Analysis and application of Aczel-Alsina aggregation operators based on bipolar complex fuzzy information in multiple attribute decision making

In recent days researchers have tried to handle the maximum information and use those techniques and methods in which there is no chance of data loss or loss of information is minimum. The structure like fuzzy set and complex fussy set cannot discuss the upper and lower approximations. Moreover, we can observe that a fuzzy rough set cannot discuss the second dimension and in this case, there is a chance of data loss. To cover all these issues in previous ideas, the notion of a complex fuzzy rough set in Cartesian form is the demand of the day because this structure can discuss the second dimension as well as upper and lower approximations. For this purpose, in this manuscript, we have developed the theory of complex fuzzy relation and complex fuzzy rough set in Cartesian form. Moreover, we have initiated the fundamental laws for complex fuzzy rough numbers based on Frank t-norm and t-conorm. The fundamental tools that can convert the overall input into a single output are called aggregation operators (AOs). So based on the characteristics of AOs, we have defined the notion of complex fuzzy rough Frank average and complex fuzzy rough Frank geometric AOs. The utilization of the developed theory is necessary to show the importance and validity of the delivered approach. So based on developed notions, we have defined an algorithm for this purpose along with an illustrative example. We have utilized the introduced structure for the classification of AI tools for civil engineering. Moreover, the comparative analysis of the delivered approach shows the advancement of the introduced structure as compared to existing notions.

In this manuscript we demonstrate interval-valued bipolar complex fuzzy set (IVBCFS) and then interval-valued bipolar complex fuzzy soft set (IVBCFSS), as a generalization of fuzzy set, interval-valued fuzzy set, bipolar fuzzy set, complex fuzzy set and soft set. We also initiate operational laws and basic results and properties for IVBCFS and IVBCFSS. Further explanation is given for the basic algebraic operations like complement, extended union, extended intersection, restricted union, and restricted intersection, AND product and OR product for IVBCFSS. Moreover, we demonstrate some fundamental aggregation operators like IVBCFS average aggregation, IVBCFS geometric and as well as their properties. To emphasize the usefulness and application of the system, we also develop the decision-making method and joint instances of the IVBCFSS (set-ups 1 and 2). In order to describe the effectiveness and influence of the approaching novel work, this study uses a comparative analysis of the new creating concept with prevailing ideas.

The interval‐valued q‐rung orthopair fuzzy set (IVq‐ROFS) and complex fuzzy set (CFS) are two generalizations of the fuzzy set (FS) to cope with uncertain information in real decision making problems. The aim of the present work is to develop the concept of complex interval‐valued q‐rung orthopair fuzzy set (CIVq‐ROFS) as a generalization of interval‐valued complex fuzzy set (IVCFS) and q‐rung orthopair fuzzy set (q‐ROFS), which can better express the time‐periodic problems and two‐dimensional information in a single set. In this article not only basic properties of CIVq‐ROFSs are discussed but also averaging aggregation operator (AAO) and geometric aggregation operator (GAO) with some desirable properties and operations on CIVq‐ROFSs are discussed. The proposed operations are the extension of the operations of IVq‐ROFS, q‐ROFS, interval‐valued Pythagorean fuzzy, Pythagorean fuzzy (PF), interval‐valued intuitionistic fuzzy, intuitionistic fuzzy, complex q‐ROFS, complex PF, and complex intuitionistic fuzzy theories. Further, the Analytic hierarchy process (AHP) and technique for order preference by similarity to ideal solution (TOPSIS) method are also examine based on CIVq‐ROFS to explore the reliability and proficiency of the work. Moreover, we discussed the advantages of CIVq‐ROFS and showed that the concepts of IVCFS and q‐ROFS are the special cases of CIVq‐ROFS. Moreover, the flexibility of proposed averaging aggregation operator and geometric aggregation operator in a multi‐attribute decision making (MADM) problem are also discussed. Finally, a comparative study of CIVq‐ROFSs with pre‐existing work is discussed in detail.

Fuzzy set (FS) theory is one of the most important tool to deasl with complicated and difficult information in real-world. Now FS has many extensions and hesitant fuzzy set (HFS) is one of them. Further generalization of FS is complex fuzzy set (CFS), which contains only the membership grade, whose range is unit disc instead of [0, 1]. The aim of this paper is to present the idea of complex hesitant fuzzy set (CHFS) and to introduce its basic properties. Basically, CHFS is the combination of CFS and HFS to deal with two dimension information in a single set. Further, the vector similarity measures (VSMs) such as Jaccard similarity measures (JSMs), Dice similarity measures (DSMs) and Cosine similarity measures (CSMs) for CHFSs are discussed. The special cases of the proposed measures are also discussed. Then, the notion of complex hesitant fuzzy hybrid vector similarity measures are utilized in the environment of pattern recognition and medical diagnosis. Further, based on these distance measures, a decision-making method has been presented for finding the best alternative under the set of the feasible one. Illustrative examples from the field of pattern recognition as well as medical diagnosis have been taken to validate the approach. Finally, the comparison between proposed approaches with existing approaches are also discussed to find the reliability and proficiency of the elaborated measures for complex hesitant fuzzy elements.

The innovative concept of Complex Fuzzy Set is introduced. The objective of the this paper to investigate the concept of Complex Fuzzy set in constraint to a traditional Fuzzy set , where the membership function ranges from [0, 1], but in the Complex fuzzy set extended to a unit circle in a complex plane, where the member ship function in the form of complex number. The Compressive study of mathematical operation of Complex Fuzzy set is presented. The basic operation like addition, subtraction, multiplication and division are described here. The Novel idea of this paper to measure the similarity between two fuzzy relations by evaluating δ -equality. Here also we introduce the probabilistic interpretation of the complex fuzzy set where we attempted to clarify the distinction between Fuzzy logic and probability. KEY WORD Complex fuzzy set, Complex fuzzy relation, complement of complex fuzzy relation, δ -equality, probability.

A complex-fuzzy approach using complex fuzzy sets is proposed in the paper to deal with the problem of adaptive image noise cancelling. A image may be corrupted by noise, resulting in the degradation of valuable image information. Complex fuzzy set (CFS) is in contrast with traditional fuzzy set in membership description. A CFS has the membership state within the complexvalued unit disc of the complex plane. Based on the membership property of CFS, we design a complex neural fuzzy system (CNFS), so that the functional mapping ability by the CNFS can be augmented. A hybrid learning method is devised for training of the proposed CNFS, including the artificial bee colony (ABC) method and the recursive least square estimator (RLSE) algorithm. Two cases for image restoration are used to test the proposed approach. Experimental results are shown with good restoration quality.

As a generalization of complex fuzzy set (CFS), complex hesitation fuzzy set (CHFS) is a powerful tool to express two-dimensional fuzzy information, but it ignores the randomness that exists widely in real life. Considering the randomness and ambiguity often simultaneously occur in real-world problems, we come up with an innovative fuzzy set called complex probabilistic hesitant fuzzy set (CPHFS) in this article, which is a joint extension of CFS and probabilistic hesitant fuzzy set (PHFS). Moreover, we establish four basic operations for complex probabilistic hesitant fuzzy element(CPHFE) and a series of useful operational laws are also developed. Secondly, several aggregation operators including complex probabilistic hesitant fuzzy weighted averaging operator(CPHFWA), complex probabilistic hesitant fuzzy weighted geometric operator(CPHFWG), complex probabilistic hesitant fuzzy Heronian Mean operator(CPHFHM) and complex probabilistic hesitant fuzzy Geometric Heronian Mean operator(CPHFGHM) as well as their weighted forms that can reflect the interrelationship among all decision attributes are proposed, which is of great use in multi-attribute decision making(MADM) or in multi-attribute group decision making(MAGDM) issues when we want to synthesize multiple evaluation values into one. Besides, we also present a number of valuable properties of these provided aggregation operators. Thirdly, a MADM process is built in the CPHF environment step by step based on the aggregation operators we have just established. Furthermore, we provide an example about purchasing a most suitable car to demonstrate the effectiveness and practicability of the proposed decision process. Finally, we point out the superiorities of MADM in working under the CPHF environment through comparing it with the existing method and also offer some possible prospects for further research. The method displayed in this work can effectively solve the MADM problems where the decision-making information is described by CPHFEs and the attributes are interactive.

Recently, complex fuzzy sets have become a powerful tool for expanding the range of fuzzy sets that are located in a unit disk within complex plane. In this article, the complex fuzzy numbers are discussed and implemented to handle complex fractional fuzzy partial differential equation involving complex time-fractional fuzzy heat equation in sense of Hukuhara differentiability. After that, an explicit finite difference scheme known as the forward time center space are implemented to solve complex time-fractional fuzzy heat equations. The uncertainty present in our problem appears in the boundary and initial conditions as well as coefficients in both amplitude and phase terms simultaneously where the Convex normalized triangular fuzzy numbers are extended to unit disk in the complex plane. The properties and benefits of complex fuzzy set theory have been employed in the proposed numerical methods. A numerical experiment is used to showcase the efficiency and practicality of the presented method. The obtained outcomes are found to be consistent with the exact solution and related theoretical concepts.

The theory of complex hesitant fuzzy set (CHFS) is a modification technique of the complex fuzzy set (CFS) to cope with awkward and unreliable information’s in daily life issues. CHFS contains the grade of truth in the form of complex number, whose real and imaginary parts are in the form of the finite subset of the unit interval. CHFS is the mixture of hesitant fuzzy set (HFS) and CFS, which handles the complex and uncertain information in real-world issues which is compared with fuzzy sets and complex fuzzy sets. The positive membership in CHFS is in the form a polar coordinate belonging to unit disc in the complex plane. The aims of this manuscript are to explore some similarity measures (SMs), weighted SMs (WSMs) such as cosine SMs, weighted cosine SMs, SMs based on cosine function, WSMs based on cosine function, SMs based on tangent function, and WSMs based on tangent function of CHFS. Some special cases of the presented measures are discussed in detail. Moreover, we use our described SMs and weighted SMs of CHFS in the environment of medical diagnosis and pattern recognition to assess the practicality and competence of the described SMs. Finally, to find the validity and proficiency of the investigated measures based on CHFSs, the comparison between explored measures with some already defined measures and their graphical representations are also discussed in detail.

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 ( F P - CMFSES ), and investigate part of its fundamental properties. This F P - 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 F P - 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.

A complex fuzzy set is an extension of the traditional fuzzy set, where traditional [0,1]-valued membership grade is extended to the complex unit disk. The aggregation operator plays an important role in many fields, and this paper presents several complex fuzzy geometric aggregation operators. We show that these operators possess the properties of rotational invariance and reflectional invariance. These operators are also closed on the upper-right quadrant of the complex unit disk. Based on the relationship between Pythagorean membership grades and complex numbers, these operators can be applied to the Pythagorean fuzzy environment.

Complex fuzzy set (CFS), as a generalization of fuzzy set (FS), is characterized by complex-valued membership degrees. By considering the complex-valued membership degree as a vector in the complex unit disk, we introduce the cosine similarity measures between CFSs. Then, we investigate some invariance properties of the cosine similarity measure. Finally, the cosine similarity measure is applied to measure the robustness of complex fuzzy connectives and complex fuzzy inference.

A new fuzzy set theory, C-fuzzy set theory, is introduced in this paper. It is a particular case of the classical set theory and satisfies all formulas of the classical set theory. To add a limitation to C-fuzzy set system, in which all fuzzy sets must be “non-uniform inclusive” to each other, then it forms a family of sub-systems, the Z-fuzzy set family. It can be proved that the Z0-fuzzy set system, one of Z-fuzzy set systems, is equivalent to Zadeh’s fuzzy set system. Analysis shows that 1) Zadeh’s fuzzy set system defines the relations A = B and A ⊆ B between two fuzzy sets A and B as “∀u ∈ U, (μA ∈ (u) = μB(u))” and “∀u ∈ U, (μA(u) ≤ μB(u))” respectively is inappropriate, because it makes all fuzzy sets be “non-uniformly inclusive”; 2) it is also inappropriate to define two fuzzy sets’ union and intersection operations as the max and min of their grades of membership, because this prevents fuzzy set’s ability to correctly reflect different kinds of fuzzy phenomenon in the natural world. Then it has to work around the problem by invent unnatural functions that are hard to understand, such as augmenting max and min for union and intersection to min{a + b, 1} and max{a + b – 1, 0}, but these functions are incorrect on inclusive case. If both pairs of definitions are used together, not only are they unnatural, but also they are still unable to cover all possible set relationships in the natural world; and 3) it is incorrect to define the set complement as 1 – μA(u), because it can be proved that set complement cannot exist in Zadeh’s fuzzy set, and it causes confusion in logic and thinking. And it is seriously mistaken to believe that logics of fuzzy sets necessarily go against classical and normal thinking, logic, and conception. The C-fuzzy set theory proposed in this paper overcomes all of the above errors and shortcomings, and more reasonably reflects fuzzy phenomenon in the natural world. It satisfies all relations, formulas, and operations of the classical set theory. It is consistent with normal, natural, and classical thinking, logic, and concepts.

A complex fuzzy set is a set whose membership grades are complex values in the unit circle in the complex plane. This paper introduces the concept of approximate parallelity between complex fuzzy sets based on the phase of complex-valued membership grade. After that, the property of approximate parallelity preserving for complex fuzzy operators and complex fuzzy inference are investigated.

<p>Complex spherical fuzzy sets (CSFSs) are a theory that addresses confusing and unreliable information in real-life decision-making contexts by integrating elements of two theories: spherical fuzzy sets (SFSs) and complex fuzzy sets (CFSs). CSFSs are classified into three categories, represented by polar coordinates: membership, nonmember, and abstention. These grades are located on a complex plane within a unit disc. It is necessary for the total squares representing the real components of the grades for abstinence, membership, and non-membership to not surpass a certain interval. Several aspects of CSFS and the corresponding operational laws were examined in this work. The key components of this article were based on CSFs, including complex spherical fuzzy Schweizer-Sklar prioritized aggregation (CSFSSPA), complex spherical fuzzy Schweizer-Sklar weighted prioritized aggregation (CSFSSWPA), complex spherical fuzzy Schweizer-Sklar prioritized geometry (CSFSSPG), and complex spherical fuzzy Schweizer-Sklar prioritized weighted geometry (CSFSSWPG). Additionally, the suggested operators' specific instances were examined. The main outcome of this work includes new aggregation techniques for CSFS information, based on t-conorm and t-norm from Schweizer-Sklar (SS). The basic characteristics of the operators were established by this study. We looked at a numerical example centered on efficient mobile e-tourism selection to show the effectiveness and viability of the recommended approaches. Additionally, we carried out a thorough comparative analysis to assess the outcomes of the suggested aggregation approaches in comparison to the current methods. Last, we offer an overview of the planned study and talk about potential directions for the future.</p>