The Total Variation of Bounded Variation Functions to Evaluate and Rank Fuzzy Quantities

In this paper, we present a different approach to introduce evaluation and ranking of fuzzy quantities. These general fuzzy sets are obtained by the union of several fuzzy sets. They are neither normal nor convex. The idea we have followed is to use the total variation and the bounded variation function definitions applied to the membership function of a fuzzy set to introduce its evaluation. This approach has produced that the well‐known method of area compensation, introduced by Fortemps and Roubens only in a geometrical framework, is now presented in a general contest and useful for any fuzzy set. Moreover, this new representation formula provides an α‐cut view. This aspect, absent in Fortemps and Roubens paper, offers an evaluation by a weighted average of alfa‐cuts values, where the weights are connected with the number of subintervals that produce every α‐cut. Following the same idea, we have introduced the ambiguity definition of a general fuzzy set. By this new definition of evaluation and the consequent ambiguity, we present a way to rank fuzzy quantities.     

Fuzzy ε-subgroups

The aim of the paper is to investigate the known classical problem of a partition of a semigroup into a family of groups in the lattice-valued (fuzzy) framework. For that purpose, a fuzzy subgroup of a fuzzy subsemigroup is introduced and characterized by cuts, using the cutworthy approach. As a main result it is proved that a fuzzy subsemigroup can be partitioned (using special type of fuzzy partitions) into a family of fuzzy ε-subgroups if and only if it is fuzzy completely regular.     

Algebraic properties on the cuts of lattice-valued regular languages

In this paper, the algebraic operations on the cuts of lattice-valued regular languages are studied. Some sufficient conditions are given to guarantee the family of the cuts of lattice-valued regular languages to be closed under such algebraic operations as union, intersection, complement, quotient, homomorphism, inverse homomorphism, concatennation, reversal, etc. This work is supported by National Science Foundation of China (Grant No.10571112), “TRAPOYT” of China and National 973 Foundation Research Program (Grant No. 2002CB312200).     

Development of fuzzy process control charts and fuzzy unnatural pattern analyses

Many problems in scientific investigation generate nonprecise data incorporating nonstatistical uncertainty. A nonprecise observation of a quantitative variable can be described by a special type of membership function defined on the set of all real numbers called a fuzzy number or a fuzzy interval. A methodology for constructing control charts is proposed when the quality characteristics are vague, uncertain, incomplete or linguistically defined. Fuzzy set theory is an inevitable tool for fuzzy control charts as well as other applications subjected to uncertainty in any form. The vagueness can be handled by transforming incomplete or nonprecise quantities to their representative scalar values such as fuzzy mode, fuzzy midrange, fuzzy median, or fuzzy average. Then crisp methods may be applied to those representative values for control chart decisions as “in control” or “out of control”. Transforming the vague data by using one of the transformation methods may result in biased decisions since the information given by the vague data is lost by the transformation. Such data needs to be investigated as fuzzy sets without transformation, and the decisions based on the vague data should not be concluded with an exact decision. A “direct fuzzy approach (DFA)” to fuzzy control charts for attributes under vague data is proposed without using any transformation method. Then, the unnatural patterns for the proposed fuzzy control charts are defined using the probabilities of fuzzy events.     

基于t-模的区间值模糊的截集性质研究

区间值模糊集的隶属度使其拥有更多的自由度。从而在处理信息不确定性和模糊性时比经典模糊集更有优势。为了更好地利用区间模糊集,研究其截集及性质具有重要的意义。本文首先定义了一种基于t-模的区间值模糊集的截集,进一步讨论了基于t-模的区间值模糊集的广义交、并、补的截集的相关性质。特别地,若T(S)为∧(∨)-可表示的,则区间模糊集的交、并、补运算与截集运算可交换。     

General IF-sets with triangular norms and their applications to group decision making

A concept of general IF-sets, i.e. “intuitionistic” fuzzy sets according to Atanassov, with triangular norm-based hesitation degrees is introduced and developed. That concept is used to construct flexible algorithms of group decision making which involve relative scalar cardinalities defined by means of generalized sigma counts of fuzzy sets. Two cases of group decisions, based on individual or social fuzzy preference relations, are considered.     

出现网的t型s切和t切

首先引入出现网的t切的转移的概念,利用t切的转移集τ,可以得到t切p[τ]。其次引入s切的伴随集E(u)和t型s切的概念。证明了出现网的t型s切和t切有对应的关系以及这种对应关系保持t型s切的转移和t切的转移。给出了在s切有向图中查找t切的算法,证明了出现网的t切都是t型s切的伴随集。     

A Bayesian approach to fuzzy hypotheses testing for the estimation of optimal age for vaccination against measles

Fuzzy Bayesian tests were performed to evaluate whether the mother’s seroprevalence and children’s seroconversion to measles vaccine could be considered as “high” or “low”. The results of the tests were aggregated into a fuzzy rule-based model structure, which would allow an expert to influence the model results. The linguistic model was developed considering four input variables. As the model output, we obtain the recommended age-specific vaccine coverage. The inputs of the fuzzy rules are fuzzy sets and the outputs are constant functions, performing the simplest Takagi–Sugeno–Kang model. This fuzzy approach is compared to a classical one, where the classical Bayes test was performed. Although the fuzzy and classical performances were similar, the fuzzy approach was more detailed and revealed important differences. In addition to taking into account subjective information in the form of fuzzy hypotheses it can be intuitively grasped by the decision maker.     

Fuzzy rule based decision trees

This paper presents a new architecture of a fuzzy decision tree based on fuzzy rules – fuzzy rule based decision tree (FRDT) and provides a learning algorithm. In contrast with “traditional” axis-parallel decision trees in which only a single feature (variable) is taken into account at each node, the node of the proposed decision trees involves a fuzzy rule which involves multiple features. Fuzzy rules are employed to produce leaves of high purity. Using multiple features for a node helps us minimize the size of the trees. The growth of the FRDT is realized by expanding an additional node composed of a mixture of data coming from different classes, which is the only non-leaf node of each layer. This gives rise to a new geometric structure endowed with linguistic terms which are quite different from the “traditional” oblique decision trees endowed with hyperplanes as decision functions. A series of numeric studies are reported using data coming from UCI machine learning data sets. The comparison is carried out with regard to “traditional” decision trees such as C4.5, LADtree, BFTree, SimpleCart, and NBTree. The results of statistical tests have shown that the proposed FRDT exhibits the best performance in terms of both accuracy and the size of the produced trees.     

Numerical solution of fuzzy differential equations by predictor-corrector method

In this paper three numerical methods to solve “The fuzzy ordinary differential equations” are discussed. These methods are Adams-Bashforth, Adams-Moulton and predictor-corrector. Predictor-corrector is obtained by combining Adams-Bashforth and Adams-Moulton methods. Convergence and stability of the proposed methods are also proved in detail. In addition, these methods are illustrated by solving two fuzzy Cauchy problems.     

Grouping fuzzy sets by similarity

The paper presents results on factorization of systems of fuzzy sets. The factorization consists in grouping those fuzzy sets which are pairwise similar at least to a prescribed degree a. An obstacle to such factorization, well known in fuzzy set theory, is the fact that “being similar at least to degree a” is not an equivalence relation because, in general, it is not transitive. As a result, ordinary factorization using equivalence classes cannot be used. This obstacle can be overcome by considering maximal blocks of fuzzy sets which are pairwise similar at least to degree a. We show that one can introduce a natural complete lattice structure on the set of all such maximal blocks and study this lattice. This lattice plays the role of a factor structure for the original system of fuzzy sets. Particular examples of our approach include factorization of fuzzy concept lattices and factorization of residuated lattices.     

Regular ordered semigroups in terms of fuzzy subsets

Given a set S, a fuzzy subset of S (or a fuzzy set in S) is, by definition, an arbitrary mapping f : S → [0, 1] where [0, 1] is the usual interval of real numbers. If the set S bears some structure, one may distinguish some fuzzy subsets of S in terms of that additional structure. This important concept of a fuzzy set was first introduced by Zadeh. Fuzzy groups have been first considered by Rosenfeld, fuzzy semigroups by Kuroki. A theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed. Some results on ordered groupoids-semigroups have been already given by the same authors in [N. Kehayopulu, M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002) 128-132; N. Kehayopulu, M. Tsingelis, The embedding of an ordered groupoid into a poe-groupoid in terms of fuzzy sets, Inform. Sci. 152 (2003) 231-236; N. Kehayopulu, M. Tsingelis, Fuzzy bi-ideals in ordered semigroups, Inform. Sci. 171 (2004) 13-28] where S has been endowed with the structure of an ordered semigroup and defined “fuzzy” analogous for several notions that have been proved to be useful in the theory of ordered semigroups. The characterization of regular rings in terms of right and left ideals is well known. The characterization of regular semigroups and regular ordered semigroups in terms of left and right ideals or in terms of left, right ideals and quasi-ideals is well known as well. The characterization of regular le-semigroups (that is lattice ordered semigroups having a greatest element) in terms of right ideal elements and left ideal elements or right, left and quasi-ideal elements is also known. In the present paper we first give the main theorem which characterizes the regular ordered semigroups by means of fuzzy right and fuzzy left ideals. Then we characterize the regular ordered semigroups in terms of fuzzy right, fuzzy left ideals and fuzzy quasi-ideals. The paper serves as an example to show that one can pass from the theory of ordered semigroups to the theory of “fuzzy” ordered semigroups. Some of our results are true for ordered groupoids in general.     

A test for determining systems with “mixed” small gain and passivity properties

“Mixedness” is a property that captures elements of the notions of passivity and small gain. In the frequency domain, a linear, time-invariant system is called “mixed” if, over some frequency bands, it is strictly passive and, over the remaining frequencies, it has a gain of less than one; there exist no frequencies over which the system has neither of the notions of these properties associated with it. In this paper, a test is developed for determining whether or not a linear, time-invariant system is “mixed”.     

Standard ML–NJ Weak Polymorphism and Imperative Constructs

Standard ML of New Jersey (SML–NJ) uses “weak type variables” to restrict the polymorphic use of functions that may allocate reference cells, manipulate continuations, or use exceptions. However, the type system used in the SML–NJ compiler has not previously been presented in a form other than source code nor proved correct. We present a set of typing rules, based on analysis of the concepts underlying “weak polymorphism”, that appears to subsume the implemented algorithm and uses type variables of only a slightly more general nature than the compiler. One insight in the analysis is that allowing a variable to occur both “ordinarily” and “weakly” in a type permits a simpler and more flexible formulation of the typing rules. In particular, we are able to treat applications of polymorphic functions to imperative arguments with greater flexibility than SML–NJ. The soundness of the type system is proved for imperative code using operational semantics, by showing that evaluation preserves typability. By incorporating assumptions about memory addresses in the type system, we avoid proofs by co-induction.     

Isotone fuzzy Galois connections with hedges

We study isotone fuzzy Galois connections and concept lattices parameterized by particular unary operators. The operators represent linguistic hedges such as “very”, “rather”, “more or less”, etc. Isotone fuzzy Galois connections and concept lattices provide an alternative to their antitone counterparts which are the fundamental structures behind formal concept analysis of data with fuzzy attributes. We show that hedges enable us to control the number of formal concepts in the associated concept lattice. We also describe the structure of the concept lattice and provide a counterpoint to the main theorem of concept lattices.     

Algorithms of discrete optimization and their application to problems with fuzzy coefficients

An approach to solving optimization problems with fuzzy coefficients in objective functions and constraints is described. It consists in formulating and solving one and the same problem within the framework of mutually related models with constructing equivalent analogs with fuzzy coefficients in objective functions alone. It enables one to maximally cut off dominated alternatives “from below” as well as “from above”. Since the approach is applied within the context of fuzzy discrete optimization problems, several modified algorithms of discrete optimization are discussed. These algorithms are associated with the method of normalized functions, are based on a combination of formal and heuristic procedures, and allow one to obtain quasi-optimal solutions after a small number of steps, thus overcoming the computational complexity posed the NP-completeness of discrete optimization problems. The subsequent contraction of the decision uncertainty regions is associated with reduction of the problem to multiobjective decision making in a fuzzy environment with using techniques based on fuzzy preference relations. The techniques are also directly applicable to situations in which the decision maker is required to choose alternatives from a set of explicitly available alternatives. The results of the paper are of a universal character and can be applied to the design and control of systems and processes of different purposes as well as the enhancement of corresponding CAD/CAM systems and intelligent decision making systems. The results of the paper are already being used to solve problems of power engineering.     

t-Norm based cuts of intuitionistic fuzzy sets

We consider the system of intuitionistic fuzzy sets (IF-sets) in a universe X and study the cuts of an IF-set. Suppose a left continuous triangular norm is given. The t-norm based cut (level set) of an IF-set is defined in a way that binds the membership and nonmembership functions via the triangular norm. This is an extension of usual cuts of IF-sets. We show that the system of these cuts fulfils analogical properties as usual systems of cuts. However, it is not possible to reconstruct an IF-set from the system of t-norm based cuts.     

Design and implementation of a fuzzy expert system for performance assessment of an integrated health, safety, environment (HSE) and ergonomics system: The case of a gas refinery

The objective of this study is to design a fuzzy expert system for performance assessment of health, safety, environment (HSE) and ergonomics system factors in a gas refinery. This will lead to a robust control system for continuous assessment and improvement of HSE and ergonomics performance. The importance of this study stems from the current lack of formal integrated methodologies for interpreting and evaluating performance data for HSE and ergonomics. Three important reasons to use fuzzy expert systems are (1) reduction of human error, (2) creation of expert knowledge and (3) interpretation of large amount of vague data. To achieve the objective of this study, standard indicators and technical tolerances for assessment of HSE and ergonomics factors are identified. Then, data is collected for all indicators and consequently, for each indicator four conditions are defined as “acceptance”, “low deviation”, “mid deviation” and “high deviation”. A membership function is defined for each fuzzy condition (set) because an indicator cannot be allocated to just one of the above conditions. The expert system uses fuzzy rules, which are structured with Data Engine. Previous studies have introduced HSE expert system whereas this study introduces an integrated HSE and ergonomics expert system through fuzzy logic.     

Interpretability of linguistic fuzzy rule-based systems: An overview of interpretability measures

Linguistic fuzzy modelling, developed by linguistic fuzzy rule-based systems, allows us to deal with the modelling of systems by building a linguistic model which could become interpretable by human beings. Linguistic fuzzy modelling comes with two contradictory requirements: interpretability and accuracy. In recent years the interest of researchers in obtaining more interpretable linguistic fuzzy models has grown.Whereas the measures of accuracy are straightforward and well-known, interpretability measures are difficult to define since interpretability depends on several factors; mainly the model structure, the number of rules, the number of features, the number of linguistic terms, the shape of the fuzzy sets, etc. Moreover, due to the subjectivity of the concept the choice of appropriate interpretability measures is still an open problem.In this paper, we present an overview of the proposed interpretability measures and techniques for obtaining more interpretable linguistic fuzzy rule-based systems. To this end, we will propose a taxonomy based on a double axis: “Complexity versus semantic interpretability” considering the two main kinds of measures; and “rule base versus fuzzy partitions” considering the different components of the knowledge base to which both kinds of measures can be applied. The main aim is to provide a well established framework in order to facilitate a better understanding of the topic and well founded future works.     

Industrial applications of type-2 fuzzy sets and systems: A concise review

Data, as being the vital input of system modelling, contain dissimilar level of imprecision that necessitates different modelling approaches for proper analysis of the systems. Numbers, words and perceptions are the forms of data that has varying levels of imprecision. Existing approaches in the literature indicate that, computation of different data forms are closely linked with the level of imprecision, which the data already have. Traditional mathematical modelling techniques have been used to compute the numbers that have the least imprecision. Type-1 fuzzy sets have been used for words and type-2 fuzzy sets have been employed for perceptions where the level of imprecision is relatively high. However, in many cases it has not been easy to decide whether a solution requires a traditional approach, i.e., type-1 fuzzy approach or type-2 fuzzy approach. It has been a difficult matter to decide what types of problems really require modelling and solution either with type-1 or type-2 fuzzy approach. It is certain that, without properly distinguishing differences between the two approaches, application of type-1 and type-2 fuzzy sets and systems would probably fail to develop robust and reliable solutions for the problems of industry. In this respect, a review of the industrial applications of type-2 fuzzy sets, which are relatively novel to model imprecision has been considered in this work. The fundamental focus of the work has been based on the basic reasons of the need for type-2 fuzzy sets for the existing studies. With this purpose in mind, type-2 fuzzy sets articles have been selected from the literature using the online databases of ISI-Web of Science, ScienceDirect, SpringerLink, Informaworld, Engineering Village, Emerald and IEEE Xplore. Both the terms “type-2 fuzzy” and “application” have been searched as the main keywords in the topics of the studies to retrieve the relevant works. The analysis on the industrial applications of type-2 fuzzy sets/systems (FSs) in different topics allowed us to summarize the existing research areas and therefore it is expected be useful to prioritize future research topics. This review shows that there are still many opportunities for application of type-2 FSs for several different problem domains. Shortcomings of type-1 FSs can also be considered as an opportunity for the application of type-2 FSs in order to provide a better solution approach for industrial problems.