Fuzzy Interval Arithmetic Research Articles

On Atangana–Baleanu fractional granular calculus and its applications to fuzzy economic models in market equilibrium

Based on relative-distance-measure fuzzy interval arithmetic (RDM-FIA), this paper presents several new concepts for fuzzy functions in fractional granular calculus with Mittag-Leffler (ML) kernels, such as Atangana–Baleanu (AB) fractional granular integrals, AB fractional granular derivatives, and AB Caputo fractional granular derivatives. Then, we present formulas for the fuzzy granular Laplace transform on Caputo fractional granular derivatives, AB fractional granular derivatives, and AB Caputo fractional granular derivatives. In addition to the relationship between AB fractional granular derivatives and AB Caputo fractional granular derivatives established using the granular Laplace transform, new fractional Newton–Leibniz-type formulas for both derivatives and integrals are proved. As applications, we obtain the fundamental solutions for the fuzzy economic models in the framework of Caputo fractional granular derivatives and AB Caputo fractional granular derivatives by utilizing the fuzzy granular Laplace transform. Several graphical explanations are given to show the dynamic behavior of the fundamental solutions. Moreover, we illustrate the advantages of using RDM-FIA to study the mathematical models of uncertain dynamical systems. The results of this study are illustrated using several numerical examples.

Multidimensional solution of fuzzy linear programming

There are several approaches to address fuzzy linear programming problems (FLPP). However, due to using standard interval arithmetic (SIA), these methods have some limitations and are not complete solutions. This article establishes a new approach to fuzzy linear programming via the theory of horizontal membership functions and the multidimensional relative-distance-measure fuzzy interval arithmetic. Furthermore, we propose a multidimensional solution based on the primal Simplex approach that satisfies any equivalent form of FLPP. The new solutions of FLPP are also compared with the results of existing methods. Some numerical examples have been illustrated to show the efficiency of the proposed method.

Tracking Control of a Class of Fuzzy Dynamical Systems under the Concept of Granular Differentiability

In this work, the tracking control of a class of uncertain linear dynamical systems is investigated. The uncertainty is considered to be represented as fuzzy numbers, and hence, these uncertain dynamical systems are referred to as fuzzy linear dynamical systems, which are presented in the form of fuzzy differential equations (FDEs). The solution of an FDE is found using an approach called relative-distance-measure fuzzy interval arithmetic and under the granular differentiability concept. The control objective is to provide a control law such that the output of the system tracks a desired reference input in the presence of uncertainties. To this end, a theorem is proposed, which suggests that the control law should take the form of a feedback of fuzzy states with fuzzy gains and a fuzzy pre-compensator. However, since the fuzzy states of the system may not always be measurable, a fuzzy observer is designed for the estimation of such fuzzy states. It is also clearly shown that the generalized Hukuhara differentiability concept is unable for solving the problem examined in this study. Finally, the efficiency of the approach is examined for a plane landing control problem.

Improving Human Reliability Analysis for Railway Systems Using Fuzzy Logic

The International Union of Railway provides an annually safety report highlighting that human factor is one of the main causes of railway accidents every year. Consequently, the study of human reliability is fundamental, and it must be included within a complete reliability assessment for every railway-related system. However, currently RARA (Railway Action Reliability Assessment) is the only approach available in literature that considers human task specifically customized for railway applications. The main disadvantages of RARA are the impact of expert’s subjectivity and the difficulty of a numerical assessment for the model parameters in absence of an exhaustive error and accident database. This manuscript introduces an innovative fuzzy method for the assessment of human factor in safety-critical systems for railway applications to address the problems highlighted above. Fuzzy logic allows to simplify the assessment of the model parameters by means of linguistic variables more resemblant to human cognitive process. Moreover, it deals with uncertain and incomplete data much better than classical deterministic approach and it minimizes the subjectivity of the analyst evaluation. The output of the proposed algorithm is the result of a fuzzy interval arithmetic, $\alpha $ -cut theory and centroid defuzzification procedure. The proposed method has been applied to the human operations carried out on a railway signaling system. Four human tasks and two scenarios have been simulated to analyze the performance of the proposed algorithm. Finally, the results of the method are compared with the classical RARA procedure underline compliant results obtain with a simpler, less complex and more intuitive approach.

Fuzzy Fractional Quadratic Regulator Problem Under Granular Fuzzy Fractional Derivatives

In this paper, a class of uncertain linear dynamical systems called fuzzy fractional linear dynamical systems is investigated. The aim is to find control inputs to keep the states of the fuzzy fractional dynamical systems near the zero in an optimal manner. The optimality criterion is in a form of a granular fuzzy integral whose integrand is a quadratic function of the state variables and control inputs. The fuzzy fractional dynamical system is described using fuzzy fractional differential equations (FFDEs). In order to achieve the aim, an effective approach for solving FFDEs should be at disposal. Due to some restrictions imposed by the previous approaches dealing with FFDEs, a new approach is proposed. The proposed approach is based on the granular derivative and the so-called relative-distance-measure fuzzy interval arithmetic. New definitions of fuzzy fractional derivatives and integral called left and right granular Riemann–Liouville fuzzy fractional derivatives, left and right granular Caputo fuzzy fractional derivatives, and the left and right granular fuzzy fractional integral are also presented. In addition, the concepts of granular fuzzy partial derivative and granular fuzzy chain rule are introduced. By the approximations of the granular fuzzy fractional integral and the granular Caputo fuzzy fractional derivative, the approximation solution to the FFDEs is obtained. Consequently, based on the new concepts and theorems, the solution to the fuzzy fractional quadratic regulator problem is given by a theorem. This paper closes with an example of regulating the motion of Boeing 747 in longitudinal direction with the presence of uncertainty in the initial conditions and the coefficients.

Fuzzy Bang-Bang control problem under granular differentiability

In this paper, a class of fuzzy optimal control problem called fuzzy Bang-Bang (FBB) control problem is revisited. The FBB control problem aims at finding control inputs which transfer the states of an uncertain dynamical system to the origin in a minimum time. Since the conditions and/or parameters of the dynamical system are uncertain, the FBB control problem is a challenge task. To address the problem, we first introduce the concepts of the granular integral and derivative of fuzzy functions whose domain is uncertain. In addition, the notion of granular partial derivative of a multivariable fuzzy function whose variables are fuzzy functions with uncertain domains is presented. Then, we propose a theorem which is proved to be applicable to the FBB control problem. Moreover, taking the relative-distance-measure fuzzy interval arithmetic and horizontal membership functions into consideration, we further give complementary theorems to ensure that if the problem has a solution, then the controller assumes its boundary values. The simulated results confirm this theoretical conclusion. These findings may enrich our insight of the behavior of the uncertain dynamical system subjected to the FBB control problem, and guide to predict the uncertain trajectories.

Sub-optimal control of fuzzy linear dynamical systems under granular differentiability concept

This paper deals with sub-optimal control of a fuzzy linear dynamical system. The aim is to keep the state variables of the fuzzy linear dynamical system close to zero in an optimal manner. In the fuzzy dynamical system, the fuzzy derivative is considered as the granular derivative; and all the coefficients and initial conditions can be uncertain. The criterion for assessing the optimality is regarded as a granular integral whose integrand is a quadratic function of the state variables and control inputs. Using the relative-distance-measure (RDM) fuzzy interval arithmetic and calculus of variations, the optimal control law is presented as the fuzzy state variables feedback. Since the optimal feedback gains are obtained as fuzzy functions, they need to be defuzzified. This will result in the sub-optimal control law. This paper also sheds light on the restrictions imposed by the approaches which are based on fuzzy standard interval arithmetic (FSIA), and use strongly generalized Hukuhara and generalized Hukuhara differentiability concepts for obtaining the optimal control law. The granular eigenvalues notion is also defined. Using an RLC circuit mathematical model, it is shown that, due to their unnatural behavior in the modeling phenomenon, the FSIA-based approaches may obtain some eigenvalues sets that might be different from the inherent eigenvalues set of the fuzzy dynamical system. This is, however, not the case with the approach proposed in this study. The notions of granular controllability and granular stabilizability of the fuzzy linear dynamical system are also presented in this paper. Moreover, a sub-optimal control for regulating a Boeing 747 in longitudinal direction with uncertain initial conditions and parameters is gained. In addition, an uncertain suspension system of one of the four wheels of a bus is regulated using the sub-optimal control introduced in this paper.

Treating design uncertainty in the application of Eco-indicator 99 with Monte Carlo simulation and fuzzy intervals

In incremental eco-design improvements, design engineers attempt to modify an existing product via some eco-design measures to reduce the product’s environmental impacts (e.g. reduction of material usage and energy consumption). In this process, several design concepts can be proposed, and concept selection is required to allocate resources sensibly to promising design concepts only. In this context, the research purpose is to estimate the environmental impacts of each concept given the uncertainty of design information. In the proposed methodology, the fuzzy interval arithmetic is used to specify and propagate imprecise design information. Then, the centroid concept is applied to model different views of imprecision (i.e. pessimistic, balanced and optimistic) associated with fuzzy impact assessment. Accordingly, a decision scheme is developed to support concept selection and suggest the potential areas for further eco-design improvements. A coffee maker is used as an example to demonstrate the proposed methodology. Also, the Monte Carlo Simulation is applied for the same example to compare the numerical outcomes by the fuzzy interval approach.

Methods for uncertainty propagation in life cycle assessment

Life cycle assessment (LCA) calculates the environmental impact of a product over its entire life cycle. Uncertainty analysis is an important aspect in LCA, and is usually performed using Monte Carlo sampling. In this study, Monte Carlo sampling, Latin hypercube sampling, quasi Monte Carlo sampling, analytical uncertainty propagation and fuzzy interval arithmetic were compared based on e.g. convergence rate and output statistics. Each method was tested on three LCA case studies, which differed in size and behaviour. Uncertainty propagation in LCA using a sampling method leads to more (directly) usable information compared to fuzzy interval arithmetic or analytical uncertainty propagation. Latin hypercube and quasi Monte Carlo sampling provide more accuracy in determining the sample mean than Monte Carlo sampling and can even converge faster than Monte Carlo sampling for some of the case studies discussed in this paper.

On the use of possibility theory in uncertainty analysis of life cycle inventory

Purpose The purpose of this paper is to enhance the mathematical and physical understanding of practitioners of uncertainty analysis of life cycle inventory (LCI), on the application of possibility theory. The main questions dealt with are (1) clear definition of the terms—“necessity–possibility,” “probability,” “belief–plausibility,” and of their mutual relationships; (2) what justifies the substitution of classical probability for possibility; (3) mutual comparison of, and transformations in both senses between probability and possibility uncertainty measures; (4) how to construct meaningful input possibility measures from available probabilistic/statistic information; and (5) comparative analysis of the solutions of the problem of data uncertainty propagation in LCI, afforded, respectively, by probabilistic Monte Carlo simulation and possibilistic fuzzy interval arithmetic.

Multi-objective supervisory flow control based on fuzzy interval arithmetic: Application for scheduling of manufacturing systems

Complex production systems can produce more than one part type under multiple and possibly conflicting objectives. This paper considers the design of the multiple objective real-time scheduling problem of a multiple-part-type production system. Based on fuzzy control theory and fuzzy arithmetic intervals, distributed and supervised continuous-flow control architecture has been proposed. The objective is to balance the production process by adjusting the continuous production rates of the machines on the basis of the average behaviour. The supervisory control aims to maintain the overall performance within acceptable limits. In the proposed approach, the problem of a real-time scheduling of jobs is considered at the shop-floor level. In this case, the actual dispatching times are determined from the continuous production rates through a discretisation procedure. To deal with conflicts between jobs at a shared machine, a decision for the actual part to be processed is taken using some criterions which represent a measure of the job’s priority. A case study demonstrates the efficiency of the proposed control approach.

Evaluation of Mn concentration provided by soil in citrus-growing regions

Manganese (Mn) is an essential nutrient element in citrus growing and Mn deficiency causes some problems related with physiological and morphological structure. Spatial evaluation of Mn obtained from soils in citrus-growing areas is the main objective of this paper. For this purpose, a citrus-growing region in Turkey has been selected and three effective estimation methods: kriging, neural–fuzzy modelling, and fuzzy interval arithmetic have been considered for the spatial evaluations. The model works primarily focus on the model accuracy and smoothing degree of estimations. In addition, error analysis and comparative assessments, which present the advantages and drawbacks of the models, are conducted in the paper. The results and performance evaluations prove the superiorities of soft computing approach in this evaluation.

IMPACT OF LEVEL OF CONTINGENCIES ON WELLBEING STATES OF GENERATING SYSTEM – A PROBABILISTIC AND FUZZY APPROACH

The required level of operating reserve to be maintained by an electric power system can be determined by deterministic and probabilistic techniques. The power system parameters such as failure and repair rates used in the probabilistic models basically come from historical operation records, which are necessarily limited and subject to errors. There is considerable data uncertainty that exists in these parameters. This paper presents the application of fuzzy sets, which is recognized as a potential tool to include data uncertainty in normal calculations, for evaluation of spinning reserve is illustrated. In conventional reliability evaluation method, loads are deterministic values. The failure rate of the generating units is represented as triangular fuzzy number. The fuzzy set is used to represent consumer's demand, which is the deterministic load, as fuzzy loads. The concept is extended for evaluating the fuzzy based well-being indices using different criterion for the generating system. The fz-forced outage rate (FOR) and fz-availability of a unit in service are evaluated using fuzzy Sets and fuzzy interval arithmetic to find the fuzzy well-being states, which are designated as healthy, marginal and at risk. The concept is illustrated for a Roy Billinton's Test System (RBTS) and IEEE RTS. Impact of outage level and deterministic criteria is also illustrated. The proposed method can be applied to the operation of deregulated power system in which operating reserve is served as an important issue with respect to stable and reliable operation.

An uncertainty oriented fuzzy methodology for grade estimation

Grade estimation is directed toward mineral inventory estimates for a mineral deposit. In this paper, a new uncertainty oriented methodology for global grade estimation is given and its stages are demonstrated in a step by step manner. The proposed fuzzy method uses a series of fuzzy algorithms and finally estimates the ore grades based on fuzzy interval arithmetic. The method is relatively simple, and practical especially for limited numbers of data. In addition, the uncertainties can be taken into account by using fuzzy numbers. The results show that the method can be applied in reserve estimation.