Complex fuzzy sets: towards new foundations

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 set (CFS) is described by a complex-valued truth membership function, which is a combination of a standard true membership function plus a phase term. In this paper, we extend the idea of a fuzzy graph (FG) to a complex fuzzy graph (CFG). The CFS complexity arises from the variety of values that its membership function can attain. In contrast to a standard fuzzy membership function, its range is expanded to the complex plane’s unit circle rather than [0,1]. As a result, the CFS provides a mathematical structure for representing membership in a set in terms of complex numbers. In recent times, a mathematical technique has been a popular way to combine several features. Using the preceding mathematical technique, we introduce strong approaches that are properties of CFG. We define the order and size of CFG. We discuss the degree of vertex and the total degree of vertex of CFG. We describe basic operations, including union, join, and the complement of CFG. We show new maximal product and symmetric difference operations on CFG, along with examples and theorems that go along with them. Lastly, at the base of a complex fuzzy graph, we show the application that would be important for measuring the symmetry or asymmetry of acquaintanceship levels of social disease: COVID-19.

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.

A novel self-organizing complex neuro-fuzzy approach to the problem of time series forecasting

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.

Self-learning complex neuro-fuzzy system with complex fuzzy sets and its application to adaptive image noise canceling

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.

Complex fuzzy sets (CFS) generalize traditional fuzzy sets (FS) since the membership functions of CFS reduces to the membership functions of FS. FS values are always at [0, 1], unlike CFS which has values in the unit disk of C. This paper merges notion and concept in group theory and presents the notion of a complex fuzzy subgroup of a group. This proposed idea represents a more general and better optional mathematical tool as one of the approaches in the fuzzy group. However, this research defines the notion of complex fuzzy subgroupiod, complex fuzzy normal subgroup, and complex fuzzy left(right) ideal. Therefore, the lattice, homomorphic preimage, and image of complex fuzzy subgroupiod and ideal are introduced and studied its properties. Finally, complex fuzzy subgroups and their properties are presented and investigated

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.

<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>

The principle of a complex interval-valued Pythagorean fuzzy set (CIVPFS) is a valuable procedure to manage inconsistent and awkward information genuine life troubles. The principle of CIVPFS is a mixture of the two separated theories such as complex fuzzy set and interval-valued Pythagorean fuzzy set which covers the truth grade (TG) and falsity grade (FG) in the form of the complex number whose real and unreal parts are the sub-interval of the unit interval. The superiority of the CIVPFS is that the sum of the square of the upper grade of the real part (also for an unreal part) of the duplet is restricted to the unit interval. The goal of this article is to explore the new principle of CIVPFS and its algebraic operational laws. By using the CIVPFSs, certain Einstein operational laws by using the t-norm and t-conorm are also developed. Additionally, we explore the complex interval-valued Pythagorean fuzzy Einstein weighted geometric (CIVPFEWG), complex interval-valued Pythagorean fuzzy Einstein ordered weighted geometric (CIVPFEOWG) operators and utilized their special cases. Moreover, a multicriteria decision-making (MCDM) technique is explored based on the elaborated operators by using the complex interval-valued Pythagorean fuzzy (CIVPF) information. To determine the consistency and reliability of the elaborated operators, we illustrated certain examples by using the explored principles. Finally, to determine the supremacy and dominance of the explored theories, the comparative analysis and graphical expressions of the developed principles are also discussed.

A systematic review of complex fuzzy sets and logic

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

The theory of /spl mu/-synthesis introduced in [1, 2] provides, in principle, a broadly applicable theory for the optimal synthesis of multiloop feedback control laws that robustly meet performance and disturbance attenuation specifications despite unknown-but-bounded nonlinearities and parameter variations. Commercial MATLAB-based software packages implementing a crude approximation to the theory are available [3, 4], but these computer packages address the complex /spl mu/-synthesis problem via a somewhat flawed implementation of the original D-K iteration algorithm which involves the repetition of the following three operations on a suitable augmented closed-loop system transfer function: 1. Optimize a diagonal scaling frequency response matrix D(jw) for a fixed control law K(s). 2. Perform an ad hoc state space curve-fit to D(jw). 3. Use H/sup /spl infin// control to compute a control law K(s) with the diagonal scaling D(s) fixed. The curve-fitting of Step 2 has, until now, been a major obstacle to the realization of the original vision of a completely automated /spl mu/-synthesis procedure for robust control design. This paper describes new theoretical results and how they enable one to bypas the difficult and awkward curve-fitting of Step 2. The result is the first reliable computational algorithm for /spl mu/-synthesis controller design. The technique has been implemented on MATLAB. An example involving real /spl mu/-synthesis for the ACC Benchmark problem is included.

In traditional image segmentation models based on single level set function, only two regions can be identified because different regions are identified by the signs of single level set function. Though several segmentation models with multiple regions have been proposed, but the largest number of regions that can be identified was limited by the number of embedded level set functions in them. Moreover, the more embedded level set functions, the higher the time cost, usually increasing linearly with the increase of embedded level set functions. In this paper, by introducing the segmentation-measure function, a new model for multi-regions image segmentation based on single level set function is proposed. At the same time, a new initialization function for the level set function is also proposed in order to reduce the time cost of the segmentation model. The experiment results show that the new Image segmentation mode with multiple regions proposed in this paper performs well and dramatically reduces the time cost compared with the popular model for multiple regions proposed by Vese and Chan.