We develop a novel mathematical tool known as complex multi-fuzzy soft set (CMFSS) which has the ability to handle uncertainties, imprecision, and vagueness of information that are inherent in the data by considering the amplitude and phase terms of the complex numbers simultaneously. This CMFSS constitutes of a hybrid structure of multi-fuzzy set and soft set which are defined in a complex setting. The structure is flexible as it allows for a greater range of values for the membership function by extending them to the unit circle in a complex plane through modification of the multi-fuzzy soft set by the inclusion of an additional term called the phase term in order to take into account of the periodic nature of the data. Accordingly, the novelty of this paper lies in the complex multi-membership functions which consider more range of values while handling the uncertainty of the periodic data. In this paper, the concept of complex multi-fuzzy soft set is introduced. We then define its basic operations of complement, union and intersection and study some related properties, with supporting proofs. Subsequently, by means of level soft sets, we present an algorithm to solve a CMFSSs decision making problem, to illustrate the effectiveness and practicality of the proposed concept. Finally, we introduce axiomatic definitions of entropy and similarity measure for CMFSSs, and some formulas have also been put forward to calculate them. Numerical examples are given to demonstrate that the proposed entropy measure for CMFSSs is an important concept for measuring uncertainty in the information/data. Furthermore, some theorems are proposed showing how the entropy of CMFSS can be found from the similarity measure of CMFSS.

In this paper, we introduce the concept of complex multi-fuzzy sets (CM ^k FSs) as a generalization of the concept of multi-fuzzy sets by adding the phase term to the definition of multi-fuzzy sets. In other words, we extend the range of multi-membership function from the interval [0,1] to unit circle in the complex plane. The novelty of CM ^k FSs lies in the ability of complex multi-membership functions to achieve more range of values while handling uncertainty of data that is periodic in nature. The basic operations on CM ^k FSs, namely complement, union, intersection, product and Cartesian product are studied along with accompanying examples. Properties of these operations are derived. Finally, we introduce the intuitive definition of the distance measure between two complex multi-fuzzy sets which are used to define δ -equalities of complex multi-fuzzy sets.

In this paper, we extend the two concepts of bipolar fuzzy sets and soft expert sets to bipolar fuzzy soft expert sets. We will define its basic theoretic operation, namely complement, union, intersection, AND and OR on bipolar fuzzy soft expert sets along with illustrative examples, and study some related properties with supporting proofs. The basic properties and relevant laws pertaining to this concept are proven. We then construct an algorithm based on this concept. Finally, we apply it to a decision-making problem to demonstrate the applicability of the proposed method. It is shown that this concept is effective in solving decision-making problems using an illustrative example.

The concept of Fermatean fuzzy set offers a more comprehensive perspective on Pythagorean fuzzy sets, with greater potential for practical applications due to its broader scope compared to Pythagorean fuzzy sets. In contrast, distance measures serve as effective information measures in decision-making using deep learning methods. This research paper introduces a novel distance measure within the Fermatean fuzzy framework. This new distance measure exhibit enhanced performance indices compared to existing distance measures discussed in previous literature. The newly proposed distance operators based on Fermatean fuzzy sets offer a fresh approach by considering all three membership functions of Fermatean fuzzy sets, which differs from existing practices. This distance operator also incorporate all membership functions to prevent errors caused by exclusions, which have been observed in other distance operators. Several theorems are provided to validate the alignment of this new Fermatean fuzzy distance technique with properties expected of distance operators. The comparative analysis of the newly introduced Fermatean fuzzy distance measure with similar existing distance techniques is conducted through numerical comparison and also in relation to their applications, aiming to highlight the superiority of the proposed Pythagorean fuzzy distance techniques over the existing ones. The application of the proposed Fermatean fuzzy distance measure is demonstrated in scenarios involving pattern classification and bidirectional approximate reasoning.

Contemporary research has refined systems with complex fuzzy sets in order to improve the design and model of real-life applications. Symmetry and antisymmetry are basic characteristics of binary relations used when modeling the decision maker’s preferences. A recent focus has been the analysis of a complex data set using the properties of fuzzy concept lattice and the complex soft set. We will introduce a new concept to represent the information which utilizes the time factor, called fuzzy parameterized complex multi-fuzzy soft expert set ( $FP$ - $\mathcal{CMFSES}$ ), and investigate part of its fundamental properties. This $FP$ - $\mathcal{CMFSES}$ model allows us to validate the information provided by an expert, at a given phase of time, using the properties of complex fuzzy sets. We then construct an algorithm based on this concept by converting it from the complex state to the real state. Eventually, we implement it to a decision-making problem to demonstrate the applicability of the suggested method. A comparison among $FP$ - $\mathcal{CMFSES}$ and other existing methods is made to expose the dominance of the suggested method. Apart from that, we also propose the weighted fuzzy parameterized complex multi-fuzzy soft expert set and investigate its application to decision-making.