Intuitionistic Fuzzy Distance Measure Research Articles

Novel strict intuitionistic fuzzy similarity measures-based on fuzzy negation and their applications

The concept of similarity measure is fundamental for evaluating the consistency between intuitionistic fuzzy sets (IFSs). Several researchers have developed various intuitionistic fuzzy similarity measures (IFSimMs) utilizing intuitionistic fuzzy distance measures (IFDisMs) or combinations of membership and non-membership degrees of IFSs. Nevertheless, there is still ample room for the improvement, as some of these measures do not satisfy the axiomatic properties of IFSimMs and produce unreasonable results. In this paper, we first establish a dual relationship between IFDisMs and IFSimMs based on fuzzy negations, enabling the construction of an infinite number of IFSimMs from a given IFDisM. We then propose two novel IFSimMs by directly manipulating the membership and non-membership degrees of IFSs and prove that they satisfy the axiomatic properties of strict IFSimMs (SIFSimMs). Furthermore, a comparative analysis reveals that the proposed SIFSimMs do not yield any unreasonable results in various scenarios. Finally, we apply these SIFSimMs to pattern recognition, medical diagnosis, and clustering analysis problems, demonstrating their superior performance compared to some existing measures.

An approach to extract topological information from Intuitionistic Fuzzy Sets and their application in obtaining a Natural Hierarchical Clustering Algorithm

Topological data analysis (TDA) is a powerful mathematical framework that extracts valuable insights about the shape and structure of complex datasets by identifying and analyzing underlying topological features, including connected components, holes, voids, and higher-dimensional counterparts. TDA achieves this by constructing a simplicial complex from the data, comprising simple shapes like points, lines, triangles, and higher-dimensional counterparts. The Vietoris–Rips complex (VRC), a widely used method, constructs simplicial complexes by measuring the distances between data points in a metric space. While the link between Intuitionistic Fuzzy Sets (IFSs) and TDA is clear through intuitionistic fuzzy distance measures, the literature has not explored the application of TDA for extracting topological features from uncertain and vague datasets using IFSs. In this paper, we propose a novel approach that leverages Intuitionistic Fuzzy Distance Measures (IFDMs) to generate the Intuitionistic Fuzzy Vietoris–Rips Complex (IFVRC) of IFSs. Specifically, we investigate the persistence of invariant relations of topological features between IFVRCs constructed using different distances, including Atanassov’s Euclidean distance (l2IFS), Boran and Akay’s distance (D), and Liu’s distance measure (dL), over artificially generated IFSs. Furthermore, we demonstrate that the generation of IFVRC can be viewed as a Natural Hierarchical Clustering Algorithm (NHCA) for IFSs. Finally, we apply the proposed IFVRC as a Natural Hierarchical Clustering Algorithm to cluster intuitionistic fuzzy datasets related to Car and Building materials. Moreover, we incorporate a relatively large dataset, the GTZAN dataset from the Kaggle datasets repository, to classify music genres based on their topological information.

Strict intuitionistic fuzzy distance/similarity measures based on Jensen-Shannon divergence

Being a pair of dual concepts, the normalized distance and similarity measures are important tools for decision-making and pattern recognition under the intuitionistic fuzzy set framework. In this paper, we first construct some counterexamples to illustrate that two existing similarity measures do not meet the axiomatic definition of intuitionistic fuzzy similarity measures. We then show that (1) these two measures cannot effectively distinguish some intuitionistic fuzzy values (IFVs); (2) except for the endpoints, there exist infinitely many pairs of IFVs, where the maximum distance “1” can be achieved under these two distances, leading to counter-intuitive results. To overcome these drawbacks, we introduce the concept of strict intuitionistic fuzzy distance measure (SIFDisM) and strict intuitionistic fuzzy similarity measure (SIFSimM), and propose an improved intuitionistic fuzzy distance measure based on Jensen-Shannon divergence. Moreover, we prove that (1) it is a SIFDisM; (2) its dual similarity measure is a SIFSimM; (3) its induced entropy is an intuitionistic fuzzy entropy. Comparative analysis and numerical examples demonstrate that our proposed distance measure is superior to the existing ones. In particular, our proposed distance measure can better distinguish and rank intuitionistic fuzzy sets.

An extended combined compromise solution framework based on novel intuitionistic fuzzy distance measure and score function with applications in sustainable biomass crop selection

Now-a-days, due to the increasing demand of the energy resources across the world, choosing the best sustainable biomass crop for the production of biofuels is a strategic decision-making problem. This is generally accepted that intuitionistic fuzzy sets (IFSs) are much more efficient in comparison with fuzzy sets at representing and processing the uncertainty in real-life problems. Proceeding on the same line, this paper attempts to introduce an extended combined compromise solution (CoCoSo) framework to analyze the sustainable biomass crop selection (SBCS) problem in an intuitionistic fuzzy environment. In this framework, we suggest a new integration function based on double normalization multiple aggregation approach to overcome the aggregation biases of the original CoCoSo approach and discuss its advantages with some numerical examples. We also develop a combined weighting strategy based on distance measure and decision experts’ (DEs) opinions to evaluate the significance of criteria . For this, we propose a novel distance measure (DM) and establish its superiority through some numerical comparisons. Also, the rationality of the suggested measure over the extant measures is justified by the use of an algorithm based on the developed measure for pattern recognition issues. In this framework, the comparison issue of IFSs is resolved by proposing a new score function. Furthermore, a case study of the SBCS is presented for the implementation of the developed CoCoSo approach, which confirms the viability and effectiveness of the new methodology. The results of the sensitivity analysis demonstrate that option “Miscanthus” consistently achieves the highest rank and is independent of variations of trade-off parameter and balancing factor. Finally, a comprehensive comparison is carried out to ensure the steadiness and reliability of the introduced framework.

Threat assessment of aerial targets based on improved GRA-TOPSIS method and three-way decisions.

Target threat assessment is a critical aspect of information warfare and can offer valuable auxiliary support to combat command decision-making. Aiming to address the shortcomings of three decision-making methods in air combat target assessment, such as the inability to effectively handle uncertain situation information and quantitatively rank the decision-making targets according to their importance, a dynamic intuitionistic fuzzy decision model based on the improved GRA-TOPSIS method and three-way decisions is proposed. First, the target attribute weight is obtained by cosine intuitionistic fuzzy entropy algorithm; then, a novel intuitionistic fuzzy distance measure is introduced, and grey incidence analysis and TOPSIS are used to build the conditional probability for three-way decisions that fully utilize the existing information and reflect the consistency of dynamic change trend; finally, the comprehensive loss function matrix is constructed and the threat classification results are obtained using the decision rules. The example analysis shows that the proposed method can not only effectively handle complex battlefield situations and dynamic uncertain information, but it can also classify targets, improving the effectiveness and rationality of decision-making and providing a reference basis for scientific command decision-making.

Topological analysis of intuitionistic fuzzy distance measures with applications in classification and clustering

The distance measure is a vital classifier used to solve classification and clustering problems in a metric space. In this paper, we discussed the types of distance measures and elaborated that they generate uniquely shaped balls. Due to their balls, they cannot guarantee a satisfactory classification outcome for a given dataset. To address this limitation, researchers used intuitionistic fuzzy sets (IFSs) to lose the boundary of belongingness of data points in a given metric space with respect to the chosen membership and non-membership functions and accomplished some novel intuitionistic fuzzy distance measures (IFDMs). IFDMs are successfully used in various applications; however, they also predict counterintuitive cases. To address this issue, in this paper, we conducted a topological analysis of intuitionistic fuzzy distance measures. To do that, we constructed three categories of distance measures; Type 1 distance measures, Type 2 distance measures, and Intuitionistic fuzzy distance measures (in this paper, we called them Intuitive Distance Measures (IDMs)). We further sub-categorized Intuitive Distance Measures into Type 1 Intuitive Distance Measures (T1IDMs) and Type 2 Intuitive Distance Measures (T2IDMs). We established a homeomorphic relation between the topological spaces generated by the T1IDMs and T2IDMs. Using the proposed homeomorphic relation and proving the necessary lemma and theorems, we presented the type 2 distance measures of well-known normable T1IDMs. We applied the proposed type 2 variants to solve some classification and clustering problems of machine learning. The result analysis shows that the proposed type 2 variants overcome the drawbacks of their type 1 counterparts.

Interval-valued intuitionistic fuzzy multi-attribute group decision-making method considering risk preference of decision-makers and its application

An improved interval-valued intuitionistic fuzzy multi-attribute group decision-making method considering the risk preference of decision-makers is proposed to solve the multi-attribute group decision-making problem with interval-valued intuitionistic fuzzy numbers and the condition that the attribute weight information is completely unknown. Firstly, the decision-maker weight of each attribute is determined by combining similarity and proximity. In order to consider the influence of the decision-maker's risk preference on the decision result and avoid the asymptotic behavior of interval-valued intuitionistic fuzzy matrix, the risk aversion coefficient of the decision-maker is introduced and combined with the determined decision-maker's weight aggregation to form a group decision matrix. Then, the information of group decision matrix is mined, and the interval-valued intuitionistic fuzzy entropy is used to determine the attribute weight and relative weight. Based on the interval-valued intuitionistic fuzzy distance measure formula and the TODIM method, the overall superiority of each scheme relative to other schemes is obtained by calculating the superiority between schemes, and the optimal scheme is determined by comparing and sequencing. Finally, the rationality and effectiveness of the proposed method are verified by an example of mechanical assembly supplier selection decision.

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In machine learning, distance measure plays an important role in defining the similarity between two data-items. In the paper, we discuss some of the drawbacks of distance measures (metrics) with their possibly induced clustering algorithms. Further, to overcome the drawbacks, we propose a novel intuitionistic fuzzy distance measure associated with generalized cesa´ro paranormed sequence space Cesq p(F). We also discuss some geometric properties of Cesq p(F). Moreover, the proposed distance measure is utilized in k-mean clustering algorithm to propose fuzzy c-mean clustering algorithm for Cesq p(F)

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.

A Divergence-Based Distance Measure for Intuitionistic Fuzzy Sets and Its Application in the Decision-Making of Innovation Management

The theory of intuitionistic fuzzy set has attracted much attention because of its stronger ability in depicting and modeling uncertainty. Recent years, numbers of distance measures of intuitionistic fuzzy sets have been proposed to distinguish the information conveyed by intuitionistic fuzzy sets. However, many existing distance measures cannot meet the requirements of a metric distance. So defining new distance measure for intuitionistic fuzzy sets is important to provide more choice in application. A new method to measure the distance between intuitionistic fuzzy sets is proposed in this paper. The proposed method is developed based on the divergence measure between intuitionistic fuzzy sets, which was firstly introduced by Shannon based on the idea of entropy. The mathematical properties of the new distance measure are discussed, and it is shown that the proposed distance measure satisfies all axiomatic properties of intuitionistic fuzzy distance measure. Numerical examples are presented to verify the effectiveness and reasonability of the new distance measure. Finally, the new distance measure is applied in innovation management to solve the decision making problems. It is illustrated that the proposed distance measure performs better than most of existing measures.

An Extension of the CODAS Approach Using Interval-Valued Intuitionistic Fuzzy Set for Sustainable Material Selection in Construction Projects with Incomplete Weight Information

Optimal selection of sustainable materials in construction projects can benefit several stakeholders in their respective industries with the triple bottom line (TBL) framework in a broader perspective of greater business value. Multiple criteria of social, environmental, and economic aspects should be essentially accounted for the optimal selection of materials involving the significant group of experts to avoid project failures. This paper proposes an evaluation framework for solving multi criteria decision making (MCDM) problems with incomplete weight information by extending the combinative distance assessment (CODAS) method with interval-valued intuitionistic fuzzy numbers. To compute the unknown weights of the evaluation criteria, this paper presents an optimization model based on the interval-valued intuitionistic fuzzy distance measure. In this study, we emphasize the importance of individual decision makers. To illustrate the proposed approach, an example of material selection in automotive parts industry is presented followed by a real case study of brick selection in sustainable building construction projects. The comparative study indicates the advantages of the proposed approach in comparison with the some relevant approaches. A sensitivity analysis of the proposed IVIF-CODAS method has been performed by changing the criteria weights, where the results show a high degree of stability.

Noise Robust Multiobjective Evolutionary Clustering Image Segmentation Motivated by the Intuitionistic Fuzzy Information

Images are always contaminated by noise, increasing uncertainty. Fuzzy set (FS) theory is a useful tool for dealing with uncertainty in images. When comparing with the FS, an intuitionistic fuzzy set (IFS) can better describe the blurred characteristic in images due to the membership, nonmembership, and hesitation degrees. However, when applied to an image segmentation, the IFS cannot completely overcome the influence of noise. With the aim of performing noisy image segmentation under several criteria, this paper defines a noise robust IFS (NR-IFS) for an image and then presents a novel noise robust multiobjective evolutionary intuitionistic fuzzy clustering algorithm (NR-MOEIFC). A majority dominated suppressed similarity measure using the neighborhood statistics and the competitive learning is proposed to obtain the NR-IFS representation for the image corrupted by noise. Then, the NR-IFS is fully used to motivate the whole process of multiobjective evolutionary clustering: first, computing a three-parameter intuitionistic fuzzy distance measure; second, constructing intuitionistic fuzzy fitness functions; third, designing a nonuniform intuitionistic fuzzy mutation operator; and forth, defining an intuitionistic fuzzy cluster validity index to select the optimal solution from the final nondominated solution set. The histogram statistics of NR-IFS are adopted in the NR-MOEIFC to greatly reduce the computational complexity. Experimental results on Berkeley and real magnetic resonance images reveal that the NR-MOEIFC behaves well in noise robustness and segmentation performance while requiring a low time cost.

Intuitionistic Fuzzy Distance Based TOPSIS Method and Application to MADM

In this paper, an intuitionistic fuzzy (IF) distance measure between two triangular intuitionistic fuzzy numbers (TIFNs) is developed. The metric properties of the proposed IF distance measure are also studied. Then, based on this IF distance, an extended TOPSIS is developed to solve multi-attribute decision making (MADM) problems with the ratings of alternatives on attributes of TIFNs. In this methodology, the IF distances between each alternative and the TIFN positive ideal-solution are calculated as well as the TIFN negative ideal-solution. Then the relative closeness degrees obtained of each alternative to the TIFN positive ideal solution are TIFNs. Based on the ranking methods of TIFNs the alternatives are ranked. A numerical example is examined to the validity and practicability of the method proposed in this paper.

A Distance Model of Intuitionistic Fuzzy Cross Entropy to Solve Preference Problem on Alternatives

In the field of decision-making, for the multiple attribute decision-making problem with the partially unknown attribute weights, the evaluation information in the form of the intuitionistic fuzzy numbers, and the preference on alternatives, this paper proposes a comprehensive decision model based on the intuitionistic fuzzy cross entropy distance and the grey correlation analysis. The creative model can make up the deficiency that the traditional intuitionistic fuzzy distance measure is easy to cause the confusion of information and can improve the accuracy of distance measure; meanwhile, the grey correlation analysis method, suitable for the small sample and the poor information decision-making, is applied in the evaluation. This paper constructs a mathematical optimization model of maximizing the synthesis grey correlation coefficient between decision-making evaluation values and decision-makers’ subjective preference values, calculates the attribute weights with the known partial weight information, and then sorts the alternatives by the grey correlation coefficient values. Taking venture capital firm as an example, through the calculation and the variable disturbance, we can see that the methodology used in this paper has good stability and rationality. This research makes the decision-making process more scientific and further improves the theory of intuitionistic fuzzy multiple attribute decision-making.

A Class of New Interval-Valued Intuitionistic Fuzzy Distance Measures and their Applications in Discriminant Analysis

In this paper we develop a class of new distance measures for interval-valued intuitionistic fuzzy sets. Then we discuss some of the special cases of it by taking different parameters. Finally we apply them for discriminant analysis with interval-valued intuitionistic fuzzy information.