Fuzzy reasoning based on a new fuzzy rough set and its application to scheduling problems

In this paper, we study the fuzzy reasoning based on a new fuzzy rough set. First, we define a broad family of new lower and upper approximation operators of fuzzy sets between different universes using a set of axioms. Then, based on the approximation operators above, we propose the fuzzy reasoning based on the new fuzzy rough set. By means of the above fuzzy reasoning based on the new fuzzy rough set, for a given premise, we can obtain the fuzzy reasoning consequence expressed by the fuzzy interval constructed by the above two approximations of fuzzy sets. Furthermore, through the defuzzification of the lower and upper approximations, we can get the corresponding two values constructing the interval used as the fuzzy reasoning consequence after defuzzification. Then, from the above interval, a suitable value can be selected as the final reasoning consequence so that some special constraints are satisfied as possibly. At last, we apply the fuzzy reasoning based on the new fuzzy rough set to the scheduling problems, and numerical computational results show that the fuzzy reasoning based on the new fuzzy rough set is more suitable for the scheduling problems compared with the fuzzy reasoning based on the CRI method and the III method.     

Reduction method based on a new fuzzy rough set in fuzzy information system and its applications to scheduling problems

In this paper, we present the concept of fuzzy information granule based on a relatively weaker fuzzy similarity relation called fuzzy T_L-similarity relation for the first time. Then, according to the fuzzy information granule, we define the lower and upper approximations of fuzzy sets and a corresponding new fuzzy rough set. Furthermore, we construct a kind of new fuzzy information system based on the fuzzy T_L-similarity relation and study its reduction using the fuzzy rough set. At last, we apply the reduction method based on the defined fuzzy rough set in the above fuzzy information system to the reduction of the redundant multiple fuzzy rule in the scheduling problems, and numerical computational results show that the reduction method based on the new fuzzy rough set is more suitable for the reduction of multiple fuzzy rules in the scheduling problems compared with the reduction methods based on the existing fuzzy rough set.     

Design of fuzzy systems using neurofuzzy networks

Introduces a systematic approach for fuzzy system design based on a class of neural fuzzy networks built upon a general neuron model. The network structure is such that it encodes the knowledge learned in the form of if-then fuzzy rules and processes data following fuzzy reasoning principles. The technique provides a mechanism to obtain rules covering the whole input/output space as well as the membership functions (including their shapes) for each input variable. Such characteristics are of utmost importance in fuzzy systems design and application. In addition, after learning, it is very simple to extract fuzzy rules in the linguistic form. The network has universal approximation capability, a property very useful in, e.g., modeling and control applications. Here we focus on function approximation problems as a vehicle to illustrate its usefulness and to evaluate its performance. Comparisons with alternative approaches are also included. Both, non-noisy and noisy data have been studied and considered in the computational experiments. The neural fuzzy network developed here and, consequently, the underlying approach, has shown to provide good results from the accuracy, complexity, and system design points of view.     

General SISO Takagi-Sugeno fuzzy systems with linear ruleconsequent are universal approximators

Takagi-Sugeno (TS) fuzzy systems have been employed as fuzzy controllers and fuzzy models in successfully solving difficult control and modeling problems in practice. Virtually all the TS fuzzy systems use linear rule consequent. At present, there exist no results (qualitative or quantitative) to answer the fundamentally important question that is especially critical to TS fuzzy systems as fuzzy controllers and models, “Are TS fuzzy systems with linear rule consequent universal approximators?” If the answer is yes, then how can they be constructed to achieve prespecified approximation accuracy and what are the sufficient renditions on systems configuration? In this paper, we provide answers to these questions for a general class of single-input single-output (SISO) fuzzy systems that use any type of continuous input fuzzy sets, TS fuzzy rules with linear consequent and a generalized defuzzifier containing the widely used centroid defuzzifier as a special case. We first constructively prove that this general class of SISO TS fuzzy systems can uniformly approximate any polynomial arbitrarily well and then prove, by utilizing the Weierstrass approximation theorem, that the general TS fuzzy systems can uniformly approximate any continuous function with arbitrarily high precision. Furthermore, we have derived a formula as part of sufficient conditions for the fuzzy approximation that can compute the minimal upper bound on the number of input fuzzy sets and rules needed for any given continuous function and prespecified approximation error bound, An illustrative numerical example is provided     

Lukasiewicz型直觉模糊推理三I方法的性质分析

直觉模糊推理的两个基本模型是Intuitionistic Fuzzy Modus Ponens（IFMP）和Intuitionistic Fuzzy Modus Tollens（IFMT）。首先利用经典模糊集之间的自然距离定义了直觉模糊集间的一种距离。其次，证明了基于Lukasiewicz直觉模糊蕴涵的IFMP和IFMT问题的三I方法关于该距离都具有连续性，并且分别给出了IFMP和IFMT问题的三I方法满足逼近性的充分条件。     

Optimal fuzzy rules cover extrema

A fuzzy system approximates a function by covering the graph of the function with fuzzy rule patches and averaging patches that overlap. But the number of rules grows exponentially with the total number of input and output variables. the best rules cover the extrema or bumps in the function—they patch the bumps. For mean-squared approximation this follows from the mean value theorem of calculus. Optimal rules can help reduce the computational burden. to find them we can find or learn the zeroes of the derivative map and then center input fuzzy sets at these points. Neural systems can then both tune these rules and add rules to improve the function approximation. © 1995 John Wiley & Sons, Inc.     

Performance evaluation of fuzzy classifier systems formultidimensional pattern classification problems

We examine the performance of a fuzzy genetics-based machine learning method for multidimensional pattern classification problems with continuous attributes. In our method, each fuzzy if-then rule is handled as an individual, and a fitness value is assigned to each rule. Thus, our method can be viewed as a classifier system. In this paper, we first describe fuzzy if-then rules and fuzzy reasoning for pattern classification problems. Then we explain a genetics-based machine learning method that automatically generates fuzzy if-then rules for pattern classification problems from numerical data. Because our method uses linguistic values with fixed membership functions as antecedent fuzzy sets, a linguistic interpretation of each fuzzy if-then rule is easily obtained. The fixed membership functions also lead to a simple implementation of our method as a computer program. The simplicity of implementation and the linguistic interpretation of the generated fuzzy if-then rules are the main characteristic features of our method. The performance of our method is evaluated by computer simulations on some well-known test problems. While our method involves no tuning mechanism of membership functions, it works very well in comparison with other classification methods such as nonfuzzy machine learning techniques and neural networks.     

An approach to fuzzy default reasoning for function approximation

This paper discusses fuzzy reasoning for approximately realizing nonlinear functions by a small number of fuzzy if-then rules with different specificity levels. Our fuzzy rule base is a mixture of general and specific rules, which overlap with each other in the input space. General rules work as default rules in our fuzzy rule base. First, we briefly describe existing approaches to the handling of default rules in the framework of possibility theory. Next, we show that standard interpolation-based fuzzy reasoning leads to counterintuitive results when general rules include specific rules with different consequents. Then, we demonstrate that intuitively acceptable results are obtained from a non-standard inclusion-based fuzzy reasoning method. Our approach is based on the preference for more specific rules, which is a commonly used idea in the field of default reasoning. When a general rule includes a specific rule and they are both compatible with an input vector, the weight of the general rule is discounted in fuzzy reasoning. We also discuss the case where general rules do not perfectly but partially include specific rules. Then we propose a genetics-based machine learning (GBML) algorithm for extracting a small number of fuzzy if-then rules with different specificity levels from numerical data using our inclusion-based fuzzy reasoning method. Finally, we describe how our approach can be applied to the approximate realization of fuzzy number-valued nonlinear functions     

Function approximation based on fuzzy rules extracted frompartitioned numerical data

We present an efficient method for extracting fuzzy rules directly from numerical input-output data for function approximation problems. First, we convert a given function approximation problem into a pattern classification problem. This is done by dividing the universe of discourse of the output variable into multiple intervals, each regarded as a class, and then by assigning a class to each of the training data according to the desired value of the output variable. Next, we partition the data of each class in the input space to achieve a higher accuracy in approximation of class regions. Partition terminates according to a given criterion to prevent excessive partition. For class region approximation, we discuss two different types of representations using hyperboxes and ellipsoidal regions, respectively. Based on a selected representation, we then extract fuzzy rules from the approximated class regions. For a given input datum, we convert, or in other words, defuzzify, the resulting vector of the class membership degrees into a single real value. This value represents the final result approximated by the method. We test the presented method on a synthetic nonlinear function approximation problem and a real-world problem in an application to a water purification plant. We also compare the presented method with a method based on neural networks.     

直觉模糊神经网络的函数逼近能力

运用直觉模糊集理论,建立了自适应神经-直觉模糊推理系统（ANIFIS）的控制模型,并证明了该模型具有全局逼近性质.首先将Zadeh模糊推理神经网络变为直觉模糊推理网络,建立一个多输入单输出的T-S型ANIFIS模型;然后设计了系统变量的属性函数和推理规则,确定了各层的输入输出计算关系,以及系统输出结果的合成计算表达式;最后通过证明所建模型的输出结果计算式满足Stone-Weirstrass定理的3个假设条件,完成了该模型的全局逼近性证明.     

Design of an adaptive fuzzy sliding mode control for uncertain discrete-time nonlinear systems based on noisy measurements

This paper presents the design of an adaptive fuzzy sliding mode control (AFSMC) for uncertain discrete-time nonlinear dynamic systems. The dynamic systems are described by a discrete-time state equation with nonlinear uncertainties, and the uncertainties include the modelling errors and the external disturbances to be unknown but nonlinear with the bounded properties. The states are measured by the restriction of measurement sensors and the contamination with independent measurement noises. The nonlinear uncertainties are approximated by using the fuzzy IF-THEN rules based on the universal approximation theorem, and the approximation error is compensated by adding an adaptive complementary term to the proposed AFSMC. The fuzzy inference approach based on the extended single input rule modules is proposed to reduce the number of the fuzzy IF-THEN rules. The estimates for the un-measurable states and the adjustable parameters are obtained by using the weighted least squares estimator and its simplified one. It is proved that under some conditions the estimation errors will remain in the vicinity of zero as time increases, and the states are ultimately bounded subject to the proposed AFSMC. The effectiveness of the proposed method is indicated through the simulation experiment of a simple numerical system.     

Avoiding exponential parameter growth in fuzzy systems

For standard fuzzy systems where the input membership functions are defined on a grid on the input space, and all possible combinations of rules are used, there is an exponential growth in the number of parameters of the fuzzy system as the number of input dimensions increases. This “curse of dimensionality” effect leads to problems with design of fuzzy controllers (e.g., how to tune all these parameters), training of fuzzy estimators (e.g., complexity of a gradient algorithm for training, and problems with “over parameterization” that lead to poor convergence properties), and with computational complexity in the implementation for practical problems. We introduce a fuzzy system whose number of parameters grows linearly depending upon the number of inputs, even though it is constructed by using all possible combinations of the membership functions in defining the rules. We prove that this fuzzy system is equivalent to the standard fuzzy system as long as its parameters are specified in a certain way. Then, we show that it still holds the universal approximator property by using the Stone-Welerstrass theorem. Finally, we illustrate the performance of the fuzzy system via an application     

Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets

Fuzzy interpolative reasoning is an important research topic of sparse fuzzy rule-based systems. In recent years, some methods have been presented for dealing with fuzzy interpolative reasoning. However, the involving fuzzy sets appearing in the antecedents of fuzzy rules of the existing fuzzy interpolative reasoning methods must be normal and non-overlapping. Moreover, the reasoning conclusions of the existing fuzzy interpolative reasoning methods sometimes become abnormal fuzzy sets. In this paper, in order to overcome the drawbacks of the existing fuzzy interpolative reasoning methods, we present a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets. The proposed fuzzy interpolative reasoning method can handle the situation of non-normal and overlapping fuzzy sets appearing in the antecedents of fuzzy rules. It can overcome the drawbacks of the existing fuzzy interpolative reasoning methods in sparse fuzzy rule-based systems.     

基于OWA算子的区间值加权模糊推理

针对如何对区间值模糊产生式规则赋予合理权值的问题,将OWA算子引入到区间值模糊推理中。介绍一种基于OWA算子的区间值赋权方法,根据此方法给出区间值模糊集上的加权模糊产生式规则的推理算法。在采用该算法的过程中,为合理地计算输入事实与规则前件的匹配程度,引入基于OWA算子的区间值模糊匹配函数值和总体贴近度的计算方法。实例分析表明了所给出的区间值模糊推理算法的有效性和可行性。     

Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value

Value and ambiguity are two parameters which were introduced to represent fuzzy numbers. In this paper, we find the nearest trapezoidal approximation and the nearest symmetric trapezoidal approximation to a given fuzzy number, with respect to the average Euclidean distance, preserving the value and ambiguity. To avoid the laborious calculus associated with the Karush–Kuhn–Tucker theorem, the working tool in some recent papers, a less sophisticated method is proposed. Algorithms for computing the approximations, many examples, proofs of continuity and two applications to ranking of fuzzy numbers and estimations of the defect of additivity for approximations are given.     

Triple I method of fuzzy reasoning

The theory of the triple I method with total inference rules of fuzzy reasoning is investigated by using Zadeh's implication operator R_z. The computational formulae for both fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) are obtained. The reversibility properties for FMP and FMT are analyzed and the reversibility criteria are given. We also investigated the generalized problem of the triple I method and obtained the formulae for the -triple I FMP and the -triple I FMT.     

Approximation Capabilities of Hierarchical Fuzzy Systems

Derived from practical application in location analysis and pricing, and based on the approach of hierarchical structure analysis of continuous functions, this paper investigates the approximation capabilities of hierarchical fuzzy systems. By first introducing the concept of the natural hierarchical structure, it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and parameters. Then, based on Kolmogorov's theorem, it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical fuzzy systems to achieve the universal approximation property. Further, the conditions under which the hierarchical fuzzy approximation is superior to the standard fuzzy approximation in overcoming the curse of dimensionality are analyzed     

Learning fuzzy rules for similarity assessment in case-based reasoning

Fundamental to case-based reasoning is the assumption that similar problems have similar solutions. The meaning of the concept of “similarity” can vary in different situations and remains an issue. This paper proposes a novel similarity model consisting of fuzzy rules to represent the semantics and evaluation criteria for similarity. We believe that fuzzy if-then rules present a more powerful and flexible means to capture domain knowledge for utility oriented similarity modeling than traditional similarity measures based on feature weighting. Fuzzy rule-based reasoning is utilized as a case matching mechanism to determine whether and to which extent a known case in the case library is similar to a given problem in query. Further, we explain that such fuzzy rules for similarity assessment can be learned from the case library using genetic algorithms. The key to this is pair-wise comparisons of cases with known solutions in the case library such that sufficient training samples can be derived for genetic-based fuzzy rule learning. The evaluations conducted have shown the superiority of the proposed method in similarity modeling over traditional schemes as well as the feasibility of learning fuzzy similarity rules from a rather small case base while still yielding competent system performance.     

A systematic approach to a self-generating fuzzy rule-table forfunction approximation

In this paper, a systematic design is proposed to determine fuzzy system structure and learning its parameters, from a set of given training examples. In particular, two fundamental problems concerning fuzzy system modeling are addressed: 1) fuzzy rule parameter optimization and 2) the identification of system structure (i.e., the number of membership functions and fuzzy rules). A four-step approach to build a fuzzy system automatically is presented: Step 1 directly obtains the optimum fuzzy rules for a given membership function configuration. Step 2 optimizes the allocation of the membership functions and the conclusion of the rules, in order to achieve a better approximation. Step 3 determines a new and more suitable topology with the information derived from the approximation error distribution; it decides which variables should increase the number of membership functions. Finally, Step 4 determines which structure should be selected to approximate the function, from the possible configurations provided by the algorithm in the three previous steps. The results of applying this method to the problem of function approximation are presented and then compared with other methodologies proposed in the bibliography.