Fuzzy topology on fuzzy sets: Product fuzzy topology and fuzzy topological groups

The notion of product fuzzy topology in the case of fuzzy topology on fuzzy sets is introduced and the product invariance of fuzzy Hausdorffness, compactness, connectedness are examined. The product fuzzy topology is used to define fuzzy group topology on a fuzzy subgroup of a group G and some properties of fuzzy topological groups are obtained.     

Fuzzy approach in psychophysical modeling of human operator-manual lifting system

A fuzzy representation of the human operator decision-making process of selecting the maximum acceptable weight of manual lift is developed. Measures of acceptability are expressed in terms of membership functions which describe the degree to which the combined effect of biomechanical and physiological stresses is acceptable to the human operator. It is assumed that the combination of these two stresses has a synergistic effect on the individual's performance, and that the perception of this effect is reflected in an operator's choice when selecting the preferable weight of lift. The combined acceptabilities of a lifting task are compared to the subjective capacity norms established by the subjects in a psychophysical experiment. The conditions under which the distance between acceptability measures of the combined and the psychophysical stresses can be minimized are established. It is concluded that at a certain level of similarity between the acceptability measures of these stresses, a decision is made, and the preferred weight of lift selected.     

Fuzzy Closure Spaces vs. Fuzzy Rough Sets

This paper investigates the relationship among fuzzy rough sets, fuzzy closure spaces and fuzzy topology. It is shown that there exists a bijective correspondence between the set of all fuzzy reflexive approximation spaces and the set of all quasi-discrete fuzzy closure spaces satisfying a certain extra condition. Similar correspondence is also obtained between the set of all fuzzy tolerance approximation spaces and the set of all symmetric quasi-discrete fuzzy closure spaces satisfying a certain extra condition.     

Fuzzified neural network based on fuzzy number operations

The fuzzified neural network based on fuzzy number operations is presented as a powerful modelling tool here. We systematically introduce ideas and concepts of a novel neural network based on fuzzy number operations. First we suggest how to compute the results of addition, subtraction, multiplication and division for two fuzzy numbers. Second we propose a learning algorithm, and present some ideas about the choice of fuzzy weights and fuzzy biases and a numerical scheme for the calculation of outputs of the fuzzified neural network. Finally, we show some results of computer simulations.     

Rule generation for hierarchical collaborative fuzzy system

A new method of rule generation for the hierarchical collaborative fuzzy system, HCFS, is proposed. This HCFS is structured like various parallel fuzzy subsystems and it overcomes the dimensionality problem and the lack of interpretability of most of the traditional fuzzy systems, when dealing with complex real-world problems. An association process of different fuzzy systems is presented in this work, through the use of a relevance concept of a fuzzy system. The result of this aggregation is a collaborative structure where all sub-models have the ability to gradually improve the overall accuracy of approximation by adding their own contributions. For this structure we propose a new algorithm to be used in the procedures of the three learning phases: the structure building, the parametric identification and the division of the learning data among the various levels of the hierarchical structure. This new fuzzy modelling technique automatically generates and tunes the sets of fuzzy rules in the hierarchical collaborative structure (HCS). The effectiveness of the proposed HCFS model in handling high-dimensional and complex problems is demonstrated through various numerical simulations.     

Attribute reduction based on generalized fuzzy evidence theory in fuzzy decision systems

Attribute reduction is viewed as an important issue in data mining and knowledge representation. This paper studies attribute reduction in fuzzy decision systems based on generalized fuzzy evidence theory. The definitions of several kinds of attribute reducts are introduced. The relationships among these reducts are then investigated. In a fuzzy decision system, it is proved that the concepts of fuzzy positive region reduct, lower approximation reduct and generalized fuzzy belief reduct are all equivalent, the concepts of fuzzy upper approximation reduct and generalized fuzzy plausibility reduct are equivalent, and a generalized fuzzy plausibility consistent set must be a generalized fuzzy belief consistent set. In a consistent fuzzy decision system, an attribute set is a generalized fuzzy belief reduct if and only if it is a generalized fuzzy plausibility reduct. But in an inconsistent fuzzy decision system, a generalized fuzzy belief reduct is not a generalized fuzzy plausibility reduct in general.     

函数空间上的模糊线性拓扑

本文给出了模糊拓扑向量空间（Ｘ，Ｗ）到（Ｙ，Ｊ）的函数族Ｆ上的模糊线性拓扑，证明了若值域空间（Ｙ，Ｊ）是（Ｑ）型的、局部凸的模糊拓扑向量空间，则（Ｆ，），也是型的、局部凸的模糊拓扑向量空间。     

Fuzzy topological vector spaces I

Some of the properties of fuzzy topological vector spaces are investigated. Also, there are given necessary and sufficient conditions for a family of fuzzy sets, in a vector space E, to be the family of all neighborhoods of zero for a fuzzy linear topology.     

普通选择函数的若干合理性条件的模糊化研究

以普通选择函数的有关结论为基础,利用模糊逻辑联结运算模糊化了Arrow的五个合理性条件,讨论了模糊化后的这些合理性条件之间的关系,研究了文献中给出的一个开问题,并对模糊情形下不成立的情况提供反例予以说明,从而推广了普通选择函数的相关结论。     

A historical review of evolutionary learning methods for Mamdani-type fuzzy rule-based systems: Designing interpretable genetic fuzzy systems

The need for trading off interpretability and accuracy is intrinsic to the use of fuzzy systems. The obtaining of accurate but also human-comprehensible fuzzy systems played a key role in Zadeh and Mamdani’s seminal ideas and system identification methodologies. Nevertheless, before the advent of soft computing, accuracy progressively became the main concern of fuzzy model builders, making the resulting fuzzy systems get closer to black-box models such as neural networks. Fortunately, the fuzzy modeling scientific community has come back to its origins by considering design techniques dealing with the interpretability-accuracy tradeoff. In particular, the use of genetic fuzzy systems has been widely extended thanks to their inherent flexibility and their capability to jointly consider different optimization criteria. The current contribution constitutes a review on the most representative genetic fuzzy systems relying on Mamdani-type fuzzy rule-based systems to obtain interpretable linguistic fuzzy models with a good accuracy.     

Regularly open sets in fuzzy topological spaces

This paper is devoted to the study of the role of fuzzy regularly open sets. We prove some properties of fuzzy almost continuous mappings and define fuzzy almost open mappings. We prove that under a fuzzy almost continuous and fuzzy almost open map, the inverse image of a fuzzy regularly open set is fuzzy regularly open. Further we define a new type of fuzzy separation axioms, fuzzy almost separation axioms. It is interesting that there are some deviations in the behaviour of these axioms as compared to those in general topology. For example, in a fuzzy almost T₁ space not every fuzzy singleton is δ-closed. Also a fuzzy space which is fuzzy almost as well as fuzzy almost T₀ is fuzzy almost regular. While in general topology we have to take an almost T₂ space in place of almost T₀ space.     

Fuzzy Perfect Mappings and Q-Compactness in Smooth Fuzzy Topological Spaces

We point out that the product of two fuzzy closed sets of smooth fuzzy topological spaces need not be fuzzy closed with respect to the the existing notion of product smooth fuzzy topology. To get this property, we introduce a new suitable product smooth fuzzy topology. We investigate whether $F_1\times F_2$ and $(F,H)$ are weakly smooth fuzzy continuity whenever $F_1$, $F_2$, $F$ and $H$ are weakly smooth fuzzy continuous. Using this new product smooth fuzzy topology, we define smooth fuzzy perfect mapping and prove that composition of two smooth fuzzy perfect mappings is smooth fuzzy perfect under some additional conditions. We also introduce two new notions of compactness called $Q$-compactness and $Q$-$\alpha$-compactness; and discuss the compactness of the image of a $Q$-compact set ($Q$-$\alpha$-compact set) under a weakly smooth fuzzy continuous function ($(\alpha ,\beta )$-weakly smooth fuzzy continuous function).     

以模糊数排序为基础的可能性线性规划分析——兼与文献[1]作者商榷

对文献 [1]提出的可能性线性规划的非模糊等价模型从模糊数排序分析的角度阐述并举证了该模型的非有效性。因而 ,以此为基础对原规划作出的变换被视为不良变换。为克服这一缺陷 ,提出相应的改正模型 ,并结合例题进行比较分析。     

Multi Fuzzy Fractal Theorems in Fuzzy Metric Spaces

The Hutchinson-Barnsley (HB) operators are studied for multi iterated function system in a general setting. Some existence and uniqueness results are obtained in fuzzy fractal and multi fuzzy fractal spaces. From them, we obtain our results, which generalize some recent results in the literature.     

Descriptions of equivalence of O-connectivity

We first study some properties of the subspace, and investigate into the relationship of separation between a fuzzy topological space (fts) and its subspace. Then we obtain the equivalence conditions for O-connectivity. The results on O-connectivity and separation are very similar to those in general topology. Finally we discuss the relationship of connectivity between an O-connected set A in the fts (X, ω ($\text{T}$)) induced by the crisp topological space (X, $\text{T}$) and the crisp set A₀ (=supp A) in (X, $\text{T}$).     

A class of linear differential dynamical systems with fuzzy matrices

This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the 2-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems.     

Compactness in fuzzy convergence spaces

Lowen and Lowen [Applications of category theory to fuzzy subsets (Kluwer, 1992) p. 153] and Lowen et al. [Fuzzy Sets and Systems 40 (1991) 347] recently introduced the category FCS of fuzzy convergence spaces, a topological quasitopos which is a supercategory of FTS, the category of fuzzy topological spaces. In this paper, compactness in FCS is examined. Doing so we found that to define compactness as an absolute property we had to generalize the definition of fuzzy convergence space to fuzzy subsets. All basic theorems are proved including the Tychonoff product theorem. Based on the theory developed here, in a following publication, a Richardson compactification for fuzzy convergence spaces will be given.     

Fuzzy control of thyristor-controlled series compensator in power system transients

Power system transient stability is one of the most challenging technical areas in electric power industry. Thyristor-controlled series compensation (TCSC) is expected to improve transient stability and damp power oscillations. TCSC control in power system transients is a nonlinear control problem. This paper presents a T–S-model-based fuzzy control scheme and a systematic design method for the TCSC fuzzy controller. The nonlinear power system containing TCSC is modelled as a fuzzy “blending” of a set of locally linearized models. A linear optimal control is designed for each local linear model. Different control requirements at different stages during power system transients can be considered in deriving the linear control rules. The resulting fuzzy controller is then a fuzzy “blending” of these linear controllers. Quadratic stability of the overall nonlinear controlled system can be checked and ensured using H^∞ control theory. Digital simulation with NETOMAC software has verified that the fuzzy control scheme can improve power system transient stability and damp power swings very quickly.     

On Inverse Limits and Direct Limits for Fuzzy Topological Spaces

Li has introduced the concepts of inverse system and direct system for fuzzy topological spaces and studied inverse limits and direct limits on such spaces by presenting the explicit constructions of these limits.In this paper some important concepts of fuzzy topology,such as,product fuzzy topology,quotient fuzzy topology,fuzzy continuity etc.,are used for further study of inverse limits and direct limits for fuzzy topological spaces.     

An inventory model with backorders with fuzzy parameters and decision variables

The paper considers an inventory model with backorders in a fuzzy situation by employing two types of fuzzy numbers, which are trapezoidal and triangular. A full-fuzzy model is developed where the input parameters and the decision variables are fuzzified. The optimal policy for the developed model is determined using the Kuhn-Tucker conditions after the defuzzification of the cost function with the graded mean integration (GMI) method. Numerical examples and a sensitivity analysis study are provided to highlight the differences between crisp and the fuzzy cases.