On fuzzy goals and maximizing decisions in stochastic optimal control

Fuzzy set theory has developed significantly in a mathematical direction during the past several years but few applications have emerged. This paper investigates the role of fuzzy set theory in certain optimal control formulations. In particular, it is shown that the well-known quadratic performance criterion in deterministic optimal control is equivalent to the exponential membership function of a certain fuzzy decision (set). In a stochastic setting, similar equivalences establish new definitions for “confluence of goals” and “maximizing decision” in fuzzy set theory. These and other definitions could lead to the development of a more applicable theory of fuzzy sets.     

AHP多路径排序度量转换中的非线性问题

由于不同测量条件下的测量结果不是线性可加,AHP用矩阵乘法实现多路径序转换值得商榷.自隶属度从只取"1或0"两个值扩展到可取[0,1]区间上一切实数,可表征界于"是"与"不是"之间所有可能"部分是"模糊状态时起,对二值逻辑的研究已拓展到研究近似推理的模糊逻辑.这是逻辑的一个新的研究方向,目的是在隶属度转换过程中,通过对人类近似推理本领进行规范,使得到的目标值是"真值"在当前条件下的最优近似.模糊逻辑的量化方法是数值计算;推理依据是区分权滤波的冗余理论;实质性计算是由冗余理论导出的、实现隶属度转换的非线性去冗算法;所建的隶属度转换模型也是不同测量条件下高维状态空间上测量结果的非线性可加模型.将一维测量数据映射到高维状态空间上表为隶属度向量,可借助隶属度转换模型解决AHP多路径序转换的非线性计算.     

Logic of agreement: Foundations,semantic system and proof theory

In this paper a multi-valued propositional logic — logic of agreement — in terms of its model theory and inference system is presented. This formal system is the natural consequence of a new way to approach concepts as commonsense knowledge, uncertainty and approximate reasoning — the point of view of agreement. Particularly, it is discussed a possible extension of the Classical Theory of Sets based on the idea that, instead of trying to conceptualize sets as “fuzzy” or “vague” entities, it is more adequate to define membership as the result of a partial agreement among a group of individual agents. Furthermore, it is shown that the concept of agreement provides a framework for the development of a formal and sound explanation for concepts (e.g. fuzzy sets) which lack formal semantics. According to the definition of agreement, an individual agent agrees or not with the fact that an object possesses a certain property. A clear distinction is then established, between an individual agent — to whom deciding whether an element belongs to a set is just a yes or no matter — and a commonsensical agent — the one who interprets the knowledge shared by a certain group of people. Finally, the logic of agreement is presented and discussed. As it is assumed the existence of several individual agents, the semantic system is based on the perspective that each individual agent defines her/his own conceptualization of reality. So the semantics of the logic of agreement can be seen as being similar to a semantics of possible worlds, one for each individual agent. The proof theory is an extension of a natural deduction system, using supported formulas and incorporating only inference rules. Moreover, the soundness and completeness of the logic of agreement are also presented.     

POWERSET OPERATOR BASED FOUNDATION FORPOINT-SET LATTICE-THEORETIC (POSLAT) FUZZY SET THEORIES and TOPOLOGIES

Abstract

This paper sets forth in detail point-set lattice-theoretic or poslat foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and pre-image of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure and probability theory, and topology. In particular, those aspects of fuzzy sets, hinging around (crisp) powersets of fuzzy subsets and around powerset operators between such powersets lifted from ordinary functions between the underlying base sets, are examined and characterized using point-set and lattice-theoretic methods. The basic goal is to uniquely derive the powerset operators and not simply stipulate them, and in doing this we explicitly distinguish between the “fixed-basis” case (where the underlying lattice of membership values is fixed for the sets in question) and the “variable-basis” case (where the underlying lattice of membership values is allowed to change). Applications to fuzzy sets/logic include: development and justification/characterization of the Zadeh Extension Principle [36], with applications for fuzzy topology and measure theory; characterizations of ground category isomorphisms; rigorous foundation for fuzzy topology in the poslat sense; and characterization of those fuzzy associative memories in the sense of Kosko [18] which are powerset operators. Some results appeared without proof in [31], some with partial proofs in [32], and some in the fixed-basis case in Johnstone [13] and Manes [22].     

Basic concepts for a theory of evaluation: The aggregative operator

Starting with an explication of the “aggregative”-concept and deducing a general structure which satisfies a number of minimal requirements (properties of clustering) the main features of a new mathematical theory — called “theory of evaluation” — are developed. The theory sheds new light on such well-known concepts as membership, conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set, and its many applications, of human decision making and of multicriteria analysis.     

Convergence of powers of a fuzzy matrix

A Boolean matrix is a matrix with elements having values of either 1 or 0; a fuzzy matrix is a matrix with elements having values in the closed interval [0, 1]. Fuzzy matrices occur in the modeling of various fuzzy systems, with products usually determined by the “max(min)” rule arising from fuzzy set theory. In this paper, some sufficient conditions for convergence under “max(min)” products of the powers of a square fuzzy matrix and of a fuzzy state process are established.     

Fuzzy events and fuzzy logics in classical information systems

Classical information systems are introduced in the framework of measure and integration theory. The measurable characteristic functions are identified with the exact events while the fuzzy events are the real measurable functions whose range is contained in the unit interval. Two orthogonality relations are introduced on fuzzy events, the first linked to the fuzzy logic and the second to the fuzzy structure of partial a Baer¹-ring. The fuzzy logic is then compared with the “empirical” fuzzy logic induced by the classical information system. In this context, quantum logics could be considered as those empirical fuzzy logics in which it is not possible to have preparation procedures which provide physical systems whose “microstate” is always exactly defined.     

Performance evaluation of fault-tolerant routing algorithms: an optimization problem

In this paper, we present a new theoretical approach for studying the behaviour and the performance of shortest paths fault-tolerant distributed algorithms of a certain class. The behaviour of each processor is modeled by means of a stochastic matrix. We show that achieving the optimal behaviour of Nprocessors is equivalent to solvingan optimization problem of a function of 2N variables under constraints; this function is neither convex nor concave. Solutions for which such a type of algorithms has an optimal behaviour are derived. Using that result, we build a fuzzy set of solutions which provides a global overview (a sort of “relief”): each solution of the fuzzy set has value ? ranging between 0 and 1, which may be regarded as its“bench-mark” so (1 -?) points out the proximity of any solution from the optimal solution     

On the suitability of minimum and product operators for the intersection of fuzzy sets

In many papers concerning fuzzy set theory it is assumed that the membership or an element in the intersection of two or more fuzzy sets is given by the minimum of product of the corresponding membership values. To use these operators in modelling aspects of the real world, such as decision making, however, it is necessary to prove their appropriateness empirically. The main question of this study is whether people rating the membership of objects in the intersection of two fuzzy sets behave in accordance with one of these models. An important problem in answering this question is how to measure membership which seems to have the characteristics of an absolute scale. No measurement structure is available at present, but a practical method for scaling is suggested. The results of our experiments indicate that neither the product nor the minimum fit the data sufficiently well, but the latter seems to be preferable.     

On membership and marginal values

In Kleinberg and Weiss, Math Soc Sci 12:21–30 (1986b), the authors used the representation theory of the symmetric groups to characterize the space of linear and symmetric values. We call such values “membership” values, as a player’s payoff depends on the worths of the coalitions to which he belongs and not necessarily on his marginal contributions. This could mean that the player would get some share of $v(N)$ regardless of whether or not he makes a marginal contribution to the welfare of society. In this paper it is demonstrated that the set of (non-marginal) membership values include those that embody numerous widely held notions of fairness, such as partial “benefit equalization”, individual rationality and “greater rewards follow from greater contributions”, where one’s contributions are not measured marginally. We also present a very simple and revealing way of interpreting all values, including those having a marginal interpretation. Finally, we obtain a mapping which effectively embeds the space of marginal values in the space of all membership values.     

A new methodology for fuzzy multi-attribute group decision making with multi-granularity and non-homogeneous information

The aim of this paper is to develop a new methodology for solving fuzzy multi-attribute group decision making problems with non-homogeneous information, including multi-granular linguistic term sets, fuzzy numbers, interval values and real numbers. In this methodology, different distances are defined to measure differences between alternatives and the ideal solution as well as the negative ideal solution. A relative closeness method is developed by introducing the multi-attribute ranking index based on the particular measure of closeness to the IS. The proposed method determines a compromise solution for the group, providing a maximum “group utility” for the “majority” and a minimum of an individual regret for the “opponent”. The implementation process, effectiveness and feasibility of the method proposed in this paper are illustrated with a real example of the missile weapon system design project selection.     

Fuzzy variables

The purpose of this study is to explore a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. The approach we propose is analogous to the sample space concept of probability theory. A fuzzy variable is a mapping from an abstract space (called the pattern space) onto the real line. The membership function is obtained as the extension of a special type of capacity (called a scale) from the pattern space to the real line via the fuzzy variable. In essence we are proposing an entirely new definition of a fuzzy set on the line as a mapping to the line rather than on the line. The current definition of a transformation of a fuzzy set is obtained as a derived result of our model. In addition, we derive the membership function of sums and products of fuzzy sets and present an example which reinforces the credibility of our approach.     

Constructing membership functions using statistical data

Determination of the membership functions is vital in practical applications of the fuzzy set theory. This paper presents a guideline to construct the membership functions for fuzzy sets whose elements have a defining feature with a known probability density function (pdf) in the universe of discourse. The method finds the smallest fuzzy set which assigns high average membership values to those objects with the defining features distributed according to the given pdf. It is show that, for any pdf, the method is capable of generating membership functions in accordance with the possibility-probability consistency principle. Membership functions derived from some of the well known pdfs and an application in solving noise contaminated linear system of equations are presented.     

Fuzzy sets as a basis for a theory of possibility

The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. More specifically, if F is a fuzzy subset of a universe of discourse U={u} which is characterized by its membership function μ_F, then a proposition of the form “X is F,” where X is a variable taking values in U, induces a possibility distribution ∏X which equates the possibility of X taking the value u to μF(u)—the compatibility of u with F. In this way, X becomes a fuzzy variable which is associated with the possibility distribution ∏_x in much the same way as a random variable is associated with a probability distribution. In general, a variable may be associated both with a possibility distribution and a probability distribution, with the weak connection between the two expressed as the possibility/probability consistency principle.A thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is, in the main, possibilistic rather than probabilistic in nature. Thus, by employing the concept of a possibility distribution, a proposition, p, in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p. Several types of conditional translation rules are discussed and, in particular, a translation rule for propositions of the form “X is F is α-possible,” where α is a number in the interval [0, 1], is formulated and illustrated by examples.     

毕达哥拉斯模糊软集的相似性测度及其应用

直觉模糊软集不能处理隶属度与非隶属度之和大于1的情况,且现有的直觉模糊软集的相似性测度只考虑了隶属度与非隶属度,忽视了犹豫度。针对以上问题,本文提出了一种基于隶属度、非隶属度以及犹豫度三个参数的毕达哥拉斯模糊软集的相似性测度和加权相似性测度。在为加权相似性测度的权重取值时,本文基于现有文献中直觉模糊熵存在的缺陷建立一种改进的直觉模糊熵,利用熵权法计算权重。分别讨论两相似性测度公式的性质,最后将两相似性侧度公式应用在建筑材料的模式识别问题中。     

Fuzzy variant of a statistical test point Kalman filter

In this paper, we propose the conceptual use of fuzzy clustering techniques as iterative spatial methods to estimate a posteriori statistics in place of the weighted averaging scheme of the Unscented Kalman filter. Specifically, instead of a linearization methodology involving the statistical linear regression of the process and measurement functions through some deterministically chosen set of test points (sigma points) contained within the “uncertainty region” around the state estimate, we present a variant of the Unscented transformation involving fuzzy clustering techniques which will be applied to the test points yielding “degrees of membership” in which Gaussian shapes can be “fit” using a least squares scheme. Implementation into the Kalman methodology will be shown along with simple state and parameter estimation examples.     

Membership functions and operational law of uncertain sets

Uncertain set is a set-valued function on an uncertainty space, and attempts to model “unsharp concepts” that are essentially sets but their boundaries are not sharply described. This paper will propose a concept of membership function and define the independence of uncertain sets. This paper will also present an operational law of uncertain sets via membership functions or inverse membership functions. Finally, the linearity of expected value operator is verified.     

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.