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THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 23, Number 4, December 2015

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On IF Soft Oscillating Relation Sharmistha Bhattacharya Halder and Piyali Debnath

Sharmistha Bhattacharya Halder and Piyali Debnath

Dept. of Mathematics, Tripura University

Abstract: The concept of soft relation is given by Molodtsov in 1999. The aim of this paper is to introduce a new concept of soft relation based on oscillation and study some of its properties.

The concept of Soft Set has a promising potential for giving an optimal solution to decision making problems. The concepts of Soft oscillating relation may also help us in solving a number of Decision Making Problem.

Key words: Soft set, Soft relation, Fuzzy Soft relation, IF Soft proximity relation.

A Crisp and Fuzzy Goal Geometric Programming Problem (G2P2) Based on Goal Attainment Method

Payel Ghosh

Department of Mathematics, Adamas Institute of Technology, Barasat P. O. ¨CJagannathpur, Barbaria, 24 Parganas (N), West Bengal-700126, India Phone: +919477608974 E-mail: ghoshpayel86@yahoo.com

Tapan Kumar Roy

Department of Mathematics, India Institute of Engineering Science and Technology Shibpur, P. O. ¨CBotanic Garden, Howrah, West Bengal-711103, India. Phone: +919477419380, E-mail: roy_t_k@yahoo.co.in

Abstract: A goal geometric programming technique is presented here for solving multi-objective optimization problems.

We take a special type of multi-objective nonlinear programming and describe it by multi-objective goal geometric programming approach both in precise and imprecise environment which is based on goal attainment method. We use Generalized Trapezoidal fuzzy number in fuzzy goal geometric programming problem (G2P2) based on goal attainment method. This approach is illustrated with the help of two real life problems.

Key words and phrases: Goal Programming, Goal Attainment, Geometric Programming, Fuzzy Number, Generalized Fuzzy Number.

Characterizations of ¦Ã-compact Fuzzy Topological Space

N. Gowrisankar

70/232 Kollupettai Street, M. Chavady, Thanjavur-613001, Tamilnadu, INDIA. E-mail: gowrisankartnj@gmail.com

N. Rajesh

Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur-613005, Tamilnadu, INDIA. E-mail: nrajesh_topology@yahoo.co.in

Abstract: In this paper, we propose to introduce a new approach to characterize the notion of ¦Á-¦Ã-compactness in terms of ordinary nets and filters.

Key words: uzzy topological spaces, ¦Á-¦Ã-compactness, ¦Á-¦Ã-cluster point of nets and filters, ¦Á-¦Ã-adherent point, ¦Á-¦Ã-limit point.

¦Õ-contractive Mixed Monotone Mappings in Coupled Fixed Point Consideration on Fuzzy Metric Spaces with Application to Integral Equations

Bhavana Deshpande

Department of Mathematics, Govt. P. G. Arts and Science College, 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 coupled fixed point theorems for ¦Õ-contractive mixed monotone mappings on partially ordered non-Archimedean 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 an example to validate our result.

Key words: Non-Archimedean fuzzy metric spaces, Mixed monotone mapping, Coupled fixed point, Contractive condition, Integral equaltion.

Type-2 Fuzzy Linear Programming Problem with Interval-Valued Linear Programming Approach

Ritika Chopra

Department of Mathematics, University of Delhi, Delhi, India. E-mail: ritikaritsin17@gmail.com

Ratnesh R. Saxena

Department of Mathematics, Deen Dayal Upadhyay College, University of Delhi, Delhi, India. E-mail:ratnesh65@gmail.com

Abstract: In real world, the parameters of a linear programming problem are seldom known exactly due to lack of knowledge or unobtainable information. In this paper, we consider a type-2 fuzzy linear programming problem and solve it using interval valued linear programming approach. We introduce the concept of ¦Á-cut of a type-2 fuzzy set and use it to convert the type-2 fuzzy set to an interval-valued fuzzy number by weighted average approach. Finally, the type-2 FLP is converted to interval-valued FLP using the ¦Á-cuts and hence the problem is solved by finding the worst and the best optimum solution. A numerical example is given to illustrate the method.

Key words: Type-2 fuzzy linear programming, interval-valued linear programming, ¦Á-cuts, weighted average approach.

Intuitionistic Cofuzzy Graphs

R. Dhavasellan

Department of Mathematics, Sona College of Technology, Salem-5, Tamil Nadu, India. E-mail: dhavaseelan.r@gmail.com

Abstract: In this paper, we introduce the notion of an intuitionistic cofuzzy graphs and study some methods of construction of new intuitionistic cofuzzy graphs. We compute degree of a vertex, strong intuitionistic cofuzzy graphs and complete intuitionistic cofuzzy graphs. We also introduced and give properties of regular and totally regular intuitionistic cofuzzy graphs.

Key words: Intuitionistic Cofuzzy Graphs; degree of a vertex; storng intuitionistic cofuzzy graphs; complete intuitionsitic cofuzzy graphs; regular and totally regular intuitionistic cofuzzy graphs.

On Fuzzy Irreducible Ideals in BCK-algebras

Biao Long Meng

Department of Mathematics, Northwest University, Xi¡¯an, P. R. China. College of Science, Xian University of Science and Technology, Xi¡¯an, P. R. China.

Xiao Long Xin

Department of Mathematics, Northwest University, Xi¡¯an, P. R. China.

Abstract: The main aim of the present paper is to investigate relation between fuzzy prime ideals and fuzzy irreducible ideals in BCK-algebras and establish the fuzzy prime ideal theorem. We prove that every fuzzy irreducible ideal in a commutative BCK -algebra must be a fuzzy prime ideal, but the converse is not true. A fuzzy prime ideal ¦Ì in a commutative BCK -algebra is a fuzzy irreducible ideal if and only if ¦Ì(0)=1 and |Im(¦Ì)|=2. Furthermore, we establish the fuzzy prime ideal theorem and give fuzzy prime representation theorem of fuzzy ideals in a commutative BCK -algebra. Also we investigate some of important properties of fuzzy obstinate ideals, fuzzy implicative ideals, fuzzy positive implicative ideals, fuzzy prime ideals, fuzzy maximal ideals and their relations.

Key words: BCK-algebra, fuzzy ideal, fuzzy prime ideal, fuzzy irreducible ideal, fuzzy obstinate ideal, fuzzy maximal ideal.

Rough Semi-Prime Fuzzy Ideals in Rings S

Paul Isaac

Centre for Post Graduate Studies and Research Department of Mathematics Bharata Mata College Thrikkakara Kochi-682021, Kerala, India. E-mail address:pibmct@gmail.com

C. A. Neelima

Department of Mathematics and Statistics SNM College Maliankara Ernakulam (Dt.)-683516, Kerala, India. E-mail address:neelimaasokan@gmail.com

Abstract: In this paper, we shall introduce the concept of rough semi-prime fuzzy ideal and rough completely semi-prime fuzzy ideal. Also we give some properties of homomorphism and anti-homomorphism on these rough fuzzy ideals.

Key words: Rough ring, rough ideal, rough fuzzy ideal, rough semi-prime fuzzy ideal, ring homomorphism and ring anti-homomorphism.

Satisfactory Solution for Salesman Problem Based on Similarity Degree Using Intuitionistic Trapezoidal Fuzzy Number

D. Stephen Dinagar

P. G. and Research Department of Mathematics, TBML College, Porayar, India.

K. Thiripurasundari

P. G. and Research Department of Mathematics, Poompuhar College, Melaiyur, India.E-mail:ksundari_1982@yahoo.com

Abstract: In this paper, the length of the salesman problem is taken as intuitionistic trapezoidal fuzzy number and it has been proposed to give an algorithm for satisfactory solution for fuzzy salesman problem using intuitionistic trapezoidal fuzzy number where the satisfactory solution is identified with the highest similarity degree.

Key words: Intuitionistic trapezoidal fuzzy number, Ranking function, Similarity Degree, Decision Maker.

Intuitionistic Fuzzy Baire Spaces

R. Dhavaseelan

Department of Mathematics, Sona College of Technology, Salem-636005, Tamil Nadu, India. E-mail:dhavaseelan.r@gmail.com

Abstract: In this paper the concepts of intuitionistic fuzzy Baire spaces are introduced and characterizations of intuitionistic fuzzy Baire spaces are studied. Several related examples are given.

Key words: Intuitioistic fuzzy first category; intuitionistic fuzzy second category; Intuitionistic fuzzy residual set; Intuitionistic fuzzy Baire space.

Intuitionistic Fuzzy Nowhere Dense Sets

R. Dhavaseelan

Department of Mathematics, Sona College of Technology, Salem-636005, Tamil Nadu, India. Email: : dhavaseelan.r@gmail.com

E. Roja and M. K. Uma

Department of Mathematics, Sri Saradha College for Women, Salem-16, Tamil Nadu, India. Email: : ar.udhay@yahoo.co.in

Abstract: In this paper the concepts of intuitionistic fuzzy nowhere dense sets are introduced and characterizations of intuitionistic fuzzy nowhere dense sets are studied. Several related examples are given.

Key words: IIntuitionistic fuzzy dense set; intuitionistic fuzzy open hereditarily irresolvable; Intuitionistic fuzzy nowhere dense set.

Common Coupled Fixed Point Theorems for Weakly Compatible Mapping under ?-contractive Condition on Intuitionistic Fuzzy Metric Spaces with Application

Bhavana Deshpande

Department of Mathematics, Govt. P. G. Arts and Science College, Ratlam-457001 (MP), India. E-mail: bhavnadeshpande@yahoo.com

Amrish Handa

Department of Mathematics, Govt. P. G. Arts and Science College, Ratlam-457001 (MP), India. E-mail: amrishhanda83@gmail.com

Abstract: We establish a common coupled fixed point theorem for weakly compatible mappings under ?-contractive conditions on intuitionistic 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 an example to validate our result.

Key words: Coupled coincidence point, coupled fixed point, intuitionistic fuzzy metric space, weakly compatible mappings.

A Study on Determinant of Type-2 Triangular Fuzzy Matrices

D. Stephen Dinagar

P. G. and Research Department of Mathematics, TBML College, Porayar, India.

K. Latha

P. G. Department of Mathematics, Poompuhar College, Melaiyur, India. E-mail: k.latha1976@yahoo.in

Abstract: Type-2 fuzzy sets are fuzzy sets whose membership values are fuzzy sets on the interval [0,1]. This concept was proposed by Zadeh, as an extension of fuzzy sets. Type-2 fuzzy sets posess a great expressive power and are conceptually quite appealing. Also fuzzy matrices play an important role in scientific developments. In this paper, determinant of type-2 triangular fuzzy matrices (T2TFM) are defined. Also some basic properties of determinant of T2TFM are verified.

Key words: Type-2 fuzzy set, Type ¨C2 triangular fuzzy number, Type-2 triangular fuzzy matrices.

Critical Blocks in Fuzzy Graphs

N. Anjali and Sunil Mathew

National Institute of Technology, Calicut, Kerala, India.E-mail:anjalin0.0@gmail.com

Abstract: In this paper, we introduce the concept of critical blocks in fuzzy graphs, analogous to critical blocks in graphs. We obtain two necessary conditions for a critical block and a sufficient condition for a block to be critical. We also introduce block-graph of a fuzzy graph and prove a necessary condition for the block-graph when G is a tree. We also mention some of the properties of blocks.

Key words: Critical block, strongest strong cycle, block-graph.

Multivariate Fuzzy-random Error Function Based Neural Network Approximation

George A. Anastassiou

Department of Mathematical Sciences university of Memphis Memphis, TN 38152, U. S. A.E-mail:ganastss@memphis.edu

Abstract: In this article we deal with the rate of multivariate pointwise and uniform convergence in the -mean to the fuzzy-random unit operator multivariate fuzzy-random quasi-interpolation error function based neural network operators. These multivariate fuzzy-random operators arise in a natural way among multivariate fuzzy-random neural networks. The rates are given through multivariate probabilistic-Jackson type inequalities involving the multivariate fuzzy-random modulus of continuity of the engaged multivariate fuzzy-random function. The plain stochastic extreme analog of this theory is also treated for the stochastic analogous to the above operators, the stochastic kantorovich type operator and the stochastic quadrature type operator.

Key words: Fuzzy-random analysis, fuzzy-random neural networks and operators, fuzzy-random modulus of continuity, fuzzy-random functions, Jackson type and probabilistic inequalities.

Common Fixed Point Results for Generalized Compatible Pair on Modified Intuitionistic Fuzzy Metric Spaces

Bhavana Deshpande and Amrish Handa

Department of Mathematics, Govt. P. G. Arts and Science College, Ratlam-457001 (MP), India. E-mail:bhavnadeshpande@yahoo.com , amrishhanda83@gmail.com

Abstract: We establish a common fixed point theorem for generalized compatible pair F,G:X¡ÁX¡úX, without mixed monotone property of any of the mappings, on a non complete modified intuitionistic fuzzy metric space, which is not partially ordered. We also give an example to validate our result. Our results generalize some recent comparable results in the literature.

Key words: Coupled coincidence point, coupled fixed point, common fixed point, modified intuitionistic fuzzy metric space, generalized compatible mapping.

High Order Multivariate Fuzzy Approximation by Neural Network Operators Based on The Error Function

George A. Anastassiou

Department of Mathematical Sciences, University of Memphis, Memphis, TN38152, U. S. A. E-mail:ganastss@memphis.edu

Abstract: Here are studied in terms of multivariate fuzzy high approximation to the multivariate unit sequences of multivariate fuzzy error function based neural network operators. These operators are multivariate fuzzy analogs of earlier considered multivariate real ones. The derived results generalize earlier real ones into the fuzzy level. Here the high order multivariate fuzzy pointwise and uniform convergences with rates to the multivariate fuzzy unit operator are given through multivariate fuzzy Jackson type inequalities involving the multivariate fuzzy moduli of continuity of the order (m¡Ý0) -fuzzy partial derivatives, of the involved multivariate fuzzy number valued function. The treated operators are of quasi-interpolation, Kantorovich and quadrature types at the multivariate fuzzy setting.

Key words: Multivariate fuzzy real analysis, multivariate fuzzy quasi-interpolation neural network operators, multivariate fuzzy Kantorovich and Quadrature operators, ultivariate fuzzy modulus of continuity and multivariate Jackson type inequalities.

Coupled Coincidence and Common Coupled Fixed Point Theorems for A Hybrid Pair of Mappings on Partially Ordered Fuzzy Metric Space

Bhavana Deshpande and Shamim Ahmad Thoker

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

Abstract: In this paper, we establish a coupled coincidence point and common coupled fixed point theorems for a hybrid pair of multivalued and single-valued nonlinear contraction mappings in partially ordered Fuzzy metric space. Our result extend, improve, and unify some well-known results in the literature.

Key words: Coupled fixed point; coupled coincidence point; multi-valued contractive mapping; Fuzzy metric space; partially ordered set.

Theory of CESFM: A Proposal to FIFA and IFAB for A New ¡®Continuous Evaluation Fuzzy Method¡¯ of Deciding The WINNER of A Football Match That Would Have Otherwise Been Drawn or Tied after 90 Minutes of Play

Ranjit Biswas

Department of Computer Science and Engineering, Jamia Hamdard University, Hamdard Nagar, New Delhi-110062, INDIA. E-mail:ranjitbiswas@yahoo.com

Abstract: In this research work the author consider only those cases of football matches for which the penalty shootout is the existing method for determining a winner, otherwise that would have been drawn or tied after 90 minutes of play. The existing method of penalty shootout is having a lot of criticisms due to its huge demerits. In this research work the author proposes an improved method to FIFA(IFAB) to select the winner by a continuous soft-computing evaluation method. The method is called by CESFM which is basically a fuzzy constructive method. But CESFM in its theory does not exclude penalty kicks by each team as followed presently after 90 minutes of play. The result of penalty kicks is one of the many components in the method of CESFM. This proposed fuzzy method can easily extract the real dominant team out of the two tied teams using a fuzzy method can easily extract the true dominant team out of the two tied teams using a software called by CESFM -software with the help of continuous real time inputs from the Referee where some of the inputs are given by the Referee using his ¡®fuzzy pocket machine¡¯ M which is an electronic wireless machine (looking like a mobile phone). This work is in fact a revised version of the work [7]. It is claimed that if FIFA (IFAB) implement this method of in world cup football matches, it will be a true justice to the football world, to the fans, to the players, to the teams (in particular to the looser team) and consequently it will give enormous justice to the ¡®football¡¯ if considered as a subject. The Theory of CESFM is not to be applicable if the football game ends after 90 minutes of play with a straightforward result (not draw).

Key words: Fuzzy set, FIFA, IFAB, CESFM, positive parameters, negative parameters, CES-Score (CS), fuzzy pocket machine, CPS, CESFM -software.