THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 4, Number 1, March 1996

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On the Fredholm Integral Equation of the First Kind and Ascoli-Arzela Theorem of Fuzzy Set-Valued Function

Liu Yi Hun

Hebei University of Technology, Tianjin, China

Abstract: In this paper, the Fredholm integral equation of the first kind of fuzzy set-valued function is discussed. It is pointed out that the solution of the equation is unstable. By a functional defined on the space of continuous fuzzy set-valued function C(I, E1), the approximate solution of the equation is given. And the well known Ascoli-Arzela theorem is generalized in C(I, E1).

Keywords: Fuzzy set-valued function, integral equation, stability, Ascoli-Arzela theorem.

A Note on Fuzzy Factorizations

Nistala V.E.S. Murthy

Department of mathematics Andhra University P.G.Center Srikakulam-532402, A.P.State, India

Abstract: In this note, we essentially show that there are no non-trivial, (unique) fuzzy factorization and principal ideal, structures on a commutative ring with unity and without zerodivisors. Further, fuzzy simple rings are determined.

Keywords: Complelte Brouwerian lattice, L-fuzzy point, semiring, semidomain.

How Fuzzy are the Fuzzy Ideals?

Nistala V.E.S. Murthy MIG-B 171, Sagar Nagar VIZAG-530043, A.P.State, India

Abstract: In this paper, we establish a one-one correspondence, between the meet complete lattice of all fuzzy ideals of a ring and the meet complete lattice of all closed (crisp) ideals of the associated fuzzy-point semiring of the given ring, in such a way that it defines a complete meet homomorphism.

Keywords: Complete Brouwerian lattice, L-fuzzy point, m-semiring, s-semiring.

On Solving Multiobjective Nonlinear Programming Problems with General Parameters in the Constraints Using the Fuzzy Mathematical Programming

Abou-Zaid H.El-Banna and El-Said E.Ammar

Department of mathematics, Faculty of Science, Tanta University, Tanta-Egypt.

Abstract: In this paper, a formulation of parametric multiobjective nonlinear programming problems in fuzzy mathematics as fuzzy nonlinear programming problems is presented. A special and general fuzzy problems will be considered. Also, some basic notions of the two fuzzy problems are defined. Moreover, the relations between these notions and the basic concepts of the ordinary problem are also introduced. An illustrative example is given.

Keywords: Vector optimization, parametric programming, efficient and weak efficient solutions, fuzzy programming, fuzzy solutions.

On Fuzzy Isomorphism

Sh.Salili And M.Mashinchi

Faculty Of Mathematics and Computer Science, Kerman University, Kerman, Iran

Abstract: In this paper we formulate a generalized definition of homomorphism (isomorphism) between T-fuzzy subgroups and compare it with the recent definition of isomorphism due to Ray(1992).

Keywords: t-norm (t-conorm), -admissibility, T-fuzzy (normal) subgroup, convex T-fuzzy subset.

Generalized Fuzzy Similitudes

Pratyayananda

Das Department of mathematics, Burdwan University Burdwan, West Bengal, Pin-713-104, India

Abstravt: Given a subinterval (a, b] of [0, 1], weak and strong (a, b)-similitudes and weak and strong (a, b)-fuzzy partition are defined. It is proved that a weak (resp. strong) (a, b)-similitude deterines a weak (resp. strong) (a, b)-fuzzy partition and conversely. Finally a generalized similitude which is a weak similitude in some subintervals and a strong similitude in some other subintervals and the corresponding fuzzy partitions are investigated.

Keywords: Fuzzy algebra, fuzzy relation, w-similitude, q-similitude, fuzzy partition, q-fuzzy partition.

A Parametric Study on Fuzzy Multiobjective Integer Linear ProgrammingProblems(Parameters in the Objective Functions and in the Constraints)

Mohamed S.Osman

Department of mathematics, Higher Technological Institue. Ramadan 10th city, Cariro, Egypt

Omar M.Saad Department of mathematics, Helwan University, Cariro, Egypt

Azza G.Hassan

Department of mathematics, Ain Shams University, Cariro, Egypt

Abstract: This paper deals with multiobjective integer linear programming problems with fuzzy parameters in the objective functions and in the constraints by using the concept of level set of fuzzy numbers. A parametric study is carried out on the problem of concern. In addition, a solution algorithm is described to soluion algorithm is described to solve the formulated model. This algorithm is based mainly on the weighting method together with the technique of Gupta and Ravindran. A numerical simple example is included to clarify the theory developed in the paper.

Keywords: Fuzzy multiobjective linear programming, Mixed-integer programming, Fuzzy parameters, Fuzzy numbers, Stability. De Morgan Algebra of Topology V Zike Deng Department of mathematics, Hunan University Changsha 410082, Hunan, China Abstract: We give a necessary and sufficient condition for a de Morgan Algebra of topology to be uniformizable. Among its special cases are traditional uniformization theorem and the fuzzy one. Keywords: De Morgan Algebra of topology, De Morgan Algebra of uniformity.

Pseudo-Linear Algebra

A.Markova

STU Radlinskeho 11 81368 Bratislava, Slovakia

Abstract: For a given additive generator g, basic notions of pseudo-linear algebra based on g are introduced, such as g-inverse matrix, g-determinant, etc. Colosed connection of some integrals for fuzzy measure (PAN-integral, Pap's integral) with generator g promisses to introduce several applications. Keywords: Additive generator, fuzzy measure, g-integral, pseudo-addition, pseudo-multiplication. On Disconnectedness In FTS Arun K.Srivastava and Anuranwaj Singh Department of mathematics, Faculty of Science, Banaras Hindu Universtiy, Varanasi-221005, India Abstract: The purpose of this paper is to initiate a study of (Preuss' concept of) 'disconnectedness' in the category of fuzzy topological spaces. Connections between this concept and certain separation axioms in fuzzy topology are also pointed out.

Keywords: Connectedness, disconnectedness, fuzzy T0 and T1-separation axioms.

Fuzzy Linear Order-Homomorphism and Its Structures

Fang Jin-Xuan

Department of mathematics, Nanjing Normal University Nanjing, Jiangsu 210097, P.R.China

Abstract: In this paper, the concept of fuzzy linear order-homomorphism is introduced and its structures are studied. We prove a decomposition theorem and a pointwise characterization theorem for fuzzy linear order-homomorphism.

Keywords: Order-homomorphism, fuzzy linear ordear order homomorphism, fuzzy linear operator.

Approximate Reasoning Based on Regular Inference Functions

Wangming Wu* and Ho Chung Lui

Institute of Systems Science, National University of Singapore, Singapore 0511

Bo Yuan

Department of Systems Science and Industrial Engineering Thomas J.Watson School, Binghamton University-SUNY, Binghamton, New York 13902 USA

Abstract: In this paper, a parametric approach based upon regular inference functions for approximate reasoning is presented. Inferences in approximate reasoning systems are realized by either the compositional rule of inference (CRI for short) or the compatibility modification inference (CMI for short). In general, CRI and conventional CMI inference methods do not satisfy the basic inference criterion, modus ponens (MP for short). It seems a disadvantage to apply these two methods. In order to overcome this anomaly, we introduce the concept of regular inference functions with which the MP criterion is satisfied. Via parametric regular inference functions introduced in this paper, we present a parametric inference method for approximate reasoning with several parallel rules. Finally, the regular inference functions with set-valued fuzzy consequences for set-valued approximate reasoning are also discussed.

Keywords: Modus ponens, compositional rule of inference, compatibility modification, parametric disjunction, regular inference functions, approximate reasoning.

Posynomial Geometric Programming

Gao Bing-Yuan

Department of Mathematics, Changsha University of ElectricPower, Hunan, 410077, China

Abstract: With a weak dual theorem extended, the author gets some corresponding theorems in terms of fuzzy subconsistency, discusses a fuzzy converse theorem and builds a fuzzy strong dual theorem as he has developed the relationship between fuzzy posynomial geometric programming and its dual programming built on a fuzzy weak dual theorem[1, 2]. Keywords: Fuzzy posynomial geometric programming, strong(or weak) dual, converse theorem. Fuzzy Weakly Pre-Irresolute and Strongly Pre-Irresolute Mapping Jin Han Park Department of Natural Sciences, Pusan National University of Technology, Yongdang-dang Nam-ku, Pusan 608-739, Korea Hoe Young Ha Department of Mathematics, In-Je University, Obang-dong, Kimhae, 621-749,Korea Abstract: In this paper, we introduce and study fuzzy weakly pre-irresolute and strongly pre-irresolute mappings between fuzzy topological spaces, which are stronger and weaker respectively than fuzzy pre-irresolute mappings.

Keywords: fuzzy weakly pre-irresolute mapping, fuzzy strongly pre-irresolute mapping, fuzzy topological spaces, fuzzy pre-irresolute mappings.

On Possibilistic and Interval Multi-Objective Programming

M.Wagenknecht

TU Berlin, Institut fur Proze und Anlagentechnik Rudower. Chaussee 5 (Geb. 10.5), 12484 Berlin, Germany

A.V.Yazenin

Tver State University, Dept. Of Algorithmic Languages Ul. Zhelyabova 33, 170000 Tver, Russia

Abstract: In the present paper we consider certain possibilistic multi-object programming problems. It shown that they can be reduced to interval programming for which powerful algorithms are available. The linear case is considered as special application.

Keywords: Possibilistic multi-object programming, interval optimization, linear programming.

Variational Inequality with Fuzzy Function

Hsiao-Fan Wang and Hsueh-Ling Liao

Department of Industrial Engineering, National Tsing-Hua University Hsinchu, taiwan, ROC

Abstract: In this study, we discuss a kind of variational inequality problem with fuzzy functions, denoted by VI ( ) . According to the existence theorm, an approach derived from the concepts of multiple objective mathematical programming problems for solving the VI( ) is proposed, and an algorithm is developed to find its fuzzy optimal solution set. Furthermore, the LCP with fuzzy will also be discussed.

Keywords: Ariational inequality, generalized complementarity problems, linear complementarity problems, fuzzy function.

Fuzzy Homomorphisms of Rings and Algebraic Structures of Fuzzy Subrings

Yao Bingxue

Department of mathematics, Liaocheng Teachers College, Shandong 252000, P.R.China

Abstract: In this paper, the concept of fuzzy homomorphism of a ring is introduced and the fuzzy subring generated by a fuzzy subset is discussed. Keywords: Fuzzy homomorphism, fuzzy subring , fuzzy ideal. Graded Mapping and Its Application Marian Matloka Department of mathematics, Academy of Economics al. Niepodleglosci 10, 60-967 Poznan, Poland Abstract: The aim of this paper is to explain a concept of fixed-point theorem for graded mapping. Next, the game is described and the equilibrium theorem is proved.

Keywords: Graded mapping, fixed point, graded game, equilibrium play.

Remarks on Convergent in Fuzzy Measure Theorem of Generalized Fuzzy Integrals on L-fuzzy Sets*

Wu Congxin

Department of Mathematics,Harbin Institute of Technology Harbin 150001,P.R.China

Abstract: In this paper,our purpose is to modify the convergent in fuzzy measure theorem of generalized fuzzy integrals on L-fuzzy sets,which is shown in [2].

Keywords: Generalized triangle norm,generalized fuzzy integrals on L-fuzzy sers,convergent in fuzzy measure theorem.

Application of Fuzzy Loglinear Regerssion Analysis toEvaluation of Efficiency for DMUs

Yoshiki Uemura

Faculty of Education,Mie University, Tsu,Mie 514 Japan

Masaki Kobayashi

Faculty of Economics,Osaka University Toyonaka,Osaka 560 Japan Kazuki Hiro Nara National College of Technology Koriyama,Nara 639-11,Japan

Abstract: The evaluation of efficiency for DMUs (Decision Making Units) in some company is very important. In this view point, the study of evaluation of efficiency is every active, on the basis of the formulation for the production function. Up to this time, loglinear method does not have the lower limit of the production set but the upper limit of one. Because we consider that the real problem has the production set from lower limit to upper limit, in this paper we construct the possibility production function by introducing the fuzzy theory into the loglinear production function. Every fuzzy parameter in the possibility production function is identified very easily by the concept of the fuzzy regression analysis. As we try to evaluate the efficiency by thes possibility production function, this method can obtain two efficiency ratings from upper limit and lower limit. By these two efficiency ratings, this methid can have the more exact ranking for DMYs than loglinear method. And the efficiency of loglinear model can be evidenved by non-efficiency ratings from the lower limit of possibility production function. We call this proposed method as the fuzzy loglinear regression anlysis.

Keywords: Loglinear model, possibility production function, possibility production set, fuzzy regression analysis, evaluation of efficiency.

Constructing Fuzzy Measures by Transformations

George J. Klir, Zhenyuan Wang and Wei Wang

Department of Systems Science and Industrial EngineeringThomas J. Wastson School of Engineering and Applied Science Binghamton University-SUNY, Binghamton, New York 13902-6000, USA

Abstract: In a similar way as obtaining quasi-measures from classical measures, fuzzy measures can be constructed by transformations from other fuzzy measures. The relation berween the given fuzzy measures and the constructed ones is shown as a necessary and some typical transformations are discussed.

Keywords: Fuzzy measures, quasi-measures, Sugeno measures, order-isomorphism, structural characteristics of set functions.

On N-dimensional Fuzzy Integrals

Ma Ming1, Menahem Friedman2 and Abraham Kandel

Computer Science and Engineering Department University of South Florisda, Tampa, FL 33620-5350 USA

Abstract: The concept of the one-dimensional fuzzy integral is extended to the -dimensional case. Twodifferent approaches which lead to an identical result are implemented and their applicability is illustrated with suitable examples. Keywords: Fuzzy measure; fuzzy integral; abstract fuzzy measure; abstract fuzzy integral; vector lattice.

-filterbase and -semi -compactness in Fuzzy Topological Spaces

A.M. Zahran

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71511, Egypt

Abstract: The concept of -semicompact fuzzy topological spaces (henceforth fts, for short) was introduced by A.S. Mashhour et al [9] and the concept of -closed fts was introduced by A.M. Zahran [14]. Inthis paper we study the cincept of -semi -compact fts which lie between the above teo concepts and we characterize -semi -compact spaces in terms of the concept of fuzzy - filterbase. Also, we show that an fts is -semi -compact iff is -compact, where is a fuzzy topology in which has as a subbase. Finally preservation of some fuzzy topological structures under some types of fuzzy continuity are studied. Keywords: Fuzzy topology, -filterbase, fuzzy semi -cluster point, -semi -compact spaces, fuzzy semi irresolute function.