Application of fuzzy logic and variable precision rough set approach in a remote monitoring manufacturing process for diagnosis rule induction

Rough set has been shown to be a valuable approach to mine rules from a remote monitoring manufacturing process. In this research, an application of the fuzzy set theory with the fuzzy variable precision rough set approach for mining the causal relationship rules from the database of a remote monitoring manufacturing process is presented. The membership function in the fuzzy set theory is used to transfer the data entries into fuzzy sets, and the fuzzy variable precision rough set approach is applied to extract rules from the fuzzy sets. It is found that the induced rules are identical to the practical knowledge and fault diagnosis thinking of human operators. The induced rules are then compared with the rules induced by the original rough set approach. The comparison shows that the rules induced by the fuzzy rough set are expressed in linguistic forms, and are evaluated by plausibility and future effectiveness measures. The fuzzy rough set approach, being less sensitive to noisy data, induces better rules than the original rough set approach.     

Fault detection in dynamic systems by a Fuzzy/Bayesian network formulation

In this paper the fault detection problem is solved using an alternative methodology based on a fuzzy/Bayesian strategy combining a Bayesian network and the fuzzy set theory. The new important issue in this proposed methodology is to address uncertainties in the input of the Bayesian Network. This contribution is possible since the fuzzy set theory is used as the knowledge representation. To illustrate the technique, the fault detection problem in induction machine stator-winding is considered. Specifically, the faults in the induction machine stator-winding are detected by a state change of stator current. Simulation results are presented to illustrate the advance of the proposed methodology when compared to standard Bayesian network.     

犹豫模糊软集

犹豫模糊集是对模糊集的一种推广,它是一类关于域中每个元素所含隶属度的集合,常应用于群决策中,但由于其本身在参数工具上的缺乏使得难于处理不确定数据。为了提高决策的精确性,将软集与犹豫模糊集结合起来,提出犹豫模糊软集的概念,并给出犹豫模糊软集的基本运算法则和性质。     

模糊粗糙集的相似度量和相似性方向

粗糙集理论是一种新的处理模糊和不确定性知识的软计算工具,在人工智能及认知科学等众多领域已经得到了广泛的应用。相似度量的研究是模糊集理论与粗糙集理论的热点问题之一。文章提出了一种更精确、更合理的相似度量方法,讨论了它的一些性质。然后,在此基础上提出了模糊粗糙集的相似性方向的概念,用于比较两个相似的模糊粗糙集所包含信息的精确性大小,并给出了一个关于相似性方向的判别函数。这在近似推理、模式识别和决策分析等领域有着广泛的应用。最后,通过一个实例,分析说明了这种相似度量方法和相似性方向的判别方法是更合理更有效的。     

覆盖多粒度梯形模糊数决策理论粗糙集模型

作为经典Pawlak粗糙集模型的推广,基于论域上的等价关系,针对风险决策分类问题,多粒度粗糙集已有研究。其特点是在力争决策的期望损失（亦称决策的条件风险）最小的条件下,比较客观地确定对象分类区域的概率描述临界值,进而进行对象的最佳分类决策。然而,在实际应用中论域上的等价关系很难把握,况且特征状态的风险损失往往带有某种不确定性。凡此,无疑在一定程度上限制了多粒度决策理论粗糙集的应用。对此进行了研究：提出了覆盖多粒度梯形模糊数决策理论粗糙集模型,分别就平均、乐观和悲观的情形进行了讨论和刻划;得到了覆盖多粒度梯形模糊数决策理论粗糙集与已有相关模型之间的关系;结果和算例表明了模型的广泛性。     

多粒化模糊软粗糙集模型

为了扩大粗糙集理论的应用,特别是在模糊环境中的应用,基于模糊软集和模糊蕴涵算子,主要研究基于软模糊近似空间的乐观多粒化模糊软粗糙集模型。该模型将参数集根据客户的不同要求或目标进行重组,只选择若干相关参数集参与计算上、下近似,这样定义的上、下近似不再由整个属性集决定,而是根据重组后的多个属性集一并生成,从而使结果更加符合实际需求。另外,还定义了乐观多粒化模糊软粗糙集模型的截集并讨论了其相关性质。最后给出了算例。     

Generalized fuzzy rough sets determined by a triangular norm

The theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications. Various fuzzy generalizations of rough approximations have been made over the years. This paper presents a general framework for the study of T-fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used. By using a pair of dual triangular norms in the constructive approach, some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations. The connections between special fuzzy relations and the T-upper and T-lower approximation operators of fuzzy sets are also examined. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, T-fuzzy approximation operators are defined by axioms. Different axiom sets of T-upper and T-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators. The independence of axioms characterizing the T-fuzzy rough approximation operators is examined. Then the minimal sets of axioms for the characterization of the T-fuzzy approximation operators are presented. Based on information theory, the entropy of the generalized fuzzy approximation space, which is similar to Shannon’s entropy, is formulated. To measure uncertainty in T-generalized fuzzy rough sets, a notion of fuzziness is introduced. Some basic properties of this measure are examined. For a special triangular norm T = min, it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable.     

粗糙模糊集的构造与公理化方法

用构造性方法和公理化研究了粗糙模糊集．由一个一般的二元经典关系出发构造性地定义了一对对偶的粗糙模糊近似算子，讨论了粗糙模糊近似算子的性质，并且由各种类型的二元关系通过构造得到了各种类型的粗糙模糊集代数．在公理化方法中，用公理形式定义了粗糙模糊近似算子，各种类型的粗糙模糊集代数可以被各种不同的公理集所刻画．阐明了近似算子的公理集可以保证找到相应的二元经典关系，使得由关系通过构造性方法定义的粗糙模糊近似算子恰好就是用公理化定义的近似算子。     

A formal approach for graphically building fuzzy XML model

eXtensible Markup Language (XML) has been the de facto standard of data representation and exchange over the Web. In addition, imprecise and uncertain data are inherent in the real world. Although fuzzy data have been extensively investigated in the context of the relational model, the classical relational database model and its fuzzy extension to date do not satisfy the need of modeling complex objects with imprecision and uncertainty on the Web. On the basis of possibility theory, this paper concentrates on fuzzy information modeling in the fuzzy XML model and the fuzzy IFO model. In particular, the formal approach to mapping a fuzzy IFO model to a fuzzy document-type definition model is developed.     

正态模糊集合——Fuzzy集理论的新拓展

直觉模糊集（intuitionistic fuzzy sets）、区间值模糊集（interval-valued fuzzy sets）以及Vague集对普通fuzzy集的扩展是给出了隶属度的上下限，把隶属度从[0，1]区间中的一个单值推广到了[0，1]的子区间。但是该子区间犹如一个黑洞，隶属度在其内部的分布情况我们无从知晓，即这个子区间中的每一个值是等可能地作为元素的隶属度还是区间中的某些值较另外的值有更大的可能性呢？为了清晰的刻画出元素的隶属度在[0，1]区间中的分布情况，本文通过对投票模型的分析及正态分布理论，提出了一种新的模糊集合——正态模糊集合，同时对正态模糊集合的交、并、补等基本运算性质进行了讨论，文章最后对正态模糊集与fuzzy集、直觉模糊集的相互关系也作出了详细阐述。正态模糊集合是模糊集合理论的进一步推广，为我们处理模糊信息提供了一种全新的思想方法。     

Fuzzy sets-a convenient fiction for modeling vagueness andpossibility

This paper is a reply to Laviolette and Seaman's critical discussion of fuzzy set theory. Rather than questioning the interest of the Bayesian approach to uncertainty, some reasons why Bayesian find the idea of a fuzzy set not palatable are laid bare. Some links between fuzzy sets and probability that Laviolette and Seaman seem not to be aware of are pointed out. These links suggest that, contrary to the claim sometimes found in the literature, probability theory is not a special case of fuzzy set theory. The major objection to Laviolette and Seaman is that they found their critique on as very limited view of fuzzy sets, including debatable papers, while they fail to account for significant works pertaining to axiomatic derivation of fuzzy set connectives, possibility theory, fuzzy random variables, among others     

Fuzzy methods for case-based recommendation and decision support

The paper proposes two case-based methods for recommending decisions to users on the basis of information stored in a database. In both approaches, fuzzy sets and related (approximate) reasoning techniques are used for modeling user preferences and decision principles in a flexible manner. The first approach, case-based decision making, can principally be seen as a case-based counterpart to classical decision principles well-known from statistical decision theory. The second approach, called case-based elicitation, combines aspects from flexible querying of databases and case-based prediction. Roughly, imagine a user who aims at choosing an optimal alternative among a given set of options. The preferences with respect to these alternatives are formalized in terms of flexible constraints, the expression of which refers to cases stored in a database. As both types of decision support might provide useful tools for recommender systems, we also place the methods in a broader context and discuss the role of fuzzy set theory in some related fields.     

模糊证据理论及其在信息融合中的应用

为了使证据理论能更加有效地应用,把证据理论向模糊集推广,利用模糊集的隶属函数提出一种构造证据理论中的基本概率赋值函数的方法,实现了模糊理论和证据理论的有效结合。不但有效地解决了证据理论中的基本概率赋值函数的不易确定问题,而且由于证据理论应用于实际更加方便和有效,融合结果也更加合理。     

基于覆盖的区间直觉模糊粗糙集

在经典的覆盖近似空间中,定义了区间直觉模糊概念的粗糙近似。通过区间直觉模糊覆盖概念,给出了一种基于区间直觉模糊覆盖的区间直觉模糊粗糙集模型。讨论了两种模型的一些相关性质。     

Statistics with fuzzy data in statistical quality control

Statistical quality control (SQC) is an important field where both theory of probability and theory of fuzzy sets may be used. In the paper we give a short overview of basic problems of SQC that have been solved using both these theories simultaneously. Some new results on the applications of fuzzy sets in SQC are presented in details. We also present problems which are still open, and whose solution should definitely increase the applicability of fuzzy sets in quality control.     

Rough set theory for the interval-valued fuzzy information systems

The notion of a rough set was originally proposed by Pawlak [Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (5) (1982) 341-356]. Later on, Dubois and Prade [D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General System 17 (2-3) (1990) 191-209] introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets. This paper deals with an interval-valued fuzzy information system by means of integrating the classical Pawlak rough set theory with the interval-valued fuzzy set theory and discusses the basic rough set theory for the interval-valued fuzzy information systems. In this paper we firstly define the rough approximation of an interval-valued fuzzy set on the universe U in the classical Pawlak approximation space and the generalized approximation space respectively, i.e., the space on which the interval-valued rough fuzzy set model is built. Secondly several interesting properties of the approximation operators are examined, and the interrelationships of the interval-valued rough fuzzy set models in the classical Pawlak approximation space and the generalized approximation space are investigated. Thirdly we discuss the attribute reduction of the interval-valued fuzzy information systems. Finally, the methods of the knowledge discovery for the interval-valued fuzzy information systems are presented with an example.     

On various eigen fuzzy sets and their application to image reconstruction

In this study, we formulate and solve a problem of image reconstruction using eigen fuzzy sets. Treating images as fuzzy relations, we propose two algorithms of generating eigen fuzzy sets that are used in the reconstruction process. The first one corresponds to a convex combination of eigen fuzzy set equations, i.e., fuzzy relational equations involving convex combination of max-min and min-max compositions. In the case of the first algorithm, various eigen fuzzy sets can be generated by changing the parameter controlling the convex combination of the corresponding equations. The second algorithm generates various eigen fuzzy sets with respect to the original fuzzy relation using a permutation matrix. A thorough comparison of the proposed algorithms and a conventional algorithm which reconstructs an image using the greatest and smallest eigen fuzzy sets is presented as well. In the experiments, 10,000 artificial images of size 5 × 5 pixels. The approximation error in the case of the first/second algorithm is decreased to 68.2%/97.9% of that of the conventional algorithm, respectively. Furthermore, through the experimentation using real images extracted from Standard Image DataBAse (SIDBA), it is confirmed that the approximation error of the first algorithm is decreased to 41.5% of that of the conventional one.     

Vague综合评判方法

给出了一种基于Vague集的模糊综合评判方法,利用Vague集的运算得到全部候选方案的总体模糊评判值,通过两两比较建立可能度矩阵,得到候选方案集的排序向量,从而实现对候选方案的最优选择。通过实例表明,该方法切实可行,可以很好地帮助用户从多个候选方案中选择最适合的方案。     

A soft computing method of performance evaluation with MCDM based on interval-valued fuzzy numbers

This study presented a new performance evaluation method for tackling fuzzy multicriteria decision-making (MCDM) problems based on combining VIKOR and interval-valued fuzzy sets. The performance evaluation problem often exists in complex administrative processes in which multiple evaluation criteria, subjective/objective assessments and fuzzy conditions have to be taken into consideration simultaneously in management. Here, the subjective, imprecise, inexact and uncertain evaluation processes are modeled as fuzzy numbers by means of linguistic terms, as fuzzy theory can provide an appropriate tool to deal with such uncertainties. However, the presentation of linguistic expressions in the form of ordinary fuzzy sets is not clear enough [15] and [21]. Interval-valued fuzzy sets can provide more flexibility [4] and [14] to represent the imprecise/vague information that results, and it can also provide a more accurate modeling. This paper presents the interval-valued fuzzy VIKOR, which aims to solve MCDM problems in which the weights and performances of criteria are unequal by using the concepts of interval-valued fuzzy sets. A case study for evaluating the performances of three major intercity bus companies from an intercity public transport system is conducted to illustrate the effectiveness of the method.     

A multistage fuzzy-stochastic programming model for supporting sustainable water-resources allocation and management

In this study, a multistage fuzzy-stochastic programming (MFSP) model is developed for tackling uncertainties presented as fuzzy sets and probability distributions. A vertex analysis approach is proposed for solving multiple fuzzy sets in the MFSP model. Solutions under a set of α-cut levels can be generated by solving a series of deterministic submodels. The developed method is applied to the planning of a case study for water-resources management. Dynamics and uncertainties of water availability (and thus water allocation and shortage) could be taken into account through generation of a set of representative scenarios within a multistage context. Moreover, penalties are exercised with recourse against any infeasibility, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic consequences when the promised water-allocation targets are violated. The modeling results can help to generate a range of alternatives under various system conditions, and thus help decision makers to identify desired water-resources management policies under uncertainty.