On Generalized Fuzzy Belief Functions in Infinite Spaces

Determined by a fuzzy implication operator, a general type of fuzzy belief structure and its induced dual pair of fuzzy belief and plausibility functions in infinite universes of discourse are first defined. Relationship between the belief-structure-based and the belief-space-based fuzzy Dempster-Shafer models is then established. It is shown that the lower and upper fuzzy probabilities induced by the fuzzy belief space yield a dual pair of fuzzy belief and plausibility functions. For any fuzzy belief structure, there must exist a fuzzy belief space such that the fuzzy belief and plausibility functions defined by the given fuzzy belief structure are just the lower and upper fuzzy probabilities induced by the fuzzy belief space, respectively. Essential properties of the fuzzy belief and plausibility functions are also examined. The fuzzy belief and plausibility functions are, respectively, a fuzzy monotone Choquet capacity and a fuzzy alternating Choquet capacity of infinite order.     

Fuzzy Reasoning as a Control Problem

Different from the dominant view of treating fuzzy reasoning as generalization of classical logical inference, in this paper fuzzy reasoning is treated as a control problem. A new fuzzy reasoning method is proposed that employs an explicit feedback mechanism to improve the robustness of fuzzy reasoning. The fuzzy rule base given a priori serves as a controlled object, and the fuzzy reasoning method serves as the corresponding controller. The fuzzy rule base and the fuzzy reasoning method constitute a control system that may be open loop or closed loop, depending on the underlying reasoning goals/constraints. The fuzzy rule base, the fuzzy reasoning method, and the corresponding reasoning goals/constraints define the three distinct ingredients of fuzzy reasoning. While various existing fuzzy reasoning methods are essentially a static mapping from the universe of single fuzzy premises to the universe of single fuzzy consequences, the new fuzzy reasoning method maps sequences of fuzzy premises to sequences of fuzzy consequences and is a function of the underlying reasoning goals/constraints. The Monte Carlo simulation shows that the new fuzzy reasoning method is much more robust than the optimal fuzzy reasoning method proposed in our previous work. The explicit feedback mechanism embedded in the fuzzy reasoning method does significantly improve the robustness of fuzzy reasoning, which is concerned with the effects of perturbations associated with given fuzzy rule bases and/or fuzzy premises on fuzzy consequences. The work presented in this paper sets a new starting point for various principles of feedback control and optimization to be applied in fuzzy reasoning or logical inference and to explore new forms of reasoning including robust reasoning and adaptive reasoning. It can be also expected that the new fuzzy reasoning method presented in this paper can be used for modeling and control of complex systems and for decision-making under complex environments.     

Fuzzy control with fuzzy inputs

This paper is concerned with the use of fuzzy inputs in fuzzy logic controllers. A precise representation of fuzzy logic controllers by means of mappings is used to introduce different ways for dealing with fuzzy inputs. Two types of fuzzy inputs are presented and their potential use in fuzzy control is discussed. The proposed concepts are applied to control a first order process with a PI controller. This simple process is chosen to clearly illustrate the behavior of the closed-loop system using fuzzy inputs for fuzzy reference and fuzzy measurement. Finally, a nonlinear process is used to illustrate the effects of fuzzy inputs on a more complex system. Although it is sometimes speculated that fuzzy inputs may improve the behavior of fuzzy controllers, experiments developed in this paper show this point is not straightforward and that the relevance of fuzzy inputs should be questioned in closed-loop fuzzy control.     

Complex fuzzy logic

A novel framework for logical reasoning, termed complex fuzzy logic, is presented in this paper. Complex fuzzy logic is a generalization of traditional fuzzy logic, based on complex fuzzy sets. In complex fuzzy logic, inference rules are constructed and "fired" in a manner that closely parallels traditional fuzzy logic. The novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets, characterized by complex-valued membership functions. The range of these membership functions is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane, thus providing a method for describing membership in a set in terms of a complex number. Several mathematical properties of complex fuzzy sets, which serve as a basis for the derivation of complex fuzzy logic, are reviewed in this paper. These properties include basic set theoretic operations on complex fuzzy sets - namely complex fuzzy union and intersection, complex fuzzy relations and their composition, and a novel form of set aggregation - vector aggregation. Complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic, while benefiting from the properties of complex numbers and complex fuzzy sets. The introduction of complex-valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties, and considerable potential for further research and application.     

Fuzzy points: algebra and application

A fuzzy point is a region representing the uncertain location of a normal Euclidean point. A fuzzy point in the plane is considered to be a closed disk (a circle and its interior). The algebra of fuzzy points (which includes fuzzy vectors and fuzzy angles) is presented. Since fuzzy points are represented as closed disks, the lengths of fuzzy vectors, and the angles between fuzzy vectors can be viewed as properties of circles in the plane. Methods to compute the magnitude of a fuzzy angle are given. An application of fuzzy point algebra to the problem of detecting and tracking storms in Doppler radar image sequences, which motivates this work, is discussed.     

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.     

A fuzzy Petri net-based expert system and its application to damageassessment of bridges

In this paper, a fuzzy Petri net approach to modeling fuzzy rule-based reasoning is proposed to bring together the possibilistic entailment and the fuzzy reasoning to handle uncertain and imprecise information. The three key components in our fuzzy rule-based reasoning-fuzzy propositions, truth-qualified fuzzy rules, and truth-qualified fuzzy facts-can be formulated as fuzzy places, uncertain transitions, and uncertain fuzzy tokens, respectively. Four types of uncertain transitions-inference, aggregation, duplication, and aggregation-duplication transitions-are introduced to fulfil the mechanism of fuzzy rule-based reasoning. A framework of integrated expert systems based on our fuzzy Petri net, called fuzzy Petri net-based expert system (FPNES), is implemented in Java. Major features of FPNES include knowledge representation through the use of hierarchical fuzzy Petri nets, a reasoning mechanism based on fuzzy Petri nets, and transformation of modularized fuzzy rule bases into hierarchical fuzzy Petri nets. An application to the damage assessment of the Da-Shi bridge in Taiwan is used as an illustrative example of FPNES.     

Fuzzy absolute error distance measure based on a generalised difference operation

The main goal of this paper is to introduce a notion of fuzzy absolute error distance measure between two fuzzy numbers. For this purpose, a notion of generalised difference operation between two fuzzy numbers and absolute value of a fuzzy number was first introduced. The proposed methods were conducted on the basis of α-values of fuzzy numbers. Main properties of the proposed fuzzy distance measure was also verified in the space of fuzzy numbers. The proposed fuzzy distance measure evaluates the fuzzy distance between the two fuzzy numbers as a fuzzy number. Notably, the main advantage of such generalised difference operation is that it always exists. Therefore, it improves the shortcoming of a well-known generalised difference operation called Hakuhara difference. Some of the main properties of the proposed fuzzy absolute error distance measure including robustness were also studied in the space of fuzzy numbers. Several fuzzy distance measures, especially fuzzy absolute error distance, have been proposed so far. However, none of them save all reasonable properties required for an absolute error distance measure in fuzzy environment. Shortcomings relevant to other methods and advantages of the proposed method were also discussed.     

二型直觉模糊集

二型模糊集和直觉模糊集都具有很强的实际应用背景.二型模糊集增强了系统处理不确定性的能力,直觉模糊集为解决人们判断问题所出现的犹豫信息提供了理论依据.本文在二型模糊集和直觉模糊集的基础上,给出了二型直觉模糊集的概念,证明了二型直觉模糊集是一型模糊集、直觉模糊集、区间值模糊集、区间值直觉模糊集的广义形式,讨论了二型直觉模糊集的基本运算和二型直觉模糊关系.最后,研究了基于二型直觉模糊理论的近似推理,并实例说明了二型直觉模糊集的实际应用背景.     

A new interpretation of complex membership grade

Complex fuzzy sets utilize a complex degree of membership, represented in polar coordinates, which is a combination of a degree of membership in a fuzzy set along with a crisp phase value that denotes position within the set. The compound value carries more information than a traditional fuzzy set and enables efficient reasoning. In this paper, we present a new and generalized interpretation of a complex grade of membership, where a complex membership grade defines a complex fuzzy class. The new definition provides rich semantics that is not readily available through traditional fuzzy sets or complex fuzzy sets and is not limited to a compound of crisp cyclical data with fuzzy data. Furthermore, the two components of the complex fuzzy class carry fuzzy information. A complex class is represented either in Cartesian or in polar coordinates where both axes induce fuzzy interpretation. Another novelty of the scheme is that it enables representing an infinite set of fuzzy sets. The paper provides the new definition of complex fuzzy classes along with axiomatic definition of basic operations on complex fuzzy classes. In addition, coordinate transformation as well as an extension from two‐dimensional fuzzy classes to n‐dimensional fuzzy classes are presented. © 2011 Wiley Periodicals, Inc.     

Fuzzy holons for intelligent multi-scale design in cloud-based design for configurations

Cloud-based design for configurations can be referred to as a service-oriented networked design for configurations model. However, cloud-based models also pose challenges such as reliability, availability, capability, ability, adaptability of resources, and services across spatial boundaries. Multi-scale design can presumably stimulate greater intelligence in cloud-based models. Using the concepts of the fuzzy holon and the fuzzy attractor, this paper proposes the fuzzy holonic approach to address multi-scale design for configurations. A fuzzy design holon is defined through two basic holons: fuzzy function holon and fuzzy solution holon. A fuzzy attractor is defined as a fuzzy function holon or fuzzy function solution toward which a design tends to evolve. The proposed fuzzy holon model is driven by two conflicting drives: (a) completeness of fuzzy function holons and fuzzy solution holons, and (b) discrimination of fuzzy function holons and fuzzy solution holons. Through simulations, four possible states of behavior of fuzzy holon design are found: (a) the impossibility state characterized by the impossibility of fuzzy holon creation; (b) the creation and destruction state sometimes characterized by the creation of fuzzy holons and sometimes the destruction of fuzzy holons, (c) the development state characterized by a natural creation and development of fuzzy holons and (d) the failure state characterized by the interruption of the development of the fuzzy design holon and the destruction of already created fuzzy design holon. The model explains that design is not an orderly and well behaved phenomenon. It shows that fuzzy holon design is a discontinuous phenomenon.     

Ranking type-2 fuzzy numbers

Type-2 fuzzy sets are a generalization of the ordinary fuzzy sets in which each type-2 fuzzy set is characterized by a fuzzy membership function. In this paper, we consider the problem of ranking a set of type-2 fuzzy numbers. We adopt a statistical viewpoint and interpret each type-2 fuzzy number as an ensemble of ordinary fuzzy numbers. This enables us to define a type-2 fuzzy rank and a type-2 rank uncertainty for each intuitionistic fuzzy number. We show the reasonableness of the results obtained by examining several test cases.     

Generalized interval-valued fuzzy variable precision rough sets determined by fuzzy logical operators

The fuzzy rough set model and interval-valued fuzzy rough set model have been introduced to handle databases with real values and interval values, respectively. Variable precision rough set was advanced by Ziarko to overcome the shortcomings of misclassification and/or perturbation in Pawlak rough sets. By combining fuzzy rough set and variable precision rough set, a variety of fuzzy variable precision rough sets were studied, which cannot only handle numerical data, but are also less sensitive to misclassification. However, fuzzy variable precision rough sets cannot effectively handle interval-valued data-sets. Research into interval-valued fuzzy rough sets for interval-valued fuzzy data-sets has commenced; however, variable precision problems have not been considered in interval-valued fuzzy rough sets and generalized interval-valued fuzzy rough sets based on fuzzy logical operators nor have interval-valued fuzzy sets been considered in variable precision rough sets and fuzzy variable precision rough sets. These current models are incapable of wide application, especially on misclassification and/or perturbation and on interval-valued fuzzy data-sets. In this paper, these models are generalized to a more integrative approach that not only considers interval-valued fuzzy sets, but also variable precision. First, we review generalized interval-valued fuzzy rough sets based on two fuzzy logical operators: interval-valued fuzzy triangular norms and interval-valued fuzzy residual implicators. Second, we propose generalized interval-valued fuzzy variable precision rough sets based on the above two fuzzy logical operators. Finally, we confirm that some existing models, including rough sets, fuzzy variable precision rough sets, interval-valued fuzzy rough sets, generalized fuzzy rough sets and generalized interval-valued fuzzy variable precision rough sets based on fuzzy logical operators, are special cases of the proposed models.     

A neural fuzzy system with fuzzy supervised learning

A neural fuzzy system learning with fuzzy training data (fuzzy if-then rules) is proposed in this paper. This system is able to process and learn numerical information as well as linguistic information. At first, we propose a five-layered neural network for the connectionist realization of a fuzzy inference system. The connectionist structure can house fuzzy logic rules and membership functions for fuzzy inference. We use alpha-level sets of fuzzy numbers to represent linguistic information. The inputs, outputs, and weights of the proposed network can be fuzzy numbers of any shape. Furthermore, they can be hybrid of fuzzy numbers and numerical numbers through the use of fuzzy singletons. Based on interval arithmetics, a fuzzy supervised learning algorithm is developed for the proposed system. It extends the normal supervised learning techniques to the learning problems where only linguistic teaching signals are available. The fuzzy supervised learning scheme can train the proposed system with desired fuzzy input-output pairs which are fuzzy numbers instead of the normal numerical values. With fuzzy supervised learning, the proposed system can be used for rule base concentration to reduce the number of rules in a fuzzy rule base. Simulation results are presented to illustrate the performance and applicability of the proposed system.     

模糊决策的基础——模糊集比较与排序

研究模糊决策中模糊集的比较与排序问题。通过引入模糊极大集和模糊极小集为参照系统并以海明距离为计量工具，定义了两个模糊效用函数和一个模糊优先关系作为模糊集的排序指标。前者适合于多个模糊集的整体分析，后者适合于两两之间的比较判别。对于两个模糊集的排序问题，模糊效用函数自动退化为相应的模糊优先关系。系统分析了3种指标的性能及关系，并举例说明了它们的应用。     

Relaxing constraints in enhanced entity-relationship models using fuzzy quantifiers

While various articles about fuzzy entity relationship (ER) and enhanced entity relationship (EER) models have recently been published, not all examine how the constraints expressed in the model may be relaxed. In this paper, our aim is to relax the constraints which can be expressed in a conceptual model using the modeling tool, so that these constraints can be made more flexible. We will also study new constraints that are not considered in classic EER models. We use the fuzzy quantifiers which have been widely studied in the context of fuzzy sets and fuzzy query systems for databases. In addition, we shall examine the representation of these constraints in an EER model and their practical repercussions. The following constraints are studied: the fuzzy participation constraint, the fuzzy cardinality constraint, the fuzzy completeness constraint to represent classes and subclasses, the fuzzy cardinality constraint on overlapping specializations, fuzzy disjoint and fuzzy overlapping constraints on specializations, fuzzy attribute-defined specializations, fuzzy constraints in union types or categories and fuzzy constraints in shared subclasses. We shall also demonstrate how fuzzy (min, max) notation can substitute the fuzzy cardinality constraint but not the fuzzy participation constraint. All these fuzzy constraints have a new meaning, they offer greater expressiveness in conceptual design, and are included in the so-called fuzzy EER model.     

Semi-parametric partially logistic regression model with exact inputs and intuitionistic fuzzy outputs

Some accounting studies have focused on logistic regression relationships between exact/fuzzy inputs/outputs. However, intuitionistic fuzzy sets find application in many real studies instead of fuzzy sets. On the other hand, semi-parametric partially linear model also has attracted attentions in recent years. This study is an investigation of intuitionistic fuzzy semi-parametric partially logistic model for such cases with exact inputs, intuitionistic fuzzy outputs, intuitionistic fuzzy smooth function and intuitionistic fuzzy coefficients. For this purpose, a hybrid procedure is suggested based on curve fitting methods and least absolutes deviations to estimate the intuitionistic fuzzy smooth function and intuitionistic fuzzy coefficients. The proposed method is also compared with a common fuzzy logistic regression model as a real fuzzy data set. It is shown that the proposed intuitionistic fuzzy logistic regression model performs better and efficient results in regard to some goodness-of-fit criteria suggest that the proposed model could be successfully applied in many practical studies of intuitionistic fuzzy logistic regression model in expert systems.     

Robust fuzzy control for a plant with fuzzy linear model

A robust complexity reduced proportional-integral-derivative (PID)-like fuzzy controllers is designed for a plant with fuzzy linear model. The plant model is described with the expert's linguistic information involved. The linguistic information for the plant model is represented as fuzzy sets. In order to design a robust fuzzy controller for a plant model with fuzzy sets, an approach is developed to implement the best crisp approximation of fuzzy sets into intervals. Then, Kharitonov's Theorem is applied to construct a robust fuzzy controller for the fuzzy uncertain plant with interval model. With the linear combination of input variables as a new input variable, the complexity of the fuzzy mechanism of PID-like fuzzy controller is significantly reduced. The parameters in the robust fuzzy controller are determined to satisfy the stability conditions. The robustness of the designed fuzzy controller is discussed. Also, with the provided definition of relative robustness, the robustness of the complexity reduced fuzzy controller is compared to the classical PID controller for a second-order plant with fuzzy linear model. The simulation results are included to show the effectiveness of the designed PID-like robust fuzzy controller with the complexity reduced fuzzy mechanism.     

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.     

A general technique for deriving analytical structure of fuzzy controllers using arbitrary trapezoidal input fuzzy sets and Zadeh AND operator

Deriving the analytical structure of fuzzy controllers is very important as it creates a solid foundation for better understanding, insightful analysis, and more effective design of fuzzy control systems. We previously developed a technique for deriving the analytical structure of the fuzzy controllers that use Zadeh fuzzy AND operator and the symmetric, identical trapezoidal or triangular input fuzzy sets. Many fuzzy controllers use arbitrary trapezoidal/triangular input fuzzy sets that are asymmetric. At present, there exists no technique capable of deriving the analytical structure of these fuzzy controllers. Extending our original technique, we now present a novel method that can accomplish rigorously the structure derivation for any fuzzy controller, Mamdani type or TS type, that employs the arbitrary trapezoidal input fuzzy sets and Zadeh fuzzy AND operator. The new technique contains our original technique as a special case. Given the importance of PID control, we focus on Mamdani fuzzy PI and PD controllers in this paper and show in detail how to use the new technique for different configurations of the fuzzy PI/PD controllers. The controllers use two arbitrary trapezoidal fuzzy sets for each input variable, four arbitrary singleton output fuzzy sets, four fuzzy rules, Zadeh fuzzy AND operator, and the centroid defuzzifier. This configuration is more general and complicated than the Mamdani fuzzy PI/PD controllers in the current literature. It actually contains them as special cases. We call this configuration the generalized fuzzy PI/PD controller.