A Method for Group Decision Making Based on Interval-Valued Intuitionistic Fuzzy Geometric Distance Measures

Intuitionistic Fuzzy set (IFS) was proposed in early 80’s. It is a well known theory. As a developer in Fuzzy Mathematics, interval – valued Intuitionistic Fuzzy sets (IVIFS) were developed afterwards by Gargo and Atanssov. It has a wide range of applications in the field of Optimization and algebra. There are many distance measure in Fuzzy such as Hamming, Normalized Hamming, Euclidean, Normalized Euclidean, Geometric, Normalized Geometric etc… to calculate the distance between two fuzzy numbers. In this paper, the comparison between Geometric distance measure in Intuitionistic Fuzzy set and interval – valued Intuitionistic Fuzzy sets is explored. The step-wise conservation of Intuitionistic Fuzzy set and interval – valued Intuitionistic Fuzzy sets is also proposed. This type of comparative analysis shows that the distance between Intuitionistic Fuzzy set and interval – valued Intuitionistic Fuzzy sets varies slightly due to boundaries of interval – valued Intuitionistic Fuzzy sets.

Methods for aggregating interval-valued intuitionistic fuzzy information are investigated. Some operational laws of interval-valued intuitionistic fuzzy numbers, score function and accuracy function of interval-valued intuitionistic fuzzy numbers are intriduced. Based on these operational laws, some aggregation operators, including interval-valued intuitionistic fuzzy ordered weighted geometric aggregation (IIFOWGA) operator and interval-valued intuitionistic fuzzy hybrid geometric aggregation (IIFHGA) operator, are proposed. An approach to multiple attribute group decision making (MAGDM) with interval-valued intuitionistic fuzzy information is developed based on the IIFWGA and the IIFHGA operators. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Distance and similarity measures have recently been investigated in-depth within the context of hesitant fuzzy sets. By analyzing the existing studies concerning distance measures for hesitant fuzzy sets, we find that they have some limitations. To address the flaws, this study develops some novel distance measures for hesitant fuzzy sets, including the normalized Euclidean distance measure, the Hausdorff metric distance measure, the normalized generalized distance measure, and their corresponding weighted distance measures. The proposals of this study not only hold many ideal characteristics but also do not consider the lengths of hesitant fuzzy elements as well as the arrangement of their possible values. To deal with the situations where both of the universe of discourse and the weight of element are continuous, some continuous hesitant fuzzy distance measures are also investigated. Based on the relationship between distance measure and similarity measure, some novel similarity measures for hesitant fuzzy sets can be further deduced from the proposed distance measures. Finally, two numerical examples are given to demonstrate the applicability and validity of the proposed hesitant fuzzy distance measures.

Multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets

Hellwig’s method is a multi-criteria decision-making technique designed to facilitate the ranking of alternatives based on their proximity to the ideal solution. Typically, this approach calculates distances using the Euclidean norm, assuming implicitly that the considered criteria are independent. However, in real-world situations, the assumption of criteria independence is rarely met. The paper aims to propose an extension of Hellwig’s method by incorporating the Mahalanobis distance. Substituting the Euclidean distance with the Mahalanobis distance has proven to be effective in handling correlations among criteria, especially in the context of asymmetrical relationships between criteria. Subsequently, we investigate the impact of the Euclidean and Mahalanobis distance measures on the several variants of Hellwig procedures, analyzing examples based on various illustrative data with 10 alternatives and 4 criteria. Additionally, we examine the influence of three normalization formulas in Hellwig’s aggregation procedures. The investigation results indicate that both the distance measure and normalization formulas have some impact on the final rankings. The evaluation and ranking of alternatives using the Euclidean distance measure are influenced by the normalization formula, albeit to a limited extent. In contrast, the Mahalanobis distance-based Hellwig’s method remains unaffected by the choice of normalization formulas. The study concludes that the ranking of alternatives is strongly dependent on the distance measure employed, whether it is Euclidean or Mahalanobis. The Mahalanobis distance-based Hellwig method is deemed a valuable tool for decision-makers in real-life situations. It enables the evaluation of alternatives by considering interactions between criteria, providing a more comprehensive perspective for decision-making.

Optimized distance metrics for differential evolution based nearest prototype classifier

Effect of feature-type in selecting distance measure for an artificial immune system as a pattern recognizer

Nine oat cultivars and experimental lines from four diverse germ plasm sources were crossed in a diallel mating design, without reciprocals. Heterosis for grain yield was evaluated in two experiments and 48 F2-derived lines from each of the 36 matings were evaluated for bundle weight, grain yield, straw yield, harvest index, and heading date in two experiments. The number of transgressive segregates per trait and generalized genetic variances were calculated for each mating. Euclidean distance between parents was calculated by using the first five principal components of the parental correlation matrix. The distance measures of W. D. Hanson and E. Casas (1968. Biometrics, 24: 867–880) and T. S. Cervantes, M. M. Goodman, E. Casas, and J. O. Rawlings (1978. Genetics, 90: 339–348) were calculated for each of two experiments. The relationships between the three distance measures and the three types of breeding behavior were examined via correlation and regression. Euclidean distance was negatively correlated with transgressive segregation and generalized genetic variances. Hanson and Casas' distance measure was positively correlated with transgressive segregation and generalized genetic variances. The distance measure of Cervantes et al. was negatively correlated with heterosis in one experiment. Regressions on the three distances measures, when significant, were linear. The single exception was the regression of heterosis on Euclidean distance, which had significant linear and quadratic coefficients and high R2. No distance measure was closely associated with all three types of breeding behavior. Using information from more than one measure improved the R2 value for polynomial regressions. Key words: oats, genetic distance, heterosis, generalized variances.

Dual hesitant fuzzy set, first proposed by Zhu et al. (Dual hesitant fuzzy sets, J Appl Math, 1–13, 2012) as an extension of hesitant fuzzy sets, which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand, these sets are more flexible than the existing fuzzy sets, and provide much more information about the situation. In this paper, the axiom definition of distance and similarity measures between dual hesitant fuzzy sets is introduced. Some new distance and similarity measures based on the geometric distance model, the set-theoretic approach, and the matching functions are proposed. The proposed distance measures are then applied to the multiple-attribute decision making under dual hesitant fuzzy environments. Through the distance measure between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, a practical example of investment alternatives is given to demonstrate the effectiveness of the developed measures. The advantages of the proposed distance measure over existing measures have been presented.

Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degrees

At their core, many time series data mining algorithms reduce to reasoning about the shapes of time series subsequences. This requires an effective distance measure, and for last two decades most algorithms use Euclidean distance or DTW as their core subroutine. We argue that these distance measures are not as robust as the community seems to believe. The undue faith in these measures perhaps derives from an overreliance on the benchmark datasets and self-selection bias. The community is simply reluctant to address more difficult domains, for which current distance measures are ill-suited. In this work, we introduce a novel distance measure MPdist. We show that our proposed distance measure is much more robust than current distance measures. For example, it can handle data with missing values or spurious regions. Furthermore, it allows us to successfully mine datasets that would defeat any Euclidean or DTW distance-based algorithm. Additionally, we show that our distance measure can be computed so efficiently as to allow analytics on very fast arriving streams.

At their core, many time series data mining algorithms can be reduced to reasoning about the shapes of time series subsequences. This requires a distance measure, and most algorithms use Euclidean Distance or Dynamic Time Warping (DTW) as their core subroutine. We argue that these distance measures are not as robust as the community believes. The undue faith in these measures derives from an overreliance on benchmark datasets and self-selection bias. The community is reluctant to address more difficult domains, for which current distance measures are ill-suited. In this work, we introduce a novel distance measure MPdist. We show that our proposed distance measure is much more robust than current distance measures. Furthermore, it allows us to successfully mine datasets that would defeat any Euclidean or DTW distance-based algorithm. Additionally, we show that our distance measure can be computed so efficiently, it allows analytics on fast streams.

BackgroundGene expression divergence is a phenotypic trait reflecting evolution of gene regulation and characterizing dissimilarity between species and between cells and tissues within the same species. Several distance measures, such as Euclidean and correlation-based distances have been proposed for measuring expression divergence.ResultsWe show that different distance measures identify different trends in gene expression patterns. When comparing orthologous genes in eight rat and human tissues, the Euclidean distance identified genes uniformly expressed in all tissues near the expression background as genes with the most conserved expression pattern. In contrast, correlation-based distance and generalized-average distance identified genes with concerted changes among homologous tissues as those most conserved. On the other hand, correlation-based distance, Euclidean distance and generalized-average distance highlight quite well the relatively high similarity of gene expression patterns in homologous tissues between species, compared to non-homologous tissues within species.ConclusionsDifferent trends exist in the high-dimensional numeric data, and to highlight a particular trend an appropriate distance measure needs to be chosen. The choice of the distance measure for measuring expression divergence can be dictated by the expression patterns that are of interest in a particular study.ReviewersThis article was reviewed by Mikhail Gelfand, Eugene Koonin and Subhajyoti De (nominated by Sarah Teichmann).

Interval-valued intuitionistic fuzzy environment is appropriate for most of the practical scenarios involving uncertainty, vagueness, and insufficient information. Entropy, similarity, distance, inclusion, and cross entropy measures are a few methods used for measuring uncertainty and classifying fuzzy sets and its generalizations. Entropy of a fuzzy set describes fuzziness degree of the set and similarity measure measures similarity between two fuzzy or members of its extended family. This paper presents generalized entropy and similarity measures for interval-valued intuitionistic fuzzy sets. Further, the proposed similarity measure is compared with some existing measure of similarity with the help of an illustrative example, and a method is used to define optimal point using the existing information. Finally, entropy and similarity measures are used to identify best alternatives to solve multi-attribute decision making.

To deal with vagueness, falsity, attributive values and inconsistency, we introduce the cubic intuitionistic fuzzy soft set (CIFS-set) which is the extension of the cubic intuitionistic fuzzy set and propose distance measure, Hamming distance, Euclidean distance and separation measures of CIFS-set. Moreover, we present aggregate operator (P-union, R-intersection) of CIFS-sets. The proposed CIFS-set is more reliable, efficient and accurate. In future MCDM and MCGDM techniques can be propoed to deal with real life issues, and also this CIFS-set can be extended for its hybrids.