Fuzzy Set, Fuzzy Logic, and its Applications
|
|
- Phyllis Powell
- 5 years ago
- Views:
Transcription
1 Sistem Cerdas (TE 4485) Fuzzy Set, Fuzzy Logic, and its pplications Instructor: Thiang Room: I.201 Phone: Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 1
2 Introduction Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 2 Group of pples Group of Oranges O O O O O O O O O Group of pples? Group of Oranges? O O O O O O O O O O O Group of pples?? Group of Oranges?? O O O O
3 Introduction Definition: If temperature is higher than 50 C then it is hot Temperature is 70 C, is it hot? Temperature is 30 C, is it hot? Temperature is 51 C, is it hot? Temperature is 40 C, is it hot?? Temperature is 45 C, is it hot?? Temperature is 49 C, is it hot???? Temperature is 50 C, is it hot?????? Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 3
4 Introduction Fuzzy Sets theory was introduced by Lotfi. Zadeh (1965) Fuzzy Sets are sets with boundaries that are not precise. The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of a degree. Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 4
5 Introduction: Crisp set versus Fuzzy set The characteristic of Crisp set assigns a value of either 1 or 0 to each individual in the universal set Fuzzy set assigns a value within a specified range to each individual in the universal set and the value indicates the membership grade of that individual in the set. Larger value denotes higher degree of set membership. Crisp Fuzzy Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 5
6 Introduction: Crisp set versus Fuzzy set Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 6
7 Fuzzy Set notation Continuous ( ) F µ x / x F Example: The set, B, of numbers near to two. Membership function of the set is defined as: µ B ( x) µ B ( ) 5( x 2) x e 1 B 5 ( x 2) e / x Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 7
8 Fuzzy Set notation ( ) Discrete F µ x / x F Example: The set, B, of numbers near to two. Membership function of the set is defined as: 1 µ B ( x) B 0 / / /1+ 0.3/ / / / / / Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 8
9 Fuzzy Set: Basic Concept Support of fuzzy set Supp Core of fuzzy set Core Height of fuzzy set h ( F ) { x / ( x) > 0} µ F ( F ) { x / ( x) 1} µ F ( F ) max{ ( x) } µ F fuzzy set F is called normal when h(f) 1; it is called subnormal when h(f) < 1 Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 9
10 Fuzzy Set: Basic Concept α-cut of fuzzy set α F { x / µ ( ) α} x Strong α-cut of fuzzy set F α + F { x / µ ( x) > α} F Complement of fuzzy set ( ) ( x) h( F ) ( x) µ µ F F F Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 10
11 Fuzzy Set: example Supp( 2 ) ( 20,60) Core h( 2 ) 1 ( 2 ) [ 35,45] [ 27.5,52.5] ( 27.5,52.5) F solid area( red color) Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 11
12 Fuzzy Set: Basic Concept Fuzzy Subset fuzzy set,, is said to be a subset of fuzzy set, B, if ( x) ( x) for all x µ µ B Fuzzy Union (Logic OR ) ( x) ( x) max[ ( x), ( x) ] µ µ µ µ + B B B commutative, associative Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 12
13 Fuzzy Set: Basic Concept Fuzzy Intersection (Logic ND ) ( x) ( x) min[ ( x) ( x) ] µ µ µ B B B µ, commutative, associative ssociativity (1) Min-Max fuzzy logic has intersection distributive over union min µ ( x) ( x) ( B+ C) ( B) + ( C ) µ [,max( B, C) ] max[ min(, B),min(, C) ] Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 13
14 Fuzzy Set: Basic Concept ssociativity (2) Min-Max fuzzy logic has union distributive over intersection max µ ( x) ( x) ( B C ) ( + B) ( + C) + µ [,min( B, C) ] min[ max(, B),max(, C) ] DeMorgan s Law (1) Min-Max fuzzy logic obeys DeMorgan s Law #1 µ ( x) ( x) B C B + C µ [(1 B),(1 )] 1 min( B, C) max C Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 14
15 Fuzzy Set: Basic Concept DeMorgan s Law (2) Min-Max fuzzy logic obeys DeMorgan s Law #2 µ ( x) ( x) µ B + C B C [(1 B),(1 )] 1 max( B, C) min C The Law of Excluded Middle Min-Max fuzzy logic fails the law of excluded middle o/ min(,1 ) 0 Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 15
16 Fuzzy Set: Basic Concept The Law of Contradiction Min-Max fuzzy logic fails the law of contradiction + U max(,1 ) 1 Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 16
17 Fuzzy Set: Basic Concept Cartesian Product The intersection and union operations can also be used to assign memberships on the Cartesian product of two sets Consider, as an example, the fuzzy membership of a set, G, of liquids that taste good and the set, L, of cities far from Los ngeles µ G 0.0/Swamp Water + 0.5/Radish Juice + 0.9/Grape Juice µ L 0.0/L + 0.5/Chicago + 0.8/New York + 0.9/London Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 17
18 Fuzzy Set: Basic Concept Cartesian Product We form the set, E, of Liquids that taste good ND cities that are far from Los ngeles E G L The following table results Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 18
19 Fuzzy Set: Example Determine: 1 3 ( 1 2 ) ( 2 3 ) Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 19
20 Fuzzy Set: nswers 1 3 Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 20
21 Fuzzy Set: nswers ( 1 2 ) ( 2 3 ) Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 21
22 Fuzzy rithmetic: Fuzzy number fuzzy set is a fuzzy number if the fuzzy set meets the following properties: The fuzzy set must be a normal fuzzy set α-cut of the fuzzy set must be a closed interval Support of the fuzzy set must be an open interval Example of fuzzy number and fuzzy interval Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 22
23 rithmetic Operation on Interval Four arithmetic operations on closed intervals: [a, b] + [d, e] [a + d, b + e] [a, b] [d, e] [a e, b d] [a, b] [d, e] [min(ad, ae, bd, be), max(ad, ae, bd, be)] [a, b] / [d, e] [min(a/d, a/e, b/d, b/e), max(a/d, a/e, b/d, b/e)] Example: [-3, 4] + [-1, 2] [-4, [?,?] 6] [-3, 3] [-4, 3] [-6, [?,?] 7] [-4, 2] 2] [-2, 4] 4] [-16, [?,?] 8] [-1, [-1, 3] 3] / [2, / [2, 4] 4] [-0.5, [?,?] 1.5] Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 23
24 rithmetic Operation on Fuzzy Number Method for developing fuzzy arithmetic is based on interval arithmetic. Let and B denote fuzzy numbers and * denotes any of four basic arithmetic. Then, α ( B) α α B Example: Fuzzy Number 0 µ ( x) ( x + 1) / 2 (3 x) / 2 Fuzzy Number B 0 µ ( x) ( x 1) / 2 (5 x) / 2 Calculate: + B, B, B, / B for x 1and for 1 < x 1 for1 < x 3 for x 1and for1 < x 3 for 3 < x 5 Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 24 x x > > 5 3
25 rithmetic Operation on Fuzzy Number α [ 2α 1,3 2α ] α B [ 2α + 1,5 2α ] ddition: α [ 4α,8 4α ] for ( 0,1] ( + B) α Membership function of fuzzy number of + B is: µ + B ( x) 0 x / 4 (8 x) / 4 for for for x 0 4 < < 0 and x 4 x 8 x > 8 Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 25
26 rithmetic Operation on Fuzzy Number Subtraction: α [ 4α 6,2 4α ] forα ( 0,1] ( B) Membership function of fuzzy number of B is: µ B ( x) 0 ( x (2 + 6) / x) / 4 4 for for for x 6 and x 6 < x 2 2 < x 2 > 2 Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 26
27 rithmetic Operation on Fuzzy Number Multiplication: α ( B) [ ] 2 2 4α + 12α 5,4α 16α + 15 forα ( 0,0.5] [ ] 2 2 4α 1,4α 16α + 15 forα ( 0.5,1] Membership function of fuzzy number of B is: µ B ( x) 0 (1 + [ 3 (4 ) ] 1/ 2 x [ 4 (1 + ) ] 1/ 2 x x) 1/ 2 / 2 / / 2 2 for for for for x < 5and 5 x < 0 0 x < 3 3 x < 15 x 15 Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 27
28 rithmetic Operation on Fuzzy Number Division: α ( / B) [ (2α 1) /(2α + 1),(3 2α ) /(2α + 1) ] forα ( 0,0.5] [ (2α 1) /(5 2α ),(3 2α ) /(2α + 1) ] forα ( 0.5,1] Membership function of fuzzy number of / B is: µ / B ( x) 0 ( x + 1) /(2 2x) (5x + 1) /(2x + 2) (3 x) /(2x + 2) for x < 1and x for 1 x < 0 for 0 x < 1/ 3 for1/ 3 x < 3 3 Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 28
29 Fuzzy Relation Example of crisp relation: Let X denotes a set of cities in Southeast sia. X {Jakarta, Singapore, Kuala Lumpur, Bangkok, Manila} Crisp relation that attempts to capture the relational concept near, is represented by the following relation Jakarta Singapore Kuala Lumpur Bangkok Manila Jakarta Singapore Kuala Lumpur Bangkok Manila Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 29
30 Fuzzy Relation Using the same example as example of crisp relation, Fuzzy relation that attempts to capture the relational concept near, is represented by the following relation Jakarta 1 Jakarta Singapore Singapore Kuala Lumpur Bangkok Kuala Lumpur Bangkok Manila 1 Manila Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 30
31 Fuzzy Relation: Representations Matrices Consider the previous example, fuzzy relation is concisely represented by the matrix: J S K B M R J S K B M Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 31
32 Fuzzy Relation: Representations Mapping Diagram Consider as an example, a set of documents D {d 1, d 2, d 3, d 4, d 5 } and a set of key terms T {t 1, t 2, t 3, t 4 }. Fuzzy relation expressing the degree of relevance of each document to each key term can be represented in the following mapping diagram Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 32
33 Fuzzy Relation: Representations Directed Graph Fuzzy relation can be represented by a directed graph. Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 33
34 Fuzzy Relation: Basic Operation Inverse of a fuzzy relation (R -1 ) Inverse (R -1 ) of a fuzzy relation (R) represented by a matrix, can be obtained by exchanging the rows of given matrix with the columns. The resulting matrix is called transpose of given matrix. Example: Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 34
35 Fuzzy Relation: Basic Operation Composition of two fuzzy relations X Y 1 Z X Z a 2 a b B b B 3 c C c C 4 P Q P Q Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 35
36 Fuzzy Relation: Basic Operation Standard composition of fuzzy relations Let P [p ij ], Q [q jk ], and R [r ik ] are matrix representations of fuzzy relations for which R P Q. Matrices relation of composition of fuzzy relations is represented by expression: [r ik ] [p ij ] [q jk ] where r ik max min(p ij, q jk ) j Previous example: P Q Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 36
37 Fuzzy Relation: Basic Operation P Q r r max[min( p 11, q 11 ),min( p 12, q ),min( p ),min( p max[min(0.7,0.5), min(1,0.3), min(0,1), min(0,0)] , q 31 14, q 41 )] R P o Q Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 37
38 Fuzzy Relation: Basic Operation Result of composition of fuzzy relation P and Q: X Z R P o Q a b c B C P Q Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 38
39 Fuzzy Inference Crisp Input ntecedent Fuzzification Input Membership Function Rules Consequent Defuzzification Output Membership Function Crisp Output Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 39
40 Fuzzy Inference Example: Student pplicants Evaluation ssume that we need to evaluate student applicants based on their GP and GRE score. For simplicity, there are three categories for each score [High (H), Medium (M), and Low (L)]. ssume that the decision should be Excellent (E), Very Good (VG), GOOD (G), Fair (F), and Poor (P). n expert will associate the decisions to the GP and GRE score. They are then tabulated in Fuzzy If-then Rules form. Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 40
41 Fuzzy Inference Example of Fuzzy If-Then Rules If the GRE is HIGH and the GP is HIGH then the STUDENT will be EXCELLENT If the GRE is LOW and the GP is HIGH then the STUDENT will be FIR ntecedent Fuzzy Linguistic Variables Consequent Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 41
42 Fuzzy Inference Fuzzy If-Then Rules Table Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 42
43 Fuzzy Inference Membership Function for GRE µ GRE 1 LOW MEDIUM HIGH Typical shapes of membership function are triangular, trapezoidal, and Gaussian Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 43
44 Fuzzy Inference Membership Function for GP µ GP 1 LOW MEDIUM HIGH Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 44
45 Fuzzy Inference Membership Function for Consequent (Student) µ C 1 P F G VG E Example: Evaluate a student who has GRE of 900 and GP of 3.6! Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 45
46 Fuzzification Convert the crisp inputs (antecedents) into vector of fuzzy membership values µ GRE LOW MEDIUM HIGH 0.25 Result: µ { µ 0.75, µ 0.25, 0} GRE L M µ H Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 46
47 Fuzzification µ GP LOW MEDIUM HIGH Result: µ { µ 0, µ 0.25, 0.75} GP L M µ H Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 47
48 Rule Evaluation: Min-Max Strategy Result: µ { µ 0.25, µ 0.75, µ 0.25, µ 0, 0} C P F G VG µ E Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 48
49 Defuzzification µ { µ 0.25, µ 0.75, µ 0.25, µ 0, 0} C P F G VG µ E µ C 1 P F G VG E Center of rea Result: Student is Fair Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 49
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 informationIntroduction 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 informationWhat 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 informationGEOG 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 informationFuzzy 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 informationChapter 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 informationFuzzy 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 informationChapter 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 informationLecture 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 informationFuzzy 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 informationWhy Fuzzy Fuzzy Logic and Sets Fuzzy Reasoning. DKS - Module 7. Why fuzzy thinking?
Fuzzy Systems Overview: Literature: Why Fuzzy Fuzzy Logic and Sets Fuzzy Reasoning chapter 4 DKS - Module 7 1 Why fuzzy thinking? Experts rely on common sense to solve problems Representation of vague,
More informationA 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 informationARTIFICIAL INTELLIGENCE - FUZZY LOGIC SYSTEMS
ARTIFICIAL INTELLIGENCE - FUZZY LOGIC SYSTEMS http://www.tutorialspoint.com/artificial_intelligence/artificial_intelligence_fuzzy_logic_systems.htm Copyright tutorialspoint.com Fuzzy Logic Systems FLS
More informationCHAPTER 5 FUZZY LOGIC CONTROL
64 CHAPTER 5 FUZZY LOGIC CONTROL 5.1 Introduction Fuzzy logic is a soft computing tool for embedding structured human knowledge into workable algorithms. The idea of fuzzy logic was introduced by Dr. Lofti
More informationIntroduction 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 informationUnit 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 informationFUZZY SYSTEMS: Basics using MATLAB Fuzzy Toolbox. Heikki N. Koivo
FUZZY SYSTEMS: Basics using MATLAB Fuzzy Toolbox By Heikki N. Koivo 200 2.. Fuzzy sets Membership functions Fuzzy set Universal discourse U set of elements, {u}. Fuzzy set F in universal discourse U: Membership
More information7. Decision Making
7. Decision Making 1 7.1. Fuzzy Inference System (FIS) Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy logic. Fuzzy inference systems have been successfully
More informationWhat 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 informationMachine Learning & Statistical Models
Astroinformatics Machine Learning & Statistical Models Neural Networks Feed Forward Hybrid Decision Analysis Decision Trees Random Decision Forests Evolving Trees Minimum Spanning Trees Perceptron Multi
More informationARTIFICIAL INTELLIGENCE. Uncertainty: fuzzy systems
INFOB2KI 2017-2018 Utrecht University The Netherlands ARTIFICIAL INTELLIGENCE Uncertainty: fuzzy systems Lecturer: Silja Renooij These slides are part of the INFOB2KI Course Notes available from www.cs.uu.nl/docs/vakken/b2ki/schema.html
More informationFuzzy 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 informationStudy of Fuzzy Set Theory and Its Applications
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 12, Issue 4 Ver. II (Jul. - Aug.2016), PP 148-154 www.iosrjournals.org Study of Fuzzy Set Theory and Its Applications
More informationIntroduction 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 informationFuzzy Reasoning. Linguistic Variables
Fuzzy Reasoning Linguistic Variables Linguistic variable is an important concept in fuzzy logic and plays a key role in its applications, especially in the fuzzy expert system Linguistic variable is a
More informationFuzzy If-Then Rules. Fuzzy If-Then Rules. Adnan Yazıcı
Fuzzy If-Then Rules Adnan Yazıcı Dept. of Computer Engineering, Middle East Technical University Ankara/Turkey Fuzzy If-Then Rules There are two different kinds of fuzzy rules: Fuzzy mapping rules and
More informationDra. Ma. del Pilar Gómez Gil Primavera 2014
C291-78 Tópicos Avanzados: Inteligencia Computacional I Introducción a la Lógica Difusa Dra. Ma. del Pilar Gómez Gil Primavera 2014 pgomez@inaoep.mx Ver: 08-Mar-2016 1 Este material ha sido tomado de varias
More informationFUZZY 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 informationCHAPTER 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 informationFuzzy Sets and Systems. Lecture 2 (Fuzzy Sets) Bu- Ali Sina University Computer Engineering Dep. Spring 2010
Fuzzy Sets and Systems Lecture 2 (Fuzzy Sets) Bu- Ali Sina University Computer Engineering Dep. Spring 2010 Fuzzy Sets Formal definition: A fuzzy set A in X (universal set) is expressed as a set of ordered
More informationSINGLE VALUED NEUTROSOPHIC SETS
Fuzzy Sets, Rough Sets and Multivalued Operations and pplications, Vol 3, No 1, (January-June 2011): 33 39; ISSN : 0974-9942 International Science Press SINGLE VLUED NEUTROSOPHIC SETS Haibin Wang, Yanqing
More informationNeural 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 informationApplication 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 informationChapter 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 informationFuzzy rule-based decision making model for classification of aquaculture farms
Chapter 6 Fuzzy rule-based decision making model for classification of aquaculture farms This chapter presents the fundamentals of fuzzy logic, and development, implementation and validation of a fuzzy
More informationCHAPTER 3 FUZZY INFERENCE SYSTEM
CHAPTER 3 FUZZY INFERENCE SYSTEM Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy logic. There are three types of fuzzy inference system that can be
More informationWhy Fuzzy? Definitions Bit of History Component of a fuzzy system Fuzzy Applications Fuzzy Sets Fuzzy Boundaries Fuzzy Representation
Contents Why Fuzzy? Definitions Bit of History Component of a fuzzy system Fuzzy Applications Fuzzy Sets Fuzzy Boundaries Fuzzy Representation Linguistic Variables and Hedges INTELLIGENT CONTROLSYSTEM
More informationDinner for Two, Reprise
Fuzzy Logic Toolbox Dinner for Two, Reprise In this section we provide the same two-input, one-output, three-rule tipping problem that you saw in the introduction, only in more detail. The basic structure
More informationFuzzy Sets and Fuzzy Logic
Fuzzy Sets and Fuzzy Logic KR Chowdhary, Professor, Department of Computer Science & Engineering, MBM Engineering College, JNV University, Jodhpur, Email: Outline traditional logic : {true,false} Crisp
More informationFUZZY SETS. Precision vs. Relevancy LOOK OUT! A 1500 Kg mass is approaching your head OUT!!
FUZZY SETS Precision vs. Relevancy A 5 Kg mass is approaching your head at at 45.3 45.3 m/sec. m/s. OUT!! LOOK OUT! 4 Introduction How to simplify very complex systems? Allow some degree of uncertainty
More informationFuzzy Mathematics. Fuzzy -Sets, -Relations, -Logic, -Graphs, -Mappings and The Extension Principle. Olaf Wolkenhauer. Control Systems Centre UMIST
Fuzzy Mathematics Fuzzy -Sets, -Relations, -Logic, -Graphs, -Mappings and The Extension Principle Olaf Wolkenhauer Control Systems Centre UMIST o.wolkenhauer@umist.ac.uk www.csc.umist.ac.uk/people/wolkenhauer.htm
More informationCHAPTER 3 MAINTENANCE STRATEGY SELECTION USING AHP AND FAHP
31 CHAPTER 3 MAINTENANCE STRATEGY SELECTION USING AHP AND FAHP 3.1 INTRODUCTION Evaluation of maintenance strategies is a complex task. The typical factors that influence the selection of maintenance strategy
More informationFuzzy Sets and Fuzzy Logic. KR Chowdhary, Professor, Department of Computer Science & Engineering, MBM Engineering College, JNV University, Jodhpur,
Fuzzy Sets and Fuzzy Logic KR Chowdhary, Professor, Department of Computer Science & Engineering, MBM Engineering College, JNV University, Jodhpur, Outline traditional logic : {true,false} Crisp Logic
More informationIntroduction 3 Fuzzy Inference. Aleksandar Rakić Contents
Beograd ETF Fuzzy logic Introduction 3 Fuzzy Inference Aleksandar Rakić rakic@etf.rs Contents Mamdani Fuzzy Inference Fuzzification of the input variables Rule evaluation Aggregation of rules output Defuzzification
More informationContents. The Definition of Fuzzy Logic Rules. Fuzzy Logic and Functions. Fuzzy Sets, Statements, and Rules
Fuzzy Logic and Functions The Definition of Fuzzy Logic Membership Function Evolutionary Algorithms Constructive Induction Fuzzy logic Neural Nets Decision Trees and other Learning A person's height membership
More informationSpeed 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 informationCHAPTER - 3 FUZZY SET THEORY AND MULTI CRITERIA DECISION MAKING
CHAPTER - 3 FUZZY SET THEORY AND MULTI CRITERIA DECISION MAKING 3.1 Introduction Construction industry consists of broad range of equipment and these are required at different points of the execution period.
More informationVHDL framework for modeling fuzzy automata
Doru Todinca Daniel Butoianu Department of Computers Politehnica University of Timisoara SYNASC 2012 Outline Motivation 1 Motivation Why fuzzy automata? Why a framework for modeling FA? Why VHDL? 2 Fuzzy
More information1. Fuzzy sets, fuzzy relational calculus, linguistic approximation
1. Fuzzy sets, fuzzy relational calculus, linguistic approximation 1.1. Fuzzy sets Let us consider a classical set U (Universum) and a real function : U --- L. As a fuzzy set A we understand a set of pairs
More informationCHAPTER 3 FUZZY RELATION and COMPOSITION
CHAPTER 3 FUZZY RELATION and COMPOSITION The concept of fuzzy set as a generalization of crisp set has been introduced in the previous chapter. Relations between elements of crisp sets can be extended
More informationApproximation 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 informationLotfi 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 informationCHAPTER 3 FUZZY RELATION and COMPOSITION
CHAPTER 3 FUZZY RELATION and COMPOSITION Crisp relation! Definition (Product set) Let A and B be two non-empty sets, the prod uct set or Cartesian product A B is defined as follows, A B = {(a, b) a A,
More informationFUZZY 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 informationCHAPTER 4 FUZZY LOGIC, K-MEANS, FUZZY C-MEANS AND BAYESIAN METHODS
CHAPTER 4 FUZZY LOGIC, K-MEANS, FUZZY C-MEANS AND BAYESIAN METHODS 4.1. INTRODUCTION This chapter includes implementation and testing of the student s academic performance evaluation to achieve the objective(s)
More informationIntroduction. Aleksandar Rakić Contents
Beograd ETF Fuzzy logic Introduction Aleksandar Rakić rakic@etf.rs Contents Definitions Bit of History Fuzzy Applications Fuzzy Sets Fuzzy Boundaries Fuzzy Representation Linguistic Variables and Hedges
More informationCHAPTER 3 ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM
33 CHAPTER 3 ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM The objective of an ANFIS (Jang 1993) is to integrate the best features of Fuzzy Systems and Neural Networks. ANFIS is one of the best tradeoffs between
More informationFUZZY 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 informationCSC Discrete Math I, Spring Sets
CSC 125 - Discrete Math I, Spring 2017 Sets Sets A set is well-defined, unordered collection of objects The objects in a set are called the elements, or members, of the set A set is said to contain its
More informationChapter 2 The Operation of Fuzzy Set
Chapter 2 The Operation of Fuzzy Set Standard operations of Fuzzy Set! Complement set! Union Ma[, ]! Intersection Min[, ]! difference between characteristics of crisp fuzzy set operator n law of contradiction
More information2.2 Set Operations. Introduction DEFINITION 1. EXAMPLE 1 The union of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 2, 3, 5}; that is, EXAMPLE 2
2.2 Set Operations 127 2.2 Set Operations Introduction Two, or more, sets can be combined in many different ways. For instance, starting with the set of mathematics majors at your school and the set of
More informationAssessment 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 informationFuzzy 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 informationOptimization with linguistic variables
Optimization with linguistic variables Christer Carlsson christer.carlsson@abo.fi Robert Fullér rfuller@abo.fi Abstract We consider fuzzy mathematical programming problems (FMP) in which the functional
More information* The terms used for grading are: - bad - good
Hybrid Neuro-Fuzzy Systems or How to Combine German Mechanics with Italian Love by Professor Michael Negnevitsky University of Tasmania Introduction Contents Heterogeneous Hybrid Systems Diagnosis of myocardial
More information[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 informationThe 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 informationSets and set operations
CS 44 Discrete Mathematics for CS Lecture Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Course administration Homework 3: Due today Homework 4: Due next week on Friday,
More informationUsing Ones Assignment Method and. Robust s Ranking Technique
Applied Mathematical Sciences, Vol. 7, 2013, no. 113, 5607-5619 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.37381 Method for Solving Fuzzy Assignment Problem Using Ones Assignment
More informationFUZZY 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 informationFuzzy Logic Controller
Fuzzy Logic Controller Debasis Samanta IIT Kharagpur dsamanta@iitkgp.ac.in 23.01.2016 Debasis Samanta (IIT Kharagpur) Soft Computing Applications 23.01.2016 1 / 34 Applications of Fuzzy Logic Debasis Samanta
More informationLecture 5 Fuzzy expert systems: Fuzzy inference Mamdani fuzzy inference Sugeno fuzzy inference Case study Summary
Lecture 5 Fuzzy expert systems: Fuzzy inference Mamdani fuzzy inference Sugeno fuzzy inference Case study Summary Negnevitsky, Pearson Education, 25 Fuzzy inference The most commonly used fuzzy inference
More informationFinal Exam. Controller, F. Expert Sys.., Solving F. Ineq.} {Hopefield, SVM, Comptetive Learning,
Final Exam Question on your Fuzzy presentation {F. Controller, F. Expert Sys.., Solving F. Ineq.} Question on your Nets Presentations {Hopefield, SVM, Comptetive Learning, Winner- take all learning for
More informationAssessment 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 informationA New and Simple Method of Solving Fully Fuzzy Linear System
Annals of Pure and Applied Mathematics Vol. 8, No. 2, 2014, 193-199 ISSN: 2279-087X (P), 2279-0888(online) Published on 17 December 2014 www.researchmathsci.org Annals of A New and Simple Method of Solving
More information11 Sets II Operations
11 Sets II Operations Tom Lewis Fall Term 2010 Tom Lewis () 11 Sets II Operations Fall Term 2010 1 / 12 Outline 1 Union and intersection 2 Set operations 3 The size of a union 4 Difference and symmetric
More informationSOLUTION: 1. First define the temperature range, e.g. [0 0,40 0 ].
2. 2. USING MATLAB Fuzzy Toolbox GUI PROBLEM 2.1. Let the room temperature T be a fuzzy variable. Characterize it with three different (fuzzy) temperatures: cold,warm, hot. SOLUTION: 1. First define the
More informationON SOLVING A MULTI-CRITERIA DECISION MAKING PROBLEM USING FUZZY SOFT SETS IN SPORTS
ISSN Print): 2320-5504 ISSN Online): 2347-4793 ON SOLVING A MULTI-CRITERIA DECISION MAKING PROBLEM USING FUZZY SOFT SETS IN SPORTS R. Sophia Porchelvi 1 and B. Snekaa 2* 1 Associate Professor, 2* Research
More informationFigure 2-1: Membership Functions for the Set of All Numbers (N = Negative, P = Positive, L = Large, M = Medium, S = Small)
Fuzzy Sets and Pattern Recognition Copyright 1998 R. Benjamin Knapp Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that
More informationCOSC 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 informationAn Application of Interval Valued Fuzzy Soft Matrix in Decision Making Problem
An Application of Interval Valued Fuzzy Soft Matrix in Decision Making Problem Dr.N.Sarala 1, M.prabhavathi 2 1 Department of mathematics, A.D.M.college for women (Auto),Nagai, India. 2 Department of mathematics,
More informationFuzzy Rules & Fuzzy Reasoning
Sistem Cerdas : PTK Pasca Sarjana - UNY Fuzzy Rules & Fuzzy Reasoning Pengampu: Fatchul rifin Referensi: Jyh-Shing Roger Jang et al., Neuro-Fuzzy and Soft Computing: Computational pproach to Learning and
More informationX : 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 informationFuzzy Logic. Sourabh Kothari. Asst. Prof. Department of Electrical Engg. Presentation By
Fuzzy Logic Presentation By Sourabh Kothari Asst. Prof. Department of Electrical Engg. Outline of the Presentation Introduction What is Fuzzy? Why Fuzzy Logic? Concept of Fuzzy Logic Fuzzy Sets Membership
More informationFigure-12 Membership Grades of x o in the Sets A and B: μ A (x o ) =0.75 and μb(xo) =0.25
Membership Functions The membership function μ A (x) describes the membership of the elements x of the base set X in the fuzzy set A, whereby for μ A (x) a large class of functions can be taken. Reasonable
More information2 Review of Set Theory
2 Review of Set Theory Example 2.1. Let Ω = {1, 2, 3, 4, 5, 6} 2.2. Venn diagram is very useful in set theory. It is often used to portray relationships between sets. Many identities can be read out simply
More information2. Sets. 2.1&2.2: Sets and Subsets. Combining Sets. c Dr Oksana Shatalov, Fall
c Dr Oksana Shatalov, Fall 2014 1 2. Sets 2.1&2.2: Sets and Subsets. Combining Sets. Set Terminology and Notation DEFINITIONS: Set is well-defined collection of objects. Elements are objects or members
More informationCPS331 Lecture: Fuzzy Logic last revised October 11, Objectives: 1. To introduce fuzzy logic as a way of handling imprecise information
CPS331 Lecture: Fuzzy Logic last revised October 11, 2016 Objectives: 1. To introduce fuzzy logic as a way of handling imprecise information Materials: 1. Projectable of young membership function 2. Projectable
More informationFuzzy logic controllers
Fuzzy logic controllers Digital fuzzy logic controllers Doru Todinca Department of Computers and Information Technology UPT Outline Hardware implementation of fuzzy inference The general scheme of the
More informationBackground Fuzzy control enables noncontrol-specialists. A fuzzy controller works with verbal rules rather than mathematical relationships.
Introduction to Fuzzy Control Background Fuzzy control enables noncontrol-specialists to design control system. A fuzzy controller works with verbal rules rather than mathematical relationships. knowledge
More informationData can be in the form of numbers, words, measurements, observations or even just descriptions of things.
+ What is Data? Data is a collection of facts. Data can be in the form of numbers, words, measurements, observations or even just descriptions of things. In most cases, data needs to be interpreted and
More informationENGR 1181 MATLAB 4: Array Operations
ENGR 81 MTL 4: rray Operations Learning Objectives 1. Explain meaning of element-by-element operations. Identify situations where the standard operators in MTL (when used with arrays) are reserved for
More informationOperations in Fuzzy Labeling Graph through Matching and Complete Matching
Operations in Fuzzy Labeling Graph through Matching and Complete Matching S. Yahya Mohamad 1 and S.Suganthi 2 1 PG & Research Department of Mathematics, Government Arts College, Trichy 620 022, Tamilnadu,
More informationFuzzy Expert Systems Lecture 8 (Fuzzy Systems)
Fuzzy Expert Systems Lecture 8 (Fuzzy Systems) Soft Computing is an emerging approach to computing which parallels the remarkable ability of the human mind to reason and learn in an environment of uncertainty
More informationREASONING UNDER UNCERTAINTY: FUZZY LOGIC
REASONING UNDER UNCERTAINTY: FUZZY LOGIC Table of Content What is Fuzzy Logic? Brief History of Fuzzy Logic Current Applications of Fuzzy Logic Overview of Fuzzy Logic Forming Fuzzy Set Fuzzy Set Representation
More informationMODELING 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 informationFuzzy 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 informationA Study on Triangular Type 2 Triangular Fuzzy Matrices
International Journal of Fuzzy Mathematics and Systems. ISSN 2248-9940 Volume 4, Number 2 (2014), pp. 145-154 Research India Publications http://www.ripublication.com A Study on Triangular Type 2 Triangular
More informationComputational Intelligence Lecture 10:Fuzzy Sets
Computational Intelligence Lecture 10:Fuzzy Sets Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 arzaneh Abdollahi Computational Intelligence Lecture
More informationINTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & TECHNOLOGY (IJCET)
INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & TECHNOLOGY (IJCET) ISSN 0976 6367(Print) ISSN 0976 6375(Online) Volume 3, Issue 2, July- September (2012), pp. 157-166 IAEME: www.iaeme.com/ijcet.html Journal
More information