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THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 24, Number 4, December 2016

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Branch and Bound Technique for Single Machine Scheduling Problem Using Type-2 Trapezoidal Fuzzy Numbers

R. Helen

PG and Research Department of Mathematics, Poompuhar College (Autonomous), Melaiyur, India. E-mail: helenranjan@gmail.com

R.Sumathi

P.G. Department of Mathematics, R.V.S.College of Engg. and Technology, Karaikal. E-mail: Jeevikaarajendran@gmail.com

Abstract: This paper deals branch and bound technique to solve single machine scheduling problem involving two processing times along with due date using Type-2 Trapezoidal fuzzy numbers. Our aim is to obtained optimal sequence of jobs and to minimize the total tardiness. The working of the algorithm has been illustrated by numerical example.

Key words: Branch and Bound Technique, Processing times P1 and P2 , Optimal sequence, Type-2 Trapezoidal fuzzy numbers.

Some Operations on Intuitionistic Fuzzy Multisets P. A. Ejegwa

P. A. Ejegwa

Department of Mathematics/Statistics/Computer Science, University of Agriculture, P.M.B. 2373, Makurdi-Nigeria. E-mail: ocholohi@gmail.com

Abstract: The concept of intuitionistic fuzzy multisets is the generalization of intuitionistic fuzzy sets. In this paper the notions of normalization, concentration and dilation were extended to intuitionistic fuzzy multisets as operations and verified to be consistent to the condition of intuitionistic fuzzy multisets. Some theorems and corollaries were deduced and proved based on the proposed notions.

Key words and phrases: intuitionistic fuzzy multisets, normalization, concentration, dilation.

Three Fixed Point Theorems on Fuzzy Metric Spaces

Nanda Ram Das

Department of Mathematics Gauhati University Guwahati 781014 Assam India. E-mail: nrd47@yahoo.co.in

Mintu Lal Saha

Department of Mathematics Handique Girls¡¯ College Guwahati 781001 Assam India. E-mail: mintulal3@rediffmail.com

Abstract: In the study of fixed point theory in fuzzy metric spaces, we very often need the condition ¡®M(x,y,t)>0 , whenever t>0¡¯, which cannot be found in Kramosil and Michaleks¡¯[14] definition. This creates problems amongst the researchers working on fixed point theory in fuzzy metric spaces. Although this condition is there in George and Veeramanis¡¯[9] definition, their definition is less general as it discards t=0 case and uses a richer continuity condition for M(x,y,¡¤). We have investigated this problem and introduced the notion of fuzzy metric spaces in a slightly different way. Our definition eliminates the above problem while retaining generality and is more general than that introduced by George and Veramani. In this paper, we state and prove three fixed point theorems in the newly introduced fuzzy metric spaces. Our results fuzzify, generalize and improve some famous fixed point theorems including those due to Boyd and Wong [1], Sehgal [19] on classical metric spaces. We also introduce the notions of sequentially convergent maps and subsequentially convergent maps in the newly introduced fuzzy metric spaces and present a generalization of the fuzzy Banach contraction theorem due to Grabiec using these notions. We further deduce some corollaries and construct examples to support our results.

Key words: Complete fuzzy metric spaces, sequentially convergent map, cluster point , fixed point.

Fuzzy concepts in Property and Object Oriented Concept Lattices

Luodan Meng and Keyun Qin

College of Mathematics, Southwest Jiao tong University, Chengdu, Sichuan 610031, China. E-mail: 1586128477@qq.com E-mail: keyunqin@263.net

Abstract: Formal concept analysis (FCA) provides a theoretical framework for learning hierarchies of knowledge clusters. This paper is devoted to the study of the fuzzy concept in property and object oriented concept lattice. Based on property/object oriented concept lattices, we propose the corresponding fuzzy relations on the universe to characterize the similarity of the objects. Furthermore, we present a kind of approximation operators to characterize the fuzzy concept and its accuracy degree in FCA. The basic properties of these operators are investigated.

Key words: Formal context, Property oriented concept lattice, Object oriented concept lattice, Fuzzy concept, Accuracy degree.

The Existence of Subgroups and New Normal Subgroups for The Group Representation of The Cayley Tree

F. H. Haydarov

National University of Uzbekistan, Tashkent, Uzbekistan. E-mail:haydarov_imc@mail.ru.

Abstract: In this paper we¡¯ll prove the existence of subgroups for the group representation of the Cayley tree and give all normal subgroups of finite index (i.e.,2s(2n+1) , s belongs to {1,2} ,n belongs to ¿Ú ) for the group representation of the Cayley tree.

Key words: G_k-group, subgroup, normal subgroup, homomorphism, epimorphism..

Uniform Strong Primeness on Fuzzy Setting

Flaulles B. Bergamaschi

Departmento do Ciencias Exatas e Tecnologicas Universidade Estadual do Sudoeste da Bahia Vitoria da Conquista, Brasil. E-mail:flaulles@yahoo.com.br

Regivan H. N. Santiago

Departamento de Informatica e Matematica Aplicada Universidade Federal do Rio Grande do Norte Natal, Brasil. E-mail:regivan@dimap.ufrn.br

Abstract: The main aim of this paper is to introduce some results discovered by Bergamaschi and Santiago about the strong and uniform strong primeness in the fuzzy environment. The study of strong primeness in fuzzy setting was initially motivated by crisp problems on ring and group-ring theory, but after a short time it became itself more interesting for instance strongly prime ideals may be defined without ??-cut dependence but compatible in a certain way, some true statements about uniform strong primeness in crisp case are not true in the fuzzy setting; the Zadeh¡¯s principle over ring¡¯s homomorphism does not preserve uniform strong primeness; the t-and m-systems may be developed to fuzzy setting.

Key words: Fuzzy prime ideal; Strong primeness; Uniform strong primeness; Noncommutative ring.

On ps-ro Fuzzy Irresolute Functions

Pankaj Chettri

Department of Mathematics, Sikkim Manipal Institute of Technology Majitar, Rangpoo, Sikkim, Pin-737136, India. E-mail:pakajct@gmail.com

Khusboo Katwal

Department of Mathematics, Sikkim Manipal Institute of Technology Majitar, Rangpoo, Sikkim, Pin-737136, India. E-mail:ongmu56wong@gmail.com

Subodh Gurung

Department of Mathematics, Sikkim Manipal Institute of Technology Majitar, Rangpoo, Sikkim, Pin-737136, India. E-mail:subodhgurungdj@live.com

Abstract: The aim of this paper is to introduce and characterize some new class of function in a fuzzy topological spaces called ps-ro fuzzy irresolute function. The interrelation of this concept with the paralleled existing allied concepts are established. It is shown that the concept of ps-ro fuzzy irresolute function is totally independent of the existing concept of both fuzzy irresolute function and ps-ro fuzzy continuous function. Also it is shown that every ps-ro fuzzy irresolute function is ps-ro fuzzy semicontinuous function but the converse is not true. Further, several characterizations of these functions are established.

Key words: ps-ro open(semiopen) fuzzy set, ps-ro fuzzy semi-nbd, ps-ro fuzzy irresolute, ps-ro fuzzy continuous(semicontinuous) function.

Fuzzy Transportation Problem with Error by Using Lagrange¡¯s Polynomial

K. L. Bondar

Department of Mathematics N. E. S Science College Nanded. E-mail:klbondar_75@rediffmail.com

Ashok S. Mhaske

Department of Mathematics Dada Patil Mahavidhyalaya, Karjat-Ahmednagar. E-mail:mhaske.math@gmail.com

Abstract: Fuzzy set theory has been used in many areas such as engineering, business, mathematics, psychology, management, medicine and image processing and pattern recognition. In operation research application many times we are faced with the problem of incompleteness uncertain data. This is due to by a lack of knowledge about the consider system or changing nature of the world. Transportation problem generally studied in operation research field which has been used to simulate different type of real life problems. In this article we consider fully fuzzy balance transportation problem by using central triangular fuzzy number. For finding the initial solution we use fuzzy matrix minima method. The main objective of this paper is to find error in minimum cost between crisp transportation problem and fuzzy transportation problem by using Lagrange¡¯s polynomial which has been fit in given data.

Key words: Fuzzy transportation problem, central triangular fuzzy number, Lagrange¡¯s interpolation polynomial.

Study on a Class of Interval Multiple Attribute Ideal Decision Method

Bao Yu-e, Guo Li and Liu Guo-feng

College of Mathematics, Inner Mongolia University fot Nationalityies Tongliao 028043, China. E-mail:byebed@163.com

Abstract: For a class of interval multiple attribute decision ideal problem whose attribute weights are completely unknown, we propose the method of determining index weights and the corresponding programs sorting method based on the distance of ideal programs. Firstly, using the distance formula between interval vectors, we give the indicators distance formula of ideal programs between each attribute interval and the positive and negative ideal intervals. Secondly, according to the principle that the program is more excellent if the distance is smaller with the positive ideal solution and is greater with the negative ideal solution, we construct a goal programming model, study the determining method of index weights about interval multi-attribute ideal decision using the Lagrangian function, give three formulas of calculating attribute weights, and establish corresponding programs sorting method based on the indicators distance of ideal programs. Then, through case analysis, we verify the rationality and effectiveness of the decision-making method.

Key words: Interval vector, the indicators distance of ideal programs, attribute weights, decision-making method.

FC 4D MI TMs in Crisp and Fuzzy Environments for Substitute and Breakable Items

Sharmistha Jana

Department of Mathematics, Midnapore College, Midnapore, India. E-mail:byebed@163.com

Barun Das

Department of Mathematics, Sidho Kanho Birsha University Purulia, West Bengal, India. E-mail:Email:bdasskbu@gmail.com

Goutam Panigrahi

Department of Mathematics, National Institute of Technology, Durgapur, India. E-mail:Email:panigrahi_goutan@rediffmail.com

Manoranjan Maiti

Department of Mathematics, Vidyasagar University, Midnapore, India. E-mail:Email: mmaity2005@tahoo.com

Abstract: Some fixed charge 4-dimensional (4D) multi-item transportation models (FC4DMITMs) are formulated and solved for substitute items in crisp and fuzzy environments. The damageability/breakability of substitute items depends on the transported amount, conveyance type and traversed distance (i.e. path). The items are substituted on manage-ment¡¯s (ad hoc quantity ) and customers¡¯(price dependent) view point. Profits of the FC4DMITPs are maximize taking transportation parameters such as demands, etc. as crisp and fuzzy and solved using possibility measures and GRG method. Their Sensi-tiveness on the degrees of substitute (DOSs) are illustrated numerically. Conventional solid (3D) and general (2D) transportation problems are derived.

Key words: 4-dimensional MITP, substitutable items, breakability, fixed charge, budget constraint.

Fixed Point Theory in Intuitionistic Fuzzy Ordered Sets for Intuitionistic Fuzzy Monotone Maps

Abdelkader Stouti

Laboratory of Mathematics and Applications, Faculty of Sciences and Techniques, University Sultan Moulay Slimane, P.O. Box 523, 23000 Beni-Mellal, MOROCCO. E-mail:stouti@yahoo.comE-mail: stout@fstbm.ac.ma

Abstract: Under suitable conditions we establish the existence of maximal, minimal, least and greatest fixed points for intuitionistic fuzzy monotone maps in intuitionistic fuzzy ordered sets. Also, we give an intuitionistic fuzzy version of Tarski¡¯s fixpoint theorem.

Key words: intuitionistic fuzzy set theory, intuitionistic fuzzy order relation, intuitionistic fuzzy monotone relation, intuitionistic fuzzy Zorn¡¯s lemma, fixed point, intuitionistic fuzzy complete lattice.

Fuzzy Ideals of Subtraction Semigroups with Respect to A t-norm and A t-conorm

Rasul Rasuli

Mathematics Department, Faculty of Science Payame Noor University (PNU), Tehran, Iran. E-mail:rasulirasul@yahoo.com

Abstract: In this paper, we introduce the concepts of fuzzy interior ideal, fuzzy bi-ideal in a subtraction semigroup with respect to a t-norm T. Also we use the t-norm and t-conorm on the intuitionistic fuzzy interior ideal and intuitionistic fuzzy bi-ideal in a subtraction semigroup and give some related properties.

Key words: Algebras, theory of groups, ideals, norms, intuitionistic mathematics, fuzzy set theory.

¿Ú-ideals in PMV-algebras Based on Fuzzy Sets

F. Forouzesh

Faculty of Mathematics and Computing Higher Education complex of Bam, Kerman, Iran. E-mail: frouzesh@bam.ac.ir

A. Borumand Saeid

Dept. of Math. Shahid Bahonar University of Kerman, Iran. E-mail:arsham@mail.uk.ac.ir

Abstract: In this paper, we introduce the notion of a fuzzy .-ideal generated by fuzzy set. We define the notion of a fuzzy .-prime ideal of a PMV-algebra and investigate some of its propreties. Also, we introduce fuzzy congruences on PMV-algebras and their quotient algebras, we prove that there exists a bijection between the set of fuzzy .-ideals and the set of fuzzy congruences of unital PMV-algebras. For each fuzzy .-ideal u, there exists an associated algebra A/u. We prove that A/u is a PMV-algebra and is isomorphic to the PMV-algebra A/u_u(0). Finally, if u is a fuzzy .-prime ideal of a unital PMV-algebra A, then A/u is a linearly ordered PMV-algebra.

Key words: Fuzzy .-ideal, fuzzy .-prime ideal, fuzzy quotient, fuzzy congruence.

Nonlinear Mixed Monotone-generalized Contractions on Partially Ordered Fuzzy metric Spaces with Application to Integral Equations

Bhavana Deshpande

Department of Mathematics, Govt. .B.S. P. G. College, Jaora, Dist: Ratlam (M. P.) India. E-mail:bhavnadeshpande@yahoo.com

Amrish Handa

Department of Mathematics Govt. P. G. Arts and Science College Ratlam (M. P.) India. E-mail:amrishhanda83@gmail.com

Abstract: We establish some coupled coincidence and coupled fixed point theorems for nonlinear mixed monotone-generalized contractions on partially ordered fuzzy metric spaces. As an application of our result, we study the existence and uniqueness of the solution to a nonlinear Fredholm integral equation. We also give examples to validate our results.

Key words: Fuzzy metric spaces, mixed monotone property, partially ordered set, coupled coincidence point, coupled fixed point, integral equation.

Interval-valued Intuitionistic Fuzzy k-ideals in Ternary Semirings

D. Krishnaswamy

Department of Mathematics, Annamalai University, Annamalainagar 608002, India. E-mail:Krishna_swamy2004@yahoo.co.in

J. Jayaraj and T. Anitha

Mathematics Wing, DDE Annamalai University, Annamalainagar 608002, India. E-mail:Joe.jayaraj@gmail.com, anitha81t@gmail.com

Abstract: In this paper we introduce the notions of interval-valued intuitionistic fuzzy right ideal, interval-valued intuitionistic fuzzy right k-ideal in ternary semirings and some of the basic properties of these ideals are investigated. We characterize regular ternary semiring through interval-valued intuitionistic fuzzy right ideal.

Key words: interval-valued fuzzy right ideal, interval-valued anti fuzzy right ideal, interval-valued intuitionistic fuzzy right ideal, interval-valued fuzzy right k-ideal, interval-valued anti fuzzy right k-ideal, interval-valued intuitionistic fuzzy right k-ideal, intrinsic product.

Common Fixed Point Results under New Condition on Modified Intuitionitic Fuzzy Metric Spaces

Bhavana Deshpande

Department of Mathematics Govt..B.S. P. G. College, Jaora, Dist: Ratlam (M.P.) India.E-mail: bhavnadeshpande@yahoo.com

Mohammad Imdad

Department of Mathematics Aligarh Muslim University Aligarh 202 002 (India).E-mail: mhimdad@yahoo.co.in

Amrish Handa

Department of Mathematics Govt. Arts & Science P. G. College Ratlam (M.P) India.E-mail: amrishhanda83@gmail.com

Abstract: We introduce the concept of generalized compatibility and generalized weakly compatibility for the pair {F,G}, of mappings F,G:X*X¡úX in the setting of modified intuitionistic fuzzy metric space and also introduce the concept of common fixed point of the mappings F,G:X*X¡úX. We establish a common fixed point result for a generalized weakly compatible pair on a noncomplete modified intuitionistic fuzzy metric space, which is not partially ordered. We also give an example to validate our result. Our result generalize some comparable results in the existing literature.

Key words: Modified intuitionistic fuzzy metric spaces, generalized compatibility, generalized weakly compatibility, coupled coincidence point, common fixed point.

A Fuzzy Linear Fractional Programming Problem with Fuzzy Homogeneous Constraints in Triangular Fuzzy Numbers

S. Sekar

Department of Mathematics Govt..B.S. P. G. College, Jaora, Dist: Ratlam (M.P.) India.E-mail: bhavnadeshpande@yahoo.com

Mohammad Imdad

Assistant Professor, Department of Mathematics, Government Arts college (Autonomous), Salem, Tamil Nadu, India.E-mail: ssekar_psg@yahoo.com, sekar_nitt@rediffmail.com

S. Mohan

Assistant Professor of Mathematics, A. V. C. College( Autonomous), Mannampandal, Mayiladuthurai, Tamil Nadu, India.E-mail:mohanmyl@rediffmail.com

Abstract: In this paper, we propose a new method for solving a Fuzzy Linear Fractional Programming Problem (FLFPP) when some of its constraints are fuzzy homogeneous in triangular fuzzy numbers. Using these fuzzy homogeneous constraints a fuzzy transformation matrix T is constructed. This T transforms the given problem in to another FLFPP with fewer fuzzy constraints. A relationship between these two problems, which ensure that the solution of the original problem can be recovered from the solution of the transformed problem. A simple numerical example explains the procedure of the proposed method.

Key words: Triangular fuzzy numbers. Fuzzy homogeneous constraints. Fuzzy transformation matrix. Fuzzy linear fractional programming problem.

Interval-valued Intuitionisitc Fuzzy R-subgroup of Near-rings

Debashree Manna

Department of Mathematics, Damda Jr. High School, Purulia, India.E-mail: dmanna66@gmail.com

Amal Kumar Adak

Department of Mathematics with Oceanology and Computer Programming, Vidyasagar university, Midnapore-721102, India.E-mail:amaladak17@gmail.com

Abstract: Interval-valued intuitionistic fuzzy sets are generalization of both interval-valued fuzzy sets and intuitionistic fuzzy sets. In this paper, we apply the concept of interval-valued intuitionistic fuzzy sets to R-subgroup of near-ring R. The notion of interval-valued intuitionistic fuzzy subnear-ring is introduced and some interesting properties are discussed. Some relations on the family of all interval-valued intuitionistic fuzzy subnear-ring are presented and investigate some related properties.

Key words: Interval-valued intuitionistic fuzzy set, interval-valued intuitionistic fuzzy R-subgroup, interval-valued intuitionistic fuzzy subnear-ring, upper and lower inter-val level set.

Fuzzy Subrings over A t-norm

Rasul Rasuli

Mathematics Department, Faculty of Science , Payame Noor University (PNU), Tehran, Iran.E-mail: rasulirasul@yahoo.com

Abstract: The aim of this paper is to introduce the concepts of T-fuzzy subrings by using a t-norm T. We obtain characterization of quotient subrings over a t-norm. Also we use a t-norm T in the finite direct product of subrings.

Key words: Ring theory, norms, fuzzy set theory, direct products.