵公理化定义;然后给出一种vague集模糊熵的新计算公式;最后通过定理证明和实例分析,指出它确实同时考虑到vague集的未知性和不确定性,从而证明所给出vague集模糊熵的定义是更加合理的.     

The generalized associative law in vague groups and its applications—I

Products of elements and integral powers of elements in vague groups have a significant concern for the development and the applications of vague algebra. Formulation of properties of these notions basically depends on a many-valued counterpart to the generalized associative law (called the generalized vague associative law in vague groups). For this reason, the present paper and the forthcoming paper [M. Demirci, The generalized associative law in vague groups and its applications—II, Information Sciences, Submitted for publication] are devoted to the formulation of the generalized vague associative law and its applications in vague groups. The generalized vague associative law in vague semigroups, which is the main contribution of this exposition, and some elementary properties of the notion of a product of a finite number of elements in vague semigroups, which cover necessary preparatory results for Part II, are the subjects of this paper. Since the present paper forms an abstract foundation of the product and sum of a finite number of real numbers in vague arithmetic, some practical applications of the notions of product and sum of a finite number of real numbers in vague arithmetic are also given.     

The generalized associative law in vague groups and its applications—II

As a continuation of Part I, vague integral powers of elements in vague groups and their representation properties are introduced in this paper. Thereafter, some rudimentary algebraic properties of vague integral powers of elements, obtained from the generalized vague associative law formulated in Part I, are established.The present paper particularly provides the abstract foundations of integral powers and multiples of real numbers in vague arithmetic. For this reason, special attention is also paid to the calculation of integral powers and multiples of real numbers in vague arithmetic, and some practical applications related to the discrete structure of measurement instruments are also given.     

Vague集向Fuzzy集转化的新方法

指出了Vague集向Fuzzy集转化已有方法存在的不足,给出了Vague集转化为Fuzzy集应满足的约束准则,提出了一种Vague集向Fuzzy集转化的新方法,并证明了该方法的有效性,通过实例说明该方法比已有方法更合理。     

模糊等价关系上的粗糙集研究

从宏观的角度研究集合不容易发现元素之间的关系,并且不可避免地带来人为的随意性和不确定性。从微观元素的相似性出发,首先建立了在模糊等价关系上的等价类以及模糊等价关系上的粗糙集,研究了相似程度参数的合理取值范围问题,提出并证明了粗糙集算子的计算定理,然后论述了模糊等价关系上的粗糙集与经典粗糙集的关系,发现并研究了经典粗糙理论处理相同元素时会出现的元素分类不一致问题,最后给出了经典粗糙集算子的计算方法。     

Constructions of fuzzy functions based on fuzzy equalities

 The strong (perfect) fuzzy function have been applied to approximate reasoning and vague algebra in the literature of fuzzy sets. The construction of strong (perfect) fuzzy functions possesses an important role for their applications. In the presented paper, some of the results on the construction of strong (perfect) fuzzy functions are improved, and several new and desirable results in this direction are obtained. Furthermore, it is also shown that how these results can be used to point out the connections between fuzzy functions in the classical sense and the strong (perfect) fuzzy functions.     

Improved method of multicriteria fuzzy decision-making based on vague sets

An improved method is presented, which provides improved score functions to measure the degree of suitability of each of a set of alternatives, with respect to a set of criteria presented with vague values. The improved algorithm for score functions is introduced by taking into account the effect of an unknown degree (hesitancy degree) of the vague values on the degree of suitability to which each alternative satisfies the decision-maker’s requirement. The meaning of the proposed function is more transparent than that of other existing functions, which are not reasonable in some cases. The proposed function illustrates that it has stronger discrimination in comparison with previous functions. The applicability of this improved multicriteria fuzzy decision-making approach is also demonstrated by means of examples. The improved method can be used to rank the decision alternatives according to the decision criteria. The functions proposed in this paper can provide a more useful technique than previous functions, in order to efficiently help the decision-maker.     

An alternative characterization of fuzzy complement functional

In this short note we propose a novel fuzzy complement functional. This functional is different from other functionals known in the literature. However, it turns out to be an alternative characterization of the well-known negation function. We sincerely thank the anonymous reviewer whose insightful suggestions have significantly improved the paper.     

Linearly ordered semigroups for fuzzy set theory

The fuzzy set theory initiated by Zadeh (Information Control 8:338–353, 1965) was based on the real unit interval [0,1] for support of membership functions with the natural product for intersection operation. This paper proposes to extend this definition by using the more general linearly ordered semigroup structure. As Moisil (Essais sur les Logiques non Chrysippiennes. Académie des Sciences de Roumanie, Bucarest, 1972, p. 162) proposed to define Lukasiewicz logics on an abelian ordered group for truth values set, we give a simple negative answer to the question on the possibility to build a Many-valued logic on a finite abelian ordered group. In a constructive way characteristic properties are step by step deduced from the corresponding set theory to the semigroup order structure. Some results of Clifford on topological semigroups (Clifford, A.H., Proc. Amer. Math. Soc. 9:682–687, 1958; Clifford, A.H., Trans. Amer. Math. Soc. 88:80–98, 1958), Paalman de Miranda work on I-semigroups (Paalman de Miranda, A.B., Topological Semigroups. Mathematical Centre Tracts, Amsterdam, 1964) and Schweitzer, Sklar on T-norms (Schweizer, B., Sklar, A., Publ. Math. Debrecen 10:69–81, 1963; Schweizer, B., Sklar, A., Pacific J. Math. 10:313–334, 1960; Schweizer, B., Sklar, A., Publ. Math. Debrecen 8:169–186, 1961) are revisited in this framework. As a simple consequence of Faucett theorems (Proc. Amer. Math. Soc. 6:741–747, 1955), we prove how canonical properties from the fuzzy set theory point of view lead to the Zadeh choice thus giving another proof of the representation theorem of T-norms. This structural approach shall give a new perspective to tackle the question of G. Moisil about the definition of discrete Many-valued logics as approximation of fuzzy continuous ones.      

关于“Vague集的新模糊熵”的注记

指出了“Vague集的新模糊熵”中模糊熵定义的错误,给出了修正后的模糊熵定义,提出了一种新的Vague集模糊熵的计算方法,并证明了该方法的有效性。     

A generalized model of fuzzy rough sets

The consideration of approximation problem of fuzzy sets in fuzzy information systems results in theory of fuzzy rough sets. This paper focuses on models of generalized fuzzy rough sets, a generalized model of fuzzy rough sets based on general fuzzy relations are studied, properties and algebraic characterization of the model are revealed, and relationships between this model and related models are also discussed.     

An novel approach to supplier selection based on vague sets group decision

In practice, in purchasing decision-making, many quantitative and qualitative factors, with vagueness and imprecision, have to be considered. This makes the decision process very complicated and unstructured. Besides the fuzzy sets theory, vague sets theory is one of the methods used to deal with uncertain information. Since vague sets can provide more information than fuzzy sets, it is considered superior in mathematical analysis of uncertain information. In this paper, a new approach based on vague sets group decision is proposed to deal with the supplier selection problem in supply chain systems. The work procedure is shown briefly, as follows: First, linguistic values are used to assess the ratings and weights for quantitative or qualitative factors. Second, degree of similarity and probability of vague sets are used to determine the ranking order of all alternatives. Finally, a numerical example of the selection problem of suppliers is shown, to highlight the procedure of the proposed approach, at the end of this paper.     

关于“Vague集向Fuzzy集的转换函数”的注记

结合Vague集理论及投票模型下的解释,对“Vague集向Fuzzy集的转化函数”进行了分析,指出了其存在的不足之处,提出了两种Vague集转化为Fuzzy集的新方法,并证明了其有效性。