The Journal of Fuzzy Mathematics
Volume 18, Number 3, September 2010
CONTENT
DETAILS
Intuitionistic Fuzzy Optimization
Lisy Cherian
Mar Thoma College for Women, Perumbavoor, Kerala, India, Pin-683542.
E-mail: thomaspt46@yahoo.co.in
Abstract:
A general concept for the optimization problems under uncertainty is proposed in this paper. Practically, in most of the optimization problems the objective function(s) constraints or the parameters are determined on the basis of certain forecasting. Here we introduce a general non-membership function to solve Intuitionistic Fuzzy optimization (IFO) problems. We claim that this give a general and richest apparatus for the formulation of optimization problem and can satisfy the objective with bigger degree of satisfaction than the analogous fuzzy or crisp optimization problem.
Keyword:
Intuitionistic fuzzy set [IFS], Intuitionistic fuzzy optimization [IFO], Intuitionistic fuzzy linear programming [IFLP], Intuitionistic fuzzy transportation problem [IFTP].
Fuzzy Real Valued Generalized Difference Sequences Space of Para-normed Type
Binod Chandra Tripathy
Mathematical Science Division; Institute of Advanced Study in Science and Technology Paschim Boragaon; GARCHUK; GUWAHATI-781035; INDIA.
E-mail: tripathybc@yahoo.com; tripathybc@rediffmail.com
B. K.Nayak
Department of Mathematics; Utkal University Vani Vihar; BHUBANSEWAR-751004; ORISSA; INDIA
E-mail:bknatuu@yahoo.co.uk
Manoranjan Mishra
Department of Mathematics; Nayagarh College, NAYAGARH-752069; ORISSA; INDIA
Abstract:
We introduce the class of sequences of fuzzy real numbers and study is different properties like solidness, symmetricity, convergence free, ect. We prove some inclusion results too. The introduced class of sequences generalize and unify several existing results too.
Keyword:
Fuzzy sequence, difference sequence, solid, symmetricity, convergence free.
Generalized Piece-wise Linear Fuzzy Number with Its Application in Redundancy for Optimum Reliability of Series-parallel System
G. S. Mathapatra and T. K. Roy
Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howarah-711103, West Bengal, India
Correspondence to: tkroy_math@yahoo.com, or roy_t_k@yahoo.co.in
Tel: +91-33-26684561(Extn. 526). Fax: 91-33-26682916
Abstract:
This paper illustrates the use of geometric programming to analyze system reliability with generalized piece-wise linear fuzzy number of system cost, system weight, reliability and cost of system components. We use geometric programming to solve redundancy allocation problem. The redundancy allocation problem, whose goal is to finding the optimal allocation of redundancy components is such a way that maximize system reliability subject to available total system cost and weight. Examples are shown to illustrate the model using generalized piece-wise linear fuzzy number.
Keyword:
Generalized piece-wise linear fuzzy number, Fuzzy system reliability, Reliability optimization, Redundancy, Geometric programming.
System Reliability Assessment with Constant Failure Rate of Components under Fuzziness
G. S. Mathapatra
Department of Mathematics, Bengal Engineering and Science University, Shibpur P. O.-Botanic Garden, Howarah, West Bengal, India-711103, Tel: +91-33-26684561 Fax:+ 91-33-26682916
E-mail: g_s_mahapatra@yahoo.com
B. S. Mahapatra
Department of Mathematics, Seacom Engineering University, JL-2, Jaladulagori, Sankrail, Howarah-711302, West Bengal
E-mail: b_s_mahapatra@yahoo.com
T. K. Roy
Department of Mathematics, Bengal Engineering and Science University, Shibpur P. O.-Botanic Garden, Howarah, West Bengal, India-711103, Tel: +91-33-26684561 Fax:+ 91-33-26682916
E-mail: tkroy_math@yahoo.com
Abstract:
Here, it is considered that constraint failure rate of the components are fuzzy in nature. The failure rate of components is taken as triangular fuzzy number. The reliability function of system reliability is expressed in terms of fuzzy number of the components reliability. Simple reliability models are taken in fuzzy environment and then its reliability is evaluated. Fuzzy system reliability of series system, parallel system, k out of n system and cold standby system are approximated. Then the corresponding left and right divergence are computed, which are occurring due to approximation of system reliability. Number examples are given to illustrate the approaches.
Keyword:
Fuzzy set, failure rate, k-out-of-n system, parallel system, Series system, Cold Standby system, Triangular fuzzy number.
Fuzzy Based GA to Multicriteria Entropy Matrix Game
Chandan Bikash Das
Department of Mathematics Tamralipta Mahavidyalaya, Tamluk, Purba Midnapore-721636, West Bengal, India
E-mail: cdas_bikash@yahoo.co.in
Sankar Kumar Roy
Department of Applied Mathematics with Oceanology and Computer Programming Vidyasagar University, Midnapore-721102, West Bengal, India
E-mail: roysank@yahoo.co.in
Abstract:
In this paper the entropy function of multicriteria matrix game has been considered to the objective function and formulate a new game model namely Multicriteria Entropy Game Model. Multiobjective non-linearprogramming model for each player has been established. It is shown that the set of efficient solutions of this problem coincides with the set of Pareto-Optimal Security Strategy (POSS) for both player in muliticriteria Entropy Matrix Game. The corresponding effective solutions via Genetic Algorithm (GA) are proposed through fuzzy programming technique. A numerical example is included to illustrate the results in this paper.
Keyword:
Multicriteria Matrix Game, POSS, Entropy, Genetic Algorithm, Fuzzy programming.
A Common Fixed Point Theorem for Sequence of Mappings in Fuzzy Metric Space
Kanhaiya Jha
Department of Mathematical Science , School of Scienc Kathmandu University, P. O. Box No.6250, Kathmandu, Nepal
E-mail: jhaknh@yahoo.co.in; jhakn@ku.edu.np (K. Jha)
Abstract:
The aim of the present paper is to prove a common fixed point theorem for a sequence of self mappings in a fuzzy metric space which generalizes and improves various well-known comparable results.
Keywords:
Common fixed point, Fuzzy metric space, Weakly compatible maps.
-closed and -closed Sets in Bitopological Spaces
Anjan Mukherjee
Department of Mathematics Tripura University Suryamaninagar, Tripura-799130 INDIA
E-mail: anjan2002_m@yahoo.co.in
Subrata Bhowmik
Department of Mathematics Tripura University Suryamaninagar, Tripura-799130 INDIA
E-mail: sbsmplectic@yahoo.co.in
Abstract:
Let be a topological space. In this paper we study the -closed and -closed sets in bitopological spaces in terms of -semi open sets in , also a new type of sets called -sclopen sets and -cluster point and -cluster point of an filterbase and nets of -semi open and -semi closed sets are also defined in the bitopologicall space . The different properties of the closedness so defined and the relation between them with the -sclopen set and the clusterings so defined of a filterbase and net and sets in the bitopological space are also studied.
Keyword:
Btopological space, -semi open, in the bitopologicall space , filterbase , net
The Graph Representation of Chaotic Topological Spaces and Their Metrics Space
M. El-Ghoul
Mathematics of Department, Faculty of Science, Tanta University, Tanta, Egypt
E-mail: m.mohammedelghoul@yahoo.com
E. El-Ahmady
Mathematics of Department, Faculty of Science, Tanta University, Tanta, Egypt
E-mail: a_elahmady@yoahoo.com
M. Sokar
Mathematics of Department, Faculty of Science, Tanta University, Tanta, Egypt
E-mail:ashrafsokar@yahoo.com
Abstract:
In this article, we will introduce the chaotic topological spaces, and we will represent the chaotic topological spaces by chaotic graphs and chaotic matrics. Theorems governing this representation are deduced.
Keyword:
Chaotic topological spaces, a graph.
Multi-objective Transportation of Problem with An Additional Entropy Objective Function in Fuzzy Environment
Sahidul Islam
Department of Mathematics, Bengal College of Engineering and Technology, Durgapur-713212, West Bengal, India
E-mail: sahidull@yahoo.ca
Tapan Kumar Roy
Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howarah-711103, West Bengal, India
Abstract:
In this paper, we propose multi-objective transportation problem where the cost coefficients of the objective functions, the source and destination parameters have been expressed as triangular fuzzy numbers. There is an additional entropy objective function in the multi-objective transportation problem besides transportation cost objective function. Using nearest interval approximation and total integral value of the fuzzy number, the said fuzzy problem is transformed into two types crisp multi-objective problems. Finally, the equivalent transformed problems have been solved by fuzzy programming technique (using Bellman and Zadeh¡¯s (1970) minimum and additive operator). Numberical examples have been provided to illustrate the problem.
Keyword:
Multi-objective programming, Transportation problem, Entropy, Fuzzy number, Interval programming, Fuzzy mathematical programming.
On Poincare Paradox
Ismat Beg and Samina Ashraf
Department of Mathematics and Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences, 54792-Lahore, PAKISTAN
E-mail: ibeg@lums.edu.pk
Abstract:
Using recently reformulated definition of fuzzy transitivity and fuzzy equivalence relations a solution of Poincare paradox is presented.
Keyword:
Poincare paradox, fuzzy transitivity, fuzzy equivalence.
Dynamics of Fuzzy Topological Spaces
M. El-Goul, A. E. El-Ahmady and A. M. Sokar
Mathematics of Department, Faculty of Science, Tanta University, Tanta, Egypt
E-mail: m.mohammedelghoul@yahoo.com
E-mail: a_elahmady@yoahoo.com
E-mail:ashrafsokar@yahoo.com
Abstract:
In this paper, we will introduce the basic concepts and notations of dynamical fuzzy sets throughout definitions and examples. Some algebraic structures of fuzzy sets are also presented.
Keyword:
Fuzzy topological, dynamic.
Invertible and Completely Invertible Fuzzy Topological Spaces
Sunil C. Mathew
Department of Mathematics, St. Thomas College Palai, Arunapuram P. O. -686574,Kottayam(dist.). Kerala, India
E-mail: sunilmathew@rediffmail.com
Anjaly Jose
Department of Mathematics, St. Thomas College Palai, Centre for Mathematical Science (Pala Campus) Arunapuram
P. O. , Kottayam(Dt), Kerala 686574, India
Abstract:
In this paper we study certain properties of invertible as well as completely invertible fuzzy topological spaces. We also examine the role of fuzzy points on the invertibility of fuzzy topological spaces. Further, some conditions for the invertibility of certain types of fuzzy topological spaces are investigated. In addition, we explore the situations under which a given set is not an inverting set for fuzzy topological space. We also study the effect of complete invertibility on the invertibility of the associated spaces.
Keyword:
Homeomorphism, H-fts, Invertible fts, Completely invertible fts.
Multi-item Inventory Model with Varying Unit Cost and Shortages under Imprecise and Stochastic Constraints via Geometric Programming
Debdulal Panda
Department of Engineering Science, Haldia Institute of Technology, P. O.-Hatiberia, Dist-Purba-Medinipur, W. B, Pin-721657, India.
E-mail: debdulalpanda@yahoo.co.in
Manas Kumar Maiti
Department of Mathematics, Mahishadal Raj College, Mahishadal, Dist-Purba-Medinipur, W. BPin-721628, India.
E-mail: manasmaiti@yahoo.co.in
Manoranjan Maiti
Department of Mathematics, Vidyasagar University, Midnapore, WB, Pin-721102, India.
E-mail: mmaiti2005@yahoo.co.in
Abstract:
In this paper a multi-item economic production quantity (EPQ) inventory model in an imperfect production process with constant demand, demand dependent unit production cost and shortage is developed under the restriction on storage area, total shortage cost and total inventory investment cost. Some of the inventory parameters are considered as fuzzy and some as stochastic in nature and model is formulated following cost minimization principle. Imprecise objective and constraints are transformed to equivalent deterministic ones using possibility/necessity measure of fuzzy events and stochastic chance constraint technique. Then the model is solved by geometric programming (GP) technique. In some complex cases, with the help of Genetic Algorithm (GA), fuzzy simulation process is used to determine the optimal decision. The model is illustrate through numerical examples and some sensitivity analyses have been presented.
Keyword:
Possibility/Necessity, Geometric Programming, Flexibility and Reliability, Chauce Constraint, Genetic Algorithm.
Comments on De Morgan¡¯s Law in Fuzzy Soft Sets
M. Irfan Ali and M. Shabir
Department of Mathematics Quaid-Azam University Islamabad Pakistan
E-mail: mirfanali13@yahoo.com, E-mail: mshabirbhatti@yahoo.cco.uk
Abstract;
The basic aim of this paper is to point out many errors in assertions given in a previous paper by Maji et al [P. K. Maji, R. Biswas and A. R. Roy, Fuzzy Softs Sets, The journal of Fuzzy Mathematics Vol9, No3, 2001(589-602)]. De Morgan¡¯s type laws as given in the above mentioned paper are not true in general. Some new definitions and operations on fuzzy soft sets make it very easy to prove De Morgan¡¯s type laws in fuzzy soft sets.
Keyword:
Soft sets, fuzzy sets, fuzzy soft sets, De Morgan¡¯s law.
Fuzzy Banach Algebra
Ramkrishna Thakur and S. K. Samanta
Department of Mathematics, Visva-Bharati Santiniketan-731235, W. Bengal, INDIA
E-mail;syamal_123@yahoo.co.in
Abstract:
In this paper we introduced a definition of Fuzzy Banach Algebra. Some results involving the relation between fuzzy Banach algebra and a family of crisp Banach algebra have been obtained. It is shown that the spaces of fuzzy strongly bounded linear operators on a fuzzy normed linear space forms a fuzzy Bnanch algebra.
Keyword:
Fuzzy norm, fuzzy normed algebra, fuzzy Banach algebra, fuzzy -Banach algebra, fuzzy bounded linear operator.
On Strict Fuzzy P-spaces
R. Sreekumar
Department of Mathematics, S. D. College, Alleppey (Dist) Pin 688003 KERALA (State) INDIA
E-mail: the class of p-spaces due to Arhangelskii in Hand book of Set-Theoretic Topology edited by K. Kunen and J. F. Vaughan, Elsevier Publishers(1984), pp 423-500, is in terms of a sequence of open covers in some compactification of the space rather than the space itself. In this paper we introduce the fuzzy analogue of p-spaces, strict p-spaces and prove some characterization for strict fuzzy p-spaces. Also we study the relationship between strict fuzzy p-spaces, fuzzy -spaces and fuzzy Moore space.
Keyword:
fuzzy p-spaces, fuzzy -spaces, fuzzy Moore space, fuzzy Stone C ech compactification.
A Preference Structure on Efficient Points in A Multiple Criteria Decision Making Problem-A Fuzzy Approach
K. Mohanty
Indian Institute of Management, LUCKNOW 226013, INDIA
Abstract:
In this paper a method has been introduced to obtain ranking of efficient points in a Multicriteria Decision Making (MCDM) problem under fuzzy environment. This procedure is based upon the concept of fuzzy sets and a set of efficient points of the MCDM problem. This set of efficient points is regularly generated along the efficient frontier. A particular objective is chosen and all the generated efficient points are ranked with respect to the objective. The numerical strength of preference amongst these set of efficient points with respect to that objectives derived using fuzzy sets. This procedure of efficient points ranking over all the given objectives lead to a consensus ranking, with numerical strength of preference is treated as a final ranking of the generated efficient points to the MCDM problem. This methodology is illustrate with a numerical example to highlight the procedure.
Keyword:
Multiobjective, Efficient Point, Agreement matrix, Fuzzy sets, Ranking.
Fixed Point Theorems in Intuitionistic Fuzzy Metric Spaces
S. Muralisankar
School of Mathematics, Madurai Kamaraj University, Madurai 625021, India
G. Kalpana
Department of Mathematics, Anna University Chennai, Chennai-600025, India
E-mail: kalpanagk@gmail.com; smurali@annauniv.edu
Abstract:
The purpose of this paper is to introduce the concept of R-weakly commuting mappings of type () and () in intuitionistic fuzzy metric space and prove R-weakly commuting mapping of type () is a necessary condition for the existence of a common fixed point of contractive type mappings.
Keyword:
Fuzzy set, Intuitionistic fuzzy set, Intuitionistic fuzzy metric space, R-weakly commuting mapping of type () and type ().
Common Fixed Point Theorems in Intuitionistic Fuzzy Metric Spaces for Non-compatible Mapping
S. Muralisankar
School of Mathematics, Madurai Kamaraj University, Madurai 625021, India
G. Kalpana
Department of Mathematics, Anna University Chennai, Chennai-600025, India
E-mail: kalpanagk@gmail.com; smurali@annauniv.edu
Abstract:
The aim of this paper is to introduce the notion of pointwise R-weakly commuting mapping of type () and type () in intuitionistic fuzzy metric spaces and prove a common fixed point theorem for a pair of noncompatible mapping without assuming either the completeness of the space or the continuity of the mappings involved. This theorem can be considered as an example of new class of common fixed point theorem which pertain to Lipschitz type mappings pairs. We also study this problem for a pair of mappings and establish a situation in which the common fixed point is a point of discontinuity.
Keyword:
Fuzzy set, Intuitionistic fuzzy set, Intuitionistic fuzzy metric space, R-weakly commuting mapping of type () and type (), non-compatible, property (E. A).
The Category of Fuzzy Normed Linear Space
Sadeqi and F. Solaty Kia
Department of Mathematics, Sahand University of Technology, Tabriz, Iran
Abstract:
In this paper we first show that the category of fuzzy normed linear spaces is isomorphic to a subcategory of topological vector spaces. Then we apply the results of topological vector spaces, specially the Hahn Banach and Open mapping theorems to fuzzy normed linear spaces in general.
Keyword:
Fuzzy normed linear space, topological vector space, category, hypograph functor.
Length of Maximal Chains and Number of Fuzzy Ideals in Commutative Rings
Ali Jaballah
Department of Mathematics, University of Sharjah, P. O.Box 27272, Sharjah, UAE
E-mail: ajaballah@sharjah.ac.ae
Fathi B. Saidi
Department of Mathematics and Computer Science, College of Science Kuwait University, P. O.Box 5969, Safat 13060, State of Kuwait
E-mail: fsaidi888@ku.edu.kw
Abstract:
We establish necessary and sufficient conditions for the set of fuzzy ideals of a commutative ring to be finite. We also provide characterizations for the number of fuzzy ideals and for the length of chains of these fuzzy ideals. We finish by presenting an algorithm for the calculation of the number of fuzzy ideals.
Keywords:
Commutative ring-Artinian ring-chain ring-composition series-fuzzy ideals-fuzzy generating set
Intuitionistic Fuzzy Topological Spaces and Frames
Rajesh. K. Thumbakara
Department of Mathematics, Mar Athanasius College, Kothamangalam-686666, Kerala, INDIA.
E-mail:rthumbakara@yahoo.com
Abstract:
Frame theory is the study of topology based on its open set lattice and the notion of intuitionistic fuzzy topology generalizes the notion of topology. In this paper the categorical link between frames and intuitionistic fuzzy topologies is established.
Keyword:
Frames, Intuitionistic fuzzy topology, Frame homomorphism, Fuzzy continuous.
Interval-valued Bifuzzy k-ideals of Semirings
Muhammad Akram
Punjab University College of Information Technology, University of the Punjab, Old Campus, P. O. Box 54000, Lahore, Pakistan.
E-mail: m.akram@pucit.edu.pk
Kar-Ping Shum
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, P. R. China(SAR).
E-mail: kpshum@maths.hku.hk
Abstract:
In this paper, we apply the concept of interval-valued bifuzzy sets to semirings. The notion of interval-valued -bifuzzy left k-ideals of semirings is introduced and some of their main properties are investigated. We also discuss the normalizations of such bifuzzy left k-ideals of a semiring. Finally, the relationship between the interval-valued -bifuzzy semirings and their left k-ideals is considered.
Keyword:
Interval-valued bifuzzy sets, Semirings, Normalization, t-norms, s-norms. |