Intersection Of Fuzzy Sets Research Articles

Application of the fuzzy sets theory in the problem of products competitiveness evaluation

The article considers an example of using the tools of fuzzy sets theory in the problem of studying competitiveness of goods and firms. The competitiveness of goods has been defined through their utility in both subjective and objective manifestations. The transformation of utility indicators value from the simplest scales to the scale of equal relations has been carried out through the graphical membership functions. The general model of multi-criteria choice of optimal alternatives based on the intersection of fuzzy sets has been defined. The construction method and non-dominant alternatives set analysis on the basis of fuzzy preference relation has been considered. The problem of ranking alternatives, that are complex technical devices, considering several comparison criteria, some of which are qualitative, has been assessed by dimensionless scales with expert evaluations. Weighting coefficients have been determined for all criteria, alternatives functions belonging to the subset of optimal and matrices of fuzzy preference relations constructed, and a subset of non-dominant alternatives formed. The optimal alternative has been determined by the maximum value of subset elements. Calculation results can be used for further informed decision making in the absence of clear numerical data and presence of weakly formalized expert statements. Thus, the effectiveness and practicality of applying fuzzy sets methodology in ranking problems and optimal choice of alternatives has been shown.

Intuitionistic fuzzy optimization method for solving multi-objective linear fractional programming problems

An iterative technique based on the use of parametric functions is proposed in this paper to obtain the best preferred optimal solution of a multi-objective linear fractional programming problem (MOLFPP). Each fractional objective is transformed into a non-fractional parametric function using certain initial values of parameters. The parametric values are iteratively calculated and the intuitionistic fuzzy optimization method is used to solve a multi-objective linear programming problem. Also, some basic properties and operations of an intuitionistic fuzzy set are considered. The development of the proposed algorithm is based on the principle of optimal decision set achieved by the intersection of various intuitionistic fuzzy decision sets which are obtained corresponding to each objective function. Additionally, as the intuitionistic fuzzy optimization method utilizes the degree of belonging and degree of non-belonging, we used the linear membership function for belonging and non-belonging to see its impact on optimization and to get insight into such an optimization process. The proposed approaches have been illustrated with numerical examples.

Fuzzy Distribution Sets

We introduce a new type of fuzzy set called Fuzzy Distribution Set (FDS). Fuzzy distribution sets are fuzzy sets defined on a finite domain subject to a sum of membership values equal to 1. Such fuzzy sets can serve as models of subjective probability distributions and subjective weight distributions. Considering these distributions as fuzzy sets gives a possibility to extend on such distributions the operations of fuzzy sets and, more generally, the calculus of fuzzy restrictions developed during the last decades. Recently Yager introduced the concept of negation of probability distributions. In our works, we studied several classes of such negations. Here we consider an involutive negation of probability distributions as a complement of FDS. We introduce the operations of union and intersection of fuzzy distribution sets. These basic operations on FDS can serve as a basis for the application of fuzzy logic methods to subjective probability and weight distributions. These operations can be used for the development of reasoning models with subjective probability distributions and subjective weighting functions. Weight distributions can be used in multi-criteria, multi-person, and multi-attribute decision-making models.

ВИКОРИСТАННЯ ТЕОРІЇ НЕЧІТКИХ МНОЖИН ТА БАГАТОКРИТЕРІЙНОГО ВИБОРУ АЛЬТЕРНАТИВ ДЛЯ ОПИСУ ДІЯЛЬНОСТІ ПІДПРИЄМСТВА В УМОВАХ НЕВИЗНАЧЕНОСТІ

The task of decision-making in conditions of uncertainty is the choice of a rational decision, formed under the influence of certain and uncertain factors, which are quantitatively and qualitatively described by the features. Methods of decision-making on fuzzy models allow you to easily and objectively evaluate alternatives by individual criteria. The analysis of scientific works and the practice of evaluating the company's activities indicate an insufficient level of its analytical and mathematical support, as well as the need for more detailed consideration of multicriteria choice of alternatives to describe the company's activities and decision-making in uncertainty. Thus, the article presents various classifications of uncertainty of the enterprise, their characteristics and sources. The classification of methods of research of activity of the enterprise in the conditions of uncertainty is developed. Key concepts and definitions of fuzzy set theory are considered. The use of fuzzy set theory and multi-criteria choice of alternatives for describing the activities of the enterprise and decision-making in conditions of uncertainty is specified. The paper considers a multi-criteria choice of alternatives based on the intersection of fuzzy sets, based on the fuzzy advantage relationship, using the fuzzy input rule, based on additive convolution. The ranking of alternatives on the set of linguistic vector estimates is also revealed. The advantages and disadvantages of using different multi-criteria choices of alternatives and fuzzy logic in general are highlighted. The main problem of multi-criteria selection using fuzzy models is the presentation of information about the relationship between the criteria and ways to calculate their integrated estimates. Methods based on different approaches give different results. Each approach has its limitations and features, and the user must get an idea of them. Before applying one or another method of decision making. The heuristic approach provides the widest opportunities for presenting information. Analysis of fuzzy decision-making methods allows us to formulate requirements for further development in this area.

On union and intersection of type-2 fuzzy sets not expressible by the sup-t-norm extension principle

This paper constructs a conjunction and a disjunction operator for type-2 fuzzy sets that cannot be expressed by the sup-t-norm extension principle applied to conjunction and disjunction of type-1 fuzzy sets by means of t-norms and t-conorms on the unit interval. This result answers affirmatively an open problem posed by Hernández et al. in 2015. The duality between the set-theoretic operators for type-2 fuzzy sets is obtained from the duality of set-theoretic operations on type-1 fuzzy sets.

The role of properties of the membership function in construction of fuzzy sets ranking

The paper presents a several new definitions of concepts regarding the properties of fuzzy sets in the aspect of their use in decision support processes. These are concepts such as the image and counter – image of the fuzzy set, the proper fuzzy set, the fuzzy support and the ranking of fuzzy set. These concepts can be important in construction decision support algorithms. Particularly a lot of space was devoted to the study of the properties of membership function of the fuzzy set as a result of operations on fuzzy sets. Two additional postulates were formulated that should be fulfilled by the membership function product of fuzzy sets in decision making.

How to choose a fuzzy similarity measure in decision-making?

Purpose The paper is dedicated to the analysis of fuzzy similarity measures in uncertainty analysis in general, and in economic decision-making in particular. The purpose of this paper is to explain how a similarity measure can be chosen to quantify a qualitative description of similarities provided by experts of a given domain, in the case where the objects to compare are described through imprecise or linguistic attribute values represented by fuzzy sets. The case of qualitative dissimilarities is also addressed and the particular case of their representation by distances is presented. Design/methodology/approach The approach is based on measurement theory, following Tversky’s well-known paradigm. Findings A list of axioms which may or may not be satisfied by a qualitative comparative similarity between fuzzy objects is proposed, as extensions of axioms satisfied by similarities between crisp objects. They enable to express necessary and sufficient conditions for a numerical similarity measure to represent a comparative similarity between fuzzy objects. The representation of comparative dissimilarities is also addressed by means of specific functions depending on the distance between attribute values. Originality/value Examples of functions satisfying certain axioms to represent comparative similarities are given. They are based on the choice of operators to compute intersection, union and difference of fuzzy sets. A simple application of this methodology to economy is given, to show how a measure of similarity can be chosen to represent intuitive similarities expressed by an economist by means of a quantitative measure easily calculable. More detailed and formal results are given in Coletti and Bouchon-Meunier (2020) for similarities and Coletti et al. (2020) for dissimilarities.

Set operations of fuzzy sets using gradual elements

The conventional set operations of fuzzy sets are based on the membership functions using the $$\max $$ and $$\min $$ functions. In this paper, we shall consider the set operations of fuzzy sets based on the concepts of gradual sets and gradual elements. When the fuzzy sets can be formulated as consisting of gradual elements like the usual set consisting of usual elements, the intersection and union of fuzzy sets can be defined as the same way as the intersection and union of usual sets. In this case, the set operations of fuzzy sets will be similar to the set operations of crisp sets.

Intersection and union of non-normal fuzzy sets using aggregation functions

Intersection and union of non-normal fuzzy sets using aggregation functions

Approximation of real data by fuzzy sets for the classification problem

The article deals with the method of classification of real data using the apparatus of fuzzy sets and fuzzy logic as a flexible tool for learning and recognition of natural objects on the example of oil and gas prospecting sections of the Dnieper-Donetsk basin. The real data in this approach are the values for the membership function that are obtained not through subjective expert judgment but from objective measurements. It is suggested to approximate the fuzzy set membership functions by using training data to use the approximation results obtained during the learning phase at the stage of identifying unknown objects. In the first step of learning, each traditional future of a learning data is matched by a primary traditional one-dimensional set whose membership function can only take values from a binary set — 0 if the learning object does not belong to the set, and 1 if the learning object belongs to the set. In the second step, the primary set is mapped to a fuzzy set, and the parameters of the membership function of this fuzzy set are determined by approximating this function of the traditional set membership. In the third step, the set of one-dimensional fuzzy sets that correspond to a single feature of the object is mapped to a fuzzy set that corresponds to all the features of the object in the training data set. Such a set is the intersection of fuzzy sets of individual features, to which the blurring and concentration operations of fuzzy set theory are applied in the last step. Thus, the function of belonging to a fuzzy set of a class is the operation of choosing a minimum value from the functions of fuzzy sets of individual features of objects, which are reduced to a certain degree corresponding to the operation of blurring or concentration. The task of assigning the object under study to a particular class is to compare the values of the membership functions of a multidimensional fuzzy set and to select the class in which the membership function takes the highest value. Additionally, after the training stage, it is possible to determine the degree of significance of an object future, which is an indistinctness index, to remove non-essential data (object futures) from the analysis.

Longitudinal measures of fuzzy poverty: a focus on Czechia, Hungary and Poland after the crisis

The main scope of this paper consists in presenting an original contribution for the estimation of measures of monetary and multi-dimensional poverty in a longitudinal context; we also provide empirical on how this could be very useful in explaining the different recovery in three Visegrad Countries from the 2008 crisis. The traditional approach of poverty, defined by the dichotomy (poor, non-poor) is replaced by a defining based on the concept “a matter of degree”, which could briefly be defined as the position of an individual in the income distribution, or in any other variable’s distribution taken into account in a multidimensional approach. Another original contribution of the paper consists in proposing rules for the intersection and union of fuzzy sets, which we demonstrate to be appropriate for the study of poverty and deprivation, especially when the approach is applied to a time sequence in a longitudinal perspective; in particular, we propose the new ‘fuzzy “At-persistent-risk-of-poverty rate”, which can be seen as the fuzzy alter ego of the well-known Eurostat Laeken indicator.

A new approach of cubic interval-valued intuitionistic fuzzy subsemigroup and ideals

The main motivation of this paper to generalize the old concept. In this paper, the concept of the cubic interval-valued intuitionistic fuzzy set which is the extension of the cubic set is applied to semigroup and their related properties such as union, intersection, product, whole function and characteristic function of cubic interval-valued intuitionistic fuzzy sets are investigated. Moreover, the concept of cubic interval-valued intuitionistic fuzzy subsemigroup and cubic interval-valued intuitionistic fuzzy ideas are proposed. Characterizations of cubic interval-valued intuitionistic fuzzy ideas are considered, and how the image or inverse image of cubic interval-valued intuitionistic fuzzy subsemigroups and cubic interval-valued intuitionistic fuzzy ideals become cubic interval-valued intuitionistic fuzzy subsemigroups and cubic interval-valued intuitionistic fuzzy ideals, respectively, is studied.

On lattice ordered intuitionistic fuzzy soft sets

Lattices, soft sets, fuzzy sets and their generalizations have always been important for Mathematicians and the researchers working on uncertaities. In this paper our aim is to introduce the concept of lattice ordered intuitionistic fuzzy soft sets. After introducing extended union, extended intersection, AND-product, OR-product, basic union, basic intersection of intuitionistic fuzzy soft sets, in this paper the affects of lattice ordered intuitionistic fuzzy soft sets and anti-lattice ordered intuitionistic fuzzy soft sets on restricted union, restricted intersection, extended union, extended intersection,AND-product, OR-product, basic union, basic intersection of intuitionistic fuzzy sets are discussed. Further a decision making problem is solved by using these concepts.

Optimizing of Linear Problems Subjected to Sugeno - Weber FRI

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated where the feasible region is formed as the intersection of two inequality fuzzy systems and Sugeno-Weber family of t-norms is considered as fuzzy composition. SugenoWeber family of t-norms and t-conorms is one of the most applied one in various fuzzy modeling problems. This family of t-norms and t-conorms was suggested by Weber for modeling intersection and union of fuzzy sets. Also, the t-conorms were suggested as addition rules by Sugeno for socalled λ –fuzzy measures. The resolution of the feasible region of the problem is firstly investigated when it is defined with max-Sugeno-Weber composition. A necessary and sufficient condition and three other necessary conditions are derived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. A method is proposed to generate random feasible max-Sugeno-Weber fuzzy relational inequalities and an algorithm is presented to solve the problem. Finally, an example is described to illustrate these algorithms.

Optimizing of Linear Problems Subjected to SugenoWeber FRI

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated where the feasible region is formed as the intersection of two inequality fuzzy systems and Sugeno-Weber family of t-norms is considered as fuzzy composition. SugenoWeber family of t-norms and t-conorms is one of the most applied one in various fuzzy modeling problems. This family of t-norms and t-conorms was suggested by Weber for modeling intersection and union of fuzzy sets. Also, the t-conorms were suggested as addition rules by Sugeno for socalled λ –fuzzy measures. The resolution of the feasible region of the problem is firstly investigated when it is defined with max-Sugeno-Weber composition. A necessary and sufficient condition and three other necessary conditions are derived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. A method is proposed to generate random feasible max-Sugeno-Weber fuzzy relational inequalities and an algorithm is presented to solve the problem. Finally, an example is described to illustrate these algorithms.

Model of the level estimation of danger situations in the problems of complex objects operation

The model of information support estimation of the decision maker (DM) based on a study of qualitative indicators of informedness, which characterizes the completeness, accuracy and timeliness of DM informedness , is presented. On the basis of these characteristics the level of danger situations in the operation of complex objects is formed. In implementing the model are used the methods of one-parameter classification with operations of fuzzy sets and strong intersections, as well as methods of interval classification with the operations of union and intersection of fuzzy sets, and also with operations of strong union and intersection of fuzzy sets. The proposed methods allow classifying the situation to inform DM about the validity and security of the decision with the available information support. The proposed model is based on a study of qualitative characteristics of informedness of DM and can be used to estimate the level of danger situations in the problems of the complex objects operation.

Cubic Hesitant Fuzzy Sets and Their Applications to Multi Criteria Decision Making

In this paper we introduced cubic hesitant fuzzy set and defined internal (external) cubic hesitant fuzzy set, P(R)-union and P(R)-intersection of cubic hesitant fuzzy sets. Furthermore we defined P(R)-addition and P(R)-multiplication of cubic hesitant fuzzy sets. By using the defined operations of cubic hesitant fuzzy sets we proved their different results. We also defined R-weighted averaging and R-weighted geometric operators for cubic hesitant fuzzy sets and practiced it in multi-criteria decision making problem.

Multi-Objective Fuzzy Optimization Design of Gear Drive

fuzzy factors in the gear reducer design are analyzed, and a multi-objective optimization model with minimum volume and maximum bearing capacity is established in this paper. Optimal level cut-set method is introduced to conduct comprehensive evaluation on constraining fuzziness and convert the fuzzy constraint into general constraint according to the actual conditions and requirements; for multi-objectives, single-objective optimal solutions are fuzzified to form fuzzy sets to establish the membership functions of the fuzzy sets by taking the fuzziness between the single objectives and between the single objectives and multiple objectives into consideration, and multi-objective optimal solutions are obtained by maximizing the membership functions of the fuzzy set intersection. For an actual gear reducer, the fuzzy theory and method are applied to obtain results, and the validness of the method is demonstrated by comparing such results with the conventional design data.

Extension of Intuitionistic Fuzzy Sets to New Universal

Atanassov [1] defined the union, intersection and inclusion of intuitionistic fuzzy sets on the same universal. In [2], Riecan and Atanassov defined new intuitionistic fuzzy sets on different universals and they examined the union, intersection and inclusion of these sets. In this paper we gave counterexamples for the results of union, intersection and inclusion of these sets. For these situations we defined new intuitionistic fuzzy sets. In addition, we studied some properties of new intuitionistic fuzzy sets to be considered in their new universals.

Web usage mining with evolutionary extraction of temporal fuzzy association rules

In Web usage mining, fuzzy association rules that have a temporal property can provide useful knowledge about when associations occur. However, there is a problem with traditional temporal fuzzy association rule mining algorithms. Some rules occur at the intersection of fuzzy sets’ boundaries where there is less support (lower membership), so the rules are lost. A genetic algorithm (GA)-based solution is described that uses the flexible nature of the 2-tuple linguistic representation to discover rules that occur at the intersection of fuzzy set boundaries. The GA-based approach is enhanced from previous work by including a graph representation and an improved fitness function. A comparison of the GA-based approach with a traditional approach on real-world Web log data discovered rules that were lost with the traditional approach. The GA-based approach is recommended as complementary to existing algorithms, because it discovers extra rules.