THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 4, Number 3, September 1996

CONTENT

DETAILS

Some Fuzzy Notions of Mathematics Programming Problems with General Parameter in the Constraints

El-Said Ebrahim Ammar, Mohamed Abd El-Hady Kassem

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

Abstract: In this paper we use the theory to solve a NLP problem with general parameter in the constraints. A special and general fuzzy problem will be considered. Some basic concepts of fuzzy problems are defined. The connection between the basic notions of the ordinary and the fuzzy problems with and without convexity assumption will be considered. Some parametric notions of the general fuzzy problem will be given. An illustrative example is presented. Keywords: Fuzzy set theory; NLP; Convexity; Stability Multi-objective Multiple Facility Location Problem: A Fuzzy Interactive Approach U. Bhattacharya and R. N. Tiwari Department of Mathematics, India Institute of Technology Kharagpur-721302, India Abstract: This paper presents an interactive fuzzy goal programming model for locating multiple facility on a plane bounded by a convex polygon under the criteria. 1. Minimize the maximum distances from the facility to the demand points. 2. Minimize the sum of all the transportation costs. It has been proved that the methodology given will always give a non-dominated solution to the original problem. Rectilinear norm has been taken as the scenario may be considered in an urban setting. A numerical example has been given to illustrate the solution procedure.

Keywords: Facility location, multiple criteria, fuzzy goal, interactive programming.

The Categories of Fuzzy Module

Zhao Jianl

i Institute of fuzzy system, Liaocheng Teacher¡¯s College, Shandong, 252000, China

Abstract: We present the concepts of the homomorphisms of fuzzy modules and the categories of fuzzy modules, and we study their important properties.

Keywords: Fuzzy submodule, homomorphism of fuzzy modules, category, subcategory, category of fuzzy modules.

Interactive Stability of Multi-objective Nonlinear Programming Problems with Stochastic Parameters

Mohammed Lotfy Hussein

Department of Mathematics, Faculty of Education, Kafr El-Sheikh Tanta University, Tanta, Egypt

Mohamed Abd El-Hady Kassem

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

Abstract: This paper deals with an interactive decision making approach for solving multi-objective nonlinear programming problems with stochastic parameters in the objective functions. The stochastic parameters in the objective functions are characterized by random variables. The concept of stochastic efficient solution is introduced in which the ordinary efficient is extended based on the stochastic parameters. The stability set of the first kind is analyzed quantitatively to associate with the interactive algorithm. A numerical example illustrates various aspects of the results developed in this paper.

Keywords: Interactive decision making, multi-objective nonlinear programming problems, random variables, probability efficient.

Fuzzy Regression Models and Their Applications

Wieslaw Luczynski

Department of Microeconomics, University of Economics, al. Niepodleglosci 10, 60-967 Poznan, Poland

Marian Matloka

Department of Mathematics, University of Economics, al. Niepodleglosci 10, 60-967 Poznan, Poland

Abstract: In this paper we consider the regression model which is characteristic by polynomial of n degree with the interval and fuzzy coefficients. Theoretical consideration are illustrated by numerical examples.

Keywords: Fuzzy regression analysis, linear programming, interval model, fuzzy model.

Guard Heuristic by Dynamic Fuzzy Reasoning Model

Mei-En Chen and Yo-Ping Huang

Department of Computer Science & Engineering Tatung Institute of Technology Taipei, Taiwan 104, R. O. C.

Abstract: Chess is one of the most popular examples of two-person zero-sum game, that is, a game in which one player¡¯s loss is his opponent¡¯s gain. This paper proposes a useful fuzzy approach to help Chinese chess player to decide the best next move. A novel technique called ¡°guard heuristic¡± is presented to show that for each intersection on the board in question how well it is guarded by one¡¯s own pieces and by those of one¡¯s opponent¡¯s. Since the traditional guard heuristic cannot predict more than one next steps, this paper applies the fuzzy cognitive map (FCM) to dynamically determining the guard value by relationship between each pair of intersections on the board. This can in turn impact more than one moves of guard values. Also, we use the rule-based reasoning model to infer the next legal move. Based on the proposed model, not only the legitimacy of the next move can be improved but also the decision time is much shorter than the game tree searching method of alpha-beta model. Example is given to verify the effectiveness of the proposed methodology.

Keywords: Guard heuristic, fuzzy reasoning, fuzzy cognitive map, guard matrix, Chinese chess.

F-polygroups (I)

M.M. Zahedi and A. Hasakhani

Department of Mathematics, Kerman University Kerman, Iran

Abstract: In this note, by using the notion of fuzzy subset, the concept of F-polygroup which is a generalization of polygroup, is defined and some basic results are obtained. In particular after defining the concepts of homomorphism of F-polygroups, the product of F-polygroups, F-subpolygroups and ( weak ) normal F-subpolygroups some theorems are proved.

Keywords: Polygroup, F-polygroup, (normal) F-subpolygroup.

On Fuzzy Submodules and Their Radicals

T.K. Mukherjee

Department of Mathematics, Jadavpur University, Calcutta-700032, India

M.K. Sen

Department of Pure Mathematics, Calcutta University, 35, Ballygunge Circular Rd Calcutta-700019, India

D. Roy

Department of Mathematics, Dinabandhu Andrews College. Garia, Calcutta-700084, India

Abstract: In this paper we make a detailed study of various properties of fuzzy submodules of an R-module (R is a commutative ring with identity) connected with fuzzy quotient modules, strong radical and radical of fuzzy submodules, characterization of Noetherian and Artinian modules in terms of the set of values of fuzzy submodules.

Keywords: Fuzzy submodules, fuzzy radical, noetherian and artinian modules.

Non-archimedean Fuzzy Topological Spaces

Francisco Gallego Lupianez

Department de Geometria Y Topologia Facultad de Mathematics, Universidad Complutense de Madrid 28040-Madrid, Spain

Abstract: In this paper we define for fuzzy topological spaces a notion corresponding to non-archimedean topological spaces. We obtain some properties of these fuzzy topological spaces, particularly we prove a result on fuzzy metrizability.

Keywords: c-zero dimensionality, fuzzy point, crisp fuzzy set, quasi-coincident fuzzy sets, fuzzy base, N-compactness, fuzzy metrizability.

Fuzzy Quasi-associative Ideals in BCI-algebras

Young Bae Jun

Department of Mathematics Education Gyeongsang National University, Chinju 660-701, Korea

Abstract: We fuzzify the concept of quasi-associative ideals in BCI-algebras, and investigate its properties. We solve the problem of classifying fuzzy QA-ideals by their level QA-ideals. Finally we construct fuzzy characteristic QA-ideals and normal fuzzy QA-ideals.

Keywords: (Fuzzy) QA-ideal, level QA-ideal, (fuzzy) characteristic QA-ideal, normal fuzzy QA-ideals.

De Morgan Algebra of Proximity

Xueyou Chen

Basic Department, Shandong Institute of Technology Zibo, 255012, Shandong P.R. China

Abstract: In this paper, we introduce de Morgan algebra of proximity which unifies the classical proximity [3] and the fuzzy one [7].

Keywords: De Morgan algebra, proximity.

Characterizing the Active-Solutions to Multiple-objective Transportation Problems with Possibilistic Supplying

Mohammad Lotfy Hussein

Department of Mathematics, Faculty of Education, Kafr El-Sheikh, Tanta University, Egypt

Abstract: This paper deals with characterizing the efficient-solutions to a multiple-objective transportation problem with possibilistic supply quantities and or possibilistic demand quantities . We consider the problem by incorporating possibilitic data into the right-hand side of the constraints. A solution concept that is attractive from the standpoint of feasibility and efficiency is specified. A necessary and sufficient condition for such a solution is established. The parametric analysis is used to characterize the parametric efficient-solution to the auxiliary problem. A numerical example is included for the sake of illustration.

Keywords: Multiple-objective transportation problem; Possibilistic variables; -possibly feasible, Parametric analysis.

Some Characteristic Theorems of Union-preserving Mapping on Lattices

Zhao Bin

Department of Mathematics, Shanxi Normal University Xi¡¯an 710062, P.R. China

Abstract: The concept of SO(L)-set family in a complete lattice L is introduced. By the relation , SO(L)-set family and SO-mapping on complete lattices, some new characterizations of union-preserving mappings on complete lattices or molecular lattices are given. It is proved that a mapping between complete lattice is an union-preserving mapping iff it is a SO-mapping; the set of all union-preserving mappings from a complete lattice L to the trivial lattice {0, 1} is a molecular lattice iff L a molecular lattice; the set of all union-preserving mappings from a strong molecular lattice to another strong molecular lattice forms a new strong molecular lattice.

Keywords: Complete lattice, molecular lattice, union-preserving mapping, strong molecular lattice.

Fuzzy Sub Klein 2n-groups

D. Talukdar

Department of Mathematics, Nalbari, Assam, India

Abstract: The inexact groupoid [5] contains Klein 2n-groups [4], which are isomorphic to each other. If K be a Klein 2n-group, then its fuzzy sub Klein 2n-group is established in this paper. Some properties of fuzzy sub-inexact groupoids and fuzzy sub Klein 2n-groups are discussed here.

Keywords: Inexact groupoid, fuzzy sub groupoid of inexact groupoid, fuzzy sub Klein 2n-group and isomorphic fuzzy sub Klein 2n-group.

L-fuzzy N-continuous Mappings

Chen Shui-Li

Department of Mathematics, Jianghan Petroleum Institute Jiangling, Hubei 434102, P.R. China

Wang Xiang-Gong

Department of Geophysical Exploration, Jianghan Petroleum Institute Jiangling, Hubei 434102, P.R. China

Abstract: In this paper we introduce a new concept of N-continuity on L-fuzzy topological spaces by means of the near N-compactness, and describe some interesting characterizations of the concept by having the aid of the N-convergence theory of L-fuzzy nets and L-fuzzy ideals.

Keywords: Fuzzy topology, near N-compactness, N-convergence, L-fuzzy, N-continuity, L-fuzzy net, L-fuzzy ideal.

The Strict Orthodoxy and Boolean Kernels of Soft Algebras (Fuzzy Lattices)

Pei Li-wen and Xu Jiang-xin

Department of Mathematics, Wuhan University Wuhan, Hubei 430072, P.R. China

Abstract: In this paper, the internal structure of soft algebras (fuzzy lattices) is further investigated on the base of papers [15-17.25]. The new notion of strict orthodoxy and Boolean kernels and so on which give expression to the specific properties of a soft algebra are firstly introduced. It is proved that a soft algebra may have no Boolean kernel, or have a unique one, or have more than one. If it has many Boolean kernels, then these Boolean kernels are isomorphic each other. Various characterizations of Boolean kernels are given, and the sufficiency-necessity conditions on which a soft algebra is strictly orthodox is also presented. If a soft algebra is strictly orthodox, then so is its quotient algebra (called to be heredity). The converse theorem is untenable. But for a special kind of convergence relation (called to be PZ-convergence relation), both are equivalent. The Boolean kernels of a soft algebra correspond to the neutral elements of its quotient one. If a soft algebra has a Boolean kernel, then it is every middle ideal needs to have a Boolean kernel, and a middle ideal is normal iff it contains a unique Boolean kernel.

Keywords: Soft algebra, fuzzy lattice, De Morgan algebra, even element, odd element, neutral element, normality, strict orthodoxy, Boolean kernel, PZ-convergence relation.

A Class of Congruence Relations and the Normality of Fuzzy Lattices

Yi-quan Zhu

Department of Mathematics, Huang-Gang Teacher College Hubei, 436100, P.R. China

Li-wen Pei

Department of Mathematics, Wuhan University Wuhan, Hubei, 430072, P.R. China

Abstract: Let L be a fuzzy lattice. The fuzzy congruence relation of L defined as iff and , is called a Particular Zonal congruence relation (or simply PZ-congruence relation), for given .In this paper we will mainly discuss their fundamental properties, and give a new description for the normal fuzzy lattices by means of the PZ-congruence relation.

Keywords: Fuzzy lattice, normal fuzzy lattice, congruence relation, PZ-congruence relation.

Commutator Fuzzy Groups

K.C. Gupta

Department of Mathematics, University of Delhi, Delhi-110007, India

B.K. Sarma

Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India

Abstract: The commutator of two fuzzy subsets of a group is defined and its properties are investigated. It is shown that a pair of admissible fuzzy subsets under an operator domain generates admissible commutator. In particular, the commutator of two normal fuzzy subgroups is a normal fuzzy subgroup contained in both. Finally, the homomorphic images and preimages of commutators are studied.

Keywords: Admissible fuzzy group, generated fuzzy group, operator domain, support.

Constructing Fuzzy Measures by Rational Transformations

Wei Wang, George J. Klir, and Zhenyuan Wang

Department of Systems Science and Industrial Engineering Thomas J. Watson School of Engineering and Applied Science Binghamton University-SUNY Binghamton, New York 13902-6000, U.S.A.

Abstract: On the basis of a discussion on constructing fuzzy measures by transformations, a detailed study of rational transformations, particularly simple rational transformations, QL transformations, and LQ transformations, is given in this paper. The qualifications for these transformations are found. A new concept of mero- -rule is introduced. It is a necessary and sufficient condition that a regular fuzzy measure can be obtained by a simple rational transformation from its dual fuzzy measure.

Keywords: Fuzzy measures, -rule, mero- -rule, Sugeno measures, transformations of set functions.

The Fuzzy Life Expectancy of Wheel Bearing

Peng Xingqian and Liu Yangsong

Architecture Engineering Department, Jianghan Petroleum Institute Jingsha, Hubei 434102, P.R. China

Abstract: In this paper, prerequisite for considering the fuzziness of work loads about wheel bearing and the required reliability, the mean value of load has been determined by essential principle of fuzzy theory. To give a method calculating the fuzzy life expectancy of wheel bearing, it has been used into the main bearing in water-faucet of drilling machine.

Keywords: Wheel bearing, fuzziness, reliability, life expectancy.

Role of Clans in the Proximities of Fuzzy Sets

K.C. Chattopadhyay and U.K. Mukherjee

Department of Mathematics, Burdwan University Burdwan-713104, India

S.K. Samanta

Department of Mathematics, VISVA-BHARATI Santiniketan-731235, India

Abstract: In this paper properties of filters, grills and clans of fuzzy sets are studied and these are used to established the results ¡®every proximity of fuzzy sets is clan generated structure.¡¯

Keywords: Fuzzy proximity, proximity of fuzzy sets, topology of fuzzy sets, filters of fuzzy sets, grills of fuzzy sets, clans of fuzzy sets.

On Fuzzy Inclusion in the Intuitionistic Sense

Dogan Coker

Department of Mathematics Education, Hacettepe University, Beytepe, 06532-Ankara, Turkey

Mustafa Demirc

i Department of Mathematics, Abant Izzet Baysal University, 14280-Bolu, Turkey

Abstract: In this paper we define fuzzy inclusion in intuitionistic fuzzy sets and then construct its fundamental properties.

Keywords: Intuitionistic fuzzy set, intuitionistic fuzzy pair, intuitionistic fuzzy inclusion, intuitionistic fuzzy family, intuitionistic fuzzy topology in Sostak¡¯s sense.