Atanassov's Intuitionistic Fuzzy Research Articles

Schweizer-Sklar power aggregation operators based on complex intuitionistic fuzzy information and their application in decision-making

In 1960, Schweizer and Sklar introduced the novel Schweizer-Sklar t-norm and t-conorm which is used in the construction of aggregation operators. Schweizer-Sklar norms are more general than algebraic norms and Einstein norms. Additionally, computing the power operators based on the Schweizer-Sklar norms for complex Atanassov intuitionistic fuzzy (CA-IF) set is very awkward and complicated. In this manuscript, firstly, we propose the Schweizer-Sklar operational laws for CA-IF values, and secondly, we develop the CA-IF Schweizer-Sklar power averaging (CA-IFSSPA) operator, CA-IF Schweizer-Sklar power ordered averaging (CA-IFSSPOA) operator, CA-IF Schweizer-Sklar power geometric (CA-IFSSPG) operator, and CA-IF Schweizer-Sklar power ordered geometric (CA-IFSSPOG) operator. Some suitable and dominant properties for the above operators are also discussed. Furthermore, to simplify the above operators, we develop the procedure of decision-making technique, called multi-attribute decision-making (MADM) methods based on the proposed operators based on CA-IF values. Finally, we compare the proposed methods with some existing methods to describe the efficiency and capability of the discovered approaches by some examples.

Power aggregation operators based on hamacher t-norm and t-conorm for complex intuitionistic fuzzy information and their application in decision-making problems

Algebraic and Einstein are two different types of norms which are the special cases of the Hamacher norm. These norms are used for evaluating or constructing three different types of aggregation operators, such as averaging/geometric, Einstein averaging/geometric, and Hamacher averaging/geometric aggregation operators. Moreover, complex Atanassov intuitionistic fuzzy (CA-IF) information is a very famous and dominant technique or tool which is used for depicting unreliable and awkward information. In this manuscript, we present the Hamacher operational laws for CA-IF values. Furthermore, we derive the power aggregation operators (PAOs) for CA-IF values, called CA-IF power Hamacher averaging (CA-IFPHA), CA-IF power Hamacher ordered averaging (CA-IFPHOA), CA-IF power Hamacher geometric (CA-IFPHG), and CA-IF power Hamacher ordered geometric (CA-IFPHOG) operators. Some dominant and valuable properties are also stated. Moreover, the multi-attribute decision-making (MADM) methods are developed based on the invented operators for CA-IF information and the detailed decision steps are given. Many prevailing operators are selected as special cases of the invented theory. Finally, the derived technique will offer many choices to the expert to evaluate the best alternatives during comparative analysis.

TOPSIS Method Based on Hamacher Choquet-Integral Aggregation Operators for Atanassov-Intuitionistic Fuzzy Sets and Their Applications in Decision-Making

The collection of Hamacher t-norms was created by Hamacher in 1970, which played a critical and significant role in computing aggregation operators. All aggregation operators that are derived based on Hamacher norms are very powerful and are beneficial because of the parameter 0≤ζ≤+∞. Choquet first posited the theory of the Choquet integral (CI) in 1953, which is used for evaluating awkward and unreliable information to address real-life problems. In this manuscript, we analyze several aggregation operators based on CI, aggregation operators, the Hamacher t-norm and t-conorm, and Atanassov intuitionistic fuzzy (A-IF) information. These are called A-IF Hamacher CI averaging (A-IFHCIA), A-IF Hamacher CI ordered averaging (A-IFHCIOA), A-IF Hamacher CI geometric (A-IFHCIG), and A-IF Hamacher CI ordered geometric (A-IFHCIOG) operators; herein, we identify their most beneficial and valuable results according to their main properties. Working continuously, we developed a multi-attribute decision-making (MADM) procedure for evaluating awkward and unreliable information, with the help of the TOPSIS technique for order performance by similarity to the ideal solution, and derive operators to enhance the worth and value of the present information. Finally, by comparing the pioneering information with some of the existing operators, we illustrate some examples for evaluating the real-life problems related to enterprises, wherein the owner of a company appointed four senior board members of the enterprise to decide what was the best Asian company in which to invest money, to show the supremacy and superiority of the invented approaches.

An Efficient Malware Classification Method Based on the AIFS-IDL and Multi-Feature Fusion

In recent years, the presence of malware has been growing exponentially, resulting in enormous demand for efficient malware classification methods. However, the existing machine learning-based classifiers have high false positive rates and cannot effectively classify malware variants, packers, and obfuscation. To address this shortcoming, this paper proposes an efficient deep learning-based method named AIFS-IDL (Atanassov Intuitionistic Fuzzy Sets-Integrated Deep Learning), which uses static features to classify malware. The proposed method first extracts six types of features from the disassembly and byte files and then fuses them to solve the single-feature problem in traditional classification methods. Next, Atanassov’s intuitionistic fuzzy set-based method is used to integrate the result of the three deep learning models, namely, GRU (Temporal Convolutional Network), TCN (Temporal Convolutional Network), and CNN (Convolutional Neural Networks), which improves the classification accuracy and generalizability of the classification model. The proposed method is verified by experiments and the results show that the proposed method can effectively improve the accuracy of malware classification compared to the existing methods. Experiments were carried out on the six types of features of malicious code and compared with traditional classification algorithms and ensemble learning algorithms. A variety of comparative experiments show that the classification accuracy rate of integrating multi-feature, multi-model aspects can reach 99.92%. The results show that, compared with other static classification methods, this method has better malware identification and classification ability.

Principal Component Analysis and Factor Analysis for an Atanassov IF Data Set

The present contribution is devoted to the theory of fuzzy sets, especially Atanassov Intuitionistic Fuzzy sets (IF sets) and their use in practice. We define the correlation between IF sets and the correlation coefficient, and we bring a new perspective to solving the problem of data file reduction in case sets where the input data come from IF sets. We present specific applications of the two best-known methods, the Principal Component Analysis and Factor Analysis, used to solve the problem of reducing the size of a data file. We examine input data from IF sets from three perspectives: through membership function, non-membership function and hesitation margin. This examination better reflects the character of the input data and also better captures and preserves the information that the input data carries. In the article, we also present and solve a specific example from practice where we show the behavior of these methods on data from IF sets. The example is solved using R programming language, which is useful for statistical analysis of data and their graphical representation.

Robust moving object detection based on fusing Atanassov's Intuitionistic 3D Fuzzy Histon Roughness Index and texture features

Background modeling is a crucial step in various computer vision applications such as video surveillance, object tracking, and moving object detection. Classifying image pixels as foreground or background is yet a challenging task particularly in complicated situations such as illumination variations, rippling water, camera jitter, and the presence of fast and slow moving objects. Therefore, for a better detection of the moving objects, the multi-modal nature of a scene in those intricate situations should be modeled by multiple models for each image pixel. To this end, in this article, we improve our previous work by fusing color features and texture features using Choquet fuzzy integral. Thereby, our proposed spatial color features that are described by Atanassov's Intuitionistic 3D Fuzzy Histon Roughness Index are fused by the texture features extracted using a covariance matrix. As handling multi-modal background updating is an arduous task, we also propose a new model updating for tackling various challenges such as model initializing with moving objects, existence of fast and slow moving objects in a scene, and existence of the moving objects that stop for a while. We intensively evaluate our proposed approach on diverse benchmark datasets. Experimental results demonstrate the robustness and supremacy of our proposed approach compared to its previous version and the state-of-the-art algorithms in the field.

Multi-criteria decision making problem for evaluating ERP system using entropy weighting approach and q-rung orthopair fuzzy TODIM

The q-rung orthopair fuzzy set theory is a flexible tool for dealing with the ambiguity of human judgements information as compared to the fuzzy, Atanassov intuitionistic fuzzy and Pythagorean fuzzy models. However, there is very few investigation for the entropy measure of q-rung orthopair fuzzy sets. Therefore, this paper firstly proposed a new entropy measure for q-rung orthopair fuzzy set along with their elegant properties. By changing the parameter q, $$q\ge 1$$ , q-rung orthopair fuzzy set can adjust the range of indication of decision information. Based on the proposed entropy measure, we proposed a new method to deal with MCDM problems under the q-rung orthopair fuzzy environment where the information about criteria weights is are partially known. We extend the TODIM (an Acronym in Portuguese of interactive and multicriteria decision making) approach for solving the multi-criteria decision-making problems (MCDM) where the behaviour of specialists are taken into account and elements of a set are interdependent. In addition, a numerical example is provided in solving real-life problems to illustrate the highlight of this study. The experimental and comparative results show the effectiveness and flexibility of the developed approach in solving real-life problems.

An Atanassov intuitionistic fuzzy programming method for group decision making with interval-valued Atanassov intuitionistic fuzzy preference relations

The focus of this paper is on group decision making (GDM) problems with interval-valued Atanassov intuitionistic fuzzy preference relations (IV-AIFPRs). A new consistency index of an AIFPR is introduced to check the additive consistency degree of an AIFPR. Then, an additive consistency definition and an acceptable additive consistency definition of an IV-AIFPR are respectively defined by splitting an IV-AIFPR into two AIFPRs. For several IV-AIFPRs with unacceptably additive consistency, a goal program-based approach is proposed to improve their consistency simultaneously. Employing consistency degrees of individual IV-AIFPRs, decision makers’ (DMs’) weights are determined objectively and applied to integrate individual IV-AIFPRs into a collective one. Further, it is proved that the collective IV-AIFPR is acceptably additive consistent if all individual IV-AIFPRs are acceptably additive consistent. To derive priority weights of alternatives, an Atanassov intuitionistic fuzzy programming model is established and solved by three approaches considering DMs’ different risk attitudes. Thus, a novel method is put forward for GDM with IV-AIFPRs. A material selection example is analyzed to verify the effectiveness of the proposed method.

Linguistic Interval-Valued Atanassov Intuitionistic Fuzzy Sets and Their Applications to Group Decision Making Problems

In this paper, we propose the concept of a linguistic interval-valued Atanassov intuitionistic fuzzy set (LIVAIFS), whose membership and nonmembership degrees are represented by the interval-valued linguistic terms, for better dealing with imprecise and uncertain information during the decision-making process. In it, first some operational laws, score, and accuracy functions of LIVAIFS are defined with a brief study of related properties. Then, based on these operational laws, several aggregating operators are proposed to aggregate the linguistic interval-valued Atanassov intuitionistic fuzzy (LIVAIF) information. Some properties and inequalities are established to show the effectiveness and validity of proposed operators. Furthermore, a group decision making (GDM) approach, based on proposed operators, has been presented to solve the multiattribute GDM problems under LIVAIF environment. Finally, an illustrative example has been presented to demonstrate the approach.

Information Measures in Atanassov's Intuitionistic Fuzzy Environment and Their Application in Decision Making

This article considers some information measures such as normalized divergence measure, similarity measure, dissimilarity measure, and normalized distance measure in an intuitionistic fuzzy environment (IFE), which measure the uncertainty and hesitancy, and which can be applied to the selection of alternatives in group decision problems. We introduce and study the continuity of considered measures. Next, we prove some results that can be used to generate measures for fuzzy sets as well as for Atanassov's intuitionistic fuzzy sets and we also prove some approaches to construct point measures from set measures in IFE. We define the weight set for one and many preference orders of alternatives. Next, we investigate the properties and results related to the weight set. Based on the weight set, we develop a model for finding the uncertain weights corresponding to attributes. Also, we develop a model to finding positive certain weights corresponding to each attribute by using uncertain weights. Finally, an algorithm for choosing the best alternative according to the preference orders of alternatives in decision-making problems is proposed and its validity is shown with the help of a numerical example.

From Fuzzy Sets to Interval-Valued and Atanassov Intuitionistic Fuzzy Sets: A Unified View of Different Axiomatic Measures

This paper examines a broad collection of axiomatic definitions from various and diverse contexts within the domain of fuzzy sets (FS) to evaluate their respective extensions to the case of interval-valued fuzzy (IVF) sets and intuitionistic fuzzy (IF) sets from a purely formal point of view. We conclude that a large number of such extensions follow similar formal procedures. This fact allows us to formulate a general procedure that encompasses all the reviewed extensions as particular cases of it. The new general formulation allows us to identify three different procedures to derive the corresponding extension to the field of IVF or IF sets from a specific real-valued measure in the context of FSs. These three processes agglutinate a multitude of particular constructions found in the literature.

An Entropy-Based Knowledge Measure for Atanassov's Intuitionistic Fuzzy Sets and Its Application to Multiple Attribute Decision Making.

As the complementary concept of intuitionistic fuzzy entropy, the knowledge measure of Atanassov’s intuitionistic fuzzy sets (AIFSs) has attracted more attention and is still an open topic. The amount of knowledge is important to evaluate intuitionistic fuzzy information. An entropy-based knowledge measure for AIFSs is defined in this paper to quantify the knowledge amount conveyed by AIFSs. An intuitive analysis on the properties of the knowledge amount in AIFSs is put forward to facilitate the introduction of axiomatic definition of the knowledge measure. Then we propose a new knowledge measure based on the entropy-based divergence measure with respect for the difference between the membership degree, the non-membership degree, and the hesitancy degree. The properties of the new knowledge measure are investigated in a mathematical viewpoint. Several examples are applied to illustrate the performance of the new knowledge measure. Comparison with several existing entropy and knowledge measures indicates that the proposed knowledge has a greater ability in discriminating different AIFSs and it is robust in quantifying the knowledge amount of different AIFSs. Lastly, the new knowledge measure is applied to the problem of multiple attribute decision making (MADM) in an intuitionistic fuzzy environment. Two models are presented to determine attribute weights in the cases that information on attribute weights is partially known and completely unknown. After obtaining attribute weights, we develop a new method to solve intuitionistic fuzzy MADM problems. An example is employed to show the effectiveness of the new MADM method.

Novel Adaptive Clustering Algorithms Based on a Probabilistic Similarity Measure Over Atanassov Intuitionistic Fuzzy Set

This paper presents a novel probabilistic similarity measure (PSM) for Atanassov intuitionistic fuzzy sets. It then exploits PSM to propose an adaptive probabilistic similarity degree and develops the novel probabilistic $\lambda$ -cutting algorithm for clustering. Further, the probabilistic distance measure (obtained from the PSM) is used to develop a new clustering technique, which we have named “probabilistic intuitionistic fuzzy c-mean (PIFCM) algorithm”. Simulation experiments have been conducted over a variety of datasets including UCI machine learning datasets and real-world car dataset. The results obtained have been thoroughly compared with other well-known clustering techniques such as fuzzy c-mean (FCM), intuitionistic fuzzy c-mean, association coefficient method, and $\lambda$ -cutting method. Based upon the experimental results, it can be concluded that our probabilistic $\lambda$ -cutting algorithm and PIFCM algorithm outperform their existing counterparts.

Three Categories of Set-Valued Generalizations From Fuzzy Sets to Interval-Valued and Atanassov Intuitionistic Fuzzy Sets

Many different notions included in the fuzzy set literature can be expressed in terms of functionals defined over collections of tuples of fuzzy sets. During the past decades, different authors have independently generalized those definitions to more general contexts, like interval-valued fuzzy sets and Atanassov intuitionistic fuzzy sets. These generalized versions can be introduced either through a list of axioms or in a constructive manner. We can divide them into two further categories: set-valued and point-valued generalized functions. Here, we deal with constructive set-valued generalizations. We review a long list of functions, sometimes defined in quite different contexts, and we show that we can group all of them into three main different categories, each of them satisfying a specific formulation. We respectively call them the set-valued extension, the max–min extension, and the max–min varied extension. We conclude that the set-valued extension admits a disjunctive interpretation, whereas the max–min extension can be interpreted under an ontic perspective. Finally, the max–min varied extension provides a kind of compromise between both approaches.

Ambika Methods for Solving Matrix Games With Atanassov's Intuitionistic Fuzzy Payoffs

In the last few years, a lot of researchers have proposed different methods, to solve the mathematical programming problem of matrix games with Atanassov's intuitionistic fuzzy payoffs. In this paper, the flaws of the existing methods for solving matrix games with Atanassov's intuitionistic fuzzy payoffs (matrix games in which payoffs are represented by Atanassov's intuitionistic fuzzy numbers) are pointed out. Also, to resolve these flaws, new methods (named as Ambika methods) are proposed to obtain the optimal strategies as well as minimum expected gain of Player I and maximum expected loss of Player II for matrix games with Atanassov's intuitionistic fuzzy payoffs. To illustrate proposed Ambika methods, some existing numerical problems of matrix games with Atanassov's intuitionistic fuzzy payoffs are solved by proposed Ambika methods.

Atanassov's intuitionistic fuzzy histon for robust moving object detection

Histon and feature extracted by it, Basic Histon Roughness Index (BHRI), have been previously employed in image segmentation and moving object detection with outstanding performances. This work incorporates Atanassov's Intuitionistic Fuzzy Sets (A-IFS) theory to the concept of histon and extends 3D Basic Histon Roughness Index (3DBHRI) to Atanassov's Intuitionistic 3D Basic Histon Roughness Index. In addition, an adaptive Gaussian membership function is proposed for constructing Atanassov's Intuitionistic 3D Fuzzy Histon Roughness Index (A-IA3DFHRI). A-IFS provides a flexible framework to deal with hesitancy along with the vagueness originating from the imperfect, imprecise and noisy information. The proposed model has been rigorously tested and compared with several previous state-of-the-art models and shows significant performance improvements.

Additive consistent interval-valued Atanassov intuitionistic fuzzy preference relation and likelihood comparison algorithm based group decision making

This paper investigates a group decision making (GDM) method based on additive consistent interval-valued Atanassov intuitionistic fuzzy (IVAIF) preference relations (IVAIFPRs) and likelihood comparison algorithm. Firstly, the likelihood of IVAIF values (IVAIFVs) is defined by the likelihood of intervals. Then a likelihood comparison algorithm is designed to rank IVAIFVs. According to the additive consistent interval fuzzy preference relation, we define the additive consistency of an IVAIFPR. Two special interval fuzzy preference relations are extracted from an IVAIFPR. They can be regarded as the lowest and highest preferred matrices of the IVAIFPR, respectively. Using a parametric linear program, the IVAIF priority weights of an IVAIFPR are generated from these two extracted special interval fuzzy preference relations. For the GDM with IVAIFPRs, the group consensus is defined by the distances between the individual IVAIFPRs and the collective one. To derive decision makers' weights, an optimization model is constructed by maximizing the group consensus and transformed into a linear program to resolve. Subsequently, utilizing the IVAIF weighted averaging operator, the collective IVAIFPR is obtained and applied to obtain the IVAIF priority weights. The order of alternatives is generated by ranking the IVAIF priority weights. At length, an enterprise resource planning system selection example is analyzed to verify the effectiveness of the proposed method.

Atanassov's Intuitionistic Fuzzy Bi-Normed KU-Subalgebras of a KU-Algebra

In this paper, by using the $t$-norm $T$ and $t$-conorm $S$, we introduce the intuitionistic fuzzy bi-normed $KU$-subalgebras of a $KU$-algebra. Some properties of intuitionistic fuzzy bi-normed $KU$-subalgebras of a $KU$-algebra under the homomorphism are discussed. The direct product and the $(T,S)$-product of intuitionistic fuzzy bi-normed $KU$-subalgebras are particularly investigated.

Analysis of fuzzy Hamacher aggregation functions for uncertain multiple attribute decision making

As generalizations of algebraic and Einstein t-norms and t-conorms, Hamacher t-norm and t-conorm have been widely applied in fuzzy multiple attribute decision making (MADM) to combine assessments on each attribute, which are generally expressed by Atanassov's intuitionistic fuzzy (AIF) numbers, interval-valued intuitionistic fuzzy (IVIF) numbers, hesitant fuzzy (HF) elements, and dual hesitant fuzzy (DHF) elements. Due to the fact that AIF numbers and HF elements are special cases of IVIF numbers and DHF elements, respectively, two propositions can be established from analyzing numerical examples and real cases concerning MADM with IVIF and DHF assessments in the literature: (1) the monotonicity of alternative scores derived from Hamacher arithmetic and geometric aggregation operators with respect to the parameter r in Hamacher t-norm and t-conorm; and (2) the relationship between alternative scores generated by Hamacher arithmetic and geometric aggregation operators, given the same r. Here, we provide the theoretical proof of these two propositions in the context of MADM with IVIF and DHF assessments. With the theoretical support of these propositions, the meaning of r in MADM is explained, and a new method is proposed to compare alternatives in MADM with consideration of all possible values of r. Two numerical examples are solved by the proposed method and the other two existing methods to demonstrate the applicability and validity of the proposed method and highlight its advantages.

Knowledge Measure for Atanassov's Intuitionistic Fuzzy Sets

A measure of knowledge is often viewed as a dual measure of entropy in a fuzzy system; thus, it appears that the less entropy may always accompany the greater amount of knowledge. Actually, this does not reflect the reality in the context of Atanassov's intuitionistic fuzzy sets (A-IFSs). In this paper, we introduce a novel axiomatic framework for measuring the amount of knowledge associated with A-IFSs, as opposed to a measure of fuzzy entropy. We present an axiomatic definition of knowledge measure for A-IFSs first and then develop a new robust model that strictly complies with these axioms. More efforts are made to form the main properties of two types of axioms (respectively, for fuzzy entropy and knowledge measure) into a unified framework, under which the numerical relationship between these two kinds of measures is discussed in considerable detail. This helps to clear up a fundamental misunderstanding aforementioned and ultimately to draw a firm conclusion on this topic. In particular, the developed model, for its excellent performance in experiments as well as ability to capture the unique features of A-IFSs, can be used to tackle some special problems that are difficult to handle by using fuzzy entropy alone, such as making a difference between such special cases in which there are a large number of arguments in favor but an equally large number of arguments in disapproval at the same time.