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THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 24, Number 3, September 2016

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A Method of Generating A Consistent Fuzzy Rule Base with Less Number of Rules Using Equivalence Classes

Sharmistha Bhattacharya (Halder)

Department of Mathematics Tripura University Agartala, India, E-mail: halder_731@rediffmail.com

Shouvik Bhattacharya

Department of Mathematics Tripura University Agartala, India, E-mail: shouvik.bagla@gmail.com

Abstract: The fuzzy rule base (FRB) is the most important part of a fuzzy rule based system. Even a most advanced fuzzy rule-based system is unable to provide correct results without a proper fuzzy rule base. The aim of this paper is to develop a method to generate a consistent fuzzy rule base with fewer rules by combining individual FRBs provided by a number of experts, using equivalence relations. The results obtained have shown that using the proposed method all the inconsistencies from the combined FRB can be removed and also an FRB with considerably lesser number of rules can be generated.

Key words: Fuzzy control system, Fuzzy rule base equivalence relation, equivalence class, etc.

Multi-objective Fuzzy Probabilistic Programming Problem Involving Multi-choice Parameters

S. Acharya

School of Applied Sciences, KIIT University, BBSR. E-mail: sacharyafma@kiit.ac.in

S. Nanda

Prof. Of Eminance and Research Chair, KIIT University, BBSR, E-mail: snanda@kiit.ac.in

P. K. Rout

School of Applied Sciences, KIIT University, BBSR, E-mail: pkrout@kiitbiotech.ac.in

Abstract: The aim of this paper is to solve a multi-objective mathematical programming problem where fuzziness and randomness are observed under one umbrella. In the present mathematical problem some parameters are considered as fuzzy random variable. In first step of the solution procedure, fuzziness is removed by using alpha-cut technique to obtain multi-objective stochastic problem. By using the chance constrained technique, the multi-objective stochastic problem is transformed to equivalent crisp multi-objective mathematical problem. Then, introducing the concept of membership function, multi-objective deterministic mathematical problem is converted into single objective mathematical programming problem. Finally, it is solved with the help of existing technique. Two numerical examples are provided in order to illustrate the methodology.

Key words and phrases: Stochastic programming, Multi-objective programming, Fuzzy programming, Cauchy random variables, Optimization techniques.

An Interactive Approach for Solving Fuzzy Multiobjective Nonlinear Programming Problems

H. A. Khalifa

Department of Operations Research, Institute of Statistical Studies and Research, Cairo University, Cairo, Egypt.

Abstract: In this paper, a multiobjective nonlinear programming problem (F-MONLP) with fuzzy parameters in the objective functions is studied. These fuzzy parameters are characterized by trapezoidal numbers. The concept of ¦Á-fuzzy efficient solution is specified in which the ordinary efficient solution is extended based on the ¦Á-level of fuzzy numbers. An interactive approach for solving the ( ¦Á-MONLP) problem is proposed. The proposed approach is combined with the Attainable Reference Point (ARP) method introduced by Wang et al. (2001) and the Reference Direction (RD) method introduced by Narula et al. (1994). The stability set of the first kind corresponding to the ¦Á-efficient solution of the ¦Á-PMONLP problem is determined. A numerical example is given in the sake of this paper to clarify the obtained results.

Key words: Multiobjective nonlinear programming problems, Fuzzy parameters, Fuzzy numbers, ¦Á-pareto optimality, Attainable reference point method, Reference direction method, Stability notions.

Fuzzy Hyper BCK-commutative Ideals of Hyper BCK-algebras Muhammad Aslam Malik and Muhammad Touqeer

Muhammad Aslam Malik and Muhammad Touqeer

Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan. E-mail: malikpu@math.pu.edu.pk E-mail: touqeer-muavia@yahoo.com

Abstract: The idea of fuzzy sets is applied to (weak, strong, reflexive) hyper BCK-commutative ideals in hyper BCK-algebras. It is shown that every fuzzy (weak, strong, reflexive) hyper BCK-commutative ideal is a fuzzy (weak, strong, reflexive) hyper BCK-ideal. Relations among (fuzzy) weak hyper BCK-commutative ideals, (fuzzy) hyper BCK-commutative ideals, (fuzzy) strong hyper BCK-commutative ideals and (fuzzy) reflexive hyper BCK-commutative ideals are given. Characterization and hyper homomorphic pre-image of fuzzy (weak, strong, reflexive) hyper BCK-commutative ideals are discussed. Lastly the properties of product of fuzzy (weak, strong, reflexive) hyper BCK-commutative ideals are discussed.

Key words: Hyper BCK-algebra, (fuzzy) hyper BCK-commutative ideal, (fuzzy) weak hyper BCK-commutative ideal, (fuzzy) strong hyper BCK-commutative ideal, (fuzzy) reflexive hyper BCK-commutative ideal.

Characterizations of ¦Â-connectedness in Fuzzy Soft Topological Spaces

S. Padmapriya, M. K. Uma and E. Roja

Department of Mathematics, Sri Sarada College for Women, Salem, Tamil Nadu, India. E-mail:priyasathi17@gmail.com

Abstract: The purpose of this paper is to introduce the concepts of fuzzy soft ¦Â-regular open sets, fuzzy soft ¦Â-cloper sets and fuzzy soft ¦Â-super connectedness are introduced and studied. Some interesting properties are discussed.

Key words: Fuzzy soft ¦Â-regular open sets, fuzzy soft ¦Â-connectedness, fuzzy soft ¦Â-clopen sets, fuzzy soft ¦Â-super connectedness.

A Study on Generalized M-fuzzy Metric Space

A. Singadurai and G. Pushpalakshmi

Department of Mathematics, TDMNS College, T. Kallikulam-627117, Tamilnadu, India. E-mail:singadurai_59@yahoo.co.in

Abstract: The aim of this paper is to study some characters of the generalized M-fuzzy metric and to prove some fixed point theorem in generalized M-fuzzy metric space.

Key words: M-fuzzy metric space-Compact space-Fixed point.

Notes on Interval Valued Intuitionistic Fuzzy Rough Relations and Their Properties

Anjan Mukherjee

Department of Matheamtics, Tripura University, Agartala-799022, Tripura, India E-mail:anjan2002_m@yahoo.co.in

Ajoy Kanti Das

Department of Mathematics, ICV College, Belonia-799155, Tripura, India E-mail:ajoykantidas@gmail.com

Abstract: In this paper, we introduce interval valued intuitionistic fuzzy rough relations on a set and then we present an order on the referential set induced by an interval valued intuitionistic fuzzy rough order relation. Finally, we introduce a partially included interval valued intuitionistic fuzzy rough relation and some basic properties of this concept are studied.

Key words: Intuitionistic fuzzy rough relations, interval valued intuitionistic fuzzy rough relations, composition of interval valued intuitionistic fuzzy rough relations, interval valued intuitionistic fuzzy rough order relations.

Nonlinear Contractive Condition for Generalized Compatible Mappings in Consideration of Common Fixed Point on Fuzzy Metric Spaces

Bhavana Deshpande

Department of Mathematics Govt B. S.. P. G. College, Jaora., Dist::Ratlam-457001(M P), India. E-mail:bhavnadeshpande@yahoo.com

Suresh Chouhan

Department of Mathematics Govt. Girls¡¯ College Ratlam (M. P.) India. E-mail:s.chouhan31@gmail.com

Amrish Handa

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

Abstract: We introduce the concept of generalized compatibility for the pair {F,G} of mappings F,G:X*X¡úX in the setting of 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 theorem under nonlinear contractive condition for generalized compatible pair F,G:X*X¡úX , without mixed monotone property of any of the mappings, on a non complete fuzzy metric space, which is not partially ordered. We also give an example to validate our result. We generalize various known results.

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

On Preserving Soft g-closed Sets

S. S. Thakur

Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur (M. P.) 482011 India. E-mail:samajh_singh@rediffmail.com

S. K. Jain

Department of Applied Mathematics, Ujjain Engineering College, Ujjain (M. P.) 456010 India. E-mail:skjain63engg@gmail.com

Alpa Singh Rajput

Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur (M. P.) 482011 India. E-mail:alpasinghrajput09@gmail.com

Abstract: In this paper we extend the concepts of a-closed and a-continuous mappings due to Baker [7] in soft topological spaces and obtain several results concerning the preservation of soft g-closed sets. Furthermore we characterize soft T_1/2-spaces due to Kannan [14] in terms of soft a-continuous and soft a-closed mappings and obtain some of the basic properties and characterization of these mappings.

Key words: Soft g-closed sets, Soft g-open sets, Soft a-closed and Soft a-continuous mappings.

On Semi Separation Axioms in L-fuzzifying Topology

Mohammed. M. Khalaf

Department of Mathematics, Faculty of Science, Al Azhar University, Assiut, Egypt.

Abstract: In this paper we introduce and study semi-T_0 , semi-T_1 , semi-T_2 (Hausdorff), semi-T_3 (regularity), semi-T_4 (normality), semi-R_0 , semi-R_1 separation axioms in L-fuzzifying topology where L is a complete residuated lattice. Sometimes we add more conditions on L such as the completely distributive law or the double negation law.

Key words: Complete residuated lattice, L-fuzzifying topology, semi- T_0.

Multi-fuzzy Subgroups: An Extension of Fuzzy Subgroups

Asit Dey and Madhumangal Pal

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, India. E-mail:asitiitk@gmail.comE-mail: mmpalvu@gmail.com

Abstract: In this paper, a new kind of multi-fuzzy subgroup theory based on multi-fuzzy binary operation is presented. We propose a method to construct more general fuzzy subgroups using ordinary fuzzy subgroups as building blocks. The concepts of multi-fuzzy subgroups, abelian multi-fuzzy subgroups and normal multi-fuzzy subgroups are introduced. Finally, the fuzzy order relative to multi-fuzzy subgroups is obtained.

Key words: Multi-fuzzy vector subgroups, Abelian multi-fuzzy subgroups, Normal multi-fuzzy subgroups, Fuzzy order.

Dependent Parameters in Soft Set Theory

Asma Khalid

Center for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan. E-mail:asmakhalid4444@gmail.com

Abstract: Soft set theory is a new discipline which is more appropriate and comprehensive to deal with uncertainty. The main reason of success of this theory is the adequacy of parametrization, which lacks in other mathematical disciples. So far, it is assumed that the parameters considered in soft set theory are independent. In this paper, taking motivation from judgment aggregation and fuzzy judgment aggregation, we ignite the concept that parameters may not necessarily be independent. There may exist an implicit rule among the parameters, in which case, consistency of an intuitionistic fuzzy soft set provided by an expert needs to be considered. This paper further studies aggregation of consistent intuitionistic fuzzy soft set.

Key words: Soft sets, intuitionistic fuzzy soft sets, aggregation function, fuzzy judgment aggregation.

Existence of Coupled Coincidence Point for A Generalized Compatible Pair on Partially Ordered Modified Intuitionistic Fuzzy Metric Spaces with Applications

Bhavana Deshpande and Amrish Handa

Department of Mathematics, Govt. B. S. P. G. College, Jaora, Dist: Ratlam (M. P.) India. E-mail: bhavnadeshpande@yahoo.com E-mail: amrishhanda83@gmail.com

Abstract: We introduce the concept of generalized compatibility for the pair {F,G} , of mappings F,G:X*X¡úX in the setting of modified intuitionistic fuzzy metric space and establish a coupled coincidence result for a generalized compatible pair on partially ordered modified intuitionistic fuzzy metric spaces. We also prove coupled fixed point theorems without mixed monotone property of F . As an application of our results 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: Modified intuitionistic fuzzy metric spaces, generalized compatibility, partially ordered set, coupled coincidence point, coupled fixed point, integral equation.

Employing Generalized Compatibility to Prove Coupled Coincidence and Fixed Point Results on Fuzzy Metric Spaces with Applications

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 introduce the concept of generalized compatibility for the pair {F,G},of mappings F,G:X*X¡úX in the setting of fuzzy metric spaces and establish a coupled coincidence point result for a generalized compatible pair on partially ordered fuzzy metric spaces. We also prove some coupled fixed point theorems without mixed monotone property of F. As an application of our results we study the existence and uniqueness of the solution to a nonlinear Fredholm integral equation. We also give an example to validate our result. We generalize several known results in the existing literature.

Key words: Fuzzy metric spaces, generalized compatibility, partially ordered set, coupled coincidence point, coupled fixed point, integral equation.

Common Fixed Point Theorem for Weakly Compatible Mappings in Non-archimedean Menger PM Spaces

Rajinder Sharma

Mathematics Section College of Applied Sciences Sohar, PO Box-135, Code-311, Sohar, Sultanate of Oman. E-mail:rajind.math@gmail.com

Abstract: The aim of this paper is to prove a common fixed point theorem with different inequality condition for weakly compatible mappings in a complete non Archimedean Menger PM spaces without using the condition of continuity.

Key words: Common fixed point; Probabilistic metric space; Weakly Compatible mappings.

Somewhat Pairwise Fuzzy ¦Ã-irresolute Continuous Mappings

A. Swaminathan

Department of Mathematics (FEAT), Annamalai University,Annamalainagar, Tamil Nadu-608002, India.E-mail: asnathanway@gmail.com

Abstract: The concept of somewhat pairwise fuzzy ¦Ã-irresolute continuous mapping and somewhat pairwise fuzzy irresolute ¦Ã-open mapping have been introduced and studied. Besides, some interesting properties of those mappings are given.

Key words: Somewhat pairwise fuzzy -irresolute continuous mapping, somewhat pairwise fuzzy irresolute ¦Ã-open mapping.