On the use of words and fuzzy sets

This paper only tries to stimulate some reflections, by showing a possible way towards rethinking fuzzy sets from their roots. A rethinking that does not change the view that fuzzy sets are mathematical entities giving extension to predicates. In such a way that, if the predicates are precise, the corresponding new entities are nothing else than classical sets.What, perhaps, is a key idea is that the use of a predicate organizes, in some way, the universe of discourse. When this organization is a preorder, the extension or L-set, appears once a degree, numerical or not, but consistent with the organization, can be defined.The ultimate goal of those above mentioned reflections, provided they would be realized in the future, is to extend the current theories of fuzzy sets to wider areas of both language and reasoning. Our objective is to reach a better knowledge of the links between language and its representations for the progress of computing with words and perceptions.     

Ergodic distribution for a fuzzy inventory model of type (s, S) with gamma distributed demands

In this study, a stochastic process (X(t)), which describes a fuzzy inventory model of type (s, S) is considered. Under some weak assumptions, the ergodic distribution of the process X(t) is expressed by a fuzzy renewal function U(x). Then, membership function of the fuzzy renewal function U(x) is obtained when the amount of demand has a Gamma distribution with fuzzy parameters. Finally, membership function and alpha cuts of fuzzy ergodic distribution of this process is derived by using extension principle of L. Zadeh.     

A novel and fast MIMO fuzzy inference system based on a class of fuzzy clustering algorithms with interpretability and complexity analysis

A data driven Fuzzy Inference System (FIS) employs Membership Functions (MFs) with adjustable parameters in its IF part to fuzzify the input data. The input space is partitioned simply by dividing universe of discourse of each input variable into some fuzzy subspaces. The MFs are then defined on the fuzzy subspaces of the input variables. Parameters of the MFs are tuned for maximum accuracy of the system (which demands high runtime) without considering the data structure which impairs interpretability of the FIS and degenerates the system into a black-box tool. Such a FIS does not represent actual structure of the data and its MFs are not necessarily in accord with the data distribution in the input space. In addition, the FIS suffers from exponential complexity of order O(T^r) where T is number of linguistic terms (number of subspaces on the universe of discourse of input variables) and r is number of input variables. This article presents a novel Multiple-Input and Multiple-Output Clustering based Fuzzy Inference System (MIMO CFIS) which is made directly from a class of fuzzy clustering algorithms to overcome these shortcomings. CFIS identifies dense regions of the input data using fuzzy clustering and then places a cluster on each of these regions. These fuzzy clusters represent actual structure of the data and serve as fuzzy rules in the rule base of CFIS and provide MFs that exactly fit the dense regions of the data that makes the system more interpretable and avoids redundant rules. These MFs are normal, convex, and continuous and have no parameter to be tuned (which makes CFIS much faster than other FISs) and fuzzify the input data according to their membership in the clusters. THEN part of CFIS is generalized form of THEN part of Takagi-Sugeno (TS) fuzzy system which accommodates any function of input variables. Despite less number of adjustable parameters, testing error of CFIS is less than that of TS system and its modified versions. Moreover, number of fuzzy rules in CFIS rule base is the same as the number of linguistic terms (or fuzzy clusters) and consequently its complexity is of orderO(T). Also, CFIS is a MIMO system and avoids inconsistent (contradictory) rules by generating well-separated fuzzy clusters whereas TS system is MISO and never guarantees generation of consistent rules. In addition, CFIS satisfies most of the interpretability criteria of FISs.     

关于二型模糊集合的一些基本问题

采用集合论的方法给出了单位模糊集合和二型模糊集合及其在一点的限制等定义,使得二型模糊集合更易于理解.通过定义嵌入单位模糊集合来描述一般二型模糊集合,并给出离散、半连通二型模糊集合的表达式.根据论域、主隶属度及隶属函数的特性将二型模糊集合分为四种类型:离散、半连通、连通及复合型,并根据连通的特点将连通二型模糊集合分为单连通及多连通两类.利用支集的闭包（Closure of support,CoS）划分法表述主隶属度及区间二型模糊集合.提出了CoS二、三次划分法分别来表述单、复连通二型模糊集合,并使每一个子区域的上下边界及次隶属函数在该子区域上的限制分别具有相同的解析表述式.最后,探讨了二型模糊集合在一点的限制、主隶属度、支集、嵌入单位模糊集合之间的关系.     

Vague controller: A generalization of fuzzy logic controller

In this research, a vague controller (VC) is synthesized by using the notion of vague sets, which are a generalization of fuzzy sets and characterized by a truth-membership function and a falsity-membership function. The vague sets follow the basic set operations and logic operations defined for fuzzy sets, and are superior to fuzzy sets in that they could deal with the uncertainty encountered in real-world applications in a more natural way. Depending on the vague sets, the VC is developed as a generalization of fuzzy logic controller (FLC). The design procedures of the VC, which allow an arbitrary number of input variables, and each variable could have a distinct number of linguistic values, are outlined in this paper. In order to compensate the effort in constructing two series of membership functions for vague sets and to ease the difficulties in designing VCs, a new means of designating membership functions for VCs is also presented in this article. This method constructs a set of membership functions systematically by using only two parameters: number of linguistic values of a linguistic variable and shrinking factor. The membership functions generated by this method, shrinking-span membership functions (SSMFs), have different spans over the universe of discourse and, therefore, are more rational and more practical from the human expert's point of view.     

Analytical structures and analysis of the simplest fuzzy PDcontrollers

This paper deals with simplest fuzzy PD controllers which employ only two fuzzy sets on the universe of discourse of each input variable, and three fuzzy sets on the universe of discourse of output variable. First, analytical structures of the simplest fuzzy PD controllers are derived via triangular membership functions for fuzzification, intersection T-norm, Lukasiewicz OR and Zadeh (1965) OR T-conorms, Mamdani's minimum, Larsen's product and drastic product inference methods, and center of area method for defuzzification. Properties of such fuzzy PD controllers are investigated. Based on these properties a comparative study is made on fuzzy controllers derived, and also on the fuzzy controllers and their counterpart-conventional linear PD controller. Finally, sufficient conditions for bounded-input bounded-output stability of fuzzy PD control systems are established using the well known small gain theorem.     

Atanassov’s intuitionistic (S, T)-fuzzy n-ary sub-hypergroups and their properties

In this paper, we define a new kind of intuitionistic fuzzy n-ary sub-hypergroups of an n-ary hypergroup. This definition, which is based on Atanassov’s intuitionistic fuzzy sets, t-norms and t-conorms, includes earlier definitions of (n-ary) sub-hypergroups, (intuitionistic) fuzzy (n-ary) sub-hypergroups. Then some related properties are investigated. Also, intuitionistic fuzzy relations with respect to t-norms and t-conorms on n-ary hypergroups are discussed.     

Minkowski‐type distance measures for generalized orthopair fuzzy sets

The generalized orthopair fuzzy set inherits the virtues of intuitionistic fuzzy set and Pythagorean fuzzy set in relaxing the restriction on the support for and support against. The very lax requirement provides decision makers great freedom in expressing their beliefs about membership grades, which makes generalized orthopair fuzzy sets having a wide scope of application in practice. In this paper, we present the Minkowski‐type distance measures, including Hamming, Euclidean, and Chebyshev distances, for q‐rung orthopair fuzzy sets. First, we introduce the Minkowski‐type distances of q‐rung orthopair membership grades, based on which we can rank orthopairs. Second, we propose several distances over q‐rung orthopair fuzzy sets on a finite discrete universe and subsequently discuss their applications to multiattribute decision‐making problems. Then we extend these results to a continuous universe, both bounded and unbounded cases are considered. Some illustrative examples are employed to substantiate the conceptual arguments.     

Processing fuzzy set membership functionals as vectors

A framework is presented for processing fuzzy sets for which the universe of discourse X = {x} is a separable Hilbert Space, which, in particular, may be a Euclidian Space. In a given application, X would constitute a feature space. The membership functions of sets in such X are then “membership functionals”, that is, mappings from a vector space to the real line. This paper considers the class Φ of fuzzy sets A, the membership functionals μ _A of which belong to a Reproducing Kernel Hilbert Space (RKHS) F(X) of bounded analytic functionals on X, and satisfy the constraint . These functionals can be expanded in abstract power series in x, commonly known as Volterra functional series in x. Because of the one-to-one relationship between the fuzzy sets A and their respective μ _A , one can process the sets A as objects using their μ _A as intermediaries. Thus the structure of the uncertainty present in the fuzzy sets can be processed in a vector space without descending to the level of processing of vectors in the feature space as usually done in the literature in the field. Also, the framework allows one to integrate human and machine judgments in the definition of fuzzy sets; and to use concepts analogous to probabilistic concepts in assigning membership values to combinations of fuzzy sets. Some analytical and interpretive consequences of this approach are presented and discussed. A result of particular interest is the best approximation of a membership functional μ _A in F(X) based on interpolation on a training set {(v ⁱ , u ⁱ ),i = 1, . . . , q} and under the positivity constraint. The optimal analytical solution comes out in the form of an Optimal Interpolative Neural Network (OINN) proposed by the author in 1990 for best approximation of pattern classification systems in a F(X) space setting. An example is briefly described of an application of this approach to the diagnosis of Alzheimer’s disease.     

Measuring incompatibility between Atanassov’s intuitionistic fuzzy sets

The aim of this paper is to establish an axiomatic definition of incompatibility measure in the framework of Atanassov’s intuitionistic fuzzy sets and use geometrical methods to build some families of such incompatibility measures. First, we construct several functions to measure incompatibility for an intuitionistic t-norm that can be represented by an adequate t-norm and t-conorm. Additionally, we establish some relations between some particular cases of these functions. Similarly, we then obtain incompatibility measures for a family of non-representable intuitionistic t-norms.     

一种新的覆盖粗糙模糊集模型

在覆盖粗糙集与模糊集结合的研究中,已有的覆盖粗糙模糊集模型存在两类问题:一类是元素的上、下近似隶属度之间的差值通常过大;另一类是元素的上、下近似隶属度与其在给定模糊集中的隶属度无关.对此,通过定义模糊覆盖粗糙隶属度,将元素的最小描述与给定模糊集建立联系,同时综合元素在给定模糊集中的隶属度,进而建立一个新的覆盖粗糙模糊集模型.理论比较和实验结果均表明该模型可以有效解决上述两类问题.     

Design of a fuzzy controller in mobile robotics using genetic algorithms

The design of fuzzy controllers for the implementation of behaviors in mobile robotics is a complex and highly time-consuming task. The use of machine learning techniques, such as evolutionary algorithms or artificial neural networks for the learning of these controllers allows to automate the design process. In this paper, the automated design of a fuzzy controller using genetic algorithms for the implementation of the wall-following behavior in a mobile robot is described. The algorithm is based on the Iterative Rule Learning (IRL) approach, and a parameter (δ) is defined with the aim of selecting the relation between the number of rules and the quality and accuracy of the controller. The designer has to define the universe of discourse and the precision of each variable, and also the scoring function. No restrictions are placed neither in the number of linguistic labels nor in the values that define the membership functions.     

t-Norm based cuts of intuitionistic fuzzy sets

We consider the system of intuitionistic fuzzy sets (IF-sets) in a universe X and study the cuts of an IF-set. Suppose a left continuous triangular norm is given. The t-norm based cut (level set) of an IF-set is defined in a way that binds the membership and nonmembership functions via the triangular norm. This is an extension of usual cuts of IF-sets. We show that the system of these cuts fulfils analogical properties as usual systems of cuts. However, it is not possible to reconstruct an IF-set from the system of t-norm based cuts.     

Atanassov’s intuitionistic fuzzy grade of hypergroups

This paper deals with connections between hypergroupoids and Atanassov’s intuitionistic fuzzy sets. First a sequence of join spaces is associated with a hypergroupoid H; the length of the sequence is called Atanassov’s intuitionistic fuzzy grade of H. Second, a theorem about the existence of a hypergroup with Atanassov’s intuitionistic fuzzy grade equal to n is proved. Furthermore, some properties of the complete hypergroups in connection with this argument are presented and discussed.     

Interval type-2 fuzzy membership function generation methods for pattern recognition

Type-2 fuzzy sets (T2 FSs) have been shown to manage uncertainty more effectively than T1 fuzzy sets (T1 FSs) in several areas of engineering [4], [6], [7], [8], [9], [10], [11], [12], [15], [16], [17], [18], [21], [22], [23], [24], [25], [26], [27] and [30]. However, computing with T2 FSs can require undesirably large amount of computations since it involves numerous embedded T2 FSs. To reduce the complexity, interval type-2 fuzzy sets (IT2 FSs) can be used, since the secondary memberships are all equal to one [21]. In this paper, three novel interval type-2 fuzzy membership function (IT2 FMF) generation methods are proposed. The methods are based on heuristics, histograms, and interval type-2 fuzzy C-means. The performance of the methods is evaluated by applying them to back-propagation neural networks (BPNNs). Experimental results for several data sets are given to show the effectiveness of the proposed membership assignments.     

Fuzzy model identification for classification of gait events inparaplegics

Fuzzy system identification was applied to a biomedical system for classification purposes. Gait achieved through functional electrical stimulation (FES) of paraplegics was divided using sensor measurements of kinematic variables as inputs to five discrete events. Two identification algorithms were used to estimate the system model. Both max-min and max-product composition were used. Membership functions were either trapezoidal or triangular and all membership functions in a particular universe of discourse had the same shape and size. The universe of discourse was varied by altering the overlap between membership functions. The classification performance was assessed quantitatively, by measuring the percentage of time steps in which the correct event was found, and qualitatively, by observing types of errors. The identification algorithm affected system performance. No difference in classification was found between max-min and max-product composition. The performance was dependent on membership function overlap. A comparison of the classification found using the fuzzy rule base versus that found using a traditional look-up table demonstrated that the fuzzy approach was superior. It is speculated that the use of fuzzy logic decreased errors stemming from sensor noise and/or small variations in the input signals. The performance of this approach was compared to that of a feedforward neural network and the fuzzy system is found superior     

Intuitionistic fuzzy Hv-submodules

After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Atanassov is one among them. In this paper, we apply the concept of an intuitionistic fuzzy set to H_v-modules. The notion of an intuitionistic fuzzy H_v-submodule of an H_v-module is introduced, and some related properties are investigated. Characterizations of intuitionistic fuzzy H_v-submodules are given.     

On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision

Intuitionistic fuzzy sets [K.T. Atanassov, Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia (deposed in Central Science-Technical Library of Bulgarian Academy of Science, 1697/84), 1983 (in Bulgarian)] are an extension of fuzzy set theory in which not only a membership degree is given, but also a non-membership degree, which is more or less independent. Considering the increasing interest in intuitionistic fuzzy sets, it is useful to determine the position of intuitionistic fuzzy set theory in the framework of the different theories modelling imprecision. In this paper we discuss the mathematical relationship between intuitionistic fuzzy sets and other models of imprecision.     

Generalized intuitionistic fuzzy geometric interaction operators and their application to decision making

Considering that there may exist some interactions between membership function and non-membership function of different intuitionistic fuzzy sets, we present some new operational laws from the probability point of view and give a geometric interpretation of the new operations. Based on which, a new class of generalized intuitionistic fuzzy aggregation operators are developed, including the generalized intuitionistic fuzzy weighted geometric interaction averaging (GIFWGIA) operator, the generalized intuitionistic fuzzy ordered weighted geometric interaction averaging (GIFOWGIA) operator and the generalized intuitionistic fuzzy hybrid geometric interaction averaging (GIFHGIA) operator. The properties of these new generalized aggregation operators are investigated. Moreover, approaches to multiple attributes decision making are given based on the generalized aggregation operators under intuitionistic fuzzy environment, and an example is illustrated to show the validity and feasibility of new approach. Finally, we give a systematic comparison between the work of this paper and that of other papers.