THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 16, Number 4, December 2008

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On Optimality Criteria of Fuzzy Programming under Fuzzy Convexity and Generalized Convexity Functions

Abo Zaid El Bana, E. E. Ammar and S. A. Hagazy

Department of Mathematics, Faculty of Science, Tanta University, Egypt. E-mail: ammar-al-saeed@hotmail.com

Abstract: In this paper, we present the concepts of epigraph and convexity of the fuzzy mappings, and describe the properties of the convex fuzzy mappings, quasiconvex mappings and pseudoconvex mappings. Differentiable fuzzy mapping will be defined based on the notion defined by Hong Yan [9] and Yu-Ru Syau [22].We also introduce the convexity with differentiable. Then, we give some theorems of optimality criteria.

Keywords: Fuzzy sets theory, fuzzy mappings, convexity, interval analysis, optimality criteria.

Maximal Filters in -algebras

Qin Feng

Mathematics Department of Jiangxi Normal University,Nanchang Jiangxi 330027 E-mail:qinfeng923@163.com

Abstract: In this paper, we prove that a filter in a -algebra is maximal iff is locally finite. In particular, if is at most countable, then is maximal iff or, where is the Boolean algebra.

Keywords: -algebras, maximal filters, locally finite,.

A Characterization of g-selfcentered Fuzzy Graphs

K.Sameena and M. S. Sunitha

Department of Mathematics National Institute of Technology, Calicut-673601 India. E-mail:y3df02@nitc.ac.in, sunitha@nitc.ac.in

Abstract: Some metric aspects of fuzzy graphs using (geodesic) g-distance is carried out. A fuzzy cycle is g-selfcentered. A necessary condition for a g-selfcentered fuzzy graph is that each node is g-eccentric .A sufficient condition for a fuzzy graph to be g-selfcentered. A characterization of g-selfcentered fuzzy graph is obtained using g-eccentric nodes. The proof technique used in this characterization is independent of the metric and hence it follows that the same characterization holds for a selfcentered fuzzy graph (in particular, for a selfcentered (crisp) graph) with any metric defined on it. Finally, analogues to the Hedetniemi¡¯s construction in the crisp case, it is shown that any fuzzy graph can be embedded as the g-center of a super fuzzy graph.

Keywords: Fuzzy graph, fuzzy cycle, strong arcs, g-distance, g-center.

A Family of Iterative Methods for Solving Fuzzy Nonlinear Equations

P. K. Parida and D. K. Gupta

Department of Mathematics, Indian Institute of Technology, Kharaagpur-721302, India

Abstract: The aim of this paper is to propose a family of iterative methods for finding the roots of fuzzy nonlinear equations. A fuzzy nonlinear equation of a single fuzzy variable is considered and transformed into its equivalent real parametric forms. The roots of these parametric equations are obtained by Newton¡¯s method and the proposed methods. It is shown that the order of convergence of these methods is cubic which is better than the quadratic convergence of Newton¡¯s method. The convergence theorems are established for two members of this family used more frequently in the crisp case. The efficacy of these two methods are tested on some numerical examples and the results obtained compared with the Newton¡¯s method [7].In all the examples, the results obtained by our methods are better than those obtained by Newton¡¯s method.

Keywords: Fuzzy nonlinear equations, Newton¡¯s method, Convergence theorems, parametric equations.

Fuzzy Quasi-associative Ideals in BCI-algebras with Respect to A t-conorm

M. Akram

Punjab University College of Information Technology, University of the Punjab Old Campus, Lahore-54000, Pakistan E-mail: m.akram@pucit.edu.pk

K. P. Shum

Department of Mathematics, The University of Hong Kong, Shatin, Hong Kong (SAR), China. E-mail: kpshum@math.hku.hk

Abstract: In this paper, we introduce the notion of quasi-associative ideals of BCI-algebras with respect to a t-conorm and investigate some of their properties. We discuss the properties of homomorphic images and inverse images of fuzzy quasi-associative ideals in a BCI-algebras with respect to a t-conorm.Some properties of direct product and S-product of fuzzy quasi-associative ideals with respect to a t-conorm are also discussed.

Keywords: BCI-algebras; t-conorms; Quasi-associative ideals with t-conorms.

Simple Expansion of Pawlak Approximation Space

M. E. Abd El-Monsef, A. A. Abo Khadraand A. A. Zedan

(1)Faculty of Science Tanta University, Tanta, Egypt

(2)Faculty of Engineering, Tanta University, Tanta, Egypt

(3) Faculty of Science South Valley University, Aswan, Egypt

Abstract: Rough set theory is a new mathematics approach to vague and uncertain data analysis and based on complete information systems .Basic concepts of rough sets depend on a special type of topological, namely quasi discrete topologies. Changes of topologies [see 1,4,and 5] play important roles in the filed of general topology and its applications, and this is connected basically of expansions of topologies. In this paper we will introduce a new approximation spaces finer than Pawlak approximation space which help us to solve some problems in rough sets theory by using changes of topologies.

Keywords: Pawlak approximation space; Expansion of approximation space; Simple expansion of approximation space; Accuracy measure.

A Multi-criteria Multi-constraint Level Economic Production and Marketing Planning Problem with Fuzzy Cost Parameters:A Geometric Programming Approach

Nirmal Kumar Mandal

Department of Mathematics, Silda Chandrasekhar College Silda Paschim Medinipur-721515, West Bengal, India E-mail: mandal_n_k@hyhoo.co.in

Tapan Kumar Roy

Department of Mathematics, Bengal Engineering and Science University Howrach-711103, West Bengal, India.

Abstract: In this paper we have formulated a multi-criteria, multi-constraint level economic production and marketing planning problem. The inventory costs are taken here as trapezoidal fuzzy numbers. The model is solved using geometric programming method. A numerical example is given to support the problem. Also the effect of change of constraint levels on the objective functions are discussed in the sensitivity analysis.

Keywords: Inventory, Multi-criteria and Multi-constraint level, Geometric Programming, Trapezoidal fuzzy numbers.

PERT on A Network with Imprecise Edge Weight

Sk. Md. Abu Nayeem and Madhumangal Pal

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102,INDIA E-mail address:madhumangal@ycos.com

Abstract: A network with its arc lengths as imprecise numbers, instead of real numbers, namely, interval numbers and triangular fuzzy numbers is considered here. Existing ideas on addition and comparison between two imprecise numbers of same type are introduced. A common algorithm for the Project Evaluation and Review Technique (PERT) problem is developed which works well on both types of imprecise numbers under consideration. The proposed algorithm is illustrated with numerical examples.

Keywords: Fuzzy PERT, interval numbers, triangular fuzzy numbers, network.

Single and Multi-objective Stochastic Inventory Problems in Fuzzy Environment

S. Banerjee and T. K. Roy

Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, West Bengal, India

Abstract: This paper considers a multi item single period inventory problem with budgetary and floor space constraints where the demand of the items follows probability distributions. Two different stochastic models are discussed where the demand follows uniform distribution and exponential distribution respectively. The problem is formulated as a single objective stochastic inventory model and introducing the impreciseness to the total budget it is considered as a fuzzy single objective stochastic inventory model. Then it is also considered as a multi objective stochastic inventory model where simultaneously total revenue earn from sales is maximized as well as total inventory related cost is minimized and is also formulated in fuzzy environment introducing fuzziness in the objectives. These multi-objective models are solved by Fuzzy Non-Linear Programming techniques. The solution is illustrated by numerical example and the results of different models are compared.

Keywords: Multi-objective optimization, Fuzzy optimization, Fuzzy non-Linear programming, Stochastic inventory model.

Ideals and Convergence Relations of MS-algebra

Wang Zun-quan

College of Science, Chain Three Gorges Univ. Yichang 443002, China

Abstract: In this paper we obtain a necessary and sufficient condition that an ideal of MS-algebra is the kernel of some congruence relation, and the characterize the smallest and the greatest congruence relation whose kernel is a congruence ideal.

Keywords: MS-algebra, kernel ideal, congruence idea, smallest congruence relation, greatest congruence relation.

Principle of Inclusion-exclusion for Finite Fuzzy Sets

V. Murali and M. Talwanga

Department of Mathematics (Pure and applied) Rhodes University, Grahamstown 6140, South Africa E-mail:v.murali@ru.ac.za

Abstract: In this note we extend the classical set-theoretic principle of inclusion and exclusion to fuzzy subsets of a finite set X. This is based on some discussions of suitable general formulations of the principle for applications to fuzzy subsets of finite sets. The counting technique is applied to level sets of fuzzy subsets to derive general identities involving fuzzy subsets. We illustrate our approach with a couple of practical example.

Keywords: Finite fuzzy set; Level set; Inclusion-exclusion; Set enumeration.

A Fuzzy Economic Production Quantity Model with Capacity Constraint:

A Fuzzy Geometric Programming Approach

Sahidul Islam

Department of Mathematics, Bengal College of Engineering and Technology Durgapur-713212, West Bengal, India.

Tapan Kumar Roy

Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, West Bengal, India.

Abstract: In this paper, an economic production quantity model with investment costs for setup reduction and quality improvement is formulated. The model has involved one capacity constraint. In most of real world situations, the objective and constraint functions of the decision makers are imprecise in nature. Hence the coefficients, indices, the objective function and constraint goals are imposed here in fuzzy environment. The model has been solved by conventional Geometric Programming (GP) and Fuzzy Geometric Programming (FGP) techniques. It should be noted that FGP technique is more efficient than Fuzzy Non-linear Programming (FNLP) method for this fuzzy model. The sensitivity analysis is discussed here for the change on the fuzzy number coefficients and indices. The model is numerically illustrated and compared by both methods.

Keywords: Setup reduction, Quality improvement, Capacity constraint, Fuzzy number, Fuzzy geometric programming.

Extensions of -topological Spaces of Fuzzy Sets

K. C. Chattopadhyay

Department of Mathematics, The University of Burdwan Burdwan-713104, W. Bengal, INDIA

H. Hazra

Department of Mathematics, Bolpur College Bolpur, Birbhum, W. Bengal, INDIA

S. K. Samanta

Department of Mathematics, Visva Bharati Santiniketan-731235, W. Bengal, INDIA E-mail: syamal_123@yahoo.co.in

Abstract: The main aim of this paper is to develop an idea of extensions of -topological spaces of fuzzy sets and to construct a principal extension.

Keywords: Fuzzy topology, Principal extension, space.

A Note on Intuitionisitc Fuzzy Analysis

Lisy Cherian

Mar Thoma College for Women, Perumbavoor, Kerala India, Pin-683542

Sunny Kuriakose A.

Uniion Christian College, Aluva, Kerala, India.

Abstract: The purpose of this note is to construct the basic concepts of the so-called ¡°intuitionstic fuzzy analysis (IFA)¡±.After giving the fundamental definitions, we define ¦Á¦Â-cuts, support of an intuitionistic fuzzy set (IFS), intuitionistic fuzzy (IF) convexity, intuitionistic fuzzy continuity and state several properties in IFS. Here we prove a general non-membership function or the degree of rejection for an IF maximizing set and prove some characterization concerning intuitionistic fuzzy function (IFF).

Keywords: Intuitionistic fuzzy convexity, intuitionistic fuzzy quantity, intuitionistic fuzzy function, intuitionistic fuzzy continuity, intuitionistic fuzzy injectivity, degree of acceptance and rejection.

Compute Eigenvalues and Corresponding Eigenvectors of Fuzzy Symmetric Matrix by Jacobian Method

S. Abbasbandy

Department of Mathematics, Imam Khomeini International University, Ghazvin, 34149=16818, Iran

M. Otadi

Department of Mathematics, Islamic Azad University, Firuozkooh Branch, Firuozkooh, Iran

M. Mosleh

Department of Mathematics, Science and Research Branch Islamic Azad University, Tehram, 14778, Iran

Abstract: There are quite a new computational algorithms of finding eigenvalues and corresponding eigenvectors of a fuzzy matrix. We study the problem of calculating the eigenvalues and corresponding eigenvectors of a fuzzy symmetric matrix. The proposed method illustrated and compared with power method by solving a numerical example.

Keywords: Jacobian method, Fuzzy eigenvalues, Fuzzy eigenvectors, Fuzzy matrix.

Related Fixed Points Theorem for Two Mappings on Two Fuzzy Metric Spaces

Sushil Sharma

Department of Mathematics Madhay Vigyan Mahavidyalaya Vikram University, Ujjain-456010 India E-mail:sksharma2005@yahoo.com

Raghavendra Singh Rathore

Department of Mathematics Govt. Girls¡¯ Post Graduate College Vikram University, Ujjain-456010, India E-mail: rsrathore@lycos.com

Abstract: In this communication we have made an attempt to prove a related fixed points theorem for two mappings on two fuzzy metric spaces.

Keywords: Fuzzy metric spaces, Related fixed point, Cauchy sequence.

On The Lattice of Fuzzy Ideals of A Lattice

M. J. Rani

Department of Mathematics, St. joseph¡¯s College, Irinjalakuda Kerala, India E-mail:krvarghese@rediffmail.com Ph: 0091-4802786682

Abstract: In this paper we prove that for a lattice X, the family of all fuzzy sublattices and the family of all fuzzy ideal are also lattices. We further claim that the lattice of all fuzzy ideals is in fact a sublattice of all fuzzy sublattices. Also we develop some properties of fuzzy ideals which establishes a correspondence between the families of fuzzy ideals of two homomorphic lattices.

Keywords: Fuzzy sets, Fuzzy sublattices, Fuzzy ideals.

Intuitionistic Fuzzy Lie Ideals of Lie Algebras

M. Akram

Punjab University College of Information Technology, University of the Punjab, Old Campus, P. O. Box 54000, Lahore, Pakistan. E-mail address: m.akram@pucit.edu.pk

Abstract: We apply the concept of intuitionistic fuzzy sets to Lie algebras. We introduce the notion of an intuitionistic fuzzy Lie ideal of a Lie algebra and investigate some of related properties. Next, we give some properties of the intuitionistic fuzzy solvable Lie ideals, nilpotent Lie ideals and quotient Lie algebras.

Keywords: Lie algebras; Lie ideals; Quotient Lie algebras; Solvable Lie ideals; Nilpotent Lie ideals; Equivalence relations.