THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 12, Number 1, March 2004

CONTENT

DETAILS

Some Difference Sequence Space of Fuzzy Numbers

Metin Basarir

Department of Mathematics, Sakarya University Sakarya-54100, Turkey E-mail: basarir@sakarya.edu.tr

Mursaleen

Department of Mathematics, Aligarh M. University Aligarh-202002, India E-mail: mursaleen@postamark.net

Abstract: In this note, we define and study some spaces of difference sequences of fuzzy numbers.

Keywords and phrases: Fuzzy numbers, metric spaces, linear spaces, differences sequence.

R-Generalized Fuzzy Closed Sets

Y.C. Kim and J. M. Ko Department of Mathematics, Kangnung National University Kangnung, Kangwondo 210-702, Korea

Abstract: We introduce generalized fuzzy closed sets in a fuzzy topological space in view of the definition of Sostak [9]. We investigate some properties of them. Moreover, we investigate the relationship between generalized fuzzy continuous maps and generalized fuzzy irresolute maps.

Keywords: Fuzzy topology, generalized fuzzy closed(open) sets, fuzzy continuous maps, generalized fuzzy continuous(irresolute) maps.

Fuzzy Approach for Solving Vector Optimization Problems

Mohamed Abd El-Hady Kassem

Mathematical Department, Faculty of Science, Tanta University Tanta, Egypt

Abstract: In this paper, we presents a fuzzy approach for solving multiobjective nonlinear programming(MONLP) problems which combines the characteristics of both the fuzzy decision making of generalized Tchebycheff norm(GTN) and the method of constraints of these fuzzy decision making. Also, it is shown that the fuzzy efficient solutions of fuzzy multiobjective nonlinear programming(FMONLP) problems can be characterized in terms of the fuzzy optimal solutions of the corresponding scalarization fuzzy multiobjective nonlinear programming (SFMONLP) problems using the presented approach. Finally , the basis notions in parametric convex programming are redefined and analyzed qualitatively for FMONLP problems. Such analysis gives us the possibilily of relating different SFMONLP¡¯s with each other .An illustrated examples are clarify the obtained results.

Keywords: Multiobjective nonlinear programming problems, fuzzy decision making, stability, parametric programming, generalized tchebycheff norm.

Prefilters in Fuzzy Spaces

K. A. Dib , A. A. Ramadan , S. N. Deeb and G. A. Kamel

Math. Dept,. Faculty of Science Cairo Universityp-Fayoum Brach Cairo University-Beni-suef Branch Cairo, Egypt

Abstract: The fuzzy topology on a fuzzy spaces was introduced by Dib in 1999 to correct the deviation in the definition of the closed fuzzy subset and the closure of fuzzy subset. In this work we study prefilters and fuzzy topology on fuzzy spaces. We introduced the definition of the prefilters in a subfamily of the power set to have a tool to make a comparison of our results with that a Chang¡¯s fuzzy topology.

Keywords: Fuzzy spaces, Fuzzy subspaces, fuzzy point, fuzzy stacks, fuzzy grills, prefilters and fuzzy ultrafilters.

On -(weakly -) irresolute Continuous and -(weakly -) irresolute Open (Closed) Order Homomorphism

S. A. El-Sheikh

Oman-College of Education-Math. Dept. P.O.Box 1376 Code 611,Nizwa

Abstract: In this paper, the properties of -closure, weakly -closure, -interior and weakly -interior of an L-fuzzy set which generalize the noitions in [1,1o,11], have obtained.We prove that in the realm of fuzzy regular spaces (where is the -closure of an L-fuzzy set A). Also, we introduce and study the concepts of -( weakly -) closure and L-fuzzy net in a L-fuzzy topological space. We investigate the relationship between these notions and the notions in [5,10,11].

Keywords: Fuzzy topology, - closure, - interior, -closure, -interior, -continuity, - continuity, order homomorphism.

Fuzzifying Syntopogenous Structures

A.K. Katsaras and C. G. petalas

Department of Mathematics, University of Ioannina, Greece

Abstract: It is showen that the fuzzifying topologies, the fuzzifying proximities and the fuzzy uniformities are special cases of the fuzzifying syntopogenous structures introduced in the paper.

Keywords: fuzzifying topology, fuzzifying proximity, fuzzy uniformity, proximal convergence, uniform convergence.

Exact and Approximation Algorithms for Scheduling Unrelated Machines Under Fuzzy Environment

Ameer AL-Salem , Omar M. Saad

Mechanical Engineering Department and Department of Mathematics University of Qatar, P.O. Box 2713, Doha, Qatar

Robert L. Armacost

Department of Industrial Engineering and Management Systems University of Central Florida, P.O. Box 162450 Orlando, FL32816-2450

Abstract: This paper presents exact and approximation algorithms for scheduling jobs on unrelated parallel machines with machine eligibility restrictions. It is considered that the processing times are those fuzzy parameters and the maximum completion time is required to be minimized. The concept of -level set together with the definition of the fuzzy number and its membership function are introduced . To the best of our knowledge , the problem has not been addressed previously in the literature. The proposed multiphase approximation algorithm incorporates new concepts from the multi-depot vehicle routing in the constructive heuristic. A computational study includes problems with two or four machines , up to 105 jobs, and three levels of a machine selection values. The results show that the heuristic can yield solutions within a few percent of the optimal solutions with performance improving an the number of jobs to be scheduled increases.

Keywords: Scheduling, unrelated machines, fuzzy parameters, -level.

Fuzzy Hahn-Banach Theorem-a New Approach-I

Binimol Punnoose and A. Sunny Kuriakose

Department of Mathematics, U.C.College Aluva, 683102, Kerala, India

Abstract: In this paper we establish an equivalence relation between two fuzzy numbers in the sense of Puri and Ralescu (JMAA 91(1983) 552-558). Here we consider the equivalence classes as fuzzy numbers. A fuzzy version of Hahn-Banach theorem is proved on a fuzzy vector space over the set fuzzy real numbers.

Keywords: Fuzzy numbers, Fuzzy vector subspace, Fuzzy linear functional, Fuzzy Hahn-Banach theorem and Fuzzy topological vector space.

Some Remarks on Heilpern¡¯s Generalization of Nadler¡¯s Fixed Point Theorem

M. Marudai

Department of Mathematics, The M.D.T.Hindu College Tirunelveli-627010, India E-mail: marudaim@hotmail.com

P.S. Srinivasan

Department of Mathematics, IIT Madras Chennai-600036, India E-mail: pymani@pallava.iitm.ernet,in

Abstract: Let be a complete metric linear space and be a fuzzy contraction mapping from to .Then the fixed point theorem of Heilpern states that there exists such that . A number of other generalizations to the above theorem in various directions follows the same approach. In this paper we have proved that this theorem of Heilpern is deduced as a corollary to the classical fixed point theorem for the crisp multi-valued contraction mapping of Nadler. It is also noted that all other subsequent results can also be deduced as corollaries to their muti-valued fixed point theorems in their crisp cases. Further ,a generalization of Nadler fixed point theorem for fuzzy mapping is obtained .The proof relies on the same technique , but does not subsume the Nadler¡¯s theorem. It is also pointed out that even under weaker settings the above results hold good.

Keywords: Fuzzy set, fuzzy mappings, fixed points, contraction fuzzy mapping , nonexpansive fuzzy mappings.

Two-phase Fuzzy Algorithms for Multi-objective Transportation Problem

Shu Ping Gao and San Yang Liu

Department of Mathematics, Xidian University Xi¡¯an 710071, P. R. China E-mail: xdgaosp@263.net

Abstract: Two-phase fuzzy algorithms are presented to solve the multi-objective transportation problem with linear membership functions and with non-linear membership functions, respectively. By converting the multi-objective transportation problem to a conventional linear programming, an optimal compromise solution is obtained. Numerical examples illustrate that the above approaches are feasible and efficient.

Keywords: Multi-objective transportation problem, two-phase fuzzy algorithm, linear membership function, non-linear membership function.

Fuzzy Decision and Encryption-group Sharing of Decision Conclusion

Guo Ping Han and Kai Quan Shi

School of Mathematics and System Sciences, Shandong University Jinan, Shandong 25100, P.R. China

Abstract: The paper studies fuzzy decision theory grafted with information encryption theory and encryption technology. In the paper, we give the general concepts of group encryption and group sharing of fuzzy decision, and present unit encryption and multi-unit sharing of fuzzy decision, multi-unit encryption and unit sharing of fuzzy decision. In addition, the paper also gives the encryption algorithm of fuzzy decision and its application.

Keywords: Fuzzy decision, decision model, unit encryption, multi-unit encryption, encryption-decryption algorithm, application.

C. T. Yang¡¯s Theorem Concerning -derived Set in -fuzzy Topological

Rui-ying Wang and Zhi-fang Ji

Department of Mathematics, Capital Normal University Beijing 100037, P. R. China

Abstract: In this paper, the concept of -derived set is introduced the C.T. Yang¡¯s theorem concerning -derived set has been proved for -fuzzy points, and moreover, an example is given, which illustrates that C.T. Yang¡¯s theorem concerning -derived set does not hold for -fuzzy molecules.

Keywords: -fuzzy topological spaces, -derived set, -fuzzy points, -fuzzy molecules.

Existence of Solutions of Fuzzy Delay Differential Equations on A Closed Subet

Jin Xuan Fang and Chuan Zhi Bai

Department of Mathematics Nanjing Normal University Nanjing Jiangsu, 210097, People¡¯s Republic of China

Abstract: In this paper, we study the existence of solutions the fuzzy delay differential equations on a closed subset of the fuzzy number space by using the Kuratowski measure of noncompactness and successive approximation method.

Keywords: Fuzzy delay differential equations, fuzzy number space , Kuratowski measure of noncompactness, compactness condition, approximate solutions.

On the Characterization of Continuous Order-homomorphism

Wu-neng Zhou Institute of Mathematics, Zhejiang Normal University Jinhua, Zhejiang, 321004, P.R. China

Abstract: In this paper, the characterizations of continuous order-homomorphism, semi-continuous order-homomorphism, irresolute order-homomorphism, almost continuous order-homomorphism and weak continuous order-homomorphism are characterized by ideals and filters.

Keywords: -fuzzy topological spaces, ideal, filters, convergence, continuous order-homomorphism.

Relationship Between Measure of Fuzziness and Measure of Similarity

Wen Yi Zeng and Hong Xing Li

Department of Mathematics, Beijing Normal University Beijing, 100875, China

Abstract: Aimed at the two important concepts, measure of fuzziness and measure of similarity, the relationships between them are discussed in detail. Two interesting results are obtained and they can be described each other. Then we point out that the measure of fuzziness is the measure of similarity on a fuzzy subset and its complement. These results can be applied in many cases, for example, pattern recognition and decision-making.

Keywords: Fuzzy subsets, measure of fuzziness, measure of similarity.

Three Step Iterative Algorithms for Multivalued Quasivariational Inclusions with Fuzzy Mappings

Jong Yeoul Park

Department of Mathematics, Pusan National University Pusan 609-735, South Korea

Jae Ug Jeong

Department of Mathematics, Dong Eui University Pusan 614-714, South Korea

Abstract: In this paper, we suggest and analyze some new classes of three iterative algorithms for solving multivalued quasivariational inclusions with fuzzy mapping by using the resolvent equation technique. New iterative algorithms include the Ishikawa, Mann iterations for solving variational inclusions (inequalities) with fuzzy mappings as special cases. The results obtained in this paper represent an improvement and a significant refinement of previously known results.

Keywords: Fuzzy mapping, variational inclusions, resolvent equations, algorithms, iterative methods.

Fuzzy Variational Inequality and Complementarity Problem

S. Nanda and S. Pani

Department of Mathematics, Indian Institute of Technology Kharagpur-721302, West Benga, India E-mail: snada@maths.iitkgp.ernet.in spani@maths.iitkgp.ernet.in

Abstract: In this paper we have defined the complementarity problem for fuzzy mapping and obtained some existence results for fuzzy variational inequality and complementarity problem in reflexive real Banach space and Hilbert space.

Keywords: complementarity problem, variational inequality, fuzzy variational inequality, fuzzy complementarity problem, fuzzy fixed point.

On uniformly Continuous Mappings in Fuzzy Metric Spaces

V. Gregori

Escuela Politecnica Superior de Gandia Universidad Politecnica de Valencia, 46730 Grau de Gandia,Valencia, Spain. E-mail: vgregori@mat.upv.es

S. Romaguera

Escuela de Caminos, Departamento de Matematica Aplicada Universidad Politecnica de Valencia, 46071 Valencia, Spain. E-mail: sromague@mat.upv.es

A. Sapena

Escuela Politecnica Superior de Alcoy, Universidad Politecnica de Valencia, Alcoy, Spain. E-mail: alsapie@mat.upv.es

Abstract: The nontion of t-uniformly continuous mapping was recently introduced by V. Gregori and A. Sapena (Fuzzy Sets and Systems, 2002), in order to state new versions of the classical Banach Contraction Principle for fuzzy metric spaces. In this paper we discuss several properties of t-uniformly continuous mappings. We show that every continuous mapping on a compact fyzzy metric is t-uniformly continuous and characterize fuzzy metric spaces for which every real-valued continuous function is t-uniformly continuous. We observe that every t-uniformly continuous mapping is uniformly continuous in the sense of George and Veeramani, but the converse does not hold in general. Nevertheless, we show that for each fuzzy metric spaces such that every real-valued continuous functions is uniformly continuous there is a fuzzy metric on compatible with the topology genetated by for which every real-valued continuous functions is t-uniformly continuous.

Keywords: Fuzzy metric, t-uniformly continuous mappings, uniformly continuous mappings, t-equinormal.

The Fuzzy Mutation :An Operator to Improve the Performance of Genetic

Chen Guolong and Guo WenZhong

Department of Computer Sciences and Technology, Fuzhou University, Fuzhou 350002, China

Abstract: Genetic algorithms are robust methods which may be used to solve search and optimization problems. The important drawbacks in use of genetic algorithms are the optimizing capability and convergence speed, which are caused by the lack of diversity in the population and the disproportionate relationship between exploitation and exploration. The mutation operator is considered one of the most determinant elements for solving the problem. However, there are many factors affecting mutation probability, Pm, which are complex and hard to be described, therefore, we use fuzzy theory to propose a new mutation operator, named fuzzy mutation operator, named fuzzy mutation operator for real-coded genetic algorithms in this paper. The results of numerical experiment proved its efficiency and stability.

Keywords: genetic algorithms, fuzzy mutation operator