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The Journal of Fuzzy Mathematics

Volume 20, Number 1, March 2012



Fuzzy Modelling of Location Problems- A New Approach

Mary George and P. G. Thomaskutty

Dept. of Mathematics, Mar Ivanios College, Trivandrum, Kerala, India

Sunny Kuriakose

Dept. of Mathematics, Union Christian College, Aluva, Kerala, India

Abstract: This paper deals with the solution of location problems which possess certain kind of impreciseness. Since the situation owes impreciseness, fuzzy models are used to model the location problems. The present literature is analysed and a different method using fuzzy tools is presented, which gives the technique of identifying the optimum point of a location from the fuzzy subsets of all proposed location points.


A Fuzzy Mathematical Measurement of Poverty

Mary George and P. G. Thomaskutty

Dept. of Mathematics, Mar Ivanios College, Trivandrum, Kerala, India

Abstract: Poverty measurement mainly deals with two problems; one to identify the poor and two to measure the depth of poverty. Since the concept of being poor is vague rather than precise, the measurement will be effective if fuzzy mathematical tools are used in the measurement. The paper introduces a fuzzy poverty index, which satisfies almost all the desirable axioms, while much simple for dealing with.

Keywords: Fuzzy sets, poverty line, poverty index, welfare.

Fuzzy Topological Structures of Low Dimensional Digital Spaces

Sayaka Hamada

Department of Mathematics Yatsushiro Campus Kumamoto National College of Technology 2627 Hirayama-shinmachi, Yatsushiro, Kumamoto 866-8501 JAPAN E-mail address:

Taichi Hayashi

Department of Mathematics Kumamoto Pref. Shoyo High School Muro 1782, Ozu Town, Kikuchi, Kumamoto 869-1235 JAPAN E-mail address:

Abstract: The set of lower semicontinuous functions from the digital space to the interval is a fuzzy topology in . We give some properties of the fuzzy topology and the decomposition for each (resp. ) such that , f1 is a fuzzy open set and f2 is a fuzzy nowhere dense set.

Keywords: Fuzzy Sets, fuzzy topology, digital line, digital plane, preopen set.

A Note on Geometrical Properties of Fuzzy Sets


Department of Mathematics, IIT Kharagpur, India

R.Guha* and R.N.Mohapatra**

School of Computer Science*, Department of Mathematics** University of Central Florida, Orlando, Florida, USA

Abstract: Fuzzy logic is often used in image analysis, personal identification from handwriting. It has also been explored as tools for synthesizing information from multiple sources. This calls for a need to deal with multiple fuzzy variables representing geometrical entities such as length, breadth, area, volume etc. This paper establishes inequality bounds for such concepts in two-dimensional fuzzy sets that has been recently introduced in the literature. It also extends these ideas to three-dimensional fuzzy sets and establishes similar bounds there.

Keywords: Fuzzy sets, fuzzylogic, perimeter, diameter, length, breadth, height, width, area, volume, compactness

A Correspondence between Lodato Fuzzy Proximities and A Class of Principal Type-II Fuzzy Extensions


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

Abstract: In this paper we have established a bijection between a class of Lodato fuzzy proximities compatible with a given strongly T1-topological space of fuzzy sets (X,c) and the class of strongly T1 principal Type-II fuzzy linkage compactifications of (X,c). Keywords: Topology of fuzzy sets, principal Type-II fuzzy extensions,T0-topological space,T1-topological space, strongly T1-topological space, fuzzy conjoint grill, fuzzy linked grill, fuzzy conjointly compact space, fuzzy linkage compact space, finitely determined collection of fuzzy grills, binary collection of fuzzy grills, basic fuzzy proximity, LO-fuzzy proximity, fuzzy -clan, maximal fuzzy -clan, fuzzy linkage compactification.

Characterizations of Some Double Fuzzy Separation Axioms

S.E.Abbas and E.El-Sanosy

Dept. of Mathematics, Faculty of Science, Sohag 82524, Egypt

Abstract: The aim of this paper is to introduce and characterize some topological separation axioms in terms of quasi-coincidence, -closure and -closure operators as initiated in [11] in double fuzzy set-ting. Also, we introduce and characterize double fuzzy weakly -closed functions between double fuzzy topological spaces and also study these functions in relation to some other types of already known functions.

Keywords: (r,s)-fuzzy rerular, (r,s)-fuzzy almost regular,(r,s)-fuzzy Uryshon, double fuzzy weakly -continuous function.

An Inconsistency Checking Method for Relational Database: A Rough Set Approach

R.N.Bhaumik and Chhaya Gangwal

Dept. of Mathematics, Tripura University Suryamaninagar, Tripura(W), India, 799130,E-mail:

Abstract: Rough set theory is an elegant and powerful methodology in dealing with inconsistency problems. In this paper a new rough set methodology based on multi approximation system is introduced to solve the problem of inconsistency in relational databases. The method measures the size of certain positive region and certain negative region to reflect the dependency between the left hand side and the right hand side of the functional dependencies between sets of attributes in relations. A simple example is given to illustrate the problem.

Keywords: Rough sets, Relational database, Functional dependency.

Fuzzy ?-Semicontinuous Mappings

Athar Kharal

National University of Sciences and Technology(NUST), Islamabad, PAKISTAN


Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, PAKISTANE Email: drbashir9@gmail.comAbstract

Abstract: In this paper, we define and present several properties of fuzzy ?-semiopen, fuzzy ?-semiclosed sets; fuzzy ?-semiopen and fuzzy ?-semiclosed mappings. We also study characterizations of fuzzy ?-semicontinuous, fuzzy almost ?-continuous and fuzzy semiweaky continuous mappings. Moreover, we establish necessary (resp. sufficient) conditions for fuzzy ?-semiopen and fuzzy ?-semiclosed (resp. fuzzy ?-semiclosed, fuzzy ?-irresolute) mappings.

Keywords: Fuzzy ?-semiopen (fuzzy ?-semiclosed) sets, Fuzzy ? -semicontinuous mappings; Fuzzy ?-semiopen (fuzzy ?-semiclosed) mappings; fuzzy semiweekly continuous, fuzzy almost-continuous mappings.

An Application of Hahn-Banach Theorem to Fuzzy Bounded Linear Functionals

Lemnaouar Zedam

Laboratory of Pure and Applied Mathematics,Msila University,P.O.Box 166 Ichbilia, Msila28105, ALGERIA. E-mail address: Fax:(+213)35 55 18 36.

Abstract: In this paper we will present an application of Hahn-Banach theorem to fuzzy bounded linear functionals by proving that if u0 is a fuzzy bounded linear functional on X0, where X0 is a fuzzy subspace of a fuzzy normed space X, then there exists a fuzzy norm preserving extension of u0 to X.

Keywords: Hahn-Banach theorem, Fuzzy normed space, Fuzzy bounded linear functional, Sublinear functional.

Several Types of Double Fuzzy Semiclosed Sets

S.E.Abbas and E.El-sanosy

Department of Mathematics, Faculty of Science, Sohag University, Sohag, Eygpt. E-mail

Abstract: In this paper, we introduce (r, s)-generalized fuzzy (semi, weakly semi, regular) closed sets in a double fuzzy topological space. Also, we introduce several types of (r, s)-fuzzy regular spaces. Moreover, we investigate the relationship between generalized fuzzy semi-continuous mappings and generalized fuzzy semi-irresolute mappings.

Keywords: Double fuzzy topological space, (r, s)-fuzzy regular spaces, generalized double fuzzy semicontinuity.

A New Method for Solving Fuzzy Critical Path Problems Based on Fuzzy Linear Programming Formulation

Amit Kumar, Parmpreet Kaur

School of Mathematics and Computer Applications, Thapar University, Patiala-147 004, India,

Abstract: To the best of our knowledge, there is no method, in the literature, to find the fuzzy optimal solution of fully fuzzy critical path (FFCP) problems i.e., critical path problems in which all the parameters are represented by fuzzy numbers. In this paper, a new method is proposed to find the fuzzy optimal solution of FFCP problems. Also, a new representation of trapezoidal fuzzy numbers is proposed. It is shown that it is better to use the proposed representation instead of existing representations of trapezoidal fuzzy numbers, to find the fuzzy optimal solution of FFCP problems. To illustrate the proposed method and to show the advantages of the pro-posed representation of trapezoidal fuzzy numbers, a numerical example is solved by representing all the parameters as existing and proposed type of trapezoidal fuzzy numbers. The proposed method is very easy to understand and to apply for finding the fuzzy optimal solution of FFCP problems occurring in real life situations.

Keywords: Fully fuzzy critical path problem Ranking function Trapezoidal fuzzy number

Method for Solving A Special Type of Fuzzy Transportation Problems

Amit Kumar, Amarpreet Kaur, Anila Gupta

School of Mathematics and Computer Applications Thapar University, Patiala-147004, India Email:,,

Abstract: For finding the fuzzy optimal solution of a fuzzy transportation problem it is assumed that the product may be supplied only from source nodes to the destination nodes and also it is assumed that the obtained fuzzy transportation cost is the minimum cost. But in real life problems this cost may also be decreased if it is allowed to transport the product through some other nodes before reaching the final destination nodes. In this paper, a new method is developed to find the fuzzy optimal solution of a fuzzy transportation problem in which the product, from a source to destination, is supplied through some other nodes. In the proposed method all the parameters are represented by trapezoidal fuzzy numbers. To illustrate the proposed method a numerical example is solved. The proposed method is easy to understand and to apply for finding the fuzzy optimal solution of fuzzy transportation problems occurring in real life situations.

Keywords: Fuzzy transportation problem Ranking function Trapezoidal fuzzy number

L-fuzzifying C-uniform Spaces


Department of Mathematics, Faculty of Science, Sohag 82524,Egypt.

Abstract: In this paper, we introduce L-fuzzifying C-uniformity in term of covering approach and investigate some of their properties. We study the relationships between L-fuzzifying C-uniform space and L-fuzzifying uniform space.

Keywords: L-fuzzifying C-uniformity, L-fuzzifying uniformity, Category.

Lattice Valued Double Fuzzy Uniform Spaces


Department of Mathematics, Faculty of Science, Sohag 82524,Egypt.

Abstract: In this paper, the concept of lattice valued double fuzzy uniform spaces is introduced. The relationships among the double fuzzy uniformity, double fuzzy topology and double fuzzy interior operators are studied. We study several double fuzzy topologies induced by double fuzzy uniform space.

Keywords: Uniformity, double fuzzy topology, double fuzzy interior operator.

Operations on Intuitionistic Fuzzy Soft Sets

Pabitra Kumar Maji

Department of Pure Mathematics, University of Calcutta,35,Ballygunge Circular Road, Kolkata-19, West Bengal, India.

Abstract: In this paper we study the concept of intuitionistic fuzzy soft sets. We have introduced some new operations on this concept. Some properties of these operations have also been investigated. An application of the newly defined operation has been given.

Keywords: soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy soft sets.

Some Aspects of Pairwise Fuzzy Pre-almost Continuous Functions in Fuzzy Bitopological Spaces


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

Jonali Sharma

Laban Assamese GirlsSecondary School, Laban Shillong 793004, Meghalaya, India.

Abstract: This paper deals with the introduction of fuzzy pre-almost continuous functions in a fuzzy topological space and it also introduces the concept of pairwise fuzzy pre-almost continuous functions in a fuzzy bitopologocal space. Using various conditions on two fuzzy topologies and their mixed fuzzy topology, some results are established on pairwsie fuzzy pre-almost continuous functions.

Keywords: Pre-neighbourhood, pre almost continuous functions, pairwise fuzzy pre-almost continuous functions, fuzzy bitopological space, mixed fuzzy topology, fuzzy regular space.

On The Lattice of Fuzzy Congruences on A Lattice

M.J.Rain and N.R.Mangalambal

Department of Mathematics St. Josephs College, Irinjalakuda,KeralaIndia-680121,

Abatract: In this paper, the concept of a fuzzy congruence on a lattice X is discussed. A necessary and sufficient condition for a reflexive, symmetric fuzzy binary relation on a lattice X to be a fuzzy congruence on X has been derived. Consequently the collection F C(X) of all fuzzy congruences on X, under suitably defined meet and join operations is obtained as a lattice.

Keywords: Fuzzy equivalence, fuzzy join compatible, fuzzy meet compatible, fuzzy congruence.

Graded Propositional Fuzzy Logic For Approximate Reasoning

Xiaodong Pan

School of MathsSouthwest Jiaotong University,Chengdu,Sichuan,610031,P.R.China Intelligent Control Development Center, Southwest Jiaotong University, Chengdu , Sichuan, 610031, P.R.China E-mail

Yang Xu

Intelligent Control Development Center, Southwest Jiaotong University, Chengdu , Sichuan, 610031, P.R.China

Abstract: For the task of approximative reasoning, in this paper, the graded semantics and syntax is developed for Lukasiewicz infinite-valued propositional logic on [0,1]. The relations among ?-tautologies in the frame of graded theory are obtained. In a natural way, the fuzzy information based on a crisp set of formulas is defined. On this basis, some propositions analogous to those in the classical logic are proved, and the corresponding graded syntax is established. Moreover, we also establish the generalized deduction theorem by extending the classical deduction theorem.

Keywords: Infinite-valued logic; ?-tautology; Graded semantical consequence; Graded syntactic consequence

Multi-Fold Fuzzy Associative Filter of Residuated Lattice Implication Algebras*

Zhu Hua, chen Shuwei , Zhao Jianbin

Department of Mathematics, Zhengzhou University Zhengzhou, Henan 450001, China

Xu Yang

Intelligent Control Development Center, Southwest Jiaotong University Chengdu, Sichuan 610031, China

Abstract: In residuated lattice implication algebras, we introduce the concept of multi-fold fuzzy associative filter and investigate some properties of the multi-fold fuzzy associative filter. Then, we discuss respectively the relationships the between multi-fold fuzzy associative filter and the fuzzy filter, the multi-fold fuzzy associative filter and the multi-fold fuzzy implicative filter, between the multi-fold fuzzy associative filter and the multi-fold associative filter. Finally, we obtain that the multi-fold fuzzy associative filters are equivalent for different n in lattice implication algebra.

Keywords: filter; multi-fold fuzzy associative filter; residuated lattice implicative algebra

A New Approach to Solve Fuzzy Transportation Problems

Thangaraj Beaula and Priyadharsini.M

Research Department, T.B.M.L College, Porayar

Abstract: A new algorithm, is proposed for finding a fuzzy optimal solution for a fuzzy transportation problem where the transportation cost, supply and demand are trapezoidal fuzzy numbers. The optimal solution for the fuzzy transportation problem by this method is a trapezoidal fuzzy number. The fuzzified version of the problem has been discussed with the help of the numerical example.

Keywords: Fuzzy transportation problem, Trapezoidal fuzzy numbers, optimal solution.

Fuzzy Goal Programming Approach to Linear Fractional Bilevel Decentralized Programming Problem Based on Taylor Series Approximation

Surapati Pramanik

Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District-North 24 Parganas, Pin code-743126, West Bengal, India E-mail:

Partha Pratim Dey and Tapan Kumar Roy

Department of Mathematics, Bengal Engineering and Science University, P.O.-Shibpur, District-Howrah, Pin code-711103, West Bengal, India

Abstract: This paper presents the fuzzy goal programming approach for linear fractional bilevel decentralized programming problem based on Taylor series approximation. The bilevel decentralized problem is an extension of the conventional bilevel programming problem. Bilevel decentralized programming problem consists of an upper level decision maker at the first level and multiple lower level decision makers at the second level. In the decision making situation, each decision maker controls a decision vector independently. The objective function of both level decision maker are linear fractional in nature. To formulate the fuzzy goal programming model of the proposed method, the fractional membership function of the fuzzy objective goals of both levels are transformed into linear membership function by first order Taylor polynomial series. The objectives of both level decision makers arepotentially conflicting in nature, a possible relaxation of upper level decision is considered for avoiding decision deadlock. Then, the fuzzy goal programming approach due to Pramanik and Roy[9] is used for achieving highest degree of each of the membership goals by minimizing the negative deviational variables. To demonstrate the efficiency of the proposed method, a bilevel linear fractional decentralized programming is solved.

Keywords: Fuzzy goal programming, fractional bilevel decentralized programming problem, membership function, negative deviational variables, Taylor series

Some New measures of Entropy Using Fuzzy Cardinality

Rogi Jacob and Sunny Kuriakose A

Department of Mathematics, U.C.College, Aluva, Cochin

Abstract: Fuzzy entropy is a fuzzy measure expressing the extent of fuzziness of fuzzy subsets. The measure of fuzziness becomes smaller when the similarity of the argument is increased. The fuzzy membership reflects the degree of ambiguity present in it. In this paper we introduce a new divergence measure based on different cardinality of fuzzy sets and obtain a relation between proposed divergence measure and koskos subsethood measure and we prove that it reduces to bhandari and pal [3] divergence measure. Also the proposed divergence measure is expressed as a distance measure. Finally we define entropy using divergence measure.

Keywords: Divergence measure, Distance measure, ?-distancemeasure, Fuzzy entropy

Sets Having Finite Fuzzy Measure in Real Hilbert Spaces

Manju Cherian

Thekkeveettil House, Iringole P.O., Perumbavoor, Eranakulam District, Kerala, India E-mail:


Department of Mathematics, Farook College,Kozhikode,Kerala-673 632,India. E-mail:

Abstract: A new type of translation invariant and lower semi continuous fuzzy measure on the class of subsets of a real Hilbert space is introduced. It measures a subset of the Hilbert space as a projection of the set along a fixed vector in the Hilbert space. It is proved that corresponding to each subset of the Hilbert space. Then it is proved that the fuzzy measure of a closed convex subset of the Hilbert space can be obtained in terms of two elements of the subset itself. It is also proved that this fuzzy measure satisfies a condition similar to the null additivity.

Keywords: Vector Generated Fuzzy Measure, Null additivity, Support space, Null space, Closed and Convex subsets of a Hilbert space.