Shokrollah Ziari - Academia.edu (original) (raw)

Papers by Shokrollah Ziari

Journal of Intelligent and Fuzzy Systems, Jun 22, 2017

In this paper, we continue a study of approximation properties of the fuzzy transform (F-transfor... more In this paper, we continue a study of approximation properties of the fuzzy transform (F-transform) with the partition generated by the Shepard kernel. We make the error estimate in terms of a modulus of continuity which has a higher order than the previously known. Also, we obtain the error estimate of the iterative F-transform-based method for linear Fredholm integral equations of the second kind.

Differential Equations and Dynamical Systems, Jul 30, 2019

In this paper, we propose a method of successive approximations for nonlinear fuzzy Fredholm inte... more In this paper, we propose a method of successive approximations for nonlinear fuzzy Fredholm integral equations of the second kind. The main approximation tool is based on fuzzy block-pulse functions. The error estimation of the proposed method is established. A number of illustrative examples that demonstrate accuracy and convergence is given as well.

Fuzzy Sets and Systems, Jul 1, 2016

In this paper, we obtain the error estimation of the iterative method based on quadrature formula... more In this paper, we obtain the error estimation of the iterative method based on quadrature formula to solve nonlinear fuzzy Fredholm integral equations of the second kind given in Fuzzy Sets & Syst. 245 (2014) 1-17, in terms of uniform and partial modulus of continuity. Moreover, we extend in the context of using the modulus of continuity, the notion of numerical stability of the solution with respect to the first iteration.

Advances in Fuzzy Integral and Differential Equations, 2021

In the present work we construct an iterative method for the numerical solution of fuzzy fraction... more In the present work we construct an iterative method for the numerical solution of fuzzy fractional Volterra integral equations, by using the technique of fuzzy product integration. The existence and uniqueness of the solution and the uniform boundedness of the terms of the Picard iterations are proved. The convergence of the iterative algorithm is obtained and the apriori error estimate is given in terms of the Lipschitz constants. A numerical example illustrates the accuracy of the method.

AIMS Mathematics, 2022

This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations... more This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations using Picard iteration method and bivariate Bernstein polynomials. We first present the way to approximate the value of the multiple integral of any fuzzy-valued function based on the two dimensional Bernstein polynomials. Then, it is used to construct the numerical iterative method for finding the approximate solutions of two dimensional fuzzy integral equations. Also, the error analysis and numerical stability of the method are established for such fuzzy integral equations considered here in terms of supplementary Lipschitz condition. Finally, some numerical examples are considered to demonstrate the accuracy and the convergence of the method.

Advances in Fuzzy Integral and Differential Equations, 2021

IEEE Transactions on Fuzzy Systems, 2019

In the present study, a new iterative numerical method for solving bivariate nonlinear fuzzy Fred... more In the present study, a new iterative numerical method for solving bivariate nonlinear fuzzy Fredholm integral equations is proposed. The method combines two well proven approaches: successive approximations and mixed trapezoidal and midpoint rules for the numerical integration. Both approaches are elaborated for fuzzy-valued functions. The main advantage of the proposed approach is that being targeting the particular equation, another subordinate problem was solved under the common constraints. By this, we mean the development of a numerical method for fuzzy integrals. As a result, the proposed method is more accurate in comparison with any other mechanical combination of two separate and independent methods. We give conditions for the existence and uniqueness of a solution and estimate the error of the obtained approximation. We prove stability and method is performed on test problems due to verify our theoretical results, and numerical results are compared with those from existing methods in the literature to confirm the accuracy and efficiency of the proposed method. Index Terms-Nonlinear two-dimensional fuzzy Fredholm integral equations, Iterative numerical method, Trapezoidal and midpoint rules.

Fuzzy Information and Engineering, 2012

In this paper, we approximate the integration of continuous fuzzy real number valued function of ... more In this paper, we approximate the integration of continuous fuzzy real number valued function of one and two variables. To do this, we use Bernstein-type fuzzy polynomials. Moreover, we obtain the error estimates for these approximations in terms of the modulus of continuity.

Journal of Mathematical Extension, 2021

Fuzzy Integral equations is a mathematical tool for mod- eling the uncertain control system and e... more Fuzzy Integral equations is a mathematical tool for mod- eling the uncertain control system and economic. In this paper, we present numerical solution of nonlinear fuzzy Volterra integral equa- tions(NFVIEs) using successive approximations scheme and block-pulse functions. Additionally, the convergence analysis of the presented ap- proach is investigated involving Lipschitz and several conditions and error bound between the approximate and the exact solution is pro- vided. Finally, to approve the outcomes concerned with the theory a numerical experiment is considered.

In many applications, discrimination among decision making units (DMUs) is a problematic technica... more In many applications, discrimination among decision making units (DMUs) is a problematic technical task procedure to decision makers in data envelopment analysis (DEA). The DEA models unable to discriminate between extremely efficient DMUs. Hence, there is a growing interest in improving discrimination power in DEA yet. The aim of this paper is ranking extreme efficient DMUs in DEA based on exploiting the leave-one out idea and combining of Manhattan and infinity norms with constant and variable returns to scale. The proposed method has been able to overcome the existing difficulties in some ranking methods.

International Journal of Industrial Mathematics, 2017

T he concept of fuzzy integral was initiated by Dubois and Prade [11] and then investigated by Ka... more T he concept of fuzzy integral was initiated by Dubois and Prade [11] and then investigated by Kaleva [21], Goetschel and Voxman [20], Nanda [23] and others. In [33], the Henstock integral of fuzzy-valued functions is defined, while the fuzzy Riemann integral and its numerical integration was investigated byWu in [34]. In [7], the authors introduced some quadrature rules for the integral of fuzzy-number-valued mappings. Kaleva [21] proposed the existence and uniqueness of the solution of fuzzy differential equations using the Banach fixed point principle. Mordeson and Newman (see [22]) started the study of the subject of fuzzy integral equations. The Banach fixed point principle is the powerful tool to investigate of the existence and uniqueness of the solution

In this paper, a new approach based on fuzzy Bernstein polynomials is proposed for solving fuzzy ... more In this paper, a new approach based on fuzzy Bernstein polynomials is proposed for solving fuzzy Fredholm integral equations of the second kind (FFIE-2). The error estimation of the proposed method for approximating the solution of FFIE-2, is proved in terms of the modulus of continuity. Finally, for illustrating the accuracy and efficiency, some numerical examples are presented.

In many applications, ranking of decision making units (DMUs) is a problematic technical task pro... more In many applications, ranking of decision making units (DMUs) is a problematic technical task procedure to decision makers in data envelopment analysis (DEA), especially when there are extremely efficient DMUs. In such cases, many DEA models may usually get the same efficiency score for different DMUs. Hence, there is a growing interest in ranking techniques yet. The purpose of this paper is ranking extreme efficient DMUs in DEA based on exploiting the leave-one out and minimizing the maximum distance between DMU under evaluation and boundary efficient in input and output directions. The proposed method has been able to overcome the lacks of infeasibility and unboundedness in some DEA ranking methods.

One of the difficulties of Data Envelopment Analysis(DEA) is the problem of de_ciency discriminat... more One of the difficulties of Data Envelopment Analysis(DEA) is the problem of de_ciency discrimination among efficient Decision Making Units(DMUs) and hence, yielding large number of DMUs as efficient ones. The main purpose of this paper is to overcome this inability. One of the methods for ranking efficient DMUs is minimizing the Coefficient of Variation (CV) for inputs-outputs weights. In this paper, it is introduced a nonlinear model for ranking efficient DMUs based on the minimizing the mean absolute deviation of weights and then we convert the nonlinear model proposed into a linear programming form.

In this paper, we propose an iterative procedure based on two dimensionalfuzzy block-pulse functi... more In this paper, we propose an iterative procedure based on two dimensionalfuzzy block-pulse functions for solving nonlinear fuzzy Fredholm integralequations of the second kind. The error estimation and numerical stabilityof the proposed method are given in terms of supplementary Lipschitz condition.Finally, illustrative examples are included in order to demonstrate the accuracyand convergence of the proposed method.

Computational and Applied Mathematics

Information Processing and Management of Uncertainty in Knowledge-Based Systems

In this paper, a method of successive approximations based on the fuzzy block-pulse functions is ... more In this paper, a method of successive approximations based on the fuzzy block-pulse functions is proposed to solve linear fuzzy Fredholm integral equations of the second kind. Moreover, the error estimation of the approximation solution is given. Finally, illustrative example is included to show the accuracy and the efficiency of the proposed method.

Computational and Applied Mathematics

An iterative method based on fuzzy Bernstein polynomials is presented for solving nonlinear fuzzy... more An iterative method based on fuzzy Bernstein polynomials is presented for solving nonlinear fuzzy Volterra integral equations. To prove the convergence of the method, an error estimate is given in terms of Lipschitz constants. The accuracy of this method is illustrated by some numerical experiments that confirm the convergence stated in the theoretical result.

Journal of Intelligent and Fuzzy Systems, Jun 22, 2017

In this paper, we continue a study of approximation properties of the fuzzy transform (F-transfor... more In this paper, we continue a study of approximation properties of the fuzzy transform (F-transform) with the partition generated by the Shepard kernel. We make the error estimate in terms of a modulus of continuity which has a higher order than the previously known. Also, we obtain the error estimate of the iterative F-transform-based method for linear Fredholm integral equations of the second kind.

Differential Equations and Dynamical Systems, Jul 30, 2019

In this paper, we propose a method of successive approximations for nonlinear fuzzy Fredholm inte... more In this paper, we propose a method of successive approximations for nonlinear fuzzy Fredholm integral equations of the second kind. The main approximation tool is based on fuzzy block-pulse functions. The error estimation of the proposed method is established. A number of illustrative examples that demonstrate accuracy and convergence is given as well.

Fuzzy Sets and Systems, Jul 1, 2016

In this paper, we obtain the error estimation of the iterative method based on quadrature formula... more In this paper, we obtain the error estimation of the iterative method based on quadrature formula to solve nonlinear fuzzy Fredholm integral equations of the second kind given in Fuzzy Sets & Syst. 245 (2014) 1-17, in terms of uniform and partial modulus of continuity. Moreover, we extend in the context of using the modulus of continuity, the notion of numerical stability of the solution with respect to the first iteration.

Advances in Fuzzy Integral and Differential Equations, 2021

In the present work we construct an iterative method for the numerical solution of fuzzy fraction... more In the present work we construct an iterative method for the numerical solution of fuzzy fractional Volterra integral equations, by using the technique of fuzzy product integration. The existence and uniqueness of the solution and the uniform boundedness of the terms of the Picard iterations are proved. The convergence of the iterative algorithm is obtained and the apriori error estimate is given in terms of the Lipschitz constants. A numerical example illustrates the accuracy of the method.

AIMS Mathematics, 2022

This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations... more This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations using Picard iteration method and bivariate Bernstein polynomials. We first present the way to approximate the value of the multiple integral of any fuzzy-valued function based on the two dimensional Bernstein polynomials. Then, it is used to construct the numerical iterative method for finding the approximate solutions of two dimensional fuzzy integral equations. Also, the error analysis and numerical stability of the method are established for such fuzzy integral equations considered here in terms of supplementary Lipschitz condition. Finally, some numerical examples are considered to demonstrate the accuracy and the convergence of the method.

Advances in Fuzzy Integral and Differential Equations, 2021

IEEE Transactions on Fuzzy Systems, 2019

In the present study, a new iterative numerical method for solving bivariate nonlinear fuzzy Fred... more In the present study, a new iterative numerical method for solving bivariate nonlinear fuzzy Fredholm integral equations is proposed. The method combines two well proven approaches: successive approximations and mixed trapezoidal and midpoint rules for the numerical integration. Both approaches are elaborated for fuzzy-valued functions. The main advantage of the proposed approach is that being targeting the particular equation, another subordinate problem was solved under the common constraints. By this, we mean the development of a numerical method for fuzzy integrals. As a result, the proposed method is more accurate in comparison with any other mechanical combination of two separate and independent methods. We give conditions for the existence and uniqueness of a solution and estimate the error of the obtained approximation. We prove stability and method is performed on test problems due to verify our theoretical results, and numerical results are compared with those from existing methods in the literature to confirm the accuracy and efficiency of the proposed method. Index Terms-Nonlinear two-dimensional fuzzy Fredholm integral equations, Iterative numerical method, Trapezoidal and midpoint rules.

Fuzzy Information and Engineering, 2012

In this paper, we approximate the integration of continuous fuzzy real number valued function of ... more In this paper, we approximate the integration of continuous fuzzy real number valued function of one and two variables. To do this, we use Bernstein-type fuzzy polynomials. Moreover, we obtain the error estimates for these approximations in terms of the modulus of continuity.

Journal of Mathematical Extension, 2021

Fuzzy Integral equations is a mathematical tool for mod- eling the uncertain control system and e... more Fuzzy Integral equations is a mathematical tool for mod- eling the uncertain control system and economic. In this paper, we present numerical solution of nonlinear fuzzy Volterra integral equa- tions(NFVIEs) using successive approximations scheme and block-pulse functions. Additionally, the convergence analysis of the presented ap- proach is investigated involving Lipschitz and several conditions and error bound between the approximate and the exact solution is pro- vided. Finally, to approve the outcomes concerned with the theory a numerical experiment is considered.

In many applications, discrimination among decision making units (DMUs) is a problematic technica... more In many applications, discrimination among decision making units (DMUs) is a problematic technical task procedure to decision makers in data envelopment analysis (DEA). The DEA models unable to discriminate between extremely efficient DMUs. Hence, there is a growing interest in improving discrimination power in DEA yet. The aim of this paper is ranking extreme efficient DMUs in DEA based on exploiting the leave-one out idea and combining of Manhattan and infinity norms with constant and variable returns to scale. The proposed method has been able to overcome the existing difficulties in some ranking methods.

International Journal of Industrial Mathematics, 2017

T he concept of fuzzy integral was initiated by Dubois and Prade [11] and then investigated by Ka... more T he concept of fuzzy integral was initiated by Dubois and Prade [11] and then investigated by Kaleva [21], Goetschel and Voxman [20], Nanda [23] and others. In [33], the Henstock integral of fuzzy-valued functions is defined, while the fuzzy Riemann integral and its numerical integration was investigated byWu in [34]. In [7], the authors introduced some quadrature rules for the integral of fuzzy-number-valued mappings. Kaleva [21] proposed the existence and uniqueness of the solution of fuzzy differential equations using the Banach fixed point principle. Mordeson and Newman (see [22]) started the study of the subject of fuzzy integral equations. The Banach fixed point principle is the powerful tool to investigate of the existence and uniqueness of the solution

In this paper, a new approach based on fuzzy Bernstein polynomials is proposed for solving fuzzy ... more In this paper, a new approach based on fuzzy Bernstein polynomials is proposed for solving fuzzy Fredholm integral equations of the second kind (FFIE-2). The error estimation of the proposed method for approximating the solution of FFIE-2, is proved in terms of the modulus of continuity. Finally, for illustrating the accuracy and efficiency, some numerical examples are presented.

In many applications, ranking of decision making units (DMUs) is a problematic technical task pro... more In many applications, ranking of decision making units (DMUs) is a problematic technical task procedure to decision makers in data envelopment analysis (DEA), especially when there are extremely efficient DMUs. In such cases, many DEA models may usually get the same efficiency score for different DMUs. Hence, there is a growing interest in ranking techniques yet. The purpose of this paper is ranking extreme efficient DMUs in DEA based on exploiting the leave-one out and minimizing the maximum distance between DMU under evaluation and boundary efficient in input and output directions. The proposed method has been able to overcome the lacks of infeasibility and unboundedness in some DEA ranking methods.

One of the difficulties of Data Envelopment Analysis(DEA) is the problem of de_ciency discriminat... more One of the difficulties of Data Envelopment Analysis(DEA) is the problem of de_ciency discrimination among efficient Decision Making Units(DMUs) and hence, yielding large number of DMUs as efficient ones. The main purpose of this paper is to overcome this inability. One of the methods for ranking efficient DMUs is minimizing the Coefficient of Variation (CV) for inputs-outputs weights. In this paper, it is introduced a nonlinear model for ranking efficient DMUs based on the minimizing the mean absolute deviation of weights and then we convert the nonlinear model proposed into a linear programming form.

In this paper, we propose an iterative procedure based on two dimensionalfuzzy block-pulse functi... more In this paper, we propose an iterative procedure based on two dimensionalfuzzy block-pulse functions for solving nonlinear fuzzy Fredholm integralequations of the second kind. The error estimation and numerical stabilityof the proposed method are given in terms of supplementary Lipschitz condition.Finally, illustrative examples are included in order to demonstrate the accuracyand convergence of the proposed method.

Computational and Applied Mathematics

Information Processing and Management of Uncertainty in Knowledge-Based Systems

In this paper, a method of successive approximations based on the fuzzy block-pulse functions is ... more In this paper, a method of successive approximations based on the fuzzy block-pulse functions is proposed to solve linear fuzzy Fredholm integral equations of the second kind. Moreover, the error estimation of the approximation solution is given. Finally, illustrative example is included to show the accuracy and the efficiency of the proposed method.

Computational and Applied Mathematics

An iterative method based on fuzzy Bernstein polynomials is presented for solving nonlinear fuzzy... more An iterative method based on fuzzy Bernstein polynomials is presented for solving nonlinear fuzzy Volterra integral equations. To prove the convergence of the method, an error estimate is given in terms of Lipschitz constants. The accuracy of this method is illustrated by some numerical experiments that confirm the convergence stated in the theoretical result.