基于加权Karnik-Mendel算法的区间二型模糊集质心计算

区间二型模糊集的质心计算(也称降型)在二型模糊逻辑系统中起着很重要的作用。Karnik-Mendel(KM)算法是完成降型的标准算法。本文介绍了区间二型模糊集相关理论,比较了离散KM算法与连续KM(continuous KM,CKM)算法中的运算,通过数值分析技术中牛顿-柯特斯求积公式将KM算法扩展成三种不同形式的加权KM(weighted KM,WKM)算法,而KM算法只是WKM算法的一种特例。计算机仿真例子用来阐述和分析WKM算法的表现,其在计算两种不对称区间二型模糊集质心时可取得比KM算法更小的绝对误差和更快的计算速度,这给二型模糊集及其模糊逻辑系统设计和应用提供了潜在的价值。     

采样离散EKM算法与连续EKM算法的关系研究

计算区间二型模糊集的质心(也称降型)是二型模糊逻辑系统的核心模块。EKM(enhanced Karnik-Mendel)算法是当前最流行的计算质心的算法。本文首先给出了区间二型模糊集及其质心的相关理论,然后比较和分析了离散版本的EKM算法中求和运算与连续版本的EKM(continuous version of EKM,CEKM)算法中的求积分运算内在联系,最后通过计算机仿真例子证实了当适度改变区间二型模糊集主变量采样个数时,离散EKM算法计算的解模糊化值结果就可以精确地逼近基准的CEKM算法。     

基于改进反向搜索算法的区间二型模糊逻辑系统

二型模糊逻辑系统是当前为学术界热点研究问题。本文介绍了区间二型模糊集相关理论,结合求解区间二型模糊集质心的改进反向搜索(EODS)算法,讨论了区间二型模糊逻辑系统的模糊推理,质心降型和解模糊化等模块。用两个计算机仿真例子来阐述和分析EODS算法的表现,与最常用的Karnik-Mendel(KM)算法相比,EODS算法在计算系统输出值时在不损失计算精度的条件下具有更快的计算速度,给二型模糊逻辑系统设计者和应用者提供了潜在的价值。     

基于改进反向搜索算法的广义二型模糊集质心计算

随着广义二型模糊集的α-平面表达理论被提出,广义二型模糊集及其模糊逻辑系统在近年来成为学术界热点研究问题。计算广义二型模糊集的质心(也称降型)在广义二型模糊逻辑系统中起着到关重要的作用。本文介绍了广义二型模糊集相关理论,扩展了求区间二型模糊集质心的改进反向搜索(enhanced opposite direction searching,EODS)算法计算完成广义二型模糊集质心。在计算两种具有非对称足迹不确定性的广义二型模糊集质心降型集和解模糊糊化值时,EODS算法可在不损失计算精度的前提下取得比最常用的Karnik-Mendel算法更快的计算速度,这给二型模糊集设计及应用提供了潜在的价值。     

基于二分搜索改进Karnik-Mendel算法的区间二型模糊逻辑系统

作为一种新兴技术,区间二型模糊逻辑系统受到当前学术界广泛关注。本文基于求解区间二型模糊集质心的二分搜索改进Karnik-Mendel(Binary-Search enhanced Karnik-Mendel,BEKM)算法,讨论了区间二型模糊逻辑系统的模糊推理,质心降型和解模糊化等模块。计算机仿真实验阐述和分析了BEKM算法在计算系统解模糊化输出时的表现,与EKM算法相比,BEKM算法有更高的计算效率,给二型模糊逻辑系统设计及应用提供了潜在的价值。     

区间二型模糊集合的表述与应用

通过引入多值映射,本文给出了二型模糊集合的新定义以便其能更好地容易理解,并在修正的不确定覆盖域(footprint of uncertainty,FOU)定义与公式的基础上,提出了FOU划分法来表示连续区间二型模糊集合,最后将该表示方法应用于区间二型模糊集合的词计算及并、交、补运算之中。     

区间二型单点Mamdani模糊逻辑系统的设计

介绍区间二型模糊集的定义及相关理论,提出一种多重多维区间二型单点Mamdani模糊逻辑系统。在KM算法的理论基础上,设计模糊推理,类型简化,解模糊化等模块。在BP算法的理论基础上,调整区间二型单点Mamdani模糊逻辑系统中前件的参数。最后给出了本文的总结和进一步研究问题的展望。     

基于α截集的改进二型模糊集合降阶算法研究

针对普通二型模糊集合降阶计算量很大的问题,提出了一种基于α截集的改进降阶算法。利用α截集表示二型普通模糊集合,将普通二型模糊集合的降阶过程简化为α-区间二型模糊集合的降阶过程。对快速二型模糊集合降阶算法进行改进,利用插半法求取左、右切换点。2种不同形式的首隶属度函数和次隶属度函数的仿真实验表明,本文算法能够有效减少求取切换点的比较次数,提高运算效率,具有较强的实用性和适应性。     

基于KM算法的区间二型单点TSK模糊逻辑系统

介绍了区间二型模糊集的定义及相关理论,提出一种多输入单输出区间二型单点TSK模糊逻辑系统。在KM算法的理论基础上,讨论了区间二型单点TSK模糊逻辑系统的模糊推理,类型简化,解模糊化等模块。用两个算例验证了设计该模糊逻辑系统的可行性。最后给出了本文的总结和进一步研究问题的展望。     

有界格上二型模糊集否运算的一些性质

本文讨论有界格上二型模糊集的否运算。首先用Zadeh的扩展原理定义了有界格上二型模糊集的否运算,然后讨论了有界格上二型模糊集否运算的一些性质,最后证明了链上二型模糊集的真值代数是De Morgan Birkhoff系统及有界分配格上二型模糊集真值代数的子代数中否运算是强否运算。     

A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets

Soft set theory, originally proposed by Molodtsov, has become an effective mathematical tool to deal with uncertainty. A type-2 fuzzy set, which is characterized by a fuzzy membership function, can provide us with more degrees of freedom to represent the uncertainty and the vagueness of the real world. Interval type-2 fuzzy sets are the most widely used type-2 fuzzy sets. In this paper, we first introduce the concept of trapezoidal interval type-2 fuzzy numbers and present some arithmetic operations between them. As a special case of interval type-2 fuzzy sets, trapezoidal interval type-2 fuzzy numbers can express linguistic assessments by transforming them into numerical variables objectively. Then, by combining trapezoidal interval type-2 fuzzy sets with soft sets, we propose the notion of trapezoidal interval type-2 fuzzy soft sets. Furthermore, some operations on trapezoidal interval type-2 fuzzy soft sets are defined and their properties are investigated. Finally, by using trapezoidal interval type-2 fuzzy soft sets, we propose a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method.     

A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets”

Zhang and Zhang (2013) proposed the arithmetic operations of trapezoidal interval type-2 fuzzy numbers having different left and right heights and hence the arithmetic operations of trapezoidal interval type-2 fuzzy soft sets having different left and right heights. In this paper, it is pointed out that the complement operation of a trapezoidal interval type-2 fuzzy number, proposed by Zhang and Zhang, is not valid and hence, the complement operation of trapezoidal interval type-2 fuzzy soft set as well as all the results, proposed by Zhang and Zhang in which complement operation is used, are not valid. The results, proposed by Zhang and Zhang, are valid only for such trapezoidal interval type-2 fuzzy numbers and trapezoidal interval type-2 fuzzy soft sets in which left and right heights are equal.     

A closed form type reduction method for piecewise linear interval type-2 fuzzy sets

In this study, a new centroid type reduction method is proposed for piecewise linear interval type-2 fuzzy sets based on geometrical approach. The main idea behind the proposed method relies on the assumption that the part of footprint of uncertainty (FOU) of an interval type-2 fuzzy set (IT2FS) has a constant width where the centroid is searched. This constant width assumption provides a way to calculate the centroid of an IT2FS in closed form by using derivative based optimization without any need of iterations. When the related part of FOU is originally constant width, the proposed method finds the accurate centroid of an IT2FS; otherwise, an enhancement can be performed in the algorithm in order to minimize the error between the accurate and the calculated centroids. Moreover, only analytical formulas are used in the proposed method utilizing geometry. This eliminates the need of using discretization of an IT2FS for the type reduction process which in return naturally improves the accuracy and the computation time. The proposed method is compared with Enhanced Karnik–Mendel Iterative Procedure (EKMIP) in terms of the accuracy and the computation time on seven test fuzzy sets. The results show that the proposed method provides more accurate results with shorter computation time than EKMIP.     

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.     

基于模糊测度与累积前景理论的区间二型模糊多准则决策方法

针对准则值为区间二型模糊数且准则间存在关联关系的风险型多准则决策问题, 本文提出一种基于模糊测度理论与累积前景理论的区间二型模糊多准则决策方法。首先, 为全面反映准则间的关联关系, 本文提出Shapley区间二型模糊Choquet积分算子, 并证明该算子的一些性质。其次, 为反映专家行为偏好, 本文定义区间二型模糊前景效应与前景价值函数, 并提出累积前景Shapley区间二型模糊Choquet积分算子。然后, 为确定准则集的模糊测度, 本文建立基于区间二型模糊双向投影与Shapley函数的权重优化模型。在此基础上, 本文给出一种用于解决准则值为区间二型模糊数, 准则间存在关联关系, 专家存在风险偏好以及准则权重部分未知的多准则决策方法。最后, 通过风险投资实例佐证所提出的方法的适用性与科学性。     

Similarity,inclusion and entropy measures between type-2 fuzzy sets based on the Sugeno integral

Similarity measures of type-2 fuzzy sets are used to indicate the similarity degree between type-2 fuzzy sets. Inclusion measures for type-2 fuzzy sets are the degrees to which a type-2 fuzzy set is a subset of another type-2 fuzzy set. The entropy of type-2 fuzzy sets is the measure of fuzziness between type-2 fuzzy sets. Although several similarity, inclusion and entropy measures for type-2 fuzzy sets have been proposed in the literatures, no one has considered the use of the Sugeno integral to define those for type-2 fuzzy sets. In this paper, new similarity, inclusion and entropy measure formulas between type-2 fuzzy sets based on the Sugeno integral are proposed. Several examples are used to present the calculation and to compare these proposed measures with several existing methods for type-2 fuzzy sets. Numerical results show that the proposed measures are more reasonable than existing measures. On the other hand, measuring the similarity between type-2 fuzzy sets is important in clustering for type-2 fuzzy data. We finally use the proposed similarity measure with a robust clustering method for clustering the patterns of type-2 fuzzy sets.     

Fuzzy sets and type 2 under algebraic product and algebraic sum

The concept of fuzzy sets of type 2 has been proposed by L.A. Zadeh as an extension of ordinary fuzzy sets. A fuzzy set of type 2 can be defined by a fuzzy membership function, the grade (or fuzzy grade) of which is taken to be a fuzzy set in the unit interval [0, 1] rather than a point in [0, 1].This paper investigates the algebraic properties of fuzzy grades (that is, fuzzy sets of type 2) under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring. Moreover, the properties of fuzzy grades are also discussed in the case where algebraic product and algebraic sum are combined with the well-known operations of join and meet for fuzzy grades and it is shown that normal convex fuzzy grades form a lattice ordered semigroup under join, meet and algebraic product.     

Indirect adaptive fuzzy observer and controller design based on interval type-2 T-S fuzzy model

This paper presents the design scheme of the indirect adaptive fuzzy observer and controller based on the interval type-2 (IT2) T-S fuzzy model. The nonlinear systems can be well approximated by IT2 T-S fuzzy model, in which the fuzzy rules’ antecedents are interval type-2 fuzzy sets and consequents are linear state equations. The proposed IT2 T-S fuzzy model is a combination of IT2 fuzzy system and T-S fuzzy model, and also inherits the benefits of type-2 fuzzy logic systems, which is able to directly handle uncertainties and can minimize the effects of uncertainties in rule-based fuzzy system. These characteristics can improve the accuracy of the system modeling and reduce the number of system rules. The proposed method using feedback control, adaptive laws, and on-line object parameters are adjusted to ensure observation error bounded. In addition, using Lyapunov synthesis approach and Lipschitz condition, the stability analysis is conducted. The simulation results show that the proposed method can handle unpredicted disturbance and data uncertainties very well in advantage of the effectiveness of observation and control.     

The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making

QUALIFLEX, a generalization of Jacquet-Lagreze’s permutation method, is a useful outranking method in decision analysis because of its flexibility with respect to cardinal and ordinal information. This paper develops an extended QUALIFLEX method for handling multiple criteria decision-making problems in the context of interval type-2 fuzzy sets. Interval type-2 fuzzy sets contain membership values that are crisp intervals, which are the most widely used of the higher order fuzzy sets because of their relative simplicity. Using the linguistic rating system converted into interval type-2 trapezoidal fuzzy numbers, the extended QUALIFLEX method investigates all possible permutations of the alternatives with respect to the level of concordance of the complete preference order. Based on a signed distance-based approach, this paper proposes the concordance/discordance index, the weighted concordance/discordance index, and the comprehensive concordance/discordance index as evaluative criteria of the chosen hypothesis for ranking the alternatives. The feasibility and applicability of the proposed methods are illustrated by a medical decision-making problem concerning acute inflammatory demyelinating disease, and a comparative analysis with another outranking approach is conducted to validate the effectiveness of the proposed methodology.