Spherical fuzzy sets, originally proposed by F.K. Gündogdu, C. Kahraman, Spherical fuzzy sets and spherical fuzzy TOPSIS method, J. Intell. Fuzzy Syst. 36 (2019), 337-352, can handle the information of type: yes, no, abstain and refusal, owing to the feature of broad space of admissible triplets. This remarkable feature of spherical fuzzy set to manage the uncertainty and vagueness distinguishes it from other fuzzy set models. In this research article, we utilize spherical fuzzy sets and prioritized weighted aggregation operators to construct some spherical fuzzy prioritized weighted aggregation operators, including spherical fuzzy prioritized weighted averaging operator and spherical fuzzy prioritized weighted geometric operator. We discuss some properties which are satisfied by these operators. Further, we establish an algorithm to the multi-criteria group decision-making problem by utilizing the aforesaid operators. To elaborate the applicability of proposed operators in decision-making, we apply the algorithm to a numerical example which is related to the appointment for the post of Finance Manager. Finally, to demonstrate the authenticity of presented operators, we conduct a comparison with existing methods.

This article takes advantage of advancements in two different fields in order to produce a novel decision-making framework. First, we contribute to the theory of aggregation operators, which are mappings that combine large amounts of data into more advantageous forms. They are extensively used in different settings from classical to fuzzy set theory alike. Secondly, we expand the literature on complex Pythagorean fuzzy model, which has an edge over other models due to its ability to handle uncertain data of periodic nature. We propose some aggregation operators for complex Pythagorean fuzzy numbers that depend on the Hamacher t-norm and t-conorm, namely, the complex Pythagorean fuzzy Hamacher weighted average operator, the complex Pythagorean fuzzy Hamacher ordered weighted average operator, and the complex Pythagorean fuzzy Hamacher hybrid average operator. We explore some properties of these operators inclusive of idempotency, monotonicity, and boundedness. Then, the operators are applied to multicriteria decision-making problems under the complex Pythagorean fuzzy environment. Furthermore, we present an algorithm along with a flow chart, and we demonstrate their applicability with the assistance of two numerical examples (selection of most favorable country for immigrants and selection of the best programming language). We investigate the adequacy of this algorithm by conducting a comparative study with the case of complex intuitionistic fuzzy aggregation operators.

A q-rung orthopair fuzzy set (q-ROFS) is more practical and powerful than intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS) to model uncertainty in various decision-making problems. In this research article, we introduce the notion of q-rung orthopair fuzzy Hamacher graphs (q-ROFHGs). We utilize the Hamacher operators because they are flexible and parameterized in decision making. We determine the energy of q-ROFHGs as well as the energy of splitting and shadow q-ROFHGs. In addition, we propose the Randić energy of q-ROFHG and its some substantial results. Further, we present the idea of q-rung orthopair fuzzy Hamacher digraphs (q-ROFHDGs). We solve a decision-making numerical example related to the selection of best housing society for investment by calculating the energy and Randić energy of q-ROFHDGs and an algorithm to exhibit the applicability of the presented concepts in decision making. Finally, we present the conclusion.

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This work aims to study the extent of the association between the numbers of COVID-19 infections among the regions of Saudi Arabia using a graph theory, especially the calculation of the minimum spanning tree. The research also aims mainly to classify the central regions of Saudi Arabia, whose number of COVID-19 virus infections is centrally linked to other provinces, i.e., when the number of infections in these central regions increases, the number of infections in the associated regions increases and when infections decrease in these central regions, infections decrease in the associated regions.

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

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