Computing the Numerical Scale of the Linguistic Term Set for the 2-Tuple Fuzzy Linguistic Representation Model

When using linguistic approaches to solve decision problems, we need the techniques for computing with words (CW). Together with the 2-tuple fuzzy linguistic representation models (i.e., the Herrera and MartÍnez model and the Wang and Hao model), some computational techniques for CW are also developed. In this paper, we define the concept of numerical scale and extend the 2-tuple fuzzy linguistic representation models under the numerical scale. We find that the key of computational techniques based on linguistic 2-tuples is to set suitable numerical scale with the purpose of making transformations between linguistic 2-tuples and numerical values. By defining the concept of the transitive calibration matrix and its consistent index, this paper develops an optimization model to compute the numerical scale of the linguistic term set. The desired properties of the optimization model are also presented. Furthermore, we discuss how to construct the transitive calibration matrix for decision problems using linguistic preference relations and analyze the linkage between the consistent index of the transitive calibration matrix and one of the linguistic preference relations. The results in this paper are pretty helpful to complete the fuzzy 2-tuple representation models for CW.      

Combined numerical and linguistic knowledge representation and its application to medical diagnosis

In this paper, we propose a novel hybrid intelligent system (HIS) which provides a unified integration of numerical and linguistic knowledge representations. The proposed HIS is a hierarchical integration of an incremental learning fuzzy neural network (ILFN) and a linguistic model, i.e., fuzzy expert system (FES), optimized via the genetic algorithm (GA). The ILFN is a self-organizing network. The linguistic model is constructed based on knowledge embedded in the trained ILFN or provided by the domain expert. The knowledge captured from the low-level ILFN can be mapped to the higher level linguistic model and vice versa. The GA is applied to optimize the linguistic model to maintain high accuracy, comprehensibility, completeness, compactness, and consistency. The resulted HIS is capable of dealing with low-level numerical computation and higher level linguistic computation. After the system is completely constructed, it can incrementally learn new information in both numerical and linguistic forms. To evaluate the system's performance, the well-known benchmark Wisconsin breast cancer data set was studied for an application to medical diagnosis. The simulation results have shown that the proposed HIS performs better than the individual standalone systems. The comparison results show that the linguistic rules extracted are competitive with or even superior to some well-known methods. Our interest is not only on improving the accuracy of the system, but also enhancing the comprehensibility of the resulted knowledge representation.     

Linguistic hesitant intuitionistic fuzzy decision-making method based on VIKOR

Linguistic hesitant intuitionistic fuzzy set, which allows an element having several linguistic evaluation values and each linguistic argument having several intuitionistic fuzzy memberships, is a power tool to model uncertain information existing in multiple attribute decision-making problems. In this paper, we propose new methods by using TOPSIS and VIKOR for multiple attribute decision-making problems, in which evaluation values are in the form of linguistic hesitant intuitionistic fuzzy elements. Different situations of attribute weight information are considered. If attribute weights are partly known, a linear programming model is set up based on the idea that reasonable weights should make the relative closeness of each alternative evaluation value to the linguistic hesitant intuitionistic fuzzy positive ideal solution as large as possible. If attribute weights are unknown completely, an optimization model is set up based on the maximum deviation method. A numerical example is presented to illustrate feasibility and practical advantages of the proposed method. We compare the alternatives’ rankings derived from the linguistic hesitant intuitionistic fuzzy TOPSIS method with those derived from the hesitant fuzzy linguistic TOPSIS and the hesitant intuitionistic fuzzy TOPSIS approach to further illustrate their advantages.

     

Delay-dependent robust asymptotic state estimation of Takagi-Sugeno fuzzy Hopfield neural networks with mixed interval time-varying delays

This paper investigates delay-dependent robust asymptotic state estimation of fuzzy neural networks with mixed interval time-varying delay. In this paper, the Takagi-Sugeno (T-S) fuzzy model representation is extended to the robust state estimation of Hopfield neural networks with mixed interval time-varying delays. The main purpose is to estimate the neuron states, through available output measurements such that for all admissible time delays, the dynamics of the estimation error is globally asymptotically stable. Based on the Lyapunov-Krasovskii functional which contains a triple-integral term, delay-dependent robust state estimation for such T-S fuzzy Hopfield neural networks can be achieved by solving a linear matrix inequality (LMI), which can be easily facilitated by using some standard numerical packages. The unknown gain matrix is determined by solving a delay-dependent LMI. Finally two numerical examples are provided to demonstrate the effectiveness of the proposed method.     

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.     

语言比例二元组Bonferroni平均算子的群决策方法

针对语言比例二元组信息集成的问题,提出了语言比例二元组Bonferroni平均算子的群决策方法。基于基本单位区间单调（BUM）函数的定义,提出了三角模糊数的中心有序加权平均算子的概念。由于三角模糊数的中心有序加权平均算子和语言术语的数值表示（NR）之间存在交互关联,提出了用三角模糊数的中心有序加权平均算子代替语言术语的数值表示的方法,并将此方法得到的NR和语言术语的NR进行了对比分析。引入了语言比例二元组Bonferroni平均算子,介绍了语言比例二元组Bonferroni平均算子的群决策方法,并用一个实例来说明其可行性。     

Interval Fuzzy Model Identification Using$l_infty$-Norm

In this paper, we present a new method of interval fuzzy model identification. The method combines a fuzzy identification methodology with some ideas from linear programming theory. We consider a finite set of measured data, and we use an optimality criterion that minimizes the maximum estimation error between the data and the proposed fuzzy model output. The idea is then extended to modeling the optimal lower and upper bound functions that define the band which contains all the measurement values. This results in lower and upper fuzzy models or a fuzzy model with a set of lower and upper parameters. The model is called the interval fuzzy model (INFUMO). We also showed that the proposed structure uniformly approximates the band of any nonlinear function. The interval fuzzy model identification is a methodology to approximate functions by taking into account a finite set of input and output measurements. This approach can also be used to compress information in the case of large amount of data and in the case of robust system identification. The method can be efficiently used in the case of the approximation of the nonlinear functions family. If the family is defined by a band containing the whole measurement set, the interval of parameters is obtained as the result. This is of great importance in the case of nonlinear circuits' modeling, especially when the parameters of the circuits vary within certain tolerance bands     

An overview of methods for linguistic summarization with fuzzy sets

While the rapid development of information technology has made easy to store and access the huge amount of data, it also brings another problem, that of how to extract potentially useful knowledge not only in an efficient way but also in a way that could be easily understandable by humans. One of the solutions to this problem is linguistic summarization, aim of which is to generate explicit and concise summaries from data that is more compatible with human cognitive mechanism. The most crucial step in linguistic summarization is certainly the evaluation of linguistic summaries since they are the most important element of fuzzy rule based systems commonly used in expert systems and intelligent systems. Therefore, the selection of appropriate method for evaluating linguistic summaries in sense of different views such as quality, quantity, relevance and simplicity becomes vital. The aim of this paper is to review the state of art on linguistic summarization in the framework of fuzzy sets, focusing on the methods for evaluating linguistic summaries and the current applications. A taxonomy is proposed to identify the existing methods depending on the type of fuzzy sets (i.e., type-1 fuzzy set and type-2 fuzzy set) and the type of cardinalities (i.e., scalar cardinality and fuzzy cardinality). The recent studies on linguistic summarization are also presented to give a comprehensive framework for the future directions. The paper ends with conclusions, addressing some important issues and open questions which can be subject for future research.     

基于MACBETH方法的犹豫模糊语言多准则决策方法

犹豫模糊语言集是语言术语集的拓展.受传统的MACBETH(measuring attractiveness by a categorical-based evaluation technique)方法的启发,构建基于MACBETH方法的犹豫模糊语言多准则决策方法.首先将语言表达式表示的决策信息通过转化函数转化为犹豫模糊语言数,进而得出犹豫模糊语言判断矩阵;然后将此结果应用于改进的MACBETH决策支持系统;最后,通过毕业生选择就职企业这一实例说明该方法的有效性和可行性.     

Design variables for bias distribution in transfer function estimation

Estimation of transfer functions of linear systems is one of the most common system identification problems. Several different design variables, chosen by the user for the identification procedure, affect the properties of the resulting estimate. In this paper it is investigated how the choices of prefilters, noise models, sampling interval, and prediction horizon (i.e., the use ofk-step ahead prediction methods) influence the estimate. An important aspect is thai the true system is not assumed to be exactly represented within the chosen model set. The estimate will thus be biased. It is shown how the distribution of bias in the frequency domain is governed by a weighting function, which emphasizes different frequency bands. The weighting function, in turn, is a result of the previously listed design variables. It is shown, e.g., thai the common least-squares method has a tendency to emphasize high frequencies, and that this can be counteracted by prefiltering. It is also shown that, asymptotically, it is only the prediction horizon itself, and not how it is split up into sampling interval times number of predicted sampling instants, that affects this weighting function.     

Selecting the advanced manufacturing technology using fuzzy multiple attributes group decision making with multiple fuzzy information

Selection of advanced manufacturing technology in manufacturing system management is very important to determining manufacturing system competitiveness. This research develops a fuzzy multiple attribute decision-making applied in the group decision-making to improving advanced manufacturing technology selection process. Since numerous attributes have been considered in evaluating the manufacturing technology suitability, most information available in this stage is subjective, imprecise and vague, fuzzy sets theory provides a mathematical framework for modeling imprecision and vagueness. In the proposed approach, a new fusion method of fuzzy information is developed to managing information assessed in different linguistic scales (multi-granularity linguistic term sets) and numerical scales. The flexible manufacturing system adopted in the Taiwanese bicycle industry is employed in this study to demonstrate the computational process of the proposed method. Finally, sensitivity analysis can be performed to examine that the solution robustness.     

State estimation and decoupling of unknown inputs in uncertain LPV systems using interval observers

This paper proposes a linear parameter varying (LPV) interval observer for state estimation and unknown inputs decoupling in uncertain continuous-time LPV systems. Two different problems are considered and solved: (1) the evaluation of the set of admissible values for the state at each instant of time; and (2) the unknown input observation, i.e. the design of the observer in such a way that some information about the nature of the unknown inputs affecting the system can be obtained. In both cases, analysis and design conditions, which rely on solving linear matrix inequalities (LMIs), are provided. The effectiveness and appeal of the proposed method is demonstrated using an illustrative application to a two-joint planar robotic manipulator.     

A neural fuzzy system with linguistic teaching signals

A neural fuzzy system learning with linguistic teaching signals is proposed. This system is able to process and learn numerical information as well as linguistic information. It can be used either as an adaptive fuzzy expert system or as an adaptive fuzzy controller. 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 α-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, two kinds of learning schemes are developed for the proposed system: fuzzy supervised learning and fuzzy reinforcement learning. Simulation results are presented to illustrate the performance and applicability of the proposed system     

A Novel Approach for Linguistic Group Decision Making Based on Generalized Interval‐Valued Intuitionistic Fuzzy Linguistic Induced Hybrid Operator and TOPSIS

In the multiple attribute linguistic group decision making analysis with interval‐valued intuitionistic fuzzy linguistic information, seeking highly efficient aggregation method and order relation play a crucial role. In this paper, we redefine an interval‐valued intuitionistic fuzzy linguistic variable that considers principal component and propose generalized interval‐valued intuitionistic fuzzy linguistic induced hybrid aggregation (GIVIFLIHA) operator with entropic order‐inducing variable and interval‐valued intuitionistic fuzzy linguistic technique for order preference by similarity to an ideal solution (TOPSIS) order relation based on interval‐valued intuitionistic fuzzy linguistic distance measure. Then, some primary properties of the GIVIFLIHA operator are discussed, and a linguistic group decision‐making approach based on GIVIFLIHA operator and interval‐valued intuitionistic fuzzy linguistic TOPSIS order relation is proposed. Finally, a numerical example concerning the investment strategy is given to illustrate the validity and applicability of the proposed method, and then the method is compared with the existing method to further illustrate its flexibility.     

A parsimony fuzzy rule-based classifier using axiomatic fuzzy set theory and support vector machines

This paper proposes a classification method that is based on easily interpretable fuzzy rules and fully capitalizes on the two key technologies, namely pruning the outliers in the training data by SVMs (support vector machines), i.e., eliminating the influence of outliers on the learning process; finding a fuzzy set with sound linguistic interpretation to describe each class based on AFS (axiomatic fuzzy set) theory. Compared with other fuzzy rule-based methods, the proposed models are usually more compact and easily understandable for the users since each class is described by much fewer rules. The proposed method also comes with two other advantages, namely, each rule obtained from the proposed algorithm is simply a conjunction of some linguistic terms, there are no parameters that are required to be tuned. The proposed classification method is compared with the previously published fuzzy rule-based classifiers by testing them on 16 UCI data sets. The results show that the fuzzy rule-based classifier presented in this paper, offers a compact, understandable and accurate classification scheme. A balance is achieved between the interpretability and the accuracy.     

Extension from a linear associative memory to a linguistic linear associative memory

We present a linguistic extension from a crisp model by using a codification model that allows us to implement a fuzzy system on a discrete decision model. The paper begins with an introduction to the representation of fuzzy information, followed by a discussion of the codification method and the extension of a linear associative memory to a linguistic linear associative memory. Finally, we enumerate the advantages and disadvantages of the obtained linguistic linear associative memory. © 1998 John Wiley & Sons, Inc.13: 41–57, 1998     

Identification of continuous-time AR processes from unevenly sampled data

When identifying a continuous-time AR process from discrete-time data, an obvious approach is to replace the derivative operator in the continuous-time model by an approximation. In some cases, a linear regression model can then be formulated. The well-known least-squares method would be very desirable to apply, since it enjoy good numerical properties and low computational complexity, in particular for fast or nonuniform sampling. The focus of this paper is the latter, i.e., nonuniform sampling. Two consistent least-squares schemes for the case of unevenly sampled data are presented. The precise choice of derivative approximation turns out to be crucial. The obtained results are compared to a prediction error method.     

A Fuzzy Linguistic Methodology to Deal With Unbalanced Linguistic Term Sets

Many real problems dealing with qualitative aspects use linguistic approaches to assess such aspects. In most of these problems, a uniform and symmetrical distribution of the linguistic term sets for linguistic modeling is assumed. However, there exist problems whose assessments need to be represented by means of unbalanced linguistic term sets, i.e., using term sets that are not uniformly and symmetrically distributed. The use of linguistic variables implies processes of computing with words (CW). Different computational approaches can be found in the literature to accomplish those processes. The 2-tuple fuzzy linguistic representation introduces a computational model that allows the possibility of dealing with linguistic terms in a precise way whenever the linguistic term set is uniformly and symmetrically distributed. In this paper, we present a fuzzy linguistic methodology in order to deal with unbalanced linguistic term sets. To do so, we first develop a representation model for unbalanced linguistic information that uses the concept of linguistic hierarchy as representation basis and afterwards an unbalanced linguistic computational model that uses the 2-tuple fuzzy linguistic computational model to accomplish processes of CW with unbalanced term sets in a precise way and without loss of information.     

Deriving the personalized individual semantics of linguistic information from flexible linguistic preference relations

In group decision making, flexible linguistic preference relations (FLPRs) are very useful with the pairwise comparisons taking the form of flexible linguistic expressions (FLEs). Due to the fact that different decision makers have different understandings of words, this paper investigates the personalized individual semantics (PISs) of the linguistic information in FLPRs. Two optimization models are constructed to compute a linguistic distribution which is closest to an incomplete FLE. The FLPRs are transformed into fuzzy preference relations by using optimization models which maximize consistency and consensus of the fuzzy preference relations. The PISs of linguistic terms and subsets of the linguistic term set are obtained in this process. A group decision making model based on FLPRs is presented and a green supplier selection problem in automotive industry is solved by using the proposed model. The comparative analysis is presented to show the feasibility of the group decision making model.     

Numerical and Linguistic Prediction of Time Series With the Use of Fuzzy Cognitive Maps

In this paper, we introduce a novel approach to time-series prediction realized both at the linguistic and numerical level. It exploits fuzzy cognitive maps (FCMs) along with a recently proposed learning method that takes advantage of real-coded genetic algorithms. FCMs are used for modeling and qualitative analysis of dynamic systems. Within the framework of FCMs, the systems are described by means of concepts and their mutual relationships. The proposed prediction method combines FCMs with granular, fuzzy-set-based model of inputs. One of their main advantages is an ability to carry out modeling and prediction at both numerical and linguistic levels. A comprehensive set of experiments has been carried out with two major goals in mind. One is to assess quality of the proposed architecture, the other to examine the influence of its parameters of the prediction technique on the quality of prediction. The obtained results, which are compared with other prediction techniques using fuzzy sets, demonstrate that the proposed architecture offers substantial accuracy expressed at both linguistic and numerical levels.