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Fuzzy Numbers

Fuzzy Numbers. Definition. Fuzzy Number Convex and normal fuzzy set defined on R Equivalently it satisfies Normal fuzzy set on R Every alpha-cut must be a closed interval Support must be bounded Applications of fuzzy number Fuzzy Control, Decision Making, Optimizations. 1. 1. 1.5.

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Fuzzy Numbers

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  1. Fuzzy Numbers

  2. Definition • Fuzzy Number • Convex and normal fuzzy set defined on R • Equivalently it satisfies • Normal fuzzy set on R • Every alpha-cut must be a closed interval • Support must be bounded • Applications of fuzzy number • Fuzzy Control, Decision Making, Optimizations

  3. 1. 1. 1.5 3. 4.5 3. 1. 1. 1.5 3. 4.5 1.5 3. 4.5 Examples

  4. Arithmetic Operations • Interval Operations A = [ a1 , a3] , B = [ b1 , b3]

  5. Examples • Addition [2,5]+[1,3]=[3,8] [0,1]+[-6,5]=[-6,6] • Subtraction [2,5]-[1,3]=[-1,4] [0,1]-[-6,5]=[-5,7] • Multiplication [-1,1]*[-2,-0.5]=[-2,2] [3,4]*[2,2]=[6,8] • Division [-1,1]/[-2,-0.5]=[-2,2] [4,10]*[1,2]=[2,10]

  6. Properties of Interval Operations

  7. Arithmetic Operation on Fuzzy Numbers • Interval operations of alpha-level sets • Note: The Result is a fuzzy number. • Example: See Text pp. 105 and Fig. 4.5

  8. Arithmetic Operation by Extension Principle • By Extension Principle • Note: • * can be any operations including arithmetic operations. • Example A = { 1/2 , 0.5/3},B = { 1/3, 0.8/4}

  9. Example • A+B = {1/5, 0.8/6, 0.5/7 }

  10. Example • Max (A,B) = { (3 , 1) , (4 , 0.5) }

  11. Typical Fuzzy Numbers • Triangular Fuzzy Number • Fig. 4.5 • Trapezoidal Fuzzy Numbers: Fig. 4.4 • Linguistic variable: ”Performance” • Linguistic values (terms): “very small”…“very large” • Semantic Rules: Terms maps on trapezoidal fuzzy numbers • Syntactic rules (grammar): Rules for other terms such as “not small”

  12. An Application of Optimal Decision • Decision Making • Which alternative are you going to choose? • weighted average. • How to make use of resulted fuzzy set • Similarity comparisons with model fuzzy sets

  13. Lattice of Fuzzy Numbers • Lattice • Partially ordered set with ordering relation • Meet(g.l.b) and Join(l.u.b) operations • Example: Real number and “less than or equal to” • Lattice of fuzzy numbers

  14. Lattice of Fuzzy Numbers • Distributive lattice • MIN[A,MAX(B,C)]=MAX[MIN[A,B],MIN[A,C]] • MAX[A,MIN(B,C)]=MIN[MAX[A,B],MAX[A,C]] • Example: See Fig. 4.6 • Example: “very small” <= “small” <= … <= “very large”

  15. Fuzzy Equations • Addition • X = B-A is not a solution because A+(B-A) is not B. • Conditions to have a solution • Solution

  16. Fuzzy Equations • Multiplication • X = B/A is not a solution. • Conditions to have a solution • Solution

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