Fuzzy qualitative trigonometry

This paper presents a fuzzy qualitative representation of conventional trigonometry with the goal of bridging the gap between symbolic cognitive functions and numerical sensing & control tasks in the domain of physical systems, especially in intelligent robotics. Fuzzy qualitative coordinates are defined by replacing a unit circle with a fuzzy qualitative circle; a Cartesian translation and orientation are defined by their normalized fuzzy partitions. Conventional trigonometric functions, rules and the extensions to triangles in Euclidean space are converted into their counterparts in fuzzy qualitative coordinates using fuzzy logic and qualitative reasoning techniques. This approach provides a promising representation transformation interface to analyze general trigonometry-related physical systems from an artificial intelligence perspective.Fuzzy qualitative trigonometry has been implemented as a MATLAB toolbox named XTRIG in terms of 4-tuple fuzzy numbers. Examples are given throughout the paper to demonstrate the characteristics of fuzzy qualitative trigonometry. One of the examples focuses on robot kinematics and also explains how contributions could be made by fuzzy qualitative trigonometry to the intelligent connection of low-level sensing & control tasks to high-level cognitive tasks.     

Fuzzy processes

In this paper, contributions to fuzzy probability and to differential equations with fuzzy parameters are made.After an introductory section, a review of fuzzy sets and fuzzy algebra is given in Section 2. The main new results of the investigation are contained in Section 3.In Section 3, Zadeh's definition of the probability of a ‘fuzzy event’ the average value of a fuzzy function are extended into the time domain. It is then shown that not only grades of membership, but also probabilistic processes with notions of fuzziness contained, can be defined which obey ordinary, matric, or integro-differential equations. Applications are also given in Section 3.     

Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach

In this paper, two new algorithms are presented to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. The membership functions for the defined fuzzy goals of all objective functions at all levels are developed in the model formulation of the problem; so also are the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels. Then the fuzzy goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtain the most satisfactory solution for all decision makers.     

Handling the valuation of the predicates in a fuzzy model

Given a fuzzy model, and therefore a graded valuation of the predicates, several handlings of such a valuation are possible. For example, we can change the set of truth values, modify the scale, cut the predicates, and identify truth values. In this exploratory paper we analyze some of the properties preserved under these handlings. To do this we refer to an approach to fuzzy semantics in which the valuation structures can vary freely inside a given type.     

基于模式匹配的模糊控制算法

本文针对模糊控制器输入输出是非模糊量的特点,在隶属函数统一定义的情况下,提出一类新的基于模式匹配的简化推理决策算法。     

A new risk criterion in fuzzy environment and its application

Since the observed values of security returns in real-world problems are sometimes imprecise or vague, an increasing effort in research is devoted to study the properties of risk measures in fuzzy portfolio optimization problems. In this paper, a new risk measure is suggested to gauge the risk resulted from fuzzy uncertainty. For this purpose, the absolute deviation and absolute semi-deviation are first defined for fuzzy variable by nonlinear fuzzy integrals. To compute effectively the absolute semi-deviations of single fuzzy variable as well as its functions, this paper discusses the methods of computing the absolute semi-deviation by classical Lebesgue–Stieltjes (L–S) integral. After that, several useful absolute deviation and absolute semi-deviation formulas are established for common triangular, trapezoidal and normal fuzzy variables. Applying the absolute semi-deviation as a new risk measure in portfolio optimization, three classes of fuzzy portfolio optimization models are developed by combining the absolute semi-deviation with expected value operator and credibility measure. Based on the analytical representation of absolute semi-deviations, the established fuzzy portfolio selection models can be turned into their equivalent piecewise linear or fractional programming problems. Since the absolute semi-deviation is a piecewise fractional function and pseudo-convex on the feasible subregions of deterministic programming models, we take advantage of the structural characteristics to design a domain decomposition method to separate a deterministic programming problem into three convex subproblems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the effectiveness of the solution method.     

Interval optimization based line sampling method for fuzzy and random reliability analysis

For structural system with fuzzy variables as well as random variables, a novel algorithm for obtaining membership function of fuzzy reliability is presented on interval optimization based Line Sampling (LS) method. In the presented algorithm, the value domain of the fuzzy variables under the given membership level is firstly obtained according to their membership functions. Then, in the value domain of the fuzzy variables, bounds of reliability of the structure are obtained by the nesting analysis of the interval optimization, which is performed by modern heuristic methods, and reliability analysis, which is achieved by the LS method in the reduced space of the random variables. In this way the uncertainties of the input variables are propagated to the safety measurement of the structure, and the membership function of the fuzzy reliability is obtained. The presented algorithm not only inherits the advantage of the direct Monte Carlo method in propagating and distinguishing the fuzzy and random uncertainties, but also can improve the computational efficiency tremendously in case of acceptable precision. Several examples are used to illustrate the advantages of the presented algorithm.     

Bi-matrix Games with Fuzzy Goals and Fuzzy

A bi-matrix game with fuzzy goal is shown to be equivalent to a (crisp) non-linear programming problem in which the objective as well as all constraint functions are linear except two constraint functions, which are quadratic. This equivalence is further extended to bi-matrix games with fuzzy pay-offs, as well as to bi-matrix games with fuzzy goals and fuzzy payoffs, whose equilibrium strategies are conceptualized by employing a suitable ranking (defuzzification) function.     

Weak Form of Fuzzy Open Sets and Their Related Continuity

The aim of this paper is to give a new class of fuzzy open sets called fuzzy pre-γ-open sets in fuzzy topological spaces. Also,a new type of fuzzy continuous functions called,fuzzy pre-γ-continuous functions is introduced and studied.     

直觉模糊关系的截集性质

定义了直觉模糊关系的截集以及并给出了它们的基本性质,同时分别讨论了自反、对称、传递直觉模糊关系与经典二元关系之间的等价刻画;其次,提出直觉模糊关系的定义域与值域的概念;最后,分析了直觉模糊关系合成的截集性质.     

求模糊关系传递闭包的一种算法

根据模糊关系的传递性的特征,文章提出了利用相应的模糊矩阵求有限论域上模糊关系的传递闭包的一种计算方法,该算法可以加快获得传递闭包的速度。通过实例说明了该算法是简便、实用的。     

The case for an interval-based representation of linguistic truth

This paper develops an interval-based approach to the concept of linguistic truth. A special-purpose interval logic is defined, and it is argued that, for many applications, this logic provides a potentially useful alternative to the conventional fuzzy logic.The key idea is to interpret the numerical truth value v(p) of a proposition p as a degree of belief in the logical certainty of p, in which case p is regarded as true, for example, if v(p) falls within a certain range, say, the interval [0.7, 1]. This leads to a logic which, although being only a special case of fuzzy logic, appears to be no less linguistically correct and at the same time offers definite advantages in terms of mathematical simplicity and computational speed.It is also shown that this same interval logic can be generalized to a lattice-based logic having the capacity to accommodate propositions p which employ fuzzy predicates of type 2.     

[-1,1]上同序单调函数的同序变换群与模糊数运算

定义对称区间[-1，1]上的同序单调有界函数的同序变换，利用文[1]提出的模糊数的结构元表示方法，得到模糊数四则运算的结构元表示以及模糊数运算结果的隶属函数的确定方法。在多数的模糊数运算问题中，结构元的单调变换形式是容易得到的，此时，模糊数的运算将变得非常简单。文中还给出了一个运算的实例。     

基于粗糙集的模糊决策算法

给出一种从连续决策表中提取模糊决策规则的规则提取算法。首先,转化连续属性值为模糊值;然后,给出两个不同对象的模糊属性值关于相应连续属性的相似度;其次,给出了λ相似关系与λ相似类的定义。根据λ相似关系,给出粗糙-模糊空间中的下近似与上近似概念;最后,结合模糊集与粗糙集理论的思想,给出一种从连续值域决策表获取决策规则的算法,并通过实例说明该算法的有效性。     

模糊向量真值修正模糊化方法

模糊化是将物理空间的实测精确量影射为模糊推理空间上的模糊集合。在模糊控制中，不同的模糊推理方法要求不同的模糊化方法，不同的模糊化方法对模糊控制性能影响很大。本文首先系统地总结了现有的模糊化方法，然后提出了模糊向量真值修正模糊化方法，最后，针对常用的CRI法，完成了不同模糊化方法的一阶惯性时滞定常系统的模糊控制仿真，结果表明，该方法能够提高CRI法的模糊控制性能，消除稳态误差。     

模糊数的一种新运算方法及其若干应用

首先给出了模糊数的一种新的函数表示定理;基于该表示定理 ,提出了模糊数的一种新运算方法并将其直接应用到模糊数理论中若干问题的讨论,包括:模糊数的差问题,绝对值问题以及模糊数的确界问题 .     

Fuzzy Ideals and Fuzzy Distributive Lattices

Our main objective is to study properties of a fuzzy ideals(fuzzy dual ideals).A study of special types of fuzzy ideals(fuzzy dual ideals) is also furnished.Some properties of a fuzzy ideals(fuzzy dual ideals) are furnished.Properties of a fuzzy lattice homomorphism are discussed.Fuzzy ideal lattice of a fuzzy lattice is defined and discussed.Some results in fuzzy distributive lattice are proved.     

基于结构元的模糊值函数的一般表示方法

文[1]提出了模糊结构元的概念，并给出了模糊数与模糊值函数的模糊结构元表示，以及一类由模糊结构元线性生成的模糊值函数的微分和积分(黎曼意义下的)的表达形式。本文在文[1]的基础上进一步给出了由模糊结构元生成的模糊值函数的一般表达形式，并得到了一般表达形式下的模糊值函数的连续性和微分、黎曼积分的定义，它们与传统模糊分析中相应定义是等价的。     

幂模糊数方程和二次模糊方程的结构元求解方法

定义了幂模糊数和幂模糊数方程,基于结构元方法研究了幂模糊数运算和幂模糊数方程的求解,给出了隶属函数的表达式.同时,利用区间[-1,1]上的单调函数将二次模糊方程的求解问题转化为经典参数方程组的求解问题,给出了二次模糊方程解存在的充要条件,并辅以数值例子.