O型模糊数及其结构元表示方法

为研究平面或空间模糊几何问题的需要,在平面或空间模糊点的背景下,给出了O型模糊数的概念,它是一类二维实数域上的模糊集,同时给出了O型模糊数的二维模糊结构元表示方法.二维模糊数的结构元方法,可以使O型模糊数的运算变成普通实数与模糊结构元之间的运算,使得过去必须依赖扩张原理和表现定理来刻画的模糊数运算变得更加简单与直观,不仅仅为模糊分析计算的简化提供了工具,也为二维实数域上模糊分析理论与应用的研究开创了一条新的途径.     

中介数学系统已建立,为什么还会产生模糊数学危机

构造了两个例子,当试图用扎德的＂隶属函数＂的定义去表述例子中的模糊信息时发现,对于列举的例子,＂隶属函数＂的定义根本＂无法确切表达＂甚至根本＂无法表达＂其中所蕴含的模糊信息.进而发现,模糊数学危机产生的原因不是因为其基础理论不完善,而是因为某些定义在当初人们定义它时,定义的不准确造成的.所以要从根本上解决模糊数学的危机...     

考虑区间值的相对隶属函数与传统模糊分布函数的比较

通过分析比较,揭示了可变集理论中考虑区间值的相对隶属函数与传统的线性模糊分布函数之间的一些区别与联系,即传统的线性模糊分布函数是考虑区间值的相对隶属函数的一些特例,考虑区间值的相对隶属函数在描述模糊现象的模糊性时更具有普适性;同时指出了应用考虑区间值的相对隶属函数时应当注意的问题.     

求解多层线性规划的模糊规划法

本文用模糊集理论中的隶属函数描述多层线性规划的各层目标,在第一层给定最小满意水平下,通过求解相应层次的模糊规划来确定各层的最小满意度,从而最终得到问题的一个满意解。提出的方法只需求解一系列线性规划问题,具有较好的计算复杂性和可行性,最后的算例进一步验证了方法的有效性。     

可变模糊集合理论与可变模型集

在对立模糊集定义基础上给出以相对隶属函数表示的模糊可变集合定义,给出可变模糊聚类迭代模型、可变模糊模式识别模型、可变模糊对立识别模型.它们是可变模糊聚类、识别、优选决策、评价相统一的理论模型集,是可变模糊集的基础模型与核心内容,可用于自然、管理、人文、社会等各种学科中关于模糊聚类、识别、优选决策、评价、预测等众多实际领域.     

Measuring the fuzziness of human thoughts: An application of fuzzy sets to sociological research

Conventionally, sociologists measure the membership of an individual to a group by a “0 or 1” characteristic function. But when the definition of that group is fuzzy and an individual is neither a full member nor a nonmember, this dichotomous characteristic function may distort the reality. Instead of the “0 or 1” characteristic function by classical set theory, fuzzy set theory introduces a membership function which is a gradation from 0 to 1 to measure the degree to which an object (an individual) belongs to a concept (a group). Based on the rationale of fuzzy set theory, we suggest some new methods of data collection and analysis. Among several noteworthy findings, two points are emphasized: 1) the fuzzy set is an appropriate way of measuring the fuzziness of human thought; and 2) it allows one to relax the conventional assumption that all individuals have identical distributions and deviations around their means.     

On Optimal Forecasting with Soft Computation for Nonlinear Time Series

In this paper we present a new approach on optimal forecasting by using the fuzzy set theory and soft computing methods for the dynamic data analysis. This research is based on the concepts of fuzzy membership function as well as the natural selection of evolution theory. Some discussions in the sensitivity of the design of fuzzy processing will be provided. Through the design of genetic evolution, the AIC criteria is used as the adjust function, and the fuzzy memberships function of each gene model are calculated. Simulation and empirical examples show that our proposed forecasting technique can give an optimal forecasting in time series analysis.     

H∞ estimation for fuzzy membership function optimization

Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a specific shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a few variables and the membership optimization problem can be reduced to a parameter optimization problem. The parameter optimization problem can then be formulated as a nonlinear filtering problem. In this paper we solve the nonlinear filtering problem using H_∞ state estimation theory. However, the membership functions that result from this approach are not (in general) sum normal. That is, the membership function values do not add up to one at each point in the domain. We therefore modify the H_∞ filter with the addition of state constraints so that the resulting membership functions are sum normal. Sum normality may be desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The methods proposed in this paper are illustrated on a fuzzy automotive cruise controller and compared to Kalman filtering based optimization.     

Fixed points for fuzzy mappings

In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset R_χ of X. An element x of X is called fixed point of R iff its membership degree to R_χ is at least equal to the membership degree to R_χ of any y?X, i.e. R_χ(χ)? R_χ(y)(?y?X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of R_χ has a ‘complementary function’. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open.     

Application of Fuzzy Linear Programming in Production Planning

In this paper, a new fuzzy linear programming based methodology using a specific membership function, named as modified logistic membership function is proposed. The modified logistic membership function is first formulated and its flexibility in taking up vagueness in parameters is established by an analytical approach. This membership function is tested for its useful performance through an illustrative example by employing fuzzy linear programming. The developed methodology of FLP has provided a confidence in applying to real life industrial production planning problem. This approach of solving industrial production planning problem can have feed back within the decision maker, the implementer and the analyst. In such case this approach can be called as IFLP (Interactive Fuzzy Linear Programming). There is a possibility to design the self organizing of fuzzy system for the mix products selection problem in order to find the satisfactory solution. The decision maker, the analyst and the implementer can incorporate their knowledge and experience to obtain the best outcome.     

Competitive algorithms for the clustering of noisy data

In this paper, we consider the issue of clustering when outliers exist. The outlier set is defined as the complement of the data set. Following this concept, a specially designed fuzzy membership weighted objective function is proposed and the corresponding optimal membership is derived. Unlike the membership of fuzzy c-means, the derived fuzzy membership does not reduce with the increase of the cluster number. With the suitable redefinition of the distance metric, we demonstrate that the objective function could be used to extract c spherical shells. A hard clustering algorithm alleviating the prototype under-utilization problem is also derived. Artificially generated data are used for comparisons.     

Fuzzy dynamical systems

A fuzzy dynamical system on an underlying complete, locally compact metric state space X is defined axiomatically in terms of a fuzzy attainability set mapping on X. This definition includes as special cases crisp single and multivalued dynamical systems on X. It is shown that the support of such a fuzzy dynamical system on X is a crisp multivalued dynamical system on X, and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of X. In addition fuzzy trajectories are defined, their existence established and various properties investigated.     

Renewal reward processes with fuzzy rewards and their applications to T-age replacement policies

The application of fuzzy set theory to renewal reward processes is proposed in this paper. The reward is modeled as a fuzzy random variable. A theorem which presents the long-run average fuzzy reward per unit time is stated. A procedure to obtain the best T-age replacement policy with fuzzy cost structure is developed. The original problem is transformed into a nonlinear program in order to evaluate the membership of the long-run average fuzzy cost per unit time.     

Evolving fuzzy rule based controllers using genetic algorithms

The synthesis of genetics-based machine learning and fuzzy logic is beginning to show promise as a potent tool in solving complex control problems in multi-variate non-linear systems. In this paper an overview of current research applying the genetic algorithm to fuzzy rule based control is presented. A novel approach to genetics-based machine learning of fuzzy controllers, called a Pittsburgh Fuzzy Classifier System # 1 (P-FCS1) is proposed. P-FCS1 is based on the Pittsburgh model of learning classifier systems and employs variable length rule-sets and simultaneously evolves fuzzy set membership functions and relations. A new crossover operator which respects the functional linkage between fuzzy rules with overlapping input fuzzy set membership functions is introduced. Experimental results using P-FCS 1 are reported and compared with other published results. Application of P-FCS1 to a distributed control problem (dynamic routing in computer networks) is also described and experimental results are presented.     

An extension to possibilistic fuzzy cluster analysis

We explore an approach to possibilistic fuzzy clustering that avoids a severe drawback of the conventional approach, namely that the objective function is truly minimized only if all cluster centers are identical. Our approach is based on the idea that this undesired property can be avoided if we introduce a mutual repulsion of the clusters, so that they are forced away from each other. We develop this approach for the possibilistic fuzzy c-means algorithm and the Gustafson–Kessel algorithm. In our experiments we found that in this way we can combine the partitioning property of the probabilistic fuzzy c-means algorithm with the advantages of a possibilistic approach w.r.t. the interpretation of the membership degrees.     

Minimization approach to membership evaluation in fuzzy sets and error analysis

Evaluation of the degree of membership in fuzzy sets is a fundamental topic in fuzzy set theory. Saaty (Ref. 1) proposes a method for solving this problem that has been widely accepted. In this paper, we examine the problem from an error minimization point of view that attempts to reflect the real intentions of the decision maker. When this approach is used, the findings reveal that fuzzy sets of different cardinalities have dramatically different requirements in the consistency level of the input data as far as the error minimization criterion is concerned.     

Optimal stopping in a fuzzy environment

The problem under consideration is that of optimally controlling and stopping either a deterministic or a stochastic system in a fuzzy environment. The optimal decision is the sequence of controls that maximizes the membership function of the intersection of the fuzzy constraints and a fuzzy goal. The fuzzy goal is a fuzzy set in the cartesian product of the state space with the set of possible stopping times. Dynamic programming is applied to yield a numerical solution. This approach yields an algorithm that corrects a result of Kacprzyk.     

On fuzzy linear projection equation and applications

This paper introduces the concept of fuzzy projection of a fuzzy number on a set of fuzzy numbers based on r-cut approach. It is proved that the projection of a fuzzy number on the set of all fuzzy numbers is itself and under a special metric, the proposed fuzzy projection is a non-expansive mapping. By using this definition, the concept of fuzzy linear projection equation is defined and to solve it, a numerical method is applied. Based on the proposed algorithm and as an important application, two different types of system of fuzzy linear equations with fuzzy variables are solved. Numerical results illustrate the applicabilities of proposed approach.     

Fuzzy disk for covering fuzzy points

In this paper, we consider an important fuzzy version of the well known smallest covering circle problem which is also called the Euclidean 1-center problem. Here we are given a set of points in the plane whose locations are crisp. However, the requirement for coverage of each point is fuzzy as represented by its grade of membership. As such we qualify the points as fuzzy. In the above framework, we wish to find a fuzzy disk with minimum fuzzy area for covering the given fuzzy points. After developing a general model, certain special cases are investigated in detail. Polynomial algorithms are presented for the special cases.