Sequential fuzzy system identification

The problem of deriving the structure of a non-deterministic system from its behaviour is a difficult one even when that behaviour is itself well-defined. When the behaviour can be described only in fuzzy terms structural inference may appear virtually impossible. However, a rigorous formulation and solution of the problem for stochastic automata has recently been given [1] and, in this paper, the results are extended to fuzzy stochastic automata and grammars. The results obtained are of interest on a number of counts. (1) They are a further step towards an integrated ‘theory of uncertainty’; (2) They give new insights into problems of inductive reasoning and processes of ‘precisiation’; (3) They are algorithmic and have been embodied in a computer program that can be applied to the modelling of sequential fuzzy data; (4) They demonstrate that sequential fuzzy data may be modelled naturally in terms of ‘possibility’ vectors.     

A discrete location model with fuzzy accessibility measures

In crisply defined discrete location problems, a number of facilities are to be located at specific points within an area, according to precisely quantified criteria. However in many location problems, especially those associated with social policies, non-crisply defined criteria are used such as, how ‘near’ or ‘accessible’ a facility is, or how ‘important’ certain issues are, etc. In these cases a fuzzy sets approach is more appropriate.This paper presents an application of the set partitioning (set covering with equality constraints) type of integer programming formulation to a discrete location problem with fuzzy accessibility criteria. The solution method suggested uses the symmetry of the objectives and the constraints introduced by Bellman and Zadeh.     

The fuzzy decision problem: An approach to the point estimation problem with fuzzy information

This paper is devoted to the extension of the Bayesian method for the point estimation, when the available information is ‘vague’.In the nonfuzzy case, the parametric estimation can be approached as a particularization in the statistical decision problem. This motivates us to accomplish the mentioned extension by looking at the parametric estimation in the fuzzy case as a special situation in the fuzzy decision problem (defined by Tanaka, Okuda and Asia).In this way, concepts in the fuzzy decision problem are first ‘expressed’ in the estimation terminology. Then, on the basis of these concepts, we shall introduce some notions and state some interesting results. Finally, several illustrative examples will be exposed.     

An extensive-form analysis for comparing fuzzy information systems by means of the worth and quietness of information

Statistical decision problems deal with the choice among several actions, whose consequences depend upon the state of nature, and this choice is usually made on the basis of the outcome of an experiment selected from a collection of experiments whose probability measures depend upon the state. The selection criteria among the experiments are generally based on some measures of the information contained in each experiment about that true state.In this paper we suggest a new selection criterion among the experiments associated with a statistical decision problem, when the experimental outcomes cannot be exactly perceived by the decision maker, but rather the available information from each experimental outcome can be regarded as an element in a fuzzy information system (as defined by Tanaka, Okuda and Asai).In a previous paper we have proposed a method for comparing fuzzy information systems on the basis of the maximization of the ‘oworth of information of a fuzzy park information system’ (Tanaka et al). When the application of such a method leads to indifference between two fuzzy information systems, this indifference can be avoided by applying the criterion in this paper.This selection criterion is an extensive-form analysis which is based on two measures of the information contained in a fuzzy information system about the true state: the ‘worth of information of a fuzzy information system’ and the ‘expected quietness of information of a fuzzy information system’.The suitability of the criterion stated above is corroborated by studying its main properties and contrasting this procedure with other ones making use of different measures of information.     

A genetic design of linguistic terms for fuzzy rule based classifiers

The determination of fuzzy information granules including the estimation of their membership functions play a significant role in fuzzy system design as well as in the design of fuzzy rule based classifiers (FRBCSs). However, although linguistic terms are fundamental elements in the process of elucidating expert’s knowledge, the problem of linguistic term design along with their fuzzy-set-based semantics has not been fully addressed, since term-sets of attributes have not been interpreted as a formalized structure. Thus, the essential relationship between linguistic terms, as syntax, and the constructed fuzzy sets, as their quantitative semantics, or in other words, the problem of the natural semantics of terms behind the linguistic literal has not been addressed. In this paper, we introduce the problem of the design of optimal linguistic terms and propose a method of the design of FRBCSs which may incorporate with the design of linguistic terms to ensure that the presence of linguistic literals are supported not only by data but also by their natural semantics. It is shown that this problem plays a primordial role in enhancing the performance and the interpretability of the designed FRBCSs and helps striking a better balance between the generality and the specificity of the desired fuzzy rule bases for fuzzy classification problems. A series of experiments concerning 17 Machine Learning datasets is reported.     

A general fuzzy control scheme for nonlinear processes with stability analysis

In this paper we develop a general fuzzy control scheme for nonlinear processes. Assuming little knowledge about the dynamics of the controlled process, the proposed scheme starts by probing the process at different points in its operating region to generate a fuzzy quantisation. A simple local controller is then designed at each fuzzy locality. A fuzzy inference mechanism then links up tje local controllers to form a global controller which can be further refined by the learning algorithm. By employing a newly developed structure-adaptive fuzzy modelling scheme, the appropriate fuzzy rule-base for the inference mechanism can be extracted stably and efficiently. The conditions for the stability of the global controller are rigourously established. Simulation results are presented to illustrate the effectiveness of the scheme.     

Redundant fuzzy rules exclusion by genetic algorithms

A genetic-algorithm-based method for exclusion of the potential redundant if-then fuzzy rules that have been extracted from numerical input-output data is proposed. The main idea is the input-space separation into activation rectangles, corresponding to certain output intervals. The generation of fuzzy rules and the membership functions are based on these activation rectangles and appropriate fuzzy rules inference mechanism is proposed. As the method usually produces too many rules, it is necessary to exclude the potential redundant if-then rules. The concept for varying the family of sensitivity parameters, defining the overlapping of the fuzzy regions is proposed. The genetic algorithms are used to resolve the following combinatorial optimization problem: the generation of families of sensitivity parameters. In this way the potential redundant if-then fuzzy rules are excluded.

The method formalizes the synthesis of the fuzzy system and could be used for function approximation, classification and control purposes. An illustrative example for implementation of the method for traffic fuzzy control is given.     

A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making

Data envelopment analysis (DEA) is a methodology for measuring the relative efficiencies of a set of decision making units (DMUs) that use multiple inputs to produce multiple outputs. Crisp input and output data are fundamentally indispensable in conventional DEA. However, the observed values of the input and output data in real-world problems are sometimes imprecise or vague. Many researchers have proposed various fuzzy methods for dealing with the imprecise and ambiguous data in DEA. In this study, we provide a taxonomy and review of the fuzzy DEA methods. We present a classification scheme with four primary categories, namely, the tolerance approach, the α-level based approach, the fuzzy ranking approach and the possibility approach. We discuss each classification scheme and group the fuzzy DEA papers published in the literature over the past 20 years. To the best of our knowledge, this paper appears to be the only review and complete source of references on fuzzy DEA.     

Enhanced adaptive fuzzy sliding mode control for uncertain nonlinear systems

In this article, a novel Adaptive Fuzzy Sliding Mode Control (AFSMC) methodology is proposed based on the integration of Sliding Mode Control (SMC) and Adaptive Fuzzy Control (AFC). Making use of the SMC design framework, we propose two fuzzy systems to be used as reaching and equivalent parts of the SMC. In this way, we make use of the fuzzy logic to handle uncertainty/disturbance in the design of the equivalent part and provide a chattering free control for the design of the reaching part. To construct the equivalent control law, an adaptive fuzzy inference engine is used to approximate the unknown parts of the system. To get rid of the chattering, a fuzzy logic model is assigned for reaching control law, which acting like the saturation function technique. The main advantage of our proposed methodology is that the structure of the system is unknown and no knowledge of the bounds of parameters, uncertainties and external disturbance are required in advance. Using Lyapunov stability theory and Barbalat’s lemma, the closed-loop system is proved to be stable and convergence properties of the system is assured. Simulation examples are presented to verify the effectiveness of the method. Results are compared with some other methods proposed in the past research.     

Alfa-cut based linear programming methodology for constrained matrix games with payoffs of trapezoidal fuzzy numbers

The purpose of this paper is to develop an effective methodology for solving constrained matrix games with payoffs of trapezoidal fuzzy numbers (TrFNs), which are a type of two-person non-cooperative games with payoffs expressed by TrFNs and players’ strategies being constrained. In this methodology, it is proven that any Alfa-constrained matrix game has an interval-type value and hereby any constrained matrix game with payoffs of TrFNs has a TrFN-type value. The auxiliary linear programming models are derived to compute the interval-type value of any Alfa-constrained matrix game and players’ optimal strategies. Thereby the TrFN-type value of any constrained matrix game with payoffs of TrFNs can be directly obtained through solving the derived four linear programming models with data taken from only 1-cut and 0-cut of TrFN-type payoffs. Validity and applicability of the models and method proposed in this paper are demonstrated with a numerical example of the market share game problem.     

An adaptive neuro‐fuzzy inference system for islanding detection in wind turbine as distributed generation

This article proposes a new integrated diagnostic system for islanding detection by means of a neuro‐fuzzy approach. Islanding detection and prevention is a mandatory requirement for grid‐connected distributed generation (DG) systems. Several methods based on passive and active detection scheme have been proposed. Although passive schemes have a large non‐detection zone (NDZ), concern has been raised on active method due to its degrading power‐quality effect. Reliably detecting this condition is regarded by many as an ongoing challenge as existing methods are not entirely satisfactory. The main emphasis of the proposed scheme is to reduce the NDZ to as close as possible and to keep the output power quality unchanged. In addition, this technique can also overcome the problem of setting the detection thresholds inherent in the existing techniques. In this study, we propose to use a hybrid intelligent system called ANFIS (the adaptive neuro‐fuzzy inference system) for islanding detection. This approach utilizes rate of change of frequency (ROCOF) at the target DG location and used as the input sets for a neuro‐fuzzy inference system for intelligent islanding detection. This approach utilizes the ANFIS as a machine learning technology and fuzzy clustering for processing and analyzing the large data sets provided from network simulations using MATLAB software. To validate the feasibility of this approach, the method has been validated through several conditions and different loading, switching operation, and network conditions. The proposed algorithm is compared with the widely used ROCOF relays and found working effectively in the situations where ROCOF fails. Simulation studies showed that the ANFIS‐based algorithm detects islanding situation accurate than other islanding detection algorithms. © 2014 Wiley Periodicals, Inc. Complexity 21: 10–20, 2015     

A way to deal with fuzzy preferences in multi-criteria decision problems

This paper presents a special multiple criteria decision making approach for solving problems in context with fuzzy individual preferences.At first we briefly expose the proposed methodology. The individual preferences are explicitly given by a complete transitive relation R on a set of reference actions. The modelling of the decision-maker's preferences is obtained by means of fuzzy outranking relations. These fuzzy relations are based on a system of additive utility functions which are estimated by means of ordinal regression methods analysing the preference relation R.This is followed by a presentation of two real multicriteria problems which the proposed methodology has been applied to, i.e. a highway plan choice problem and a problem in marketing research dealing with the launching of a new product. In each application we tried to specify this method according to the specific structure of the problem considered.     

Determining best practice production in an aluminium smelter involving sub-processes based substantially on tacit knowledge: an application of Communities of Practice

This paper considers the difficulties associated with a production process that contains a sub-process that is not fully understood and for which data for many parameters are only able to be approximately obtained. The aluminium smelting industry epitomizes such a situation. Here, the critical sub-process that exemplifies these difficulties is the actual heart of the smelter, the electrolytic processing of alumina. This sub-process of aluminium production is at best ‘fuzzy’ and relies on the smelter operators to use their experience and tacit knowledge on a day-to-day basis, that is, the sub-process involves ‘alchemy’. In this paper, this is referred to as the tacit knowledge problem. The impact of such sub-processes on production is significant and the development of a methodology that will lead to a reduced reliance on uncertain alchemy associated with them, highly beneficial. The role of Communities of Practice in finding a solution to the tacit knowledge problem is discussed together with its integration into a mixed-mode model for the determination of best practice production for the smelter.     

Fixed point theorems for fuzzy numbers

In this paper we study the problem of the existence of fixed points for fuzzy maps. We first define uniform structures on sets of fuzzy subsets and describe the ‘closed convergence’ and ‘myope convergence’ structures which have properties close to the corresponding traditional ones. We then give two ways in order to obtain fixed point results. We finally give applications of these results, particularly to fuzzy Markov processes.     

一种资源投入不确定情形下的合作博弈形式及收益分配策略

首先,将经典合作博弈进行扩展,提出了一类模糊联盟合作博弈的通用形式,涵盖常见三种模糊联盟合作博弈,即多线性扩展博弈、比例模糊博弈与Choquet积分模糊博弈.比例模糊博弈、Choquet积分模糊博弈的Shapley值均可以作为一种特定形式下模糊联盟合作博弈的收益分配策略,但是对于多线性扩展博弈的Shapley值一直关注较少,因此利用经典Shapley值构造出多线性扩展博弈的Shapley值,以此作为一种收益分配策略.最后,通过实例分析了常见三类模糊联盟合作博弈的形式及其对应的分配策略,分析收益最大的模糊联盟合作对策形式及最优分配策略,为不确定情形下的合作问题提供了一定的收益分配依据.     

An architecture of fuzzy neural networks for linguistic processing

Fuzzy sets have been used successfully in order to deal with imprecise data, linguistic terms or not well-defined concepts. Recently, considerable effort has been made in the direction of combining the neural network approach with fuzzy sets. In this paper a fuzzy feed-forward neural network, able to process trapezoidal fuzzy sets, has been investigated. Normalized trapezoidal fuzzy sets have been considered. The fuzzy generalized delta rule with different back-propagation algorithms is discussed. The more interesting and characteristic property of the proposed architecture is the ability of each node to process fuzzy sets or linguistic terms, preserving the simplicity of the back-propagation algorithm. Consequently, the resulting architecture is able to cope with problems in which the input parameters and the desired targets are described by linguistic terms. This methodology has the further interesting characteristic of being able to operate at the linguistic level rather than at the numerical level, that is it can work at a higher data abstraction level. An example in computerized electrocardiography will be illustrated in order to test the proposed approach.     

An interval-coefficient fuzzy binary linear programming,the solution,and its application under uncertainties

The common difficulty in solving a Binary Linear Programming (BLP) problem is uncertainties in the parameters and the model structure. The previous studies of BLP problems normally focus on parameter uncertainty or model structure uncertainty, but not on both types of uncertainties. This paper develops an interval-coefficient Fuzzy Binary Linear Programming (IFBLP) and its solution for BLP problems under uncertainties both on parameters and model structure. In the IFBLP, the parameter uncertainty is represented by the interval coefficients, and the model structure uncertainty is reflected by the fuzzy constraints and a fuzzy goal. A novel and efficient methodology is used to solve the IFLBP into two extreme crisp-coefficient BLPs, which are called the ‘best optimum model’ and the ‘worst optimum model’. The results of these two crisp-coefficient extreme models can bound all outcomes of the IFBLP. One of the contributions in this paper is that it provides a mathematical sound approach (based on some mathematical developments) to find the boundaries of optimal alpha values, so that the linearity of model can be maintained during the conversions. The proposed approach is applied to a traffic noise control plan to demonstrate its capability of dealing with uncertainties.     

Towards a new strategy for solving fuzzy optimization problems

Fuzzy Optimization models and methods has been one of the most and well studied topics inside the broad area of Soft Computing. Particularly relevant is the field of fuzzy linear programming (FLP). Its applications as well as practical realizations can be found in all the real world areas. As FLP problems constitute the basis for solving fuzzy optimization problems, in this paper a basic introduction to the main models and methods in FLP is presented and, as a whole, Linear Programming problems with fuzzy costs, fuzzy constraints and fuzzy coefficients in the technological matrix are analyzed. But fuzzy sets and systems based optimization methods do not end with FLP, and hence in order to solve more complex optimization problems, fuzzy sets based Meta-heuristics are considered, and two main operative approaches described. Provided that these techniques obtain efficient and/or effective solutions, we present a fuzzy rule based methodology for coordinating Meta-heuristics and in addition, to provide intelligence, we propose a process of extraction of the knowledge to conduct the coordination of the system.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

Exact inversion of decomposable interval type-2 fuzzy logic systems

It has been demonstrated that type-2 fuzzy logic systems are much more powerful tools than ordinary (type-1) fuzzy logic systems to represent highly nonlinear and/or uncertain systems. As a consequence, type-2 fuzzy logic systems have been applied in various areas especially in control system design and modelling. In this study, an exact inversion methodology is developed for decomposable interval type-2 fuzzy logic system. In this context, the decomposition property is extended and generalized to interval type-2 fuzzy logic sets. Based on this property, the interval type-2 fuzzy logic system is decomposed into several interval type-2 fuzzy logic subsystems under a certain condition on the input space of the fuzzy logic system. Then, the analytical formulation of the inverse interval type-2 fuzzy logic subsystem output is explicitly driven for certain switching points of the Karnik–Mendel type reduction method. The proposed exact inversion methodology driven for the interval type-2 fuzzy logic subsystem is generalized to the overall interval type-2 fuzzy logic system via the decomposition property. In order to demonstrate the feasibility of the proposed methodology, a simulation study is given where the beneficial sides of the proposed exact inversion methodology are shown clearly.