Comparing Fuzzy Graphs

Size: px
Start display at page:

Download "Comparing Fuzzy Graphs"

Transcription

1 Comparing Fuzzy Graphs Michael R. Berthold! and Klaus-Peter Huber 2 1 Department of EECS Computer Science Division University of California at Berkeley Berkeley, CA 94720, USA bertholdclca. berkeley. edu 2 IRF (Prof. D. Schmid) Fakultiit fur Informatik U niversitat Karlsruhe P.O. Box 6980, Karlsruhe kphuberclira.ulta.de Abstract. This paper investigates techniques to compare two fuzzy graphs. Fuzzy graphs have raised increasing attention, because they not only allow an easy interpretation but in addition finding regions of interest or tolerating missing attributes can be done computationally very efficiently. This is of great interest in numerous applications, for example Intelligent Data Analysis, Metamodeling and others. In this paper a computationally efficient way to compare two fuzzy graphs is presented. 1 Introduction Fuzzy graphs [9J have recently raised increasing attention, especially in areas where understandable models are needed. In contrast to grid-based approaches fuzzy graphs allow to represent also complex relationships using only few elements, so-called fuzzy points. Since those models are often built from large amounts of data, a manual analysis is not feasible anymore. Also for applications where data is recorded online there is a need for continuous, automatic analysis. Recently an algorithm was proposed [1,41 that automatically builds a fuzzy graph from example data. Several properties of this approach make it well-suited for the automatic analysis of data. The usage of fuzzy concepts allows to hide information the user is not interested in (in/ormation granulation) and also offers an easy way to interpret the resulting fuzzy function in terms of fuzzy points or understandable if- then-rules. Also several operations on these fuzzy graphs can be done in computationally very efficient ways. In [2] for example, it was shown how missing values can be tolerated with almost no computational cost in a fuzzy environment. 234

2 In real applications it is very desirable to use the generated models for further analysis. For example finding extrema of the model function can be useful to determine parameter settings that should be avoided. Also the comparison of different models is of interest, to either judge the influence of modifications or simply the quality of the generated models itself. In theory the analysis of the models does not cause problems, but finding ways for the actual implementation is crucial for real applications. In the following it is discussed how fuzzy graphs can be compared to each other in a computationally efficient way. 2 Fuzzy Graphs In the following the concept of fuzzy graphs will be used to represent a system to approximate one output variable y depending on n input variables x, (1 ::; i ::; n). The typical fuzzy graph builds upon fuzzy points of the type if XI is Al and... and Xn is An then y is B where some of the input membership functions A, can be constant; that is, /-la, == 1. Thus different fuzzy points may only depend on a small subset of input variables. Simplifying this equation, using A = Al X. x An (x denotes the cartesian product), leads to: if ~ is A then y is B which 'can also be expressed as a fuzzy constraint on a joint variable (~,y); that is, (~,y) is A x B The membership function of Ax B is given using II as a conjunction operator which is usually defined as the minimum: or, more precisely I-'AxB(~' y) = J-IAI (xd /\... /\ PAn(Xn) /\ J-IB(Y) = min {J-IAI (XI)" ", PAn (Xn), /-lb{y)} Zadeh [8J uses the term fuzzy point to denote A x B. Often trapezoidal fuzzy points are used, which can be specified through the comers of their support- and core-region: < a, b, c, d >. The core-rectangle is defined through the lower left corner b and the upper right comer c. Similar the support-region is defined through a and d. 235

3 Figure 1: Intersection of a fuzzy graph with the cylindrical extension of A.. A collection of fuzzy points can now be regarded as forming a superposition of n fuzzy points: (x, y) is (AI X Bl Am X Bm) where + denotes the disjunction operator (usually defined as maximum). Zadeh calls this characterization of a dependency Fuzzy Graph, because the collection of rules can be seen as a coarse representation of a functional dependency f of y on x. This fuzzy graph f is thus defined through: m f = LAj x Bj ;=1 In the following we will concentrate on Fuzzy Graphs for the approximate representation of functions. The task of interpolation; that is, deriving a linguistic value B for y given an arbitrary linguistic value A for x and a fuzzy graph f: x is A f is Ei=IAj x Bj y is B This results in an intersection of the fuzzy graph f with a cylindrical extension of the input fuzzy set A. Figure 1 shows an example. The corresponding functional dependency can be computed through: J.LB ==f(a)(y) = sup{j.l/(x, y) /\ J.LA(X)} z = sup {(J.LAt XBI (x, y) V... V J.LA.xB. (x, V)) /\ J.LA(X)}., Note that individual rules do not depend on a common set of membership functions on the input variables because each fuzzy point is described through an individual set of membership functions. Therefore each fuzzy point can closely adapt to the underlying model function, making the resulting fuzzy graphs a very compact description, especially 236

4 compared to the approaches that use a "global grid", defined through only one set of membership functions for each input variable [3,5-7]. An algorithm that automatically constructs this kind of fuzzy graphs based on trapezoidal fuzzy points from example data was proposed in [1]. 3 Similarity of Fuzzy Graphs In technical applications fuzzy graphs may be used for model analysis or process control tasks. Often the generated fuzzy graph or fuzzy model is only validated or tested by some data sets using cross-validation, but this approach does not deliver insights into the behaviour. Therefore there is a need for structural methods to describe, compare and analyze several different fuzzy graphs, that may originate from different experiments or datasets. Here a way is presented to compare two fuzzy graphs. Comparing two fuzzy graphs should result in a degree of match. In most cases, however, it is also of interest if and how much the two graphs overlap. Therefore three different measurements are defined to compare two fuzzy graphs 11 and 12, based on their volume vol(j) := f 1l,{X,y)dxdy: - similarity: - overlap: - coverage: 2,vo/(11 nh) 8(11, h) := vo/(jd + vol(h) 0(1!:).= vol(fi n h) h 2 vol (h).= vol(fl n h) C(f J) h 2 vol(ft) where obviously O(Jb h) = C(h. 11); that is, if h overlaps 12, 12 is covered by It. Methods to efficiently compute the volume of one fuzzy point exist when trapezoidal membership functions are used, as is the case for the type of fuzzy graphs considered here: To compute the volume of the entire fuzzy graph the volumes of all fuzzy points are added up, ignoring possible overlaps. This is a similar approach to the Center of Gravity 237

5 Defuzzification, because regions of overlap are weighted proportional to the amount of overlap. The remaining problem, that is, the efficient computation of the intersection of two fuzzy graphs is relatively complicated, because the resulting fuzzy graph consists of fuzzy points that are not necessarily normalized or even trapezoidal. But for comparison of two fuzzy graphs it is sufficient to compute the volume of the intersection of both: For practical applications it is neccesary to compute this volume efficiently. For the precise calculation no efficient method is known, but a simple approximation can be used to compute the volume of the intersection of two fuzzy points, based on a granularization in several alpha-cuts. The alpha-cut for two intersecting fuzzy points p =< a, b, e, d > and p' =< a', b', c', d' > is equal to the rectangle: na(p,p') = [max{a + a(b - a), a' + a(b' - a ' )}, max{d - a(d - e), d' + a(d' - c')}] The correct solution for the volume of the intersection can be expressed as: 1/<-1 1 ) vol(p np') = lim L f (n(k+l)<(p,p') + -2 (n(k+l)'(p,p') - nk,(p,p'» <-+0 "",0 where the volume of the intersection is divided into l/f slices, whose volume is then added together. The correct solution is achieved by letting f go towards zero. For practical purposes a finite f can be chosen, depending on the desired accuracy. A nice property of.this approach is the ability to compute an upper bound for the resulting error, depending on the granularization f: 1/,-1 err.(pnp')::: L f (n(k+~).(p,p') - nk«p,p')) "=0 which approaches zero for f going towards zero, as should be expected. This approximation can then be used to compare two fuzzy graphs by adding up the intersection of all pairwise combinations of fuzzy points. Again, as in the computation of the volume of a fuzzy graph, areas of overlap are included multiple times. This methodology allows to compute measurements to compare two fuzzy graphs based on trapezoidal fuzzy points computationally efficient with an additional indicator of the maximum error of the used approximation. 238

6 4 Conclusions Since fuzzy graphs can be constructed from data and allow a compact representation of functions a structure based analysis is needed. Here, it was discussed how fuzzy graphs can be compared in a computationally efficient way. In addition to tolerating missing values which was presented in previous work, it is also possible to compute the volume of the. intersection of two fuzzy graphs efficiently when both graphs are based on trapezoidal fuzzy points. This allows to compute three different measures to compare two fuzzy graphs, namely similarity, overlap, and coverage. Acknowledgments M. Berthold was supported by a stipend of the "Gemeinsame Hochschulsonderprogramm III von Bund und Liindern" through the DAAD. Literature [1] Michael R. Berthold and Klaus-Peter Huber. Building fuzzy graphs from examples. In IEEE International Conference on Fuzzy Systems, 1, pages , September [2) Michael R. Berthold and Klaus-Peter Huber. Missing values and learning of fuzzy rules. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, [3] Charles M. Higgins and Rodney M. Goodman. Learning fuzzy rule-based neural networks for control. In Advances in Neural Information Processing Systems, 5, pages , California, Morgan Kaufmann. [4] Klaus-Peter Huber and Michael R. Berthold. Simulation data analysis using fuzzy graphs. In Xiaohui Liu, Paul Cohen, and Michael Berthold, editors, Advances in Intelligent Data Analysis - Reasoning about Data, LNCSI280, pages Springer Verlag, [5] Patrick K. Simpson. Fuzzy min-max neural networks - part 1: Classification. IEEE 1ransactions on Neural Networks, 3(5): , September [6J Patrick K. Simpson. Fuzzy min-max neural networks - part 2: Clustering. IEEE 7ransactions on Fuzzy Systems, 1(1):32-45, January [7J Li-Xin Wang and Jerry M. Mendel. Generating rules by learning from examples. In International Symposium on Intelligent Control, pages IEEE,

7 [8] Lotfi A. Zadeh. Soft computing and fuzzy logic. IEEE Software, pages 48~ 56, November [9] Lotfi A. Zadeh. Fuzzy logic and the calculi of fuzzy rules and fuzzy graphs: A precis. Multi. Val. Logic, 1:1-38,

MISSING VALUES AND LEARNING OF FUZZY RULES MICHAEL R. BERTHOLD. Computer Science Division, Department of EECS

MISSING VALUES AND LEARNING OF FUZZY RULES MICHAEL R. BERTHOLD. Computer Science Division, Department of EECS MISSING VALUES AND LEARNING OF FUZZY RULES MICHAEL R. BERTHOLD Computer Science Division, Department of EECS University of California, Bereley, CA 94720, USA email: Michael.Berthold@Informati.Uni-Karlsruhe.DE

More information

What is all the Fuzz about?

What is all the Fuzz about? What is all the Fuzz about? Fuzzy Systems CPSC 433 Christian Jacob Dept. of Computer Science Dept. of Biochemistry & Molecular Biology University of Calgary Fuzzy Systems in Knowledge Engineering Fuzzy

More information

Constructing fuzzy graphs from examples

Constructing fuzzy graphs from examples Constructing fuzzy graphs from examples Michael R. Berthold a, *, Klaus-Peter Huber b,1 a Berkeley Initiative in Soft Computing, Computer Science Division, Department of EECS, 329 Soda Hall, University

More information

What is all the Fuzz about?

What is all the Fuzz about? What is all the Fuzz about? Fuzzy Systems: Introduction CPSC 533 Christian Jacob Dept. of Computer Science Dept. of Biochemistry & Molecular Biology University of Calgary Fuzzy Systems in Knowledge Engineering

More information

Fuzzy Systems. Fuzzy Systems in Knowledge Engineering. Chapter 4. Christian Jacob. 4. Fuzzy Systems. Fuzzy Systems in Knowledge Engineering

Fuzzy Systems. Fuzzy Systems in Knowledge Engineering. Chapter 4. Christian Jacob. 4. Fuzzy Systems. Fuzzy Systems in Knowledge Engineering Chapter 4 Fuzzy Systems Knowledge Engeerg Fuzzy Systems Christian Jacob jacob@cpsc.ucalgary.ca Department of Computer Science University of Calgary [Kasabov, 1996] Fuzzy Systems Knowledge Engeerg [Kasabov,

More information

Learning Fuzzy Rule-Based Neural Networks for Control

Learning Fuzzy Rule-Based Neural Networks for Control Learning Fuzzy Rule-Based Neural Networks for Control Charles M. Higgins and Rodney M. Goodman Department of Electrical Engineering, 116-81 California Institute of Technology Pasadena, CA 91125 Abstract

More information

Networks for Control. California Institute of Technology. Pasadena, CA Abstract

Networks for Control. California Institute of Technology. Pasadena, CA Abstract Learning Fuzzy Rule-Based Neural Networks for Control Charles M. Higgins and Rodney M. Goodman Department of Electrical Engineering, 116-81 California Institute of Technology Pasadena, CA 91125 Abstract

More information

FUZZY BOOLEAN ALGEBRAS AND LUKASIEWICZ LOGIC. Angel Garrido

FUZZY BOOLEAN ALGEBRAS AND LUKASIEWICZ LOGIC. Angel Garrido Acta Universitatis Apulensis ISSN: 1582-5329 No. 22/2010 pp. 101-111 FUZZY BOOLEAN ALGEBRAS AND LUKASIEWICZ LOGIC Angel Garrido Abstract. In this paper, we analyze the more adequate tools to solve many

More information

Lecture notes. Com Page 1

Lecture notes. Com Page 1 Lecture notes Com Page 1 Contents Lectures 1. Introduction to Computational Intelligence 2. Traditional computation 2.1. Sorting algorithms 2.2. Graph search algorithms 3. Supervised neural computation

More information

Rough Sets, Neighborhood Systems, and Granular Computing

Rough Sets, Neighborhood Systems, and Granular Computing Rough Sets, Neighborhood Systems, and Granular Computing Y.Y. Yao Department of Computer Science University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: yyao@cs.uregina.ca Abstract Granulation

More information

Classification with Diffuse or Incomplete Information

Classification with Diffuse or Incomplete Information Classification with Diffuse or Incomplete Information AMAURY CABALLERO, KANG YEN Florida International University Abstract. In many different fields like finance, business, pattern recognition, communication

More information

Review of Fuzzy Logical Database Models

Review of Fuzzy Logical Database Models IOSR Journal of Computer Engineering (IOSRJCE) ISSN: 2278-0661, ISBN: 2278-8727Volume 8, Issue 4 (Jan. - Feb. 2013), PP 24-30 Review of Fuzzy Logical Database Models Anupriya 1, Prof. Rahul Rishi 2 1 (Department

More information

Some Properties of Intuitionistic. (T, S)-Fuzzy Filters on. Lattice Implication Algebras

Some Properties of Intuitionistic. (T, S)-Fuzzy Filters on. Lattice Implication Algebras Theoretical Mathematics & Applications, vol.3, no.2, 2013, 79-89 ISSN: 1792-9687 (print), 1792-9709 (online) Scienpress Ltd, 2013 Some Properties of Intuitionistic (T, S)-Fuzzy Filters on Lattice Implication

More information

Chapter 4 Fuzzy Logic

Chapter 4 Fuzzy Logic 4.1 Introduction Chapter 4 Fuzzy Logic The human brain interprets the sensory information provided by organs. Fuzzy set theory focus on processing the information. Numerical computation can be performed

More information

MODELING FOR RESIDUAL STRESS, SURFACE ROUGHNESS AND TOOL WEAR USING AN ADAPTIVE NEURO FUZZY INFERENCE SYSTEM

MODELING FOR RESIDUAL STRESS, SURFACE ROUGHNESS AND TOOL WEAR USING AN ADAPTIVE NEURO FUZZY INFERENCE SYSTEM CHAPTER-7 MODELING FOR RESIDUAL STRESS, SURFACE ROUGHNESS AND TOOL WEAR USING AN ADAPTIVE NEURO FUZZY INFERENCE SYSTEM 7.1 Introduction To improve the overall efficiency of turning, it is necessary to

More information

Fuzzy Reasoning. Outline

Fuzzy Reasoning. Outline Fuzzy Reasoning Outline Introduction Bivalent & Multivalent Logics Fundamental fuzzy concepts Fuzzification Defuzzification Fuzzy Expert System Neuro-fuzzy System Introduction Fuzzy concept first introduced

More information

Introduction 2 Fuzzy Sets & Fuzzy Rules. Aleksandar Rakić Contents

Introduction 2 Fuzzy Sets & Fuzzy Rules. Aleksandar Rakić Contents Beograd ETF Fuzzy logic Introduction 2 Fuzzy Sets & Fuzzy Rules Aleksandar Rakić rakic@etf.rs Contents Characteristics of Fuzzy Sets Operations Properties Fuzzy Rules Examples 2 1 Characteristics of Fuzzy

More information

FUZZY INFERENCE SYSTEMS

FUZZY INFERENCE SYSTEMS CHAPTER-IV FUZZY INFERENCE SYSTEMS Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy logic. The mapping then provides a basis from which decisions can

More information

FUZZY INFERENCE. Siti Zaiton Mohd Hashim, PhD

FUZZY INFERENCE. Siti Zaiton Mohd Hashim, PhD FUZZY INFERENCE Siti Zaiton Mohd Hashim, PhD Fuzzy Inference Introduction Mamdani-style inference Sugeno-style inference Building a fuzzy expert system 9/29/20 2 Introduction Fuzzy inference is the process

More information

International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February ISSN

International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February ISSN International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February-2016 149 KEY PROPERTIES OF HESITANT FUZZY SOFT TOPOLOGICAL SPACES ASREEDEVI, DRNRAVI SHANKAR Abstract In this paper,

More information

FUNDAMENTALS OF FUZZY SETS

FUNDAMENTALS OF FUZZY SETS FUNDAMENTALS OF FUZZY SETS edited by Didier Dubois and Henri Prade IRIT, CNRS & University of Toulouse III Foreword by LotfiA. Zadeh 14 Kluwer Academic Publishers Boston//London/Dordrecht Contents Foreword

More information

Chapter 7 Fuzzy Logic Controller

Chapter 7 Fuzzy Logic Controller Chapter 7 Fuzzy Logic Controller 7.1 Objective The objective of this section is to present the output of the system considered with a fuzzy logic controller to tune the firing angle of the SCRs present

More information

Similarity Measures of Pentagonal Fuzzy Numbers

Similarity Measures of Pentagonal Fuzzy Numbers Volume 119 No. 9 2018, 165-175 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Similarity Measures of Pentagonal Fuzzy Numbers T. Pathinathan 1 and

More information

A Brief Idea on Fuzzy and Crisp Sets

A Brief Idea on Fuzzy and Crisp Sets International OPEN ACCESS Journal Of Modern Engineering Research (IJMER) A Brief Idea on Fuzzy and Crisp Sets Rednam SS Jyothi 1, Eswar Patnala 2, K.Asish Vardhan 3 (Asst.Prof(c),Information Technology,

More information

The Travelling Salesman Problem. in Fuzzy Membership Functions 1. Abstract

The Travelling Salesman Problem. in Fuzzy Membership Functions 1. Abstract Chapter 7 The Travelling Salesman Problem in Fuzzy Membership Functions 1 Abstract In this chapter, the fuzzification of travelling salesman problem in the way of trapezoidal fuzzy membership functions

More information

CHAPTER 4 FREQUENCY STABILIZATION USING FUZZY LOGIC CONTROLLER

CHAPTER 4 FREQUENCY STABILIZATION USING FUZZY LOGIC CONTROLLER 60 CHAPTER 4 FREQUENCY STABILIZATION USING FUZZY LOGIC CONTROLLER 4.1 INTRODUCTION Problems in the real world quite often turn out to be complex owing to an element of uncertainty either in the parameters

More information

Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers

Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers Advances in Fuzzy Mathematics ISSN 973-533X Volume, Number 3 (7, pp 75-735 Research India Publications http://wwwripublicationcom Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers

More information

ANFIS: ADAPTIVE-NETWORK-BASED FUZZY INFERENCE SYSTEMS (J.S.R. Jang 1993,1995) bell x; a, b, c = 1 a

ANFIS: ADAPTIVE-NETWORK-BASED FUZZY INFERENCE SYSTEMS (J.S.R. Jang 1993,1995) bell x; a, b, c = 1 a ANFIS: ADAPTIVE-NETWORK-ASED FUZZ INFERENCE SSTEMS (J.S.R. Jang 993,995) Membership Functions triangular triangle( ; a, a b, c c) ma min = b a, c b, 0, trapezoidal trapezoid( ; a, b, a c, d d) ma min =

More information

FUZZY SPECIFICATION IN SOFTWARE ENGINEERING

FUZZY SPECIFICATION IN SOFTWARE ENGINEERING 1 FUZZY SPECIFICATION IN SOFTWARE ENGINEERING V. LOPEZ Faculty of Informatics, Complutense University Madrid, Spain E-mail: ab vlopez@fdi.ucm.es www.fdi.ucm.es J. MONTERO Faculty of Mathematics, Complutense

More information

fuzzylite a fuzzy logic control library in C++

fuzzylite a fuzzy logic control library in C++ fuzzylite a fuzzy logic control library in C++ Juan Rada-Vilela jcrada@fuzzylite.com Abstract Fuzzy Logic Controllers (FLCs) are software components found nowadays within well-known home appliances such

More information

Fuzzy Systems (1/2) Francesco Masulli

Fuzzy Systems (1/2) Francesco Masulli (1/2) Francesco Masulli DIBRIS - University of Genova, ITALY & S.H.R.O. - Sbarro Institute for Cancer Research and Molecular Medicine Temple University, Philadelphia, PA, USA email: francesco.masulli@unige.it

More information

AN APPROXIMATION APPROACH FOR RANKING FUZZY NUMBERS BASED ON WEIGHTED INTERVAL - VALUE 1.INTRODUCTION

AN APPROXIMATION APPROACH FOR RANKING FUZZY NUMBERS BASED ON WEIGHTED INTERVAL - VALUE 1.INTRODUCTION Mathematical and Computational Applications, Vol. 16, No. 3, pp. 588-597, 2011. Association for Scientific Research AN APPROXIMATION APPROACH FOR RANKING FUZZY NUMBERS BASED ON WEIGHTED INTERVAL - VALUE

More information

American International Journal of Research in Science, Technology, Engineering & Mathematics

American International Journal of Research in Science, Technology, Engineering & Mathematics American International Journal of Research in Science, Technology, Engineering & Mathematics Available online at http://www.iasir.net ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 2328-3629

More information

Integration of Fuzzy Shannon s Entropy with fuzzy TOPSIS for industrial robotic system selection

Integration of Fuzzy Shannon s Entropy with fuzzy TOPSIS for industrial robotic system selection JIEM, 2012 5(1):102-114 Online ISSN: 2013-0953 Print ISSN: 2013-8423 http://dx.doi.org/10.3926/jiem.397 Integration of Fuzzy Shannon s Entropy with fuzzy TOPSIS for industrial robotic system selection

More information

A New Method For Forecasting Enrolments Combining Time-Variant Fuzzy Logical Relationship Groups And K-Means Clustering

A New Method For Forecasting Enrolments Combining Time-Variant Fuzzy Logical Relationship Groups And K-Means Clustering A New Method For Forecasting Enrolments Combining Time-Variant Fuzzy Logical Relationship Groups And K-Means Clustering Nghiem Van Tinh 1, Vu Viet Vu 1, Tran Thi Ngoc Linh 1 1 Thai Nguyen University of

More information

Available online at ScienceDirect. Procedia Computer Science 96 (2016 )

Available online at   ScienceDirect. Procedia Computer Science 96 (2016 ) Available online at www.sciencedirect.com ScienceDirect Procedia Computer Science 96 (2016 ) 179 186 20th International Conference on Knowledge Based and Intelligent Information and Engineering Systems,

More information

Fuzzy Transportation Problems with New Kind of Ranking Function

Fuzzy Transportation Problems with New Kind of Ranking Function The International Journal of Engineering and Science (IJES) Volume 6 Issue 11 Pages PP 15-19 2017 ISSN (e): 2319 1813 ISSN (p): 2319 1805 Fuzzy Transportation Problems with New Kind of Ranking Function

More information

Simultaneous Perturbation Stochastic Approximation Algorithm Combined with Neural Network and Fuzzy Simulation

Simultaneous Perturbation Stochastic Approximation Algorithm Combined with Neural Network and Fuzzy Simulation .--- Simultaneous Perturbation Stochastic Approximation Algorithm Combined with Neural Networ and Fuzzy Simulation Abstract - - - - Keywords: Many optimization problems contain fuzzy information. Possibility

More information

Granular Computing: A Paradigm in Information Processing Saroj K. Meher Center for Soft Computing Research Indian Statistical Institute, Kolkata

Granular Computing: A Paradigm in Information Processing Saroj K. Meher Center for Soft Computing Research Indian Statistical Institute, Kolkata Granular Computing: A Paradigm in Information Processing Saroj K. Meher Center for Soft Computing Research Indian Statistical Institute, Kolkata Granular computing (GrC): Outline Introduction Definitions

More information

Neural Networks Lesson 9 - Fuzzy Logic

Neural Networks Lesson 9 - Fuzzy Logic Neural Networks Lesson 9 - Prof. Michele Scarpiniti INFOCOM Dpt. - Sapienza University of Rome http://ispac.ing.uniroma1.it/scarpiniti/index.htm michele.scarpiniti@uniroma1.it Rome, 26 November 2009 M.

More information

Using Decision Boundary to Analyze Classifiers

Using Decision Boundary to Analyze Classifiers Using Decision Boundary to Analyze Classifiers Zhiyong Yan Congfu Xu College of Computer Science, Zhejiang University, Hangzhou, China yanzhiyong@zju.edu.cn Abstract In this paper we propose to use decision

More information

A Logic Language of Granular Computing

A Logic Language of Granular Computing A Logic Language of Granular Computing Yiyu Yao and Bing Zhou Department of Computer Science University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: {yyao, zhou200b}@cs.uregina.ca Abstract Granular

More information

Modeling the Real World for Data Mining: Granular Computing Approach

Modeling the Real World for Data Mining: Granular Computing Approach Modeling the Real World for Data Mining: Granular Computing Approach T. Y. Lin Department of Mathematics and Computer Science San Jose State University San Jose California 95192-0103 and Berkeley Initiative

More information

Genetic Tuning for Improving Wang and Mendel s Fuzzy Database

Genetic Tuning for Improving Wang and Mendel s Fuzzy Database Proceedings of the 2009 IEEE International Conference on Systems, Man, and Cybernetics San Antonio, TX, USA - October 2009 Genetic Tuning for Improving Wang and Mendel s Fuzzy Database E. R. R. Kato, O.

More information

Assessment of Human Skills Using Trapezoidal Fuzzy Numbers (Part II)

Assessment of Human Skills Using Trapezoidal Fuzzy Numbers (Part II) American Journal of Computational and Applied Mathematics 05, 5(5): 54-58 DOI: 0.593/j.ajcam.050505.04 Assessment of Human Skills Using Trapezoidal Fuzzy Numbers (Part II) Michael Gr. Voskoglou Department

More information

Chapter 2: FUZZY SETS

Chapter 2: FUZZY SETS Ch.2: Fuzzy sets 1 Chapter 2: FUZZY SETS Introduction (2.1) Basic Definitions &Terminology (2.2) Set-theoretic Operations (2.3) Membership Function (MF) Formulation & Parameterization (2.4) Complement

More information

IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 1 Issue 3, May

IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 1 Issue 3, May Optimization of fuzzy assignment model with triangular fuzzy numbers using Robust Ranking technique Dr. K. Kalaiarasi 1,Prof. S.Sindhu 2, Dr. M. Arunadevi 3 1 Associate Professor Dept. of Mathematics 2

More information

Application of fuzzy set theory in image analysis. Nataša Sladoje Centre for Image Analysis

Application of fuzzy set theory in image analysis. Nataša Sladoje Centre for Image Analysis Application of fuzzy set theory in image analysis Nataša Sladoje Centre for Image Analysis Our topics for today Crisp vs fuzzy Fuzzy sets and fuzzy membership functions Fuzzy set operators Approximate

More information

Subsampling Conflicts to Construct Better Fuzzy Rules

Subsampling Conflicts to Construct Better Fuzzy Rules Subsampling Conflicts to Construct Better Fuzzy Rules Michael R. Berthold Tripos, Inc. 601 Gateway Blvd., Suite 720 South San Francisco, CA 94080, USA email: berthold@tripos. com Abstract Many fuzzy rule

More information

Rough Approximations under Level Fuzzy Sets

Rough Approximations under Level Fuzzy Sets Rough Approximations under Level Fuzzy Sets W.-N. Liu J.T. Yao Y.Y.Yao Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: [liuwe200, jtyao, yyao]@cs.uregina.ca

More information

Introduction to Intelligent Control Part 3

Introduction to Intelligent Control Part 3 ECE 4951 - Spring 2010 Introduction to Part 3 Prof. Marian S. Stachowicz Laboratory for Intelligent Systems ECE Department, University of Minnesota Duluth January 26-29, 2010 Part 1: Outline TYPES OF UNCERTAINTY

More information

RESULTS ON HESITANT FUZZY SOFT TOPOLOGICAL SPACES

RESULTS ON HESITANT FUZZY SOFT TOPOLOGICAL SPACES ISSN 2320-9143 1 International Journal of Advance Research, IJOAR.org Volume 4, Issue 3, March 2016, Online: ISSN 2320-9143 RESULTS ON HESITANT FUZZY SOFT TOPOLOGICAL SPACES A. Sreedevi, Dr.N.Ravi Shankar

More information

Assessment of Human Skills Using Trapezoidal Fuzzy Numbers

Assessment of Human Skills Using Trapezoidal Fuzzy Numbers American Journal of Computational and Applied Mathematics 2015, 5(4): 111-116 DOI: 10.5923/j.ajcam.20150504.03 Assessment of Human Skills Using Trapezoidal Fuzzy Numbers Michael Gr. Voskoglou Department

More information

Table of Contents. Recognition of Facial Gestures... 1 Attila Fazekas

Table of Contents. Recognition of Facial Gestures... 1 Attila Fazekas Table of Contents Recognition of Facial Gestures...................................... 1 Attila Fazekas II Recognition of Facial Gestures Attila Fazekas University of Debrecen, Institute of Informatics

More information

On Generalizing Rough Set Theory

On Generalizing Rough Set Theory On Generalizing Rough Set Theory Y.Y. Yao Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: yyao@cs.uregina.ca Abstract. This paper summarizes various formulations

More information

Ranking of Generalized Exponential Fuzzy Numbers using Integral Value Approach

Ranking of Generalized Exponential Fuzzy Numbers using Integral Value Approach Int. J. Advance. Soft Comput. Appl., Vol., No., July 010 ISSN 074-853; Copyright ICSRS Publication, 010.i-csrs.org Ranking of Generalized Exponential Fuzzy Numbers using Integral Value Approach Amit Kumar,

More information

A Triangular Fuzzy Model for Assessing Problem Solving Skills

A Triangular Fuzzy Model for Assessing Problem Solving Skills Annals of Pure and Applied Mathematics Vol. 7, No., 04, 3-8 ISSN: 79-087X (P), 79-0888(online) Published on 9 September 04 www.researchmathsci.org Annals of A Triangular Fuzzy Model for Assessing Problem

More information

Introduction to Fuzzy Logic and Fuzzy Systems Adel Nadjaran Toosi

Introduction to Fuzzy Logic and Fuzzy Systems Adel Nadjaran Toosi Introduction to Fuzzy Logic and Fuzzy Systems Adel Nadjaran Toosi Fuzzy Slide 1 Objectives What Is Fuzzy Logic? Fuzzy sets Membership function Differences between Fuzzy and Probability? Fuzzy Inference.

More information

Speed regulation in fan rotation using fuzzy inference system

Speed regulation in fan rotation using fuzzy inference system 58 Scientific Journal of Maritime Research 29 (2015) 58-63 Faculty of Maritime Studies Rijeka, 2015 Multidisciplinary SCIENTIFIC JOURNAL OF MARITIME RESEARCH Multidisciplinarni znanstveni časopis POMORSTVO

More information

Fuzzy Logic : Introduction

Fuzzy Logic : Introduction Fuzzy Logic : Introduction Debasis Samanta IIT Kharagpur dsamanta@iitkgp.ac.in 23.01.2018 Debasis Samanta (IIT Kharagpur) Soft Computing Applications 23.01.2018 1 / 69 What is Fuzzy logic? Fuzzy logic

More information

[Ch 6] Set Theory. 1. Basic Concepts and Definitions. 400 lecture note #4. 1) Basics

[Ch 6] Set Theory. 1. Basic Concepts and Definitions. 400 lecture note #4. 1) Basics 400 lecture note #4 [Ch 6] Set Theory 1. Basic Concepts and Definitions 1) Basics Element: ; A is a set consisting of elements x which is in a/another set S such that P(x) is true. Empty set: notated {

More information

Fuzzy Sets and Systems. Lecture 1 (Introduction) Bu- Ali Sina University Computer Engineering Dep. Spring 2010

Fuzzy Sets and Systems. Lecture 1 (Introduction) Bu- Ali Sina University Computer Engineering Dep. Spring 2010 Fuzzy Sets and Systems Lecture 1 (Introduction) Bu- Ali Sina University Computer Engineering Dep. Spring 2010 Fuzzy sets and system Introduction and syllabus References Grading Fuzzy sets and system Syllabus

More information

ON THEORY OF INTUITIONISTIC FUZZY SETS (OR VAGUE SETS)

ON THEORY OF INTUITIONISTIC FUZZY SETS (OR VAGUE SETS) International Journal of Fuzzy Systems On Theory and Rough of Intuitionistic Systems Fuzzy Sets (Or Vague Sets) 113 4(2), December 2011, pp. 113-117, Serials Publications, ISSN: 0974-858X ON THEORY OF

More information

Lotfi Zadeh (professor at UC Berkeley) wrote his original paper on fuzzy set theory. In various occasions, this is what he said

Lotfi Zadeh (professor at UC Berkeley) wrote his original paper on fuzzy set theory. In various occasions, this is what he said FUZZY LOGIC Fuzzy Logic Lotfi Zadeh (professor at UC Berkeley) wrote his original paper on fuzzy set theory. In various occasions, this is what he said Fuzzy logic is a means of presenting problems to

More information

GEOG 5113 Special Topics in GIScience. Why is Classical set theory restricted? Contradiction & Excluded Middle. Fuzzy Set Theory in GIScience

GEOG 5113 Special Topics in GIScience. Why is Classical set theory restricted? Contradiction & Excluded Middle. Fuzzy Set Theory in GIScience GEOG 5113 Special Topics in GIScience Fuzzy Set Theory in GIScience -Basic Properties and Concepts of Fuzzy Sets- Why is Classical set theory restricted? Boundaries of classical sets are required to be

More information

MultiGrid-Based Fuzzy Systems for Function Approximation

MultiGrid-Based Fuzzy Systems for Function Approximation MultiGrid-Based Fuzzy Systems for Function Approximation Luis Javier Herrera 1,Héctor Pomares 1, Ignacio Rojas 1, Olga Valenzuela 2, and Mohammed Awad 1 1 University of Granada, Department of Computer

More information

Data Fusion for Magnetic Sensor Based on Fuzzy Logic Theory

Data Fusion for Magnetic Sensor Based on Fuzzy Logic Theory 2 Fourth International Conference on Intelligent Computation Technology and Automation Data Fusion for Magnetic Sensor Based on Fuzzy Logic Theory ZHU Jian, CAO Hongbing, SHEN Jie, LIU Haitao Shanghai

More information

Fuzzy Set-Theoretical Approach for Comparing Objects with Fuzzy Attributes

Fuzzy Set-Theoretical Approach for Comparing Objects with Fuzzy Attributes Fuzzy Set-Theoretical Approach for Comparing Objects with Fuzzy Attributes Y. Bashon, D. Neagu, M.J. Ridley Department of Computing University of Bradford Bradford, BD7 DP, UK e-mail: {Y.Bashon, D.Neagu,

More information

Introduction to Fuzzy Logic. IJCAI2018 Tutorial

Introduction to Fuzzy Logic. IJCAI2018 Tutorial Introduction to Fuzzy Logic IJCAI2018 Tutorial 1 Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set 2 Crisp set vs. Fuzzy set 3 Crisp Logic Example I Crisp logic is concerned with absolutes-true

More information

Irregular Interval Valued Fuzzy Graphs

Irregular Interval Valued Fuzzy Graphs nnals of Pure and pplied Mathematics Vol 3, No, 03, 56-66 ISSN: 79-087X (P), 79-0888(online) Published on 0 May 03 wwwresearchmathsciorg nnals of Irregular Interval Valued Fuzzy Graphs Madhumangal Pal

More information

NETWORK FLOW WITH FUZZY ARC LENGTHS USING HAAR RANKING

NETWORK FLOW WITH FUZZY ARC LENGTHS USING HAAR RANKING NETWORK FLOW WITH FUZZY ARC LENGTHS USING HAAR RANKING S. Dhanasekar 1, S. Hariharan, P. Sekar and Kalyani Desikan 3 1 Vellore Institute of Technology, Chennai Campus, Chennai, India CKN College for Men,

More information

HARD, SOFT AND FUZZY C-MEANS CLUSTERING TECHNIQUES FOR TEXT CLASSIFICATION

HARD, SOFT AND FUZZY C-MEANS CLUSTERING TECHNIQUES FOR TEXT CLASSIFICATION HARD, SOFT AND FUZZY C-MEANS CLUSTERING TECHNIQUES FOR TEXT CLASSIFICATION 1 M.S.Rekha, 2 S.G.Nawaz 1 PG SCALOR, CSE, SRI KRISHNADEVARAYA ENGINEERING COLLEGE, GOOTY 2 ASSOCIATE PROFESSOR, SRI KRISHNADEVARAYA

More information

Unit V. Neural Fuzzy System

Unit V. Neural Fuzzy System Unit V Neural Fuzzy System 1 Fuzzy Set In the classical set, its characteristic function assigns a value of either 1 or 0 to each individual in the universal set, There by discriminating between members

More information

Vague Congruence Relation Induced by VLI Ideals of Lattice Implication Algebras

Vague Congruence Relation Induced by VLI Ideals of Lattice Implication Algebras American Journal of Mathematics and Statistics 2016, 6(3): 89-93 DOI: 10.5923/j.ajms.20160603.01 Vague Congruence Relation Induced by VLI Ideals of Lattice Implication Algebras T. Anitha 1,*, V. Amarendra

More information

Discrete Optimization. Lecture Notes 2

Discrete Optimization. Lecture Notes 2 Discrete Optimization. Lecture Notes 2 Disjunctive Constraints Defining variables and formulating linear constraints can be straightforward or more sophisticated, depending on the problem structure. The

More information

A Learning Algorithm for Tuning Fuzzy Rules Based on the Gradient Descent Method

A Learning Algorithm for Tuning Fuzzy Rules Based on the Gradient Descent Method A Learning Algorithm for Tuning Fuzzy Rules Based on the Gradient Descent Method Yan Shi*, Masaharu Mizumoto*, Naoyoshi Yubazaki** and Masayuki Otani** *Division of Information and Computer Sciences Osaka

More information

An Application of Fuzzy Matrices in Medical Diagnosis

An Application of Fuzzy Matrices in Medical Diagnosis Intern. J. Fuzzy Mathematical Archive Vol. 9, No. 2, 2015, 211-216 ISSN: 2320 3242 (P), 2320 3250 (online) Published on 8 October 2015 www.researchmathsci.org International Journal of An Application of

More information

Intuitionistic Fuzzy Petri Nets for Knowledge Representation and Reasoning

Intuitionistic Fuzzy Petri Nets for Knowledge Representation and Reasoning Intuitionistic Fuzzy Petri Nets for Knowledge Representation and Reasoning Meng Fei-xiang 1 Lei Ying-jie 1 Zhang Bo 1 Shen Xiao-yong 1 Zhao Jing-yu 2 1 Air and Missile Defense College Air Force Engineering

More information

Generalized Implicative Model of a Fuzzy Rule Base and its Properties

Generalized Implicative Model of a Fuzzy Rule Base and its Properties University of Ostrava Institute for Research and Applications of Fuzzy Modeling Generalized Implicative Model of a Fuzzy Rule Base and its Properties Martina Daňková Research report No. 55 2 Submitted/to

More information

The Type-1 OWA Operator and the Centroid of Type-2 Fuzzy Sets

The Type-1 OWA Operator and the Centroid of Type-2 Fuzzy Sets EUSFLAT-LFA 20 July 20 Aix-les-Bains, France The Type- OWA Operator and the Centroid of Type-2 Fuzzy Sets Francisco Chiclana Shang-Ming Zhou 2 Centre for Computational Intelligence, De Montfort University,

More information

2 Dept. of Computer Applications 3 Associate Professor Dept. of Computer Applications

2 Dept. of Computer Applications 3 Associate Professor Dept. of Computer Applications International Journal of Computing Science and Information Technology, 2014, Vol.2(2), 15-19 ISSN: 2278-9669, April 2014 (http://ijcsit.org) Optimization of trapezoidal balanced Transportation problem

More information

Dynamic Clustering of Data with Modified K-Means Algorithm

Dynamic Clustering of Data with Modified K-Means Algorithm 2012 International Conference on Information and Computer Networks (ICICN 2012) IPCSIT vol. 27 (2012) (2012) IACSIT Press, Singapore Dynamic Clustering of Data with Modified K-Means Algorithm Ahamed Shafeeq

More information

XI International PhD Workshop OWD 2009, October Fuzzy Sets as Metasets

XI International PhD Workshop OWD 2009, October Fuzzy Sets as Metasets XI International PhD Workshop OWD 2009, 17 20 October 2009 Fuzzy Sets as Metasets Bartłomiej Starosta, Polsko-Japońska WyŜsza Szkoła Technik Komputerowych (24.01.2008, prof. Witold Kosiński, Polsko-Japońska

More information

Ranking Fuzzy Numbers Using Targets

Ranking Fuzzy Numbers Using Targets Ranking Fuzzy Numbers Using Targets V.N. Huynh, Y. Nakamori School of Knowledge Science Japan Adv. Inst. of Sci. and Tech. e-mail: {huynh, nakamori}@jaist.ac.jp J. Lawry Dept. of Engineering Mathematics

More information

Solving Fuzzy Travelling Salesman Problem Using Octagon Fuzzy Numbers with α-cut and Ranking Technique

Solving Fuzzy Travelling Salesman Problem Using Octagon Fuzzy Numbers with α-cut and Ranking Technique IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 239-765X. Volume 2, Issue 6 Ver. III (Nov. - Dec.26), PP 52-56 www.iosrjournals.org Solving Fuzzy Travelling Salesman Problem Using Octagon

More information

Optimization of fuzzy multi-company workers assignment problem with penalty using genetic algorithm

Optimization of fuzzy multi-company workers assignment problem with penalty using genetic algorithm Optimization of fuzzy multi-company workers assignment problem with penalty using genetic algorithm N. Shahsavari Pour Department of Industrial Engineering, Science and Research Branch, Islamic Azad University,

More information

Fuzzy Modeling for Control.,,i.

Fuzzy Modeling for Control.,,i. Fuzzy Modeling for Control,,i. INTERNATIONAL SERIES IN INTELLIGENT TECHNOLOGIES Prof. Dr. Dr. h.c. Hans-Jiirgen Zimmermann, Editor European Laboratory for Intelligent Techniques Engineering Aachen, Germany

More information

X : U -> [0, 1] R : U x V -> [0, 1]

X : U -> [0, 1] R : U x V -> [0, 1] A Fuzzy Logic 2000 educational package for Mathematica Marian S. Stachowicz and Lance Beall Electrical and Computer Engineering University of Minnesota Duluth, Minnesota 55812-2496, USA http://www.d.umn.edu/ece/lis

More information

COSC 6397 Big Data Analytics. Fuzzy Clustering. Some slides based on a lecture by Prof. Shishir Shah. Edgar Gabriel Spring 2015.

COSC 6397 Big Data Analytics. Fuzzy Clustering. Some slides based on a lecture by Prof. Shishir Shah. Edgar Gabriel Spring 2015. COSC 6397 Big Data Analytics Fuzzy Clustering Some slides based on a lecture by Prof. Shishir Shah Edgar Gabriel Spring 215 Clustering Clustering is a technique for finding similarity groups in data, called

More information

Dejong Function Optimization by means of a Parallel Approach to Fuzzified Genetic Algorithm

Dejong Function Optimization by means of a Parallel Approach to Fuzzified Genetic Algorithm Dejong Function Optimization by means of a Parallel Approach to Fuzzified Genetic Algorithm Ebrahim Bagheri, Hossein Deldari Department of Computer Science, University of New Bruswick Department of Computer

More information

Exploring Domains of Approximation in R 2 : Expository Essay

Exploring Domains of Approximation in R 2 : Expository Essay Exploring Domains of Approximation in R 2 : Expository Essay Nicolay Postarnakevich August 12, 2013 1 Introduction In this paper I explore the concept of the Domains of Best Approximations. These structures

More information

General network with four nodes and four activities with triangular fuzzy number as activity times

General network with four nodes and four activities with triangular fuzzy number as activity times International Journal of Engineering Research and General Science Volume 3, Issue, Part, March-April, 05 ISSN 09-730 General network with four nodes and four activities with triangular fuzzy number as

More information

ESSENTIALLY, system modeling is the task of building

ESSENTIALLY, system modeling is the task of building IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 53, NO. 4, AUGUST 2006 1269 An Algorithm for Extracting Fuzzy Rules Based on RBF Neural Network Wen Li and Yoichi Hori, Fellow, IEEE Abstract A four-layer

More information

Intelligent Methods in Modelling and Simulation of Complex Systems

Intelligent Methods in Modelling and Simulation of Complex Systems SNE O V E R V I E W N OTE Intelligent Methods in Modelling and Simulation of Complex Systems Esko K. Juuso * Control Engineering Laboratory Department of Process and Environmental Engineering, P.O.Box

More information

Best proximation of fuzzy real numbers

Best proximation of fuzzy real numbers 214 (214) 1-6 Available online at www.ispacs.com/jfsva Volume 214, Year 214 Article ID jfsva-23, 6 Pages doi:1.5899/214/jfsva-23 Research Article Best proximation of fuzzy real numbers Z. Rohani 1, H.

More information

Multiple-Criteria Fuzzy Evaluation: The FuzzME Software Package

Multiple-Criteria Fuzzy Evaluation: The FuzzME Software Package Multiple-Criteria Fuzzy Evaluation: The FuzzME Software Package Jana Talašová 1 Pavel Holeček 2 1., 2. Faculty of Science, Palacký University Olomouc tř. Svobody 26, 771 46 Olomouc, Czech Republic Email:

More information

Influence of fuzzy norms and other heuristics on Mixed fuzzy rule formation

Influence of fuzzy norms and other heuristics on Mixed fuzzy rule formation International Journal of Approximate Reasoning 35 (2004) 195 202 www.elsevier.com/locate/ijar Influence of fuzzy norms and other heuristics on Mixed fuzzy rule formation Thomas R. Gabriel, Michael R. Berthold

More information

Approximation of a Fuzzy Function by Using Radial Basis Functions Interpolation

Approximation of a Fuzzy Function by Using Radial Basis Functions Interpolation International Journal of Mathematical Modelling & Computations Vol. 07, No. 03, Summer 2017, 299-307 Approximation of a Fuzzy Function by Using Radial Basis Functions Interpolation R. Firouzdor a and M.

More information

FUZZY LOGIC TECHNIQUES. on random processes. In such situations, fuzzy logic exhibits immense potential for

FUZZY LOGIC TECHNIQUES. on random processes. In such situations, fuzzy logic exhibits immense potential for FUZZY LOGIC TECHNIQUES 4.1: BASIC CONCEPT Problems in the real world are quite often very complex due to the element of uncertainty. Although probability theory has been an age old and effective tool to

More information

RPKM: The Rough Possibilistic K-Modes

RPKM: The Rough Possibilistic K-Modes RPKM: The Rough Possibilistic K-Modes Asma Ammar 1, Zied Elouedi 1, and Pawan Lingras 2 1 LARODEC, Institut Supérieur de Gestion de Tunis, Université de Tunis 41 Avenue de la Liberté, 2000 Le Bardo, Tunisie

More information