Fuzzy-valued Functions Research Articles

Solving Intuitionistic Fuzzy Unconstrained Optimization Problems Using Interval Newton’s Method

This research work studies the optimization of intuitionistic fuzzy valued functions in unconstrained problems. The Interval Newton's Method using an Intuitionistic approach addresses both single and multivariable optimization problems. The study incorporates a mathematical comprehension of interval intuitionistic fuzzy valued problems, as well as real-world examples to demonstrate their effectiveness. Furthermore, MATLAB code is provided to demonstrate the implementation of the Interval Newton's Method.

Solution Strategy and Associated Results for Fuzzy Mellin Transformation

The advancement of the Mellin transformation is that it exhibits a scale-invariant nature, and thus, it is widely used in computer science. Fuzzy Mellin transformation can provide the purposes of such problems in uncertain phenomena. The existing studies on fuzzy Mellin transformation consider the fuzzy level cut approach, which makes the fuzzy valued functions into their crisp couplets regarding lower and uppercuts. This paper reconstructs a distinct mathematical frame for the Mellin transformation in a completely fuzzy environment in the sense of a modified Hukuhara derivative. In this paper, we mean the complete fuzzy environment to describe fuzzy phenomena where the conversion of the fuzzy function to its crisp counterpart is annulled. The proposed fuzzy Mellin transformation’s superiority over existing approaches is that it can be used directly to fuzzy valued functions without converting them to their crisp counter version. The challenges to dealing with fuzzy valued mappings and inputs without its level cut representation have been tackled in this manuscript. Furthermore, numerical examples, possible applications, and further extensions of this work are hinted at in this paper.

The uncertain Malthusian model on time scales

This work offers a real-world application for the study of fuzzy dynamic equations. First, we propose the novel concept of granular delta differentiability for fuzzy-valued functions defined on time scales with the help of the relative distance measure fuzzy arithmetic and horizontal membership functions. Then, fundamental foundations of fuzzy calculus on time scales are provided. Discussion on the Malthusian model defined on particular time scales to illustrate the proposed approach is presented.

The Investigation of Some Essential Concepts of Extended Fuzzy-Valued Convex Functions and Their Applications

In this paper, we are thus motivated to define and introduce the extended fuzzy-valued convex functions that can take the singleton fuzzy values −∞˜ and +∞˜ at some points. Such functions can be characterized using the notions of effective domain and epigraph. In this way, we study important concepts such as fuzzy indicator function and fuzzy infimal convolution for extended fuzzy-valued functions. Finally, we introduce the concept of directional generalized derivative for extended above functions and its properties. Eventually, we give a practical example that will illustrate well the directional g-derivative for the extended fuzzy-valued convex function.

Generalized fractional model of heat transfer in uncertain hybrid nanofluid with entropy optimization in fuzzy-Caputo sense

In this paper, we present a new fuzzy-fractional (FF) transformation to recover FF differential model of hybrid nanofluid. The current study focuses on FF modeling of nanofluid with engine oil as base fluid, while ferrous oxide Fe2O3 and alumina Al2O3 are considered nanoparticles. In accordance to the real industrial phenomena, the flow is simulated between two squeezing plates with thermal radiation and magnetic effects. A generalized fuzzy-fraction flow problem is modeled by introducing new similarity transforms. Obtained model is validated both theoretically and numerically. At integer order Υ=1, the FF model reduces to the integer order fluid model existing in literature, proving theoretical validity. Fuzzy-valued functions are discriminated through triangular fuzzy numbers using r-cut approach. In order to solve, obtained highly non-linear FF nanofluid system, we apply He–Laplace–Carson (HLC) algorithm. Differential and convolution properties of Laplace–Carson Transform (LCT) are utilized for solution purpose. Error and convergence analysis is performed numerically to verify obtained results. Furthermore, graphical illustrations for upper and lower bound analysis on FF profiles is also presented. Analysis reveals that heat transfer in engine oil enhances with an increase in radiation at upper and lower bound in fuzzy-fractional environment. Moreover, entropy decreases with an increase in nanoparticle concentration of Fe2O3 and Al2O3 in engine oil.

A two-compartment drug concentration model using intuitionistic fuzzy sets

The intuitionistic fuzzy differential equation (IFDE) is a robust mathematical tool that can deal with problems arising from model uncertainties. Intuitionistic sets are generalized sets of fuzzy sets. They establish that IFDE comprises an intuitionistic number characterized by membership and non-membership functions. This paper deals with generalized Hukuhara derivatives for intuitionistic fuzzy-valued functions. A compartment drug concentration system has been described in a fuzzy, intuitionistic setting to get a more accurate mathematical model. The initial conditions of the proposed drug system are considered triangular intuitionistic fuzzy numbers. The proposed model’s critical points of stability analysis have been explored in an intuitionistic environment. All intuitionistic fuzzy solutions of the proposed system support robust solutions. All results and formulas are confirmed graphically and numerically by MATLAB software.

Lagrangian dual theory and stability analysis for fuzzy optimization problems

This paper investigates the Lagrangian dual theory for a class of fuzzy optimization problems with equality and inequality constraints. To begin with, the Lagrangian dual problem corresponding to the primal fuzzy optimization problem is established by introducing the Lagrangian fuzzy-valued function, and the weak dual theorem is derived. Subsequently, a necessary and sufficient condition for the existence of saddle-points is formulated, which serves as the fundamental basis for proving the strong dual theorem. The stability of the optimization model when its constraints are perturbed by parameters is then analyzed. Finally, the developed Lagrangian dual theory is applied to support vector machines to address the binary classification problem of triangular fuzzy number datasets.

Incommensurate non-homogeneous system of fuzzy linear fractional differential equations using the fuzzy bunch of real functions

This article aims to introduce and investigate the analytical fuzzy solution of the incommensurate non-homogeneous system of fuzzy linear fractional differential equations (INS-FLFDEs) using trivariate Mittag-Leffler functions. Entries of the coefficient matrix of the given system are treated as real numbers, initial-values are triangular fuzzy numbers (TFNs), and the forcing function is a fuzzy set (or a bunch) of real function. We extract the potential solution in the form of a fuzzy bunch of real functions (FBoRFs) rather than the solution of fuzzy-valued functions. We formulate the fuzzy initial value problem as a set of classical initial value problems by taking the forcing function from the class of FBoRFs and the initial value from the collection of TFNs (as a special case). The solution of this system is in the form of a trivariate Mittag-Leffler function. We interpret this solution as an element of the fuzzy solution set and assign the minimum value of membership that takes from the forcing function and the initial value in the fuzzy set. The originality of the proposed technique is that the uncertainty is smaller compared to the uncertainty extracted from other techniques. In addition, generalized derivatives increase the order and dimension of the system. Therefore, the proposed technique is better in terms of complexity because it reduces the order and dimension of the system. Finally, to grasp the proposed technique, we solve the electrical network and multiple mass-spring systems as applications and analyze their graphs to visualize and support theoretical results.

Linear correlated fuzzy solutions of nonlocal problems for multi-order fractional differential systems

This paper introduces the standard Riemann-Liouville-LC fractional derivative D0+βLCRLv(t) for the linear correlated fuzzy-valued function v:J⊂R→RF(E), where β>0, E is a fixed arbitrary fuzzy number. The focus of this study lies in addressing nonlocal problems for multi-order fractional differential systems in the linear correlated fuzzy space RF(E), where the highest order derivative may be greater than one. The solvability of these nonlocal problems is investigated comprehensively, encompassing both cases where E is a symmetric or non-symmetric fuzzy number. The contribution to the calculus literature and the solvability of multi-order fractional differential problems in the fuzzy spaces RF(E) have been presented with some illustrated examples.

Korneichuk-Stechkin Lemma, Ostrowski and Landau Inequalities, and Optimal Recovery Problems for L-Space Valued Functions

We prove an analogue of the Korneichuk–Stechkin lemma for functions with values in L-spaces. As applications, we obtain sharp Ostrowski type inequalities and solve problems of optimal recovery of identity and convexifying operators, as well as the problem of integral recovery on the classes of L-space valued functions with given majorant of modulus of continuity. The recovery is done based on n mean values of the functions over intervals. Moreover, on the classes of functions with given majorant of modulus of continuity of their Hukuhara type derivative, we solve the problem of optimal recovery of the function and the Hukuhara type derivative. The recovery is done based on n values of the function. We obtain sharp Landau type inequalities and solve an analogue of the Stechkin problem about approximation of unbounded operators by bounded ones and the problem of optimal recovery of an unbounded operator on a class of elements, known with error. Consideration of L-space valued functions gives a unified approach to solution of the mentioned above extremal problems for the classes of multi- and fuzzy-valued functions, and for the classes of functions with values in Banach spaces, in particular random processes, and many other classes of functions.

Integrating multivariate fuzzy neural networks into fuzzy inference system for enhanced decision making

In this paper, we introduce a novel family of operators for single-hidden layer feed-forward neural networks that operate on multivariate fuzzy-valued functions. These operators are based on the fractional mean value of the approximating function. The construction of these operators involves the use of sigmoidal functions, providing multiple choices for their implementation. We demonstrate the modeling capabilities of our operators employing a novel combined neural network system that incorporates the basic fuzzy inference mechanism and fuzzy neural networks. This system represents the connection strength among the output neurons using multivariate fuzzy-valued functions, enhancing decision-making processes. Additionally, we prove the convergence properties of these operators with respect to the Pompeiu-Hausdorff metric. By controlling the asymptotic decay of the sigmoidal function, we achieve accurate Jackson-type estimations using the modulus of continuity of fuzzy-valued functions.

Fuzzy Structured Element Integrated Decision of Metamorphism on Coal Dust Ignition Hazard Rating

To discuss the coal dust ignition hazard rating of different metamorphism, the fuzzy structure element integrated decision model about “no danger”, “weak danger”, “general danger”, and “strong danger” of coal dust ignition is established. 40 kinds of coal samples are selected to carry out coal quality analysis test, overpressure test, temperature test, flame propagation properties test and particle size distribution test. Based on this, coal dust ignition hazard rating decision-making index system is built. The triangular structure element is added to fuzzy comprehensive decision model, by sequencing weighted Hamming distance between fuzzy index value function and fuzzy maximal function, the coal dust ignition hazard rating of different coal quality can be decided. The result shows that: with increasing metamorphic grade, coal dust ignition hazard rating is decreasing, which provides study basis for relation between metamorphism and ignition hazard rating.

A NOVEL NUMERICAL METHOD FOR SOLVING FUZZY VARIABLE-ORDER DIFFERENTIAL EQUATIONS WITH MITTAG-LEFFLER KERNELS

In the fuzzy calculus, the study of fuzzy differential equations (FDEs) created a proper setting to model real problems which contain vagueness or uncertainties factors. In this paper, we consider a class fuzzy differential equations (FFDEs) with non-integer or variable order (VO). The variable order derivative is defined in the Atangana–Baleanu–Caputo sense on fuzzy set-valued functions. The main problem under the fuzzy initial condition is converted to a new problem by the [Formula: see text]-cut representation of fuzzy-valued function. For solving the new problem, we use the operational matrices (OMs) based on the shifted Legendre polynomials (SLPs). By approximating the unknown function and its derivative in terms of the SLPs and substituting these approximations into the equation, the main problem is converted to a system of nonlinear algebraic equations. An error estimate of the numerical solution is proved. Finally, an example is considered to confirm the accuracy of the proposed technique.

Fractional sampling operators of multivariate fuzzy functions and applications to image processing

In this work, we introduce a novel family of sampling-type operators of multivariate fuzzy-valued functions based on the fractional mean values of the approximated function. We discuss some convergence properties of these operators concerning an Lp-type fuzzy metric and examine the rate of convergence of the operators in Lipschitz classes of multivariate fuzzy valued functions. Some illustrative examples and graphs are given to demonstrate the convergence behavior of the operators in both one and two-dimensional cases. Finally, we present an algorithm for fuzzy image processing with numerical findings using the rule-based contrast enhancement method including a multidimensional fuzzy image matrix.

The Karush–Kuhn–Tucker (KKT) optimality conditions for fuzzy-valued fractional optimization problems

The Karush–Kuhn–Tucker (KKT) optimality conditions have a crucial role in finding the efficient solution of any optimization problem. Many researchers have been focussing on the establishment of KKT optimality conditions for deterministic fractional optimization problems (FOP). Also, solution algorithms for fuzzy-valued fractional optimization problem have been proposed by the researchers. However as far as authors are aware, there is no study available for the establishment of optimality conditions for the fuzzy-valued fractional optimization problem (FVFOP). In this paper, KKT optimality conditions are derived for FVFOP. Differentiation of fuzzy-valued functions is obtained using Hausdorff metric which find the distance of two fuzzy numbers and Hukuhara difference which is used to determine the difference between two fuzzy-valued functions. Considered FVFOP is transformed into non-fractional objective using the modified objective approach. The solution concept is developed using Lagrange multipliers, α-cuts and Dinkelbach algorithm. The proposed optimality conditions are verified by the numerical problems and the results are shown graphically for different values of α.

Optimality Guidelines for the Fuzzy Multi-Objective Optimization under the Assumptions of Vector Granular Convexity and Differentiability

In this research, we investigate a novel class of granular type optimality guidelines for the fuzzy multi-objective optimizations based on guidelines of vector granular convexity and granular differentiability. Firstly, the concepts of vector granular convexity is introduced to the vector fuzzy-valued function. Secondly, several properties of vector granular convex fuzzy-valued functions are provided. Thirdly, the granular type Karush-Kuhn-Tucker(KKT) optimality guidelines are derived for the fuzzy multi-objective optimizations.

Preliminary Observations on the Novel Approaches in Fuzzy Conformable Differential Calculus

This paper presents a novel extension of fuzzy calculus by integrating it with conformable calculus to introduce the fuzzy conformable derivative, a mathematical tool designed to handle the complexities of systems characterized by both uncertainty and fractional-order dynamics. The study begins by defining the fuzzy conformable derivative of order Ψ, which combines the ability of fuzzy calculus to model vagueness with the flexibility of conformable derivatives to capture non-integer order behaviors. This concept is further extended to higher orders, specifically the second order (2Ψ) and arbitrary order pΨ, enabling the modeling of more complex, multi-order dynamics in fuzzy-valued functions. Key contributions include the formal definitions of these derivatives, the establishment of their properties, and the development of operational rules that extend classical calculus operations to fuzzy systems. Additionally, the paper demonstrates how these derivatives can be applied to solve fuzzy differential equations and approximate functions using Taylor series expansions. While the fuzzy conformable derivative offers a powerful framework for modeling complex systems, the study also identifies certain limitations, such as potential unboundedness of solutions and the non-adherence to classical mathematical laws in higher-order cases. Overall, this work provides a comprehensive approach to differentiating fuzzy-valued functions in systems where uncertainty and fractional-order dynamics play a critical role. The proposed methods open up new avenues for research and application in fields such as control systems, economics, and engineering, where traditional calculus methods may not suffice. Future research directions include refining these methods, exploring computational techniques, and applying the framework to a broader range of real-world problems.

Fuzzy Form of Euler Method to Solve Fuzzy Differential Equations

Background: The Euler method is a elementary method to solve fuzzy differential equations numerically. Several authors have explored the Euler method by applying various approaches and derivative concepts. Methods: This paper proposes a fuzzy form of the Euler method to solve fuzzy initial value problems. Novelty of this approach is that the method developed based on fuzzy arithmetic. The solution by this method is readily available in the form of fuzzy-valued function. The method does not require to re-write fuzzy differential equation into a system of two crisp ordinary differential equations. Results: The algorithm of proposed method and local error expression are discussed. An illustration and solution of the fuzzy Riccati equation are provided for the applicability of the method. Conclusion: The proposed Euler method is a natural generalization of crisp Euler method. It is very efficient in solving linear and nonlinear fuzzy differential equations. One of the advantage of this method is that the solution obtained by the method is always a fuzzy-valued function due to well-defined fuzzy arithmetic.

On fuzzy numerical model dealing with the control of glucose in insulin therapies for diabetes via nonsingular kernel in the fuzzy sense

<abstract><p>Very recently, several novel conceptions of fractional derivatives have been proposed and employed to develop numerical simulations for a wide range of real-world configurations with memory, background, or non-local effects via an uncertainty parameter $ [0, 1] $ as a confidence degree of belief. Under the complexities of the uncertainty parameter, the major goal of this paper is to develop and examine the Atangana-Baleanu derivative in the Caputo sense for a convoluted glucose-insulin regulating mechanism that possesses a memory and enables one to recall all foreknowledge. However, as compared to other existing derivatives, this is a vitally important point, and the convenience of employing this derivative lessens the intricacy of numerical findings. The Atangana-Baleanu derivative in the Caputo sense of fuzzy valued functions (FVF) in parameterized interval representation is established initially in this study. Then, it is leveraged to demonstrate that the existence and uniqueness of solutions were verified using the theorem suggesting the Banach fixed point and Lipschitz conditions under generalized Hukuhara differentiability. In order to explore the regulation of plasma glucose in diabetic patients with impulsive insulin injections and by monitoring the glucose level that returns to normal in a finite amount of time, we propose an impulsive differential equation model. It is a deterministic mathematical framework that is connected to diabetes mellitus and fractional derivatives. The framework for this research and simulations was numerically solved using a numerical approach based on the Adams-Bashforth-Moulton technique. The findings of this case study indicate that the fractional-order model's plasma glucose management is a suitable choice.</p></abstract>

Analysis on determining the solution of fourth-order fuzzy initial value problem with Laplace operator.

This study presents a new analytical method to extract the fuzzy solution of the fuzzy initial value problem (FIVP) of fourth-order fuzzy ordinary differential equations (FODEs) using the Laplace operator under the strongly generalized Hukuhara differentiability (SGH-differentiability). To this end, firstly the fourth-order derivative of the fuzzy valued function (FVF) according to the type of the SGH-differentiability is introduced, and then the relationships between the fourth-order derivative of the FVF and its Laplace transform are established. Furthermore, considering the types of differentiabilities and switching points, some fundamental theorems related to the Laplace transform of the fourth-order derivative of the FVF are stated and proved in detail and a method to solve FIVP by the fuzzy Laplace transform is presented in detail. An application of our proposed method in Resistance-Inductance circuit (RL circuit) is presented. Finally, FIVP's solution is graphically analyzed to visualize and support theoretical results.