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.     

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     

Min–max formulæ for the speeds of pulsating travelling fronts in periodic excitable media

This paper is concerned with some nonlinear propagation phenomena for reaction–advection–diffusion equations in a periodic framework. It deals with travelling wave solutions of the equation $u_t =\nabla\cdot(A(z)\nabla u)\;+q(z)\cdot\nabla u+\,f(z,u),\qquad t\in\mathbb{R},\quad z\in\Omega,$ propagating with a speed c. In the case of a “combustion” nonlinearity, the speed c exists and it is unique, while the front u is unique up to a translation in t. We give a min–max and a max–min formula for this speed c. On the other hand, in the case of a “ZFK” or a “KPP” nonlinearity, there exists a minimal speed of propagation c*. In this situation, we give a min–max formula for c*. Finally, we apply this min–max formula to prove a variational formula involving eigenvalue problems for the minimal speed c* in the “KPP” case.     

Values and cores of fuzzy games with infinitely many players

The problem of the existence ofvalues (FA-valued, linear, positive, symmetric and efficient operators) on symmetric spaces of “fuzzy games” (that is, ideal set functions of bounded variation) arises naturally from [8], [18], [23] and [2], [3], [4] where it is implicitely approached for technical purposes. In our present work, this problem is approached in itself for the main reason that it is essentially related with the problem of the existence of significant countable additive measures lying in the cores of the “market games”. In fact, it is shown here that there exists a continuous value on the closed subspacebv′ICA ofIBV spanned by thebv′ functions of “fuzzy probability measures” ([9]), this values is “diagonal” onpICA, the closed subspace ofbv′ICA spanned by the natural powers of the fuzzy measures and this is used to prove the main result stating that the cooperative markets contained inpICA have unique fuzzy measures in their cores which are exactly the corresponding diagonal values. This result is of interest because it is providing a tool of determiningCA measures lying in the cores of large classes of games which are not necessarily “non-atomic” and, specially, because it is opening a way toward a new approach of the “Value Equivalence Principle” for differentiable markets with a continuum of traders which are not “perfectly competitive”.     

基于广义判断形式的模糊排序方法

广义判断下的ＡＨＰ（ＧＪＡＨＰ）是一种广义ＡＨＰ。它是在研究不完全信息下的决策排序问题时，通过构造广义判断矩阵的数学模型而建立的一种广义ＡＨＰ。本文应用集值统计的方法，在区间判断标度基础上确定模糊判断矩阵元素的正模糊数表示。给出了基于模糊区间数排序权值向量的特征根算法。讨论了Ｆｕｚｚｙ环境下求解各种判断形式的模糊排序权值向量的方法。     

A new t-norm

We define a new t-norm, and t-conorm, whose domain is usually a restricted subset of [0, 1] x [0, 1], with one parameter, the correlation coefficient between the membership values, which ranges from −1 to 1. We then investigate their properties showing at one extreme they equal the Zadeh operators (max, min), and at the other extreme they produce the Lukasiewicz operators (max(a + b - l, 0), min(a + b,1)).     

On fuzzy cliques in fuzzy networks

A definition of fuzzy clique in social networks is suggested which overcomes five limitations of current definitions. This definition is based on the networks in which the 0–1 strengths, the weighted strengths, and fuzzy strengths are all allowed. The fuzzy distance in such a network is defined. The node‐clique and clique‐clique coefficients are suggested. The core and the periphery of fuzzy cliques are discussed formally. A “cone like” property of the cores is discovered. The network structures are discussed using the new definition. A “no circle” property of networks is found. Basic fuzzy tools and the related algorithms are also discussed. Some examples are analyzed to demonstrate the theory.     

Unifying local–global type properties in vector optimization

It is well-known that all local minimum points of a semistrictly quasiconvex real-valued function are global minimum points. Also, any local maximum point of an explicitly quasiconvex real-valued function is a global minimum point, provided that it belongs to the intrinsic core of the function’s domain. The aim of this paper is to show that these “local min–global min” and “local max–global min” type properties can be extended and unified by a single general local–global extremality principle for certain generalized convex vector-valued functions with respect to two proper subsets of the outcome space. For particular choices of these two sets, we recover and refine several local–global properties known in the literature, concerning unified vector optimization (where optimality is defined with respect to an arbitrary set, not necessarily a convex cone) and, in particular, classical vector/multicriteria optimization.     

Additive fuzzy measures and integrals I

A point of view concerning “fuzzy measures” is explained. To this end, a new concept of “disjointness” for fuzzy is introduced and studied. Also, a concept of an “additive class of fuzzy sets” is defined to be a class of fuzzy sets closed under some “additive operations.” The fuzzy measures are defined to be sum-preserving real functions over such additive classes. Some basic properties of the fuzzy measures are derived. In contrast with other homonymous concepts studied in literature, our fuzzy measures lead to an additive fuzzy integral (see the part II of the paper).     

Fuzzy part coding for group technology

Contemporary Group Technology (GT) methods apply coding schemes as a popular method for capturing the design and manufacturing information pertinent to the parts to be grouped. Coding schemes are very popular and many different coding systems are commercially available. The main disadvantage of current coding systems, however, is their generality and lack of informative representation of the parts.This paper presents a new methodology for coding parts using fuzzy codes. The methodology is general and applies to attributes that have a crisp value (e.g., “length”, “ratio of length to diameter”), an interval value (e.g., “tolerance”, “surface roughness”) or a fuzzy value (e.g., “primary shape”). The methodology considers the range of attributes' values relevant for the grouping, and therefore, is tuned and adjusted to the specific collection of parts of interest. This method creates a more informative coding scheme which leads to improved variant process planning methods, scheduling and inventory control as well as other manufacturing functions that utilize GT.     

Steiner triple systems with disjoint or intersecting subsystems

The existence of incomplete Steiner triple systems of order υ having holes of orders w and u meeting in z elements is examined, with emphasis on the disjoint (z = 0) and intersecting (z = 1) cases. When and , the elementary necessary conditions are shown to be sufficient for all values of z. Then for and υ “near” the minimum of , the conditions are again shown to be sufficient. Consequences for larger orders are also discussed, in particular the proof that when one hole is at least three times as large as the other, the conditions are again sufficient. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 58–77, 2000     

Solving fuzzy multi-objective linear programming problems using deviation degree measures and weighted max–min method

This paper proposes a method for solving fuzzy multi-objective linear programming (FMOLP) problems where all the coefficients are triangular fuzzy numbers and all the constraints are fuzzy equality or inequality. Using the deviation degree measures and weighted max–min method, the FMOLP problem is transformed into crisp linear programming (CLP) problem. If decision makers fix the values of deviation degrees of two side fuzzy numbers in each constraint, then the δ-pareto-optimal solution of the FMOLP problems can be obtained by solving the CLP problem. The bigger the values of the deviation degrees are, the better the objectives function values will be. So we also propose an algorithm to find a balance-pareto-optimal solution between two goals in conflict: to improve the objectives function values and to decrease the values of the deviation degrees. Finally, to illustrate our method, we solve a numerical example.     

基于∨-·运算的Fuzzy矩阵方程的摄动问题

定义基于∨ - .运算的 Fuzzy矩阵 A的同解简化矩阵 ,利用 A的同解简化矩阵 A(1) 给出基于∨ - .运算的 Fuzzy矩阵方程的矩阵解法 ,并研究这类 Fuzzy矩阵方程的摄动问题。     

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.     

输入采用标准模糊分划的模糊控制系统性质及稳定性分析

具体定义模糊控制系统输入变量的标准模糊分划，研究和证明输入采用标准模糊分划的模糊控制系统的有关性质。在此基础上，采用Lyapunov直接法研究该类模糊控制系统的稳定性，提出一个判定模糊控制系统稳定性的充分条件。该条件将以往方法要在所有的子系统中寻找一个公共的正定矩阵满足Lyapunov不等式，放宽为在各最大交叠规则组内分别寻找公共的正定矩阵，从而减小稳定性判定的保守性和难度。     

Convexity cuts for multiple choice problems

The purpose of this note is to show that the convexity (or intensection) cut ideas can be exploited to special advantage for integer programs with certain structures. In particular, we show how to obtain now cuts for “multiple choice” and other related “combinatorial” problems. Our approach can usefully be applied to problems involving logical alternatives without having to reformulate (and enlarge) such problems by the addition of 0–1 variables and associated linear constraints.     

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].     

Coordinated cutting plane generation via multi-objective separation

In cutting plane methods, the question of how to generate the “best possible” set of cuts is both central and crucial. We propose a lexicographic multi-objective cutting plane generation scheme that generates, among all the maximally violated valid inequalities of a given family, an inequality that is undominated and maximally diverse w.r.t. the cuts that were previously found. By optimizing a diversity measure, we introduce a form of coordination between successive cuts. Our focus is on valid inequalities with 0–1 coefficients in the left-hand side and a constant right-hand side, which encompasses several families of valid inequalities. As cut diversity measure, we consider an aggregate of the 1-norm distances w.r.t. the normal vectors of the previous cuts. In this case, our lexicographic multi-objective separation problem reduces to the standard separation problem with different values for the objective function coefficients. The impact of our coordinated cutting plane generation scheme is assessed in a pure cutting plane setting when separating stable set and cut set inequalities for, respectively, the max clique and min Steiner tree problems. Compared to the standard separation of undominated maximally violated cuts, we close the same fraction of the duality gap in a considerably smaller number of rounds and cuts. The potential of our scheme is also indicated by the results obtained in a cut-and-branch setting for max clique, where cut coordination allows for a substantial reduction, on average, of the number of branch-and-bound nodes.     

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.