1. Fuzzy Numbers

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

3. Examples 3. 1. 3. 1. 1.54.5 3. 1. 4.51.5 3. 1. 4.51.5

4. Arithmetic Operations Interval Operations Interval Operations A = [ a 1, a 3 ], B = [ b 1, b 3 ]

5. Examples Addition [2,5]+[1,3]=[3,8] [0,1]+[-6,5]=[-6,6] [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. * can be any operations including arithmetic operations. Example Example A = { 1/2, 0.5/3}, B = { 1/3, 0.8/4}

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

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

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

12. An Application of Optimal Decision Decision Making Which alternative are you going to choose? Which alternative are you going to choose? weighted average. How to make use of resulted fuzzy set Similarity comparisons with model fuzzy sets Criterion 1 Criterion 2 Criterion 3 Alternative 1 LowHighHigh Alternative 2 HighHighLow Alternative 3 LowHighMedium Weight of Attribute highmediumLow

13. Lattice of Fuzzy Numbers Lattice Lattice Partially ordered set with ordering relation Partially ordered set with ordering relation Meet(g.l.b) and Join(l.u.b) operations Meet(g.l.b) and Join(l.u.b) operations Example: 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]] MIN[A,MAX(B,C)]=MAX[MIN[A,B],MIN[A,C]] MAX[A,MIN(B,C)]=MIN[MAX[A,B],MAX[A,C]] MAX[A,MIN(B,C)]=MIN[MAX[A,B],MAX[A,C]] Example: See Fig. 4.6 Example: See Fig. 4.6 Example: “very small” <= “small” <= … <= “very large” Example: “very small” <= “small” <= … <= “very large”

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

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