An enhanced VIKOR method for multi-criteria group decision-making with complex Fermatean fuzzy sets.

This paper aims to propose a new decision-making approach retaining the fascinating traits of the conventional VIKOR method in the context of the enrich multidimensional complex Fermatean fuzzy $ N $-soft set. The VIKOR technique is contemplated as the most reliable decision-making approach among others which employs a strategy to identify the compromise solution with advantageous distance from the positive ideal solution possesses maximum majority utility and minimum individual regret. At the same time, the paramount characteristic of the complex Fermatean fuzzy $ N $-soft set considers the proclivity to capture two-dimensional uncertain and imprecise information along with the multi-valued parameters. This article expands the literature to handle the multi-attribute group decision-making strategy by introducing a technique, namely, the complex Fermatean fuzzy $ N $-soft VIKOR method that amalgamates the unconventional traits of complex Fermatean fuzzy $ N $-soft with the capability of the VIKOR method. The proposed technique permits the assignment of the $ N $-soft grades to the decision-makers, alternatives, and attributes based on their performances. Firstly, we unify these individual opinions of all decision-makers about the alternatives by employing the complex Fermatean fuzzy $ N $-soft weighted average operator. After that, all entities of the aggregated decision matrix are converted into crisp data by utilizing the score function. Furthermore, we calculate the ranking measures of the group utility and the individual regret by assigning the weight of strategy belongs to the interval $ [0, 1]. $ To find the compromise solution, we arrange the ranking measures in ascending order, and the alternative that possesses the conditions of compromise solution is selected. We demonstrate the presented multi-attribute group decision-making technique by selecting the best location for a nuclear power plant. We conduct the comparative analysis of the presented technique with Fermatean fuzzy TOPSIS to endorse the veracity and accuracy of our method. Finally, we explain the merits and limitations of our strategy and give some concluding remarks.

Multifractal analysis on age-based discrimination in X-ray images for sensing the severity of COVID-19 disease.

The coronavirus, also known as COVID-19, which has been considered one of the deadliest diseases in the world, has become highly contagious, it also implants directly in the human lungs and causes severe damage to the lungs. In such case, X-ray images are widely used to analyze, detect and treat the COVID-19 patients quickly. The X-ray images without any filtering are more complex to identify the affected areas of lungs and to estimate the level of severity of various diseases. The paper analyzes the normal and filtered X-ray images through the multifractal theory and describes the effects of the infection on COVID-19 patients at different ages are classified significantly in processed X-ray images. In this study, the mean absolute error and peak signal-to-noise ratio values are calculated for comparing the noisy and denoised X-ray images using the median filter method and analyzed for comparing the severity of lung affection in X-ray images at different noise levels. Finally, the three-dimensional visualization is constructed for representative images for analyzing and comparing the fever and oxygen levels based on the ages using the corresponding Generalized Fractal Dimensions values. It is observed that the Generalized Fractal Dimensions analyze the different sets of age people's X-ray images and shows clearly that the older people have higher complexity and the younger people have lower complexity in the infected lungs.

Concave soft sets, critical soft points, and union-soft ideals of ordered semigroups.

The notions of union-soft semigroups, union-soft l-ideals, and union-soft r-ideals are introduced, and related properties are investigated. Characterizations of a union-soft semigroup, a union-soft l-ideal, and a union-soft r-ideal are provided. The concepts of union-soft products and union-soft semiprime soft sets are introduced, and their properties related to union-soft l-ideals and union-soft r-ideals are investigated. Using the notions of union-soft l-ideals and union-soft r-ideals, conditions for an ordered semigroup to be regular are considered. The concepts of concave soft sets and critical soft points are introduced, and their properties are discussed.

Hesitant fuzzy soft subalgebras and ideals in BCK/BCI-algebras.

As a link between classical soft sets and hesitant fuzzy sets, the notion of hesitant fuzzy soft sets is introduced and applied to a decision making problem in the papers by Babitha and John (2013) and Wang et al. (2014). The aim of this paper is to apply hesitant fuzzy soft set for dealing with several kinds of theories in BCK/BCI-algebras. The notions of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals are introduced, and related properties are investigated. Relations between a hesitant fuzzy soft subalgebra and a (closed) hesitant fuzzy soft ideal are discussed. Conditions for a hesitant fuzzy soft set to be a hesitant fuzzy soft subalgebra are given, and conditions for a hesitant fuzzy soft subalgebra to be a hesitant fuzzy soft ideal are provided. Characterizations of a (closed) hesitant fuzzy soft ideal are considered.

Double-framed soft hypervector spaces.

The notions of double-framed soft subfields, double-framed soft algebras over double-framed soft subfields, and double-framed soft hypervector spaces are introduced, and their properties and characterizations are considered.

Subalgebras of BCK/BCI-algebras based on cubic soft sets.

Operations of cubic soft sets including "AND" operation and "OR" operation based on P-orders and R-orders are introduced and some related properties are investigated. An example is presented to show that the R-union of two internal cubic soft sets might not be internal. A sufficient condition is provided, which ensure that the R-union of two internal cubic soft sets is also internal. Moreover, some properties of cubic soft subalgebras of BCK/BCI-algebras based on a given parameter are discussed.