Stock portfolio problems are one of the most relevant real-world problems. In this study, we discuss the portfolio's risk amount, rate of risk-return, and expected return rate under a Fermatean fuzzy environment. A linear programming problem is used to formulate a Fermatean fuzzy portfolio. The Fermatean fuzzy portfolio is converted to a deterministic form using the score function. Lingo software is used to solve these deterministic portfolio problems. The main feature of this model is that investors can select a risk coefficient to enhance predicted returns and customize their strategies according to their circumstances. An example is offered that illustrates the effectiveness and dependability of the proposed approach.

This paper aims to study Pythagorean and Fermatean Fuzzy Subgroups (FFSG)  in the context of -norm and  -conorm functions. The paper examines the extensions of fuzzy subgroups, specifically "Pythagorean Fuzzy Subgroups (PFSG)" and "FFSG", along with their properties. In the existing literature on Pythagorean and FFSG, the standard properties for membership and non-membership functions are based on the "min" and "max" operations, respectively. However, in this work, we develop a theory that utilizes the -norm for "min" and the -conorm for "max", providing definitions of Pythagorean and FFSG with these functions, along with relevant examples. By incorporating this approach, we introduce multiple options for selecting the minimum and maximum values. Additionally, we prove several results related to Pythagorean and FFSG using the -norm and -conorm, and discuss important properties associated with them.

In this paper, we , rst study fuzzy subgroups and fuzzy multigroups of a group G and we obtain some chains of fuzzy subgroups of a fuzzy subgroup and by these sequences, we construct fuzzy multi-groups. We show that there is a relationship between a fuzzy multigroup with underlying group G and a fuzzy multigroup with underlying group Aut(A). Moreover, we generate a code by using the de, ned special fuzzy multigroup automorphisms.

In the classical group theory there is an open question: Is every torsion free n-Engel group (for n  4), nilpotent? . To answer the question, Traustason [11] showed that with some additional conditions all 4-Engel groups are locally nilpotent. Here, we gave some partial answer to this question on Engel fuzzy subgroups...

Fermatean Fuzzy Sets (FFSs) provide an effective way to handle uncertainty and vagueness by expanding the scope of membership and Non-Membership Degrees (NMDs) of Intuitionistic Fuzzy Set (IFS) and Pythagorean Fuzzy Set (PFS), respectively. FFS handles uncertain information more easily in the process of decision making. The concept of composite relation is an operational information measure for decision making. This study establishes Fermatean fuzzy composite relation based on max-average rule to enhance the viability of FFSs in machine learning via soft computing approach. Some numerical illustrations are provided to show the merit of the proposed max-average approach over existing the max-min-max computational process. To demonstrate the application of the approach, we discuss some pattern recognition problems of building materials and mineral fields with the aid of the Fermatean fuzzy modified composite relation and Fermatean fuzzy max-min-max approach to underscore comparative analyses. In recap, the objectives of the paper include: 1) discussion of FFS and its composite relations, 2) numerical demonstration of Fermatean fuzzy composite relations, 3) establishment of a decision application framework under FFS in pattern recognition cases, and 4) comparative analyses to showcase the merit of the new approach of Fermatean fuzzy composite relation. In future, this Fermatean fuzzy modified composite relation could be studied in different environments like picture fuzzy sets, spherical fuzzy sets, and so on.

Optimizing airport security screening processes is imperative in the modern age, ensuring a delicate balance between stringent security measures and efficient, seamless travel experiences for passengers. This research presents an innovative paradigm for decision-making that optimizes airport security screening procedures. By utilizing sophisticated fuzzy concepts, particularly Fermatean Probabilistic Hesitant Fuzzy Sets, the research presents new hybrid aggregation techniques known as Fermatean Probabilistic Hesitant Hybrid Weighted Averaging FePHHWA and Hybrid Weighted Geometric FePHHWG. These procedures are intended to improve the decision-making process by providing a thorough method of addressing the intricacies of security screening. In addition to showcasing the flexibility and effectiveness of Fermatean Probabilistic Hesitant Fuzzy Sets, the suggested technique creates a strong foundation for resolving issues with airport security screening by utilizing the recently developed hybrid aggregation processes.

Optimizing airport security screening processes is imperative in the modern age, ensuring a delicate balance between stringent security measures and efficient, seamless travel experiences for passengers. This research presents an innovative paradigm for decision-making that optimizes airport security screening procedures. By utilizing sophisticated fuzzy concepts, particularly Fermatean Probabilistic Hesitant Fuzzy Sets, the research presents new hybrid aggregation techniques known as Fermatean Probabilistic Hesitant Hybrid Weighted Averaging FePHHWA and Hybrid Weighted Geometric FePHHWG. These procedures are intended to improve the decision-making process by providing a thorough method of addressing the intricacies of security screening. In addition to showcasing the flexibility and effectiveness of Fermatean Probabilistic Hesitant Fuzzy Sets, the suggested technique creates a strong foundation for resolving issues with airport security screening by utilizing the recently developed hybrid aggregation processes.

The target of this article is to peruse bipolar complex fuzzy subhypergroups (Hv−subgroups) by employing bipolar complex fuzzy sets and hyperstructures besides of this some related properties are debated. Furthermore, the notion of bipolar complex anti-fuzzy subhypergroups (Hv−subgroups) is presented. Their characteristics and their connections with bipolar complex fuzzy and anti-fuzzy subhypergroups (Hv−subgroups) are proposed, with several examples demonstrating these concepts.