Fuzzy Order Research Articles

Fifty years of similarity relations: a survey of foundations and applications

On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed.

An approach to optimize the cost of transportation problem based on triangular fuzzy programming problem

In this article, we address a fully fuzzy triangular linear fractional programming (FFLFP) problem under the condition that all the parameters and decision variables are characterized by triangular fuzzy numbers. Utilizing the computation of triangular fuzzy numbers and Lexicographic order (LO), the FFLFP problem is changed over to a multi-objective function. Consequently, the problem is changed into a multi-objective crisp problem. This paper outfits another idea for diminishing the computational complexity, in any case without losing its viability crisp LFP issues. Lead from real-life problems, a couple of mathematical models are considered to survey the legitimacy, usefulness and applicability of our method. Finally, some mathematical analysis along with one case study is given to show the novel strategies are superior to the current techniques.

Identification of Shipyard Priorities in a Multi-Criteria Decision-Making Environment through a Transdisciplinary Energy Management Framework: A Real Case Study for a Turkish Shipyard

Ship building, as an energy-intensive sector, produces significant amounts of air emissions, including greenhouse gases. Most research in greenhouse gas reductions from shipping concentrates on the reduction in emissions during the operational phase. However, as emissions during ship operation are reduced, the construction and dismantling phases of ships are becoming increasingly important in the assessment of the life-cycle impact of ships. In this study, priorities for a Turkish shipyard to become energy efficient were identified by means of a semi-structured questionnaire and an interview. This was undertaken using Fuzzy Multi-Criteria Decision-Making methods, including the Fuzzy Analytical Hierarchy Process and Fuzzy Order of Preference by Similarity to Ideal Solution, which are part of a proposed systematic and transdisciplinary Energy Management Framework and System. By applying Multi-Criteria Decision-Making methods, this framework supports the shipyard’s decision makers to make rational and optimized decisions regarding energy sectors within their activities. Applying the framework has significant potential to help achieve good product quality while reducing costs and environmental impacts, and can thereby enhance the sustainability of shipping. Moreover, the framework can boost both business and socio-economic perspectives for the shipyard, and improve its reputation and competitiveness, in alignment with achieving the Nationally Determined Contributions of States under the Paris Agreement.

Recursive higher order fuzzy transform method for numerical solution of Volterra integral equation with singular and nonsingular kernels

In the present paper, the continuous form of the higher degree Fuzzy transform (Fm-transform) technique is used for the numerical solution of the second kind Volterra integral equations with singular and nonsingular kernels. By employing this method, the equation is converted to the system of linear algebraic equations with a lower Hessenberg coefficient matrix. It is shown that the coefficient matrix is invertible. The error analysis is carried out for the solution procedure. The results show that for singular kernels, the accuracy decreases with an increasing degree of fuzzy transform due to the non-differentiability of the kernel. Experiments on nonsingular kernels show convergence with an increasing degree of fuzzy transform. As expected, the accuracy of the method improves with partition numbers for both singular and nonsingular kernels. Some examples are given to illustrate the theoretical results.

Optimal homotopy asymptotic method for solving several models of first order fuzzy fractional IVPs

In this work, the Optimal Homotopy Asymptotic Method (OHAM) is prolifically implemented to find the optimal solutions of fractional order of fuzzy differential equations. We inspect the competence of the method by examining fractional order time-dependent first order fuzzy initial value problems in the Caputo derivative sense. The concepts of fuzzy sets theory and fractional calculus were used to present a new general fuzzification formulation of the OHAM method followed by fuzzy analysis for the proposed fractional problems are presented in detail. OHAM is recognized as a feasible and convergent method for solving linear and nonlinear models, and it provides a convenient technique for controlling the convergence of approximate solutions optimally. The method's capability is demonstrated by obtaining approximate solutions to linear and nonlinear fuzzy initial value problems in various orders of fractional derivative. The numerical results acknowledge that OHAM is a feasible and transformation mechanism for solving linear and nonlinear fuzzy fractional initial value problems. Also, the results ensure that the method used was succinct, efficient, and simple to use, to deal with a more comprehensive fuzzy fractional differential equation.

Some Integral Inequalities for Generalized Convex Fuzzy-Interval-Valued Functions via Fuzzy Riemann Integrals

In this study, we introduce the new concept of h-convex fuzzy-interval-valued functions. Under the new concept, we present new versions of Hermite–Hadamard inequalities (H–H inequalities) are called fuzzy-interval Hermite–Hadamard type inequalities for h-convex fuzzy-interval-valued functions (h-convex FIVF) by means of fuzzy order relation. This fuzzy order relation is defined level wise through Kulisch–Miranker order relation defined on fuzzy-interval space. Fuzzy order relation and inclusion relation are two different concepts. With the help of fuzzy order relation, we also present some H–H type inequalities for the product of h-convex FIVFs. Moreover, we have also established strong relationship between Hermite–Hadamard–Fej´er (H–H–Fej´er) type inequality and h-convex FIVF. There are also some special cases presented that can be considered applications. There are useful examples provided to demonstrate the applicability of the concepts proposed in this study. This paper's thoughts and methodologies could serve as a springboard for more research in this field.

New fuzzy transportation algorithm without converting fuzzy numbers

The ranking function is widely used to convert fuzzy numbers to be crisp on solving fuzzy transportation problems. The converting process can indeed make it easier to play the fuzzy transportation method, but from the convenience, it causes failed in interpreting the results of converting fuzzy numbers. This is because the converting process of fuzzy numbers still has subjectivity values, so it cannot be eliminated, moreover, the ordering can cause incompatible input and output fuzzy numbers resulted. Therefore, the new fuzzy transportation method is proposed by fuzzy Analytical Hierarchy Process to order fuzzy parameters on fuzzy transportation problem without converting fuzzy numbers to crisp numbers, then Algorithm 2 until 6 is used to obtain a fuzzy optimal solution. The advantages of the new proposed method can improve the shortcomings of the existing methods, as well as relevant to solve fuzzy transportation problems in real life

ON FUZZY PARTIAL FRACTIONAL ORDER EQUATIONS UNDER FUZZIFIED CONDITIONS

This paper is devoted to investigating or computing the solution to one-dimensional partial fuzzy fractional order heat equation. In particular, one-dimensional fuzzy partial heat equation is hybridized into two equations by hybrid techniques along with the concept of parametric fuzzy number. For this investigation, a hybrid method of decomposition due to Adomian and Laplace transform is used. The considered techniques are presented for the computation of series of solutions of partial fractional order heat equation. The applied techniques have also provided the accuracy, simplicity and efficiency as compared to other existing methods. Finally, some illustrations are solved for the justification of our theoretical solution.

Dynamic stability of a class of fractional‐order nonlinear systems via fixed point theory

In this paper, the problem of the uniform stability for a class of fuzzy fractional‐order genetic regulatory networks with random discrete delays, distributed delays, and parameter uncertainties is studied. Although there is a portion of literature on using fixed point theorems to study the stability of fractional neural networks, most of them required the fractional order to be in . However, the case of the fractional‐order belonging to has not been discussed. To solve it, this work proposes a novel idea of using fixed point theory to study the stability of fuzzy (0,1) order neural networks, the problem of the uniqueness of the solution of the considered genetic regulatory networks is resolved, and a novel sufficient condition to guarantee the uniform stability of above genetic regulatory networks is also derived. Eventually, an example is given to demonstrate that the obtained result is effective.

Frequency Regulation of Multi Area Renewable Energy Source system with Practical Constraints under Fractional-order Fuzzy Controller

This paper introduces a novel control strategy of fractional order (FO) fuzzy (F) PID (FOFPID) controller optimized with the latest soft computing technique of seagull optimization algorithm (SOA) for power system frequency regulation. Initially, a simple and widely accepted power system of dual area photovoltaic (PV) and reheat thermal (RT) (PVRT) system is perceived and named as test system-1 in this paper. The performance of FOFPID fine-tuned with SOA mechanism is tested on PVRT system for a step load disturbance of 10% (SLD) on area-2 along with other controllers reported in the literature. Dynamical analysis of the PVRT system reveals the potency of the proposed controller over others. Further, the SOA based FOFPID controller is extended to frequency regulation of multi-area system with hybrid generating sources (MAHS) named as test system-2 in this paper for 10% SLD on area-1. MAHS system is constituted with realistic constraints to conduct research close to realistic practice. The potentiality of SOA based FOFPID is demonstrated by comparing it with traditional controllers of PID/FPID/FOPID on MAHS system. Finally, robustness analysis is perpetuated to reveal the presented control scheme robustness.

Two-person game with hesitant fuzzy payoff: An application in MADM

Hesitant Fuzzy Set (HFS) permits the membership function having a collection of probable values which are more effective for modelling the real-life problems. Multiple Attribute Decision Making (MADM) process apparently assesses multiple conflicting attribute in decision making. In traditional decision making problems, each player is moving independently whereas in reality it is seen that each player aims to maximize personal profit which causes a negative impact on other player. MADM problem treats with candidate to the best alternative corresponding to the several attributes. Here, we present an MADM problem under hesitant fuzzy information and then transforming it into two-person matrix game, referred to herein as MADM game. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is one of the prominent approach for solving the MADM problems. In this work, we develop the TOPSIS based on Ordered Weighted Aggregation (OWA) operator and hybrid hesitant fuzzy normalized Euclidean distance.Please check whether short title on odd pages have been set correctly. Then the two approaches, namely Hybrid Hesitant Fuzzy Ordered Weighted Aggregation-TOPSIS (HHFOWA-TOPSIS) and the Linear Programming Problem (LPP) are applied to solve the formulated MADM game. For solving MADM game, we apply LPP by considering the various values of α,ψ, and HHFOWA-TOPSIS for finding the optimal alternative according to their scores.Please provide missing AMS classification codes. An investment selection problem is included to explain the feasibility and superiority of our formulated approaches. A comparison analysis is drawn among the obtained results which are derived from the two approaches. LPP and HHFOWA-TOPSIS provide the best alternative for the proposed problem. Finally, conclusions about our findings and outlooks are described.

Tracking control with disturbance rejection of nonlinear fractional order fuzzy systems: Modified repetitive control approach

This paper addresses the problem of tracking control and disturbance rejection for nonlinear fractional order systems. To this end, based on Takagi-Sugeno (T-S) fuzzy modeling approach and a fractional order modified repetitive controller, an output feedback strategy is constructed for a class of nonlinear fractional order systems. T-S fuzzy model represents an accurate model of nonlinear fractional order systems and simplifies the procedure of control system design. By introducing a novel Fuzzy Lyapunov Function (FLF), sufficient conditions are established in the framework of Linear Matrix Inequalities (LMIs) to ensure the stability of the closed-loop system. Based on the achieved conditions, it is shown that the external disturbance is attenuated and the output of the system tracks the periodic reference input with an ignorable error. Lastly, three examples including fractional order Chua’s circuit, Lorenz-like and Chen systems are performed to show the efficiency of the method proposed in this paper.

A technology of invariant feature extraction of Uav remote sensing image based on fuzzy fractional order function

A technology of invariant feature extraction of Uav remote sensing image based on fuzzy fractional order function

Control and Robust Stabilization at Unstable Equilibrium by Fractional Controller for Magnetic Levitation Systems

The problem of control and stabilizing inherently non-linear and unstable magnetic levitation (Maglev) systems with uncertain equilibrium states has been studied. Accordingly, some significant works related to different control approaches have been highlighted to provide robust control and enhance the performance of the Maglev system. This work examines a method to control and stabilize the levitation system in the presence of disturbance and parameter variations to minimize the magnet gap deviation from the equilibrium position. To fulfill the stabilization and disturbance rejection for this non-linear dynamic system, the fractional order PID, fractional order sliding mode, and fractional order Fuzzy control approaches are conducted. In order to design the suitable control outlines based on fractional order controllers, a tuning hybrid method of GWO–PSO algorithms is applied by using the different performance criteria as Integrated Absolute Error (IAE), Integrated Time Weighted Absolute Error (ITAE), Integrated Squared Error (ISE), and Integrated Time Weighted Squared Error (ITSE). In general, these objectives are used by targeting the best tuning of specified control parameters. Finally, the simulation results are presented to determine which fractional controllers demonstrate better control performance, achieve fast and robust stability of the closed-loop system, and provide excellent disturbance suppression effect under nonlinear and uncertainty existing in the processing system.

New Hermite\u2013Hadamard and Jensen Inequalities for Log-h-Convex Fuzzy-Interval-Valued Functions

In the preset study, we introduce the new class of convex fuzzy-interval-valued functions which is called log-h-convex fuzzy-interval-valued functions (log-h-convex FIVFs) by means of fuzzy order relation. We have also investigated some properties of log-h-convex FIVFs. Using this class, we present Jensen and Hermite–Hadamard inequalities (HH-inequalities). Moreover, some useful examples are presented to verify HH-inequalities for log-h-convex FIVFs. Several new and known special results are also discussed which can be viewed as an application of this concept.

A process mining approach in big data analysis and modeling decision making risks for measuring environmental health in institutions

This paper aimed to introduce a process-mining framework for measuring the status of environmental health in institutions. The methodology developed a new software-based index namely Institutional Environmental Health Index (IEHI) that was integrated from ontology-based Multi-Criteria Group Decision-Making models based on the principles of fuzzy modeling and consensus evaluation. Fuzzy Ordered Weighting Average (OWA) with the capability of modeling the uncertainties and decision-making risks along with Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) were employed as the computation engine. The performance of the extended index was examined through an applied example on 20 mosques as public institutions. IEHI could analyze big data collected by environmental health investigators and convert them to a single and interpretable number. The index detected the mosques with very unsuitable health conditions that should be in priority of sanitation and suitable ones as well. Due to the capability of defining the type and numbers of criteria and benefitting from specific and user-friendly software namely Group Fuzzy Decision-Making, this index is highly flexible and practical. The methodology could be used for numerating the environmental health conditions in any intended institution or occupation. The proposed index would provide e-health assessment by more efficient analysis of big data and risks that make more realistic decisions in environmental health system.

Fuzzy Order Acceptance and Scheduling on Identical Parallel Machines

In this study, we focus on the fuzzy order acceptance and scheduling problem in identical parallel machines (FOASIPM), which is a scheduling and optimization problem to decide whether the firm should accept or outsource the order. In general, symmetry is a fundamental property of optimization models used to represent binary relations such as the FOASIPM problem. Symmetry in optimization problems can be considered as an engineering tool to support decision-making. We develop a fuzzy mathematical model (FMM) and a Genetic Algorithm (GA) with two crossover operators. The FOASIPM is formulated as an FMM where the objective is to maximize the total net profit, which includes the revenue, the penalty of tardiness, and the outsourcing. The performance of the proposed methods is tested on the sets of data with orders that are defined by fuzzy durations. We use the signed distance method to handle the fuzzy parameters. While FMM reaches the optimal solution in a reasonable time for datasets with a small number of orders, it cannot find a solution for datasets with a large number of orders due to the NP-hard nature of the problem. Genetic algorithms provide fast solutions for datasets with a medium and large number of orders.

Design of optimal fractional order fuzzy PID controller based on cuckoo search algorithm for core power control in molten salt reactors

In case of the adverse effects of unforeseeable or uncertain disturbances on control systems, stability robustness of control performance is crucial for controller design. Accordingly, in this study, an optimal design scheme for Fractional Order Fuzzy Proportional Integral Derivative (FOFPID) controller with adoption of Cuckoo Search (CS) optimization algorithm is presented in order to examine the application of the FOFPID in the core power control of molten salt reactors (MSRs) as an example. By using the proposed scheme, PID, FOPID and fuzzy PID controllers are also designed by the CS for the purpose of showing the superiority of the FOFPID controller. Firstly, a nonlinear dynamic model of the MSR system is introduced by means of point kinetic equations considering the time delay in the fluid's transit connecting the core and the heat exchanger. Then, the nonlinear model is linearized for obtaining a linear model of MSR core system. Also, for the purpose of performing the sensitivity analysis, transit behaviors for both the linear and nonlinear approaches are shown and compared for different reactivity inputs. The core power control of molten salt reactor system is examined for each tuned controller in terms of stability, time domain and the robustness analysis under instantaneous and continuous disturbances. The results showed that the proposed CS based FOFPID controller has better control performance as compared to the PID, FOPID and fuzzy PID controllers tuned by the CS, especially in point of capability of disturbance suppression, improving tracking performance, and robustness against parameter uncertainty.

Fuzzy Homotopy Analysis Method For Solving Fuzzy Autonomous Differential Equation

In this paper, we have presented the theory and applications of the fuzzy homotopy analysis method to find the fuzzy semi-analytical solutions of the second order fuzzy autonomous differential equation. This method allows for the solution of the Fuzzy initial value problems to be calculated in the form of an infinite fuzzy series in which the fuzzy components can be easily calculated. Some numerical results are given to illustrate the used method. The obtained numerical results are compared with the fuzzy exact-analytical solutions.

Approximate Solution of Higher Order Fuzzy Initial Value Problems of Ordinary Differential Equations Using Bezier Curve Representation

The Bezier curve is a parametric curve used in the graphics of a computer and related areas. This curve, connected to the polynomials of Bernstein, is named after the design curves of Renault's cars by Pierre Bézier in the 1960s. There has recently been considerable focus on finding reliable and more effective approximate methods for solving different mathematical problems with differential equations. Fuzzy differential equations (known as FDEs) make extensive use of various scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under uncertainty. This article discusses the use of Bezier curves for solving elevated order fuzzy initial value problems (FIVPs) in the form of ordinary differential equation. A Bezier curve approach is analyzed and updated with concepts and properties of the fuzzy set theory for solving fuzzy linear problems. The control points on Bezier curve are obtained by minimizing the residual function based on the least square method. Numerical examples involving the second and third order linear FIVPs are presented and compared with the exact solution to show the capability of the method in the form of tables and two dimensional shapes. Such findings show that the proposed method is exceptionally viable and is straightforward to apply.