Fuzzy Set, Fuzzy Logic, and its Applications

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