Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets.We discuss here the corresponding situation for fuzzy set theory. Our emphasis will be on various approaches toward (more or less naively formed) universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.What we give is a (critical) survey of quite a lot of such approaches which have been offered in the last approximately 35 years.Part I was devoted to model based and to axiomatic approaches; the present Part II is devoted to category theoretic approaches.This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005.     

模糊偏序关系在心理学研究中的应用

本文首次给出建立模糊偏序关系矩阵的可行途径。以大学生的颜色爱好实验为例,论证了采用本文所建立的模糊偏序关系矩阵进行排序的方法是制作顺序量表的一种较佳方法。     

Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets. What we give is a (critical)survey of quite a lot of such approaches which have been offered in the last approximately 35 years. The present Part I is devoted to model based and to axiomatic approaches; the forth-coming Part II will be devoted to category theoretic approaches. This paper is a version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005. Presented by Jacek Malinowski     

A Fuzzy Topsis Method for Prioritized Aggregation in Multi‐Criteria Decision Making Problems

Aggregation in a decision making environment requires the fusion of opinions of a group of decision makers. The group of decision makers are required to analyse a set of interrelated criteria that are usually measured on a linguistic scale. This process requires, in many instances, to capture experts experience, intuition and thinking that are traditionally expressed in a linguistic fashion rather than a numerical fashion. Furthermore, the necessity of considering the relationship between the criteria to the overall decision must be considered by the group of decision makers. This paper extends the application of fuzzy numbers, fuzzy relative importance scores (FRIS), fuzzy relative weights (FRW) and the fuzzy technique of order preference by similarity to ideal solution (TOPSIS) in prioritized aggregation. This extension provides a mean to systematically aggregate a group of decision makers' views for a set of interrelated criteria that are measured on a linguistic scale. First, an overview of the application of fuzzy numbers and the characteristics of aggregating fuzzy numbers in multi‐criteria decision making problems are presented. Then, the application of TOPSIS in fuzzy environments is presented. Next, past research is highlighted to present prioritized aggregation and the different aggregation operators' classes. Subsequently, a new prioritized aggregation method is presented. This method utilizes fuzzy TOPSIS with prioritized aggregation in fuzzy environments. Finally, the fuzzy prioritized aggregation method presented in this paper is applied on an actual case study. According to the results, the method presented in this paper provides a systematic approach to capture the uncertainty and imprecision associated with quantifying linguistic measurements in multi‐criteria decision making problems. Furthermore, it considers the relationship between the set of linguistically measured criteria undergoing prioritized aggregation in a fuzzy environment. Lastly, findings, conclusions and future work are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Fuzzy MCDM based on the concept of α‐cut

In a fuzzy multiple criteria decision‐making (MCDM) problem, with a hierarchical structure of more than two levels and involving multiple decision‐makers (DMs), to find the exact membership functions of the final aggregation ratings of all feasible alternatives is almost impossible. Thus, ranking methods based on exact membership functions cannot be utilized to rank the feasible alternatives and complete the optimal selection. To resolve the above‐mentioned complexity and to incorporate assessments of all DMs' viewpoints, in this paper a fuzzy MCDM method with multiple DMs, based on the concepts of fuzzy set theory and α‐cut, is developed. This method incorporates a number of perspectives on how to approach the fuzzy MCDM problem with multiple DMs, as follows: (1) combining quantitative and qualitative criteria as well as negative and positive ones; (2) using the generalized means to develop the aggregation method of multiple DMs' opinions; (3) incorporating the risk attitude index β to convey the total risk attitude of all DMs by using the estimation data obtained at the data input stage; (4) employing the algebraic operations of fuzzy numbers based on the concept of α‐cut to calculate the final aggregation ratings and develop a matching ranking method for proposed fuzzy MCDM method with multiple DMs. Furthermore, we use this method to survey the site selection for free port zone (FPZ) in Taiwan as an empirical study to demonstrate the proposed fuzzy MCDM algorithm. The result of this empirical investigation shows that the port of Kaohsiung, the largest international port of Taiwan as well as the sixth container port in the world in 2004, is optimal for the Taiwan government in enacting the plan of FPZ. Copyright © 2005 John Wiley & Sons, Ltd.     

儿童理解程度副词的特征

本实验用模糊数学方法研究4—9岁儿童对程度副词的理解特征。结果说明9岁儿童的理解达到了成熟水平;幼儿偏向于夸大理解程度副词的词义;7、8岁是发展的转折期。     

情绪形容词词义的模糊赋值

353名大学生被试用模糊赋值方法对表示喜、怒、哀、惧四种基本情绪的48个形容词的强度和复杂度作语义经验赋值。结果表明：①强度和复杂度是两个不同的情绪评定维度,两者间的情绪形容词模糊语义赋值结果没有必然联系。②在强度和复杂度赋值上,各词的取值基本符合人们日常对这些词所表达情绪体验的理解,性别的量表值存在较大一致性。③男、女生在赋值上的差异主要表现在各词位次和把握度的不同,这种差异可能与性别差异的情绪经验及个体差异有关。④大学生情绪形容词模糊语义赋值的把握度总体上都较高,各词的语义在两个维度上都存在一定的模糊性。⑤根据各词强度赋值隶属度数据计算的模糊距离测度和Phi方关联测度得到了四类形容词相似的聚类分析结果。     

Vagueness and Blurry Sets

This paper presents a new theory of vagueness, which is designed to retain the virtues of the fuzzy theory, while avoiding the problem of higher-order vagueness. The theory presented here accommodates the idea that for any statement S ₁ to the effect that Bob is bald is x true, for x in [0,1], there should be a further statement S ₂ which tells us how true S ₁ is, and so on – that is, it accommodates higher-order vagueness – without resorting to the claim that the metalanguage in which the semantics of vagueness is presented is itself vague, and without requiring us to abandon the idea that the logic – as opposed to the semantics – of vague discourse is classical. I model the extension of a vague predicate P as a blurry set, this being a function which assigns a degree of membership or degree function to each object o, where a degree function in turn assigns an element of [0,1] to each finite sequence of elements of [0,1]. The idea is that the assignment to the sequence 0.3,0.2, for example, represents the degree to which it is true to say that it is 0.2 true that o is P to degree 0.3. The philosophical merits of my theory are discussed in detail, and the theory is compared with other extensions and generalisations of fuzzy logic in the literature.     

刻板印象的自动激活实验研究

本研究应用模糊主次评判方法配合语义启动技术进行实验。结果表明;在年龄刻板印象上存在自动激活,而且没有性别差异;用不同性质的启动词来激活与年龄有关的刻板印象,激活后的刻板印象对年龄的判断有不同的影响,而且被试没有一个人意识到这点,表明内隐刻板印象的存在;人的判断具有模糊性,模糊主次评判方法有助于社会认知领域的量化研究,这是一个全新的尝试。     

Fuzzy Logic and Arithmetical Hierarchy,II

A very simple many-valued predicate calculus is presented; a completeness theorem is proved and the arithmetical complexity of some notions concerning provability is determined.     

On Vagueness,Truth Values and Fuzzy Logics

Some aspects of vagueness as presented in Shapiro’s book Vagueness in Context [23] are analyzed from the point of fuzzy logic. Presented are some generalizations of Shapiro’s formal apparatus. Presented by Daniele Mundici     

Attribute Partitioning in Multiple Attribute Decision Making Problems for a Decision with a Purpose – a Fuzzy Approach

This work introduces a methodology to find solutions corresponding to different purposes in a multiple attribute decision‐making problem under fuzzy environment. The discernment of purpose‐based solutions becomes important when the problem is defined vaguely and solution is targeted to heterogeneous population. Depending on the purpose, for which the solution is sought, the attributes are identified and weighted in an appropriate proportion. The level of similarity between a pair of attributes plays an important role to determine the aggregated value of attributes specific to a purpose. Our work determines the similarity levels between a pair of attributes by calculating their maximum attainability in presence of each other. The achievement of an attribute in presence of another is represented as a fuzzy set in the unit interval. The crisp equivalents of the fuzzy sets in the unit interval are used to define their simultaneous satisfaction denoted as 1‐step relation. The 1‐step relation is extended to (m‐1)‐step relation to calculate the degree of attainability of the same pair of attributes in the presence of m (all) attributes. The different levels of (m‐1)‐step relations generate several partitions of the attributes corresponding to multiple purposes in the multiple attribute decision‐making problems. The degree of fulfilment of the purposes in the alternatives are numerically derived by first taking weighted average of attributes within the equivalence classes of a partition and then aggregating the values corresponding to equivalence classes through ordered weighted averaging. The methodology is illustrated with a numerical example. Copyright © 2016 John Wiley & Sons, Ltd.     

Component Models for Fuzzy Data

The fuzzy perspective in statistical analysis is first illustrated with reference to the “Informational Paradigm” allowing us to deal with different types of uncertainties related to the various informational ingredients (data, model, assumptions). The fuzzy empirical data are then introduced, referring to J LR fuzzy variables as observed on I observation units. Each observation is characterized by its center and its left and right spreads (LR₁ fuzzy number) or by its left and right “centers” and its left and right spreads (LR₂ fuzzy number). Two types of component models for LR₁ and LR₂ fuzzy data are proposed. The estimation of the parameters of these models is based on a Least Squares approach, exploiting an appropriately introduced distance measure for fuzzy data. A simulation study is carried out in order to assess the efficacy of the suggested models as compared with traditional Principal Component Analysis on the centers and with existing methods for fuzzy and interval valued data. An application to real fuzzy data is finally performed. We would like to express our gratitude to the Editor, the Associate Editor, and the Referees whose comments and suggestions improved significantly the quality of the paper.     

Measuring logistics performance of OECD countries via fuzzy linear regression

In logistics, performance measurement has been considered as a key competency to acquire world class performance. In light of this, we presented a robust methodology to establish an analysis framework for measuring logistics performance. The proposed hybrid methodology is a combination of criteria importance through intercritera correlation (CRITIC), simple additive weighting (SAW), and Peters' fuzzy regression methods. To the best of our knowledge, country‐based logistics performance is seldom studied in the literature. Therefore, we measured the logistics performance of Organization for Economic Cooperation and Development (OECD) countries using the devised model based on the data of Logistics Performance Index 2014 provided by the World Bank. The introduced methodology, which is suitable to model imprecise relationships among system parameters, appears to be a practical alternative approach for the assessment of logistics performance. It should be noted that the evaluation framework presented in this paper is not confined to performance measurement case and can also be exploited in addressing other multiple criteria decision‐making problems incorporating uncertainty.     

Kripke Frame with Graded Accessibility and Fuzzy Possible World Semantics

A possible world structure consist of a set W of possible worlds and an accessibility relation R. We take a partial function r(·,·) to the unit interval [0, 1] instead of R and obtain a Kripke frame with graded accessibility r Intuitively, r(x, y) can be regarded as the reliability factor of y from x We deal with multimodal logics corresponding to Kripke frames with graded accessibility in a fairly general setting. This setting provides us with a framework for fuzzy possible world semantics. The basic propositional multimodal logic gK (grated K) is defined syntactically. We prove that gK is sound and complete with respect to this semantics. We discuss some extensions of gK including logics of similarity relations and of fuzzy orderings. We present a modified filtration method and prove that gK and its extensions introduced here are decidable.     

Mathematical Fuzzy Logic – What It Can Learn from Mostowski and Rasiowa

Important works of Mostowski and Rasiowa dealing with many-valued logic are analyzed from the point of view of contemporary mathematical fuzzy logic.A version of this paper has been presented during the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005.     

On Retaining Classical Truths and Classical Deducibility in Many-Valued and Fuzzy Logics

In this paper, I identify the source of the differences between classical logic and many-valued logics (including fuzzy logics) with respect to the set of valid formulas and the set of inferences sanctioned. In the course of doing so, we find the conditions that are individually necessary and jointly sufficient for any many-valued semantics (again including fuzzy logics) to validate exactly the classically valid formulas, while sanctioning exactly the same set of inferences as classical logic. This in turn shows, contrary to what has sometimes been claimed, that at least one class of infinite-valued semantics is axiomatizable.