Atanassov's Intuitionistic Fuzzy Sets 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.

Identification of a sufficient number of the best attributes in the intuitionistic fuzzy models

Dimension reduction of the models, i.e., pointing out only the necessary number of input variables (attributes, features) is an important task enabling the efficient performance of different algorithms. This paper is a continuation of our previous works on a new method of selection of the attributes in the models making use of Atanassov's intuitionistic fuzzy sets. We consider classification problems trying to point out the reduced number of the attributes and still obtain satisfactory results. We investigate the previously proposed method in more details comparing its performance with a well-known method of extraction parameters, namely Principal Component Analysis (PCA), and with a well-known method of selecting the attributes in which the so-called Gain Ratio is used. We illustrate our considerations using benchmark data from UCI Machine Learning Repository.

Global intuitionistic fuzzy weighted C-ordered means clustering algorithm

The paper proposes a novel approach to address the challenges of clustering datasets and identifying outliers by utilizing the Atanassov intuitionistic fuzzy sets (AIFS) environment. The approach provides a more flexible and nuanced solution by incorporating a new function called the typicality function for outlier detection and tuning the parameters used in defining it to improve clustering. To optimize the parameter tuning process and reduce time complexity, the paper introduces a global error search approach (k-GESA) within a finite space. The paper also presents a new clustering algorithm, called Global Intuitionistic Fuzzy Weighted C-Ordered Means (Global-IFWCOM), which utilizes k-GESA to improve the clustering results. The proposed approach is evaluated against various C-ordered means algorithms, including Fuzzy C-ordered means (FCOM), Intuitionistic fuzzy C-ordered means (IFCOM), Intuitionistic fuzzy weighted C-ordered means (IFWCOM), Fuzzy weighted C-ordered means (FWCOM), and Hesitant-FWCOM, on a two-dimensional synthetic dataset with outliers. Furthermore, the effectiveness of Global-IFWCOM is demonstrated using a six-dimensional synthetic dataset with noise and outliers compared to FWCOM.

Power Aggregation Operators of Interval-Valued Atanassov-Intuitionistic Fuzzy Sets Based on Aczel–Alsina t-Norm and t-Conorm and Their Applications in Decision Making

Aczel–Alsina t-norm and t-conorm are important t-norm and t-conorm, and they are extended from algebraic t-norm and t-conorm. Obviously, Aczel–Alsina t-norm and t-conorm are more general than some existing t-norm and t-conorm. Furthermore, the power aggregation (PA) operator is also a very famous and valuable operator which can consider the power relation between any two input parameters. In addition, Interval-valued Atanassov-intuitionistic fuzzy set (IVA-IFS) can easily express uncertain information. In order to fully use their advantages, in this analysis, we extend the PA operators based on Aczel–Alsina t-norm and t-conorm to IVA-IFS and propose the interval-valued Atanassov-intuitionistic fuzzy Aczel–Alsina power averaging (IVA-IFAAPA), interval-valued Atanassov-intuitionistic fuzzy Aczel–Alsina power ordered averaging (IVA-IFAAPOA), interval-valued Atanassov-intuitionistic fuzzy Aczel–Alsina power geometric (IVA-IFAAPG) and interval-valued Atanassov-intuitionistic fuzzy Aczel–Alsina power ordered geometric (IVA-IFAAPOG) operators. Moreover, we discuss the properties of the presented operators such as idempotency, monotonicity, and boundedness. In addition, a multi-attribute decision-making (MADM) procedure is proposed to process the IVA-IF information. Finally, a practical example is used to show the effectiveness and superiority of the proposed method by comparing it with some existing operators.

A new situation assessment method for aerial targets based on linguistic fuzzy sets and trapezium clouds

Situation assessment has attracted much attention. The chief aim of this paper is to research the problem from the perspective of multicriteria group decision-making (MCGDM). First of all, this paper expresses the threat information with linguistic fuzzy sets, including Atanassov’s intuitionistic fuzzy sets (AIFSs), Atanassov’s interval-valued intuitionistic fuzzy sets (AIVIFSs), and interval intuitionistic uncertain linguistic sets (IIULSs). Then, some relevant concepts and aggregation operators of trapezium clouds are proposed, meanwhile, a new approach that calculates the weights considering subjectivity and objectivity is presented with multi-objective programming. Next, the conversion methods between AIFSs, AIVIFSs, IIULSs and trapezium clouds are derived. Finally, the algorithm process is presented and an illustrative example is employed to prove the validity. Also, the comparison analysis with other widely used methods is conducted to verify the feasibility of the proposed algorithm.

Some novel point operators and multiple rounds voting process based decision-making algorithm under picture fuzzy set environment

Picture fuzzy set (PFS), a generalization of the Atanassov-intuitionistic fuzzy set, is one of the most trustworthy tool to handle the uncertainties in the data during the decision-making process. The objective of this paper is to present some different kinds of the point operators for the PFS. The key feature of the presented point operators is to provide a smaller degree of uncertainty which is equivalent to the corresponding PFS. The desirable properties of the proposed operators are also derived in details. Also, we provide a memoryless multiple rounds voting process under the PFS environment to strength the result, and hence deduce a relationship between the proposed PFS and the existing set. In addition to this, we address the MADM (multiple attribute decision-making) problems by presented an algorithm to solve the problem based on the proposed point operators. The applicability and the superiority of the proposed algorithm are shown through two numerical examples. At last, the validity of the proposed algorithm is tested using different test criteria.

Another view on knowledge measures in atanassov intuitionistic fuzzy sets

Another view on knowledge measures in atanassov intuitionistic fuzzy sets

Positron Emission Tomography Image Segmentation Based on Atanassov’s Intuitionistic Fuzzy Sets

In this paper, we present an approach to fully automate tumor delineation in positron emission tomography (PET) images. PET images play a major role in medicine for in vivo imaging in oncology (PET images are used to evaluate oncology patients, detecting emitted photons from a radiotracer localized in abnormal cells). PET image tumor delineation plays a vital role both in pre- and post-treatment stages. The low spatial resolution and high noise characteristics of PET images increase the challenge in PET image segmentation. Despite the difficulties and known limitations, several image segmentation approaches have been proposed. This paper introduces a new unsupervised approach to perform tumor delineation in PET images using Atanassov’s intuitionistic fuzzy sets (A-IFSs) and restricted dissimilarity functions. Moreover, the implementation of this methodology is presented and tested against other existing methodologies. The proposed algorithm increases the accuracy of tumor delineation in PET images, and the experimental results show that the proposed method outperformed all methods tested.

An extension of rule base evidential reasoning in the interval-valued intuitionistic fuzzy setting applied to the type 2 diabetes diagnostic

This paper presents an interval-valued intuitionistic fuzzy extension of rule base evidential reasoning which generally is a synthesis of fuzzy logic, the Dempster–Shafer theory of evidence (DST) and Atanassov’s intuitionistic fuzzy sets (A-IFS) theory redefined in the framework of DST. A lot of attention is paid to situations when in the solution of decision making problem the competing fuzzy classes with intersecting membership functions in antecedent parts of fuzzy rules, e.g., such as Low and Moderate play an important role in the problem formulation. As a result, a new mathematical object “belief interval bounded belief interval” (BIBBI) is obtained. The properties of BIBBI make it possible to use it as a more reliable representation of interval-valued intuitionistic fuzzy value than that formulated in terms of A-IFS. Using the introduced DST based definition of interval-valued A-IFS, the corresponding operations with BIBBIs and the DST representation of rule base evidential reasoning in the intuitionistic fuzzy setting, a new approach to the interval-valued intuitionistic fuzzy extension of rule base evidential reasoning is developed. To prove the correctness and applicability of this approach to the solution of decision making problems, a case study of its use for the type 2 diabetes diagnostics is analyzed.

Intuitionistic Fuzzy c-Ordered Means Clustering Algorithm

Atanassov intuitionistic fuzzy set (AIFS) has the capability to deal with various uncertain situations, so its popularity among researchers is quite high. It has been observed that Euclidean distance measure based AIFS clustering algorithms perform well on imprecise datasets. The performance of Euclidean measure based clustering algorithms deteriorates due to the presence of outliers/noise within a dataset. In the paper, an extension of the algorithm given by Leski, Jacek M. [Fuzzy Sets and Systems, 286 (2016): 114-133] is proposed as intuitionistic fuzzy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> -ordered means clustering algorithm. This paper analyses the functionality of clustering algorithms over outliers/noises based datasets. In IFCOM, an alternate of Euclidean distance known as Loss function is used. Moreover, IFCOM uses intuitionistic fuzzy OWA to combat the ill effects of the noises and outliers. The proposed algorithm exploits a typicality function based weighing ordering approach. The approach assigins lower weights to the outliers. Hence, the catastrophic behavior is not observed in IFCOM while dealing outliers possessing synthetic and UCI machine learning datasets.

Distance-Based Knowledge Measure for Intuitionistic Fuzzy Sets with Its Application in Decision Making.

Much attention has been paid to construct an applicable knowledge measure or uncertainty measure for Atanassov’s intuitionistic fuzzy set (AIFS). However, many of these measures were developed from intuitionistic fuzzy entropy, which cannot really reflect the knowledge amount associated with an AIFS well. Some knowledge measures were constructed based on the distinction between an AIFS and its complementary set, which may lead to information loss in decision making. In this paper, knowledge amount of an AIFS is quantified by calculating the distance from an AIFS to the AIFS with maximum uncertainty. Axiomatic properties for the definition of knowledge measure are extended to a more general level. Then the new knowledge measure is developed based on an intuitionistic fuzzy distance measure. The properties of the proposed distance-based knowledge measure are investigated based on mathematical analysis and numerical examples. The proposed knowledge measure is finally applied to solve the multi-attribute group decision-making (MAGDM) problem with intuitionistic fuzzy information. The new MAGDM method is used to evaluate the threat level of malicious code. Experimental results in malicious code threat evaluation demonstrate the effectiveness and validity of proposed method.

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.

A three-way decision based construction of shadowed sets from Atanassov intuitionistic fuzzy sets

By combining ideas from three-way decision and Pedrycz shadowed sets, this paper investigates two possible solutions to the problem of constructing a shadowed set from an Atanassov intuitionistic fuzzy set. The first solution treats the pair of membership and nonmembership functions as two evaluations and directly applies the two-evaluation based model of three-way decision. The second solution transforms the pair of membership and nonmembership functions into an evaluation and applies a single-evaluation based model of three-way decision. To address a fundamental issue of the determination of a pair of thresholds on evaluations, we present a general optimization based formulation and use Chebyshev distance to calculate the required thresholds. The results are useful for three-way decision with fuzzy sets, Atanassov intuitionistic fuzzy sets, and shadowed sets.

An Intuitionistic Fuzzy Clustering Approach for Detection of Abnormal Regions in Mammogram Images.

Breast cancer is one of the leading causes of mortality in the world and it occurs in high frequency among women that carries away many lives. To detect cancer, extraction or segmentation of lesions/tumors is required. Segmentation process is very crucial if the mammogram images are blurred or low contrast. This paper suggests a novel clustering approach for segmenting lesions/tumors in the mammogram images using Atanassov's intuitionistic fuzzy set theory. The algorithm initially converts an image to an intuitionistic fuzzy image using a novel intuitionistic fuzzy generator. From the intuitionistic fuzzy image, two membership intervals are computed. Then, using Zadeh's min t-conorm, a new membership function is computed. Using the new membership function, an interval type 2 fuzzy image is constructed. Two types of distance functions are used in clustering-intuitionistic fuzzy divergence and a fuzzy exponential type distance function. Further, in each iteration, membership matrix is updated using a hesitation degree and a clustered image is obtained. Tumors/lesions are then segmented from the clustered image. The proposed method is compared with existing methods both quantitatively and qualitatively and it is observed that the proposed method performs better than the existing methods.

Generalized intuitionistic fuzzy c-means clustering algorithm using an adaptive intuitionistic fuzzification technique

In real-world scenario, mostly, the datasets are either imprecise or uncertain in their original form. Due to this reason, the clustering of such datasets is unsatisfactory and we often get compromised results. The information present in the dataset can be made precise and useful with the popular generalization of fuzzy sets known as Atanassov intuitionistic fuzzy sets (AIFSs). Therefore, the AIFS-based c-means clustering algorithm becomes a convenient way for clustering uncertain and vague datasets. In the paper, we propose an intuitionistic fuzzy-based algorithm, namely Generalized Intuitionistic Fuzzy c-Means (G-IFCM) clustering algorithm which uses an adaptive AIFS Euclidean distance measure in its criterion function to cluster the dataset under intuitionistic fuzzy environment. The proposed intuitionistic fuzzification method incorporates a technique to transform the dataset into AIFS and maintains its original structure that tends to change during any fuzzification process. Further, simulation experiments are also conducted on few UCI machine learning repository datasets using G-IFCM and its performance is compared with some known clustering algorithms. The efficacy of the algorithm is tested with some popular benchmark indexes that check the cluster validity and clustering performance of G-IFCM. Finally, the computation suggests efficient performance of G-IFCM where the best choice of fuzzy factor, m probably lies in the interval [1, 2] that indicates reduction in the expense of computation.

Interplay between symmetry, convexity and negation of a probability distribution

Negation in general represents the contradiction and denial of anything. In classical logic, negation is identified as the logical connective that takes truthfulness to falsehood. However, there can be situations in which we have to deal with the possibility of any truth value besides true or false. The probability theory has been effective in modelling such situations. Therefore exploring the concept of probability functions from the negation point of view should be helpful in investigating new horizons related with a random phenomena. The negation of a probability distribution results from a transformation which reallocates the probability of each event equally among the other alternatives in a finite sample space. Proposed by Yager, this transformation possesses the maximum entropy allocation. Since any type of mixing increases the disorder(uncertainty) in a system, intuitively the uncertainty associated with the obtained negation must be greater than the uncertainty associated with the original probability distribution. A closer look at Yager's negation reveals that negation of probabilities are convex combinations of the probabilities in the original distribution and we have utilized this for estimating the unpredictability associated with the negation and the implications have been discussed in detail. Also, we have shown that the concept of negation can be helpful for estimation of symmetry and biasness in probability distributions. The binomial distribution has been taken as an example. An extension in case of Atanassov's intuitionistic fuzzy sets has also been discussed.

Multi-attribute decision making with Pythagorean fuzzy sets via conversions to intuitionistic fuzzy sets and ORESTE method

The aim of this paper is to study the conversions between Pythagorean fuzzy sets and Atanassov's intuitionistic fuzzy sets. Besides, an ORESTE method based on multi-attribute decision making with Pythagorean fuzzy sets is developed by utilising the developed conversions. In this paper, according to the geometric representations of Pythagorean fuzzy sets and Atanassov's intuitionistic fuzzy sets, two types of conversions between the two fuzzy sets are constructed, which are further used to derive information measures include entropy and cross-entropy measures of Pythagorean fuzzy sets. Then, by combining with the ORESTE method, a direct decision procedure for multi-attribute decision making with Pythagorean fuzzy information is developed. Finally, a numerical example of the evaluation of regional energy efficiency is shown to illustrate the feasibility and validity of the developed decision procedure.

Intuitionistic fuzzy approach for enhancement of low contrast mammogram images

AbstractMammogram image enhancement is very much necessary in diagnosing breast cancer or tumor at an early stage. Nonuniform illumination and low contrast images are commonly encountered in mammogram images. Conventional enhancement algorithms produce either some artifacts or cannot highlight minute details present in the images, particularly when dealing with mammogram images. In this article, we propose a new mammogram image enhancement scheme using Atanassov's intuitionistic fuzzy set (IFS) theory. IFS considers two uncertainties—membership and nonmembership degree apart from membership degree as in fuzzy set theory. As mammogram images are low contrast images and many of the image definitions are vague/unclear, so IFS theory may be suitable for better image enhancement. Initially, the image is transformed to an intuitionistic fuzzy image using a novel intuitionistic fuzzy generator. Hesitation degree is computed and using the hesitation degree, two membership levels are computed to form an interval type 2 fuzzy set. These two membership functions are then combined using Zadeh's fuzzy t‐conorm to form a new membership function. Threshold of interval type 2 fuzzy image is obtained using restricted equivalence function. Using the threshold, modified fuzzy hyperbolization is carried out. Real data experiments demonstrate that the proposed algorithm has better performance on contrast and visual quality of the images both quantitatively and qualitatively when compared with different existing methods.

Modified transform‐based gamma correction for MRI tumor image denoising and segmentation by optimized histon‐based elephant herding algorithm

AbstractMedical image processing is typically performed to diagnose a patient's brain tumor prior to surgery. In this study, a technique in denoising and segmentation was developed to improve medical image processing. The proposed approach employs multiple modules. In the first module, the noisy brain tumor image is transformed into multiple low‐ and high‐pass tetrolet coefficients. In the second module, multiple low‐pass tetrolet coefficients are applied through a modified transform‐based gamma correction method. Generalized cross‐validation is used on multiple high‐pass tetrolet coefficients to obtain the best threshold value. In the third module, all enhanced coefficients are applied to the partial differential equation method. In the final module, the denoised image is applied to Atanassov's intuitionistic fuzzy set histon‐based fuzzy clustering method with centroid optimization using an elephant herding method. Accordingly, the tumor part is segmented from the nontumor part in the magnetic resonance imaging brain images. The method was assessed in terms of peak signal‐to‐noise ratio, mean square error, specificity, sensitivity, and accuracy. The experimental results showed that the suggested method is superior to traditional methods.

Density estimates on the unit simplex and calculation of the mode of a sample

This paper addresses reliable and efficient calculation of the mode of a multivariate sample, which is a classical fusion function. In particular, we focus on the inputs given on the unit simplex, when aggregating elements of Atanassov intuitionistic fuzzy sets, interval-valued fuzzy sets and their extensions, as well as compositional data. We outline the use of a specially designed 2-additive fuzzy measures and the Choquet integral for the purposes of reducing computational complexity in higher dimensions. We present computational analysis and benchmark four different methods of density-based mode estimation.