Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

An extension of Sharpe's single-index model: portfolio selection with expert betas

This paper presents an approach to the portfolio selection problem based on Sharpe's single-index model and on Fuzzy Sets Theory. In this sense, expert estimations about future Betas of each financial asset have been included in the portfolio selection model denoted as ‘Expert Betas’ and modelled as trapezoidal fuzzy numbers. Value, ambiguity and fuzziness are three basic concepts involved in the model which provide enough information about fuzzy numbers representing ‘Expert Betas’ and that are simple to handle. In order to select an optimal portfolio, a Goal Programming model has been proposed including imprecise investor's aspirations concerning asset's proportions of both, high-and low-risk assets. Semantics of these goals are based on the fuzzy membership of a goal satisfaction set. To illustrate the proposed model a real portfolio selection problem is presented.     

Fuzzy sets-the approximation of semisets

This paper deals with alternative set theory which substantially departs from classical set theory. The main notion of it -a semiset-is proposed to model the same intuitive notion as fuzzy set but it is more general. The reasonable tool for ‘grasping’ semisets could be provided by fuzzy sets whose applicability is beyond discussion. In the paper reasons for such an approximation are given and two ways how to provide it are proposed. The fact that the membership function is modeled within the theory should be stressed. At the end, some problems and an indication for further investigation are discussed.     

Stability and Shapley value for an n-persons fuzzy game

The aim of the paper is to explain new concepts of solutions for n-persons fuzzy games. Precisely, it contains new definitions for ‘core’ and ‘Shapley value’ in the case of the n-persons fuzzy games. The basic mathematical results contained in the paper are these which assert the consistency of the ‘core’ and of the ‘Shapley value’. It is proved that the core (defined in the paper) is consistent for any n-persons fuzzy game and that the Shapley values exists and it is unique for any fuzzy game with proportional values.     

Fuzzy proximity spaces redefined

We use the concept ‘L-quasi-coincident’ in fuzzy set theory, to redefine L-fuzzy proximity space, and we study the following problems: proximity of L-fuzzy normal spaces, L-fuzzy filters, proximity neighborhoods and the L-fuzzy topology induced by the L-fuzzy proximity.     

Quantity of information; Comparison between information systems: 1. Non-fuzzy states

In statistical theory, experiments or probabilistic information systems are supposed to be informative, since they reduce the amount of uncertainty associated with the states of nature. For the case that the available information systems are vague (fuzzy information systems), H. Tanaka, T. Okuda and K. Asai have proven, using the ‘measure of information’ as provided by ‘entropy’, that the fuzzy information systems are informative too.Now, we wish to state and to study a criterion in order to compare fuzzy information systems by the ‘quantity of information of a fuzzy information system’ (defined by Tanaka et al.).In this first paper we consider the situation where we require information about the original state space (non-fuzzy state space).The second paper will deal with the situation where we require only information on certain vague states (fuzzy states).     

Quantity of information; comparison between information systems: 2. Fuzzy states

In statistical theory, experiments or probabilistic information systems are supposed to be informative, since they reduce the amount of uncertainty associated with the states of nature. For the case that the available information systems are vague (fuzzy information systems), H. Tanaka, T. Okuda and K. Asai have proven, using the ‘measure of information’ as provided by ‘entropy’, that the fuzzy information systems are informative too.Now, we wish to state and to study a criterion in order to compare fuzzy information systems by the ‘quantity of information of a fuzzy information system’ (defined by Tanaka et al.).In the first paper we considered the situation where we require information about the original state space (non-fuzzy state space).This second paper deals with the situation where we require only information on certain vague states (fuzzy states).     

Multivariate analysis using ‘and’ and ‘or’

The connectives ‘and’ and ‘or’ are potentially useful in multivariate analysis and theory construction. They are simple, logical ways to connect two or more variables together. However, until recently there has been no framework for operationalizing these connectives for continuous variables, and this lack has severely limited their use. Using fuzzy set theory as a basis for such a framework, this paper lays out the necessary tools and models to permit the use of ‘and’ and ‘or’ in multivariate analysis.Section 1 introduces conventional operators for ‘and’ and ‘or’, and Section 2 provides suitable extensions and generalizations of them. Section 3 sets out the required least-squares techniques for fitting these generalized operators to data, first in the context of ANOVA problems and then in regression contexts, for single-connective (three-variable) models. The theoretical developments and examples from real data-sets demonstrate the utility of ‘and’ and ‘or’ as a means to cell-specific interpretations of interaction effects which can also readily be translated into English. Section 4 extends these developments to multivariate, multiple-connective models and discusses issues of generalizability. The paper concludes (Section 5) with a brief discussion of remaining unsolved problems, future prospects for more sophisticated models, and computer programs.     

Design of group decisions II. In spaces of partial order fuzzy relations

For the case of initial data in the problem of group choice represented as fuzzy partial orderings two problems are solved: (1) design of a set group decisions which satisfy the Pareto unanimity principlle and stay ‘halfway’ between initial relations and (2) design of a unique group decision.     

Some considerations on fuzzy conditional inference

In much of human reasoning, the form of reasoning is approximate rather than exact as in ‘A red apple is ripe and this apple is more or less red. Then this apple is more or less ripe.’L.A. Zadeh and E.H. Mamdani suggested methods for such a fuzzy reasoning as an application of fuzzy set theory. The method involves an inference rule and a conditional proposition which contains fuzzy concepts.In this paper we point out that the consequence inferred by their methods does not always fit our intuitions and we suggest the improved methods which fit our intuitions under several criteria.     

Transportation policies for single and multi-objective transportation problem using fuzzy logic

In this paper, single and multi-objective transportation models are formulated with fuzzy relations under the fuzzy logic. In the single-objective model, objective is to minimize the transportation cost. In this case, the amount of quantities transported from an origin to a destination depends on the corresponding transportation cost and this relation is verbally expressed in an imprecise sense i.e., by the words ‘low’, ‘medium’, ‘high’. For the multi-objective model, objectives are minimization of (i) total transportation cost and (ii) total time for transportation required for the system. Here, also the transported quantity from a source to a destination is determined on the basis of minimum total transportation cost as well as minimum transportation time. These relations are imprecise and stated by verbal words such as ‘very high’, ‘high’, ‘medium’, ‘low’ and ‘very low’. Both single objective and multi-objective problems using Real coded Genetic Algorithms (GA and MOGA) are developed and used to solve the single level and bi-level logical relations respectively. The models are illustrated with numerical data and optimum results are presented.     

Fixed point theorems for fuzzy numbers

In this paper we study the problem of the existence of fixed points for fuzzy maps. We first define uniform structures on sets of fuzzy subsets and describe the ‘closed convergence’ and ‘myope convergence’ structures which have properties close to the corresponding traditional ones. We then give two ways in order to obtain fixed point results. We finally give applications of these results, particularly to fuzzy Markov processes.     

Sequential fuzzy system identification

The problem of deriving the structure of a non-deterministic system from its behaviour is a difficult one even when that behaviour is itself well-defined. When the behaviour can be described only in fuzzy terms structural inference may appear virtually impossible. However, a rigorous formulation and solution of the problem for stochastic automata has recently been given [1] and, in this paper, the results are extended to fuzzy stochastic automata and grammars. The results obtained are of interest on a number of counts. (1) They are a further step towards an integrated ‘theory of uncertainty’; (2) They give new insights into problems of inductive reasoning and processes of ‘precisiation’; (3) They are algorithmic and have been embodied in a computer program that can be applied to the modelling of sequential fuzzy data; (4) They demonstrate that sequential fuzzy data may be modelled naturally in terms of ‘possibility’ vectors.     

Functions,equivalence relations,quotient spaces and subsets in fuzzy set theory

In the usual set theory, the power set of a set X is lattice isomorphicto 2^x. The theory of fuzzy sets, or fuzzy set theory, involves generalizing the lattice 2 = {0, 1} to a more general, usually non-Boolean, lattice. A ‘point’ in this lattice need not have Boolean properties. Thus proper generalizations must be based on notions which are defined without explicitly using points. In the light of this we give equivalent definitions of function, relation and quotient set involving the members of a power set but not specifically the singletons.A number of authors [1,2,5] have worked with generalizations of functions and quotient sets which are not strictly point free. It emerges that these are special cases of the ones we produce.     

Fuzzy real algebra: Some results

In this paper, the possibility to perform easily most of the extended n-ary operations on fuzzy subsets of the real line is shown. A general algorithm is given. These results are particularized for usual operations such as addition, subtraction, multiplication, division, ‘max’ and ‘min’ operations for normalized convex fuzzy subsets of the real line, i.e. fuzzy numbers. A three parameters representation for fuzzy numbers is shown to be very convenient to perform usual operations. Lastly, interpretative comments about fuzzy real algebra are given and possible applications pointed out.     

Measures of fuzzy sets and measures of fuzziness

The measures presented in this paper are defined by using Weber's concept of decomposable measures m of crisp sets, having in particular the Archimedean decomposable operations in view (Section 2). Measures $\text{m}$ of fuzzy sets are introduced as integrals with respect to m. For the Archimedean cases, Weber's integral will be used as alternative to Sugeno's and Choquet's concepts (Section 3). What ‘fuzziness’ means will be described by functions of fuzziness F (another name: entropy N-functions) with respect to a negation. In addition to the types of functions of fuzziness which are induced by concave functions, we discuss also the ones which are induced by fuzzy connectives (Section 4). Now, using m for measuring the ‘importance of items’ and F for the ‘fuzziness’ of the possible values of a fuzzy set ?, m?(F ° ?) serves us as a measure of the fuzziness $\text{F}$? of ?. The concepts of De Luca and Termini, Capocelli and De Luca, Kaufmann, Knopfmacher, Loo, Gottwald, Dombi and, under the restriction to the Archimedean cases, also the concepts of Trillas and Riera and Yager turn out to be special cases (Section 5).     

Production and payment policies for an imperfect manufacturing system with discount cash flows analysis in fuzzy random environments

ABSTRACT

This paper considers an imperfect manufacturing system with credit policies in fuzzy random environments. The supplier simultaneously offers the retailer either a permissible delay in payments or a cash discount and retailer in turn provides its customer a permissible delay period. We used an alternate approach – discount cash flow analysis to establish an inventory problem. It is assumed that the elapsed time until the machine shifts from ‘in-control’ state to ‘out-of-control’ state is characterized as a fuzzy random variable. As a function of this parameter, the profit function is also a random fuzzy variable. Based on the credibility measure of fuzzy event, the model with fuzzy random elapsed time can be transformed into a crisp model . We establish several theoretical results to obtain the solution that provides the largest present value of all future cash flows. Finally, numerical example is given to illustrate the results and obtain some managerial insights.     

直觉模糊熵与加权决策算法

直觉模糊熵是直觉模糊集理论中的一个重要概念,反映了直觉模糊集的模糊程度和不确定程度.首先给出一种新的直觉模糊熵,并运用到多属性直觉模糊决策问题中.决策时根据直觉模糊熵计算属性权重,再综合决策者的偏好对各属性权重进行修正,然后使用直觉模糊集结算子和得分函数对方案进行排序,从而获得最优方案.     

Fuzzy uniqueness of fuzzy mappings

For fuzzy mappings between fuzzy sets of higher level we study a notion of fuzzy uniqueness. i.e. of fuzzy functions, which formalizes the intuition that ‘almost identical’ images under f.Characterizations are given for such fuzzy mappings which are really (i.e. with truth value 1) fuzzy unique or fuzzy biunique, and for those fuzzy sets which can be domains of really fuzzy unique fuzzy mappings.Finally, as an application fuzzy cardinals, their ordering, sum, and product are introduced, and some properties proved.     

The fuzzy decision problem: An approach to the point estimation problem with fuzzy information

This paper is devoted to the extension of the Bayesian method for the point estimation, when the available information is ‘vague’.In the nonfuzzy case, the parametric estimation can be approached as a particularization in the statistical decision problem. This motivates us to accomplish the mentioned extension by looking at the parametric estimation in the fuzzy case as a special situation in the fuzzy decision problem (defined by Tanaka, Okuda and Asia).In this way, concepts in the fuzzy decision problem are first ‘expressed’ in the estimation terminology. Then, on the basis of these concepts, we shall introduce some notions and state some interesting results. Finally, several illustrative examples will be exposed.