The Concept of Fuzzy Theory

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The Concept of Fuzzy Theory

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????

• ???????????,?????????????????????????????????????
  ??
• ?????????????????????????????????,??????????
• ??????????????????????
• ??????????????????????????????????????????????????
  ???????????????????

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?????

• ?????(Fuzzy Set Theory)?1965???????????????(L.A.Za
  deh)?????
• ???????????????????(uncertainty)????(fuzziness)??,
  ???????????????
• ??????FA(X)???0?1??,??????X????A???????????

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???

• ??????????,?????? (x)U?0,1,??????
• ???????
• ???????
• ?????x0,?? (x0)1?
• ?????????????,???????,????????????(Triangular
  Fuzzy Number,TFN),??????(Trapezoidal Fuzzy
  Number,TrFN)?

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?????
µ(x)  
               
 

1
x
0
a
b
c
d
?????
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FEATURES OF THE MEMBERSHIP FUNCTION(1)

• All information contained in a fuzzy set is
  described by its membership function.
• Core
• Support
• Boundaries
• Normal

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FEATURES OF THE MEMBERSHIP FUNCTION(2)

• Convex
• Intersection
• Crossover points
• Define fuzzy set A has value equal to 0.5
• Height
• Fuzzy number
• Most common forms of MF are those that are normal
  and convex.

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FUZZIFICATION(1)

• Fuzzification is the process of making a crisp
  quantity fuzzy.
• Digital voltmeter(???)
• Range of errors

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MEMBERSHIP VALUE ASSIGNMENTS

• Intuition
• Inference
• Rank ordering
• Angular fuzzy sets
• Neural networks
• Genetic algorithms
• Inductive reasoning
• Soft partitioning (ch11)
• Meta rules (not addressed in this book)
• Fuzzy statistics (not addressed in this book)

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Intuition

• It is simply derived from the capacity of humans
  to develop membership functions through their own
  innate intelligence and understanding.

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Inference

• In the inference method we use knowledge to
  perform deductive reasoning.
• Triangle
• 4.2 identification of a triangle
• 4.3 approximate isosceles triangle (?????)
• 4.4 right triangle(?????)
• 4.5 intersection of the isosceles and right
  triangle
• 4.6 equilateral triangle (????)
• 4.7 all triangular shapes other than I,R and E
• Example 4.1.

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Rank Ordering(1)

• Assessing preference
• Example4.2
• 1000 people
• Pairwise preference among five colors.
• Xred,organge,yellow,green,blue

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Rank Ordering(2)

• Ex4.2 Membership function for best color

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For example, the membership function of
linguistic variable S
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????????

• ???????
• ????? (a,b,c)
• ????? (a,b,c,d)
• ?????
• ???????

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???????(1/2)

1. ???????
2. ???????
3. ????????

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???????(2/2)

1. ??????? (OR Operation)
2. ??????? (AND Operation)

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?????????

• ??????? (a1,a2,a3)? (b1,b2,b3)???????
• ???? ( )
• (a1,a2,a3)()(b1,b2,b3)(a1 b1 , a2 b2 , a3
  b3)
• ???? ( - )
• (a1,a2,a3)( - ) (b1,b2,b3)(a1 - b1 , a2 - b2 ,
  a3 - b3)
• ???? ( )
• (a1,a2,a3)( )(b1,b2,b3)(a1 b1 , a2 b2 , a3
  b3)
• ???? ( ) ?b1,b2,b3? 0
• (a1,a2,a3)( )(b1,b2,b3)(a1b3, a2 b2 , a3b1)

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?????a-cut

• ?????????a-cut?? ,???? ,???X??,
• ?????a-cut,???X????????
• ?????????a??????????
• a??????(confidence level),???????,a-cut???????,???
  ??????,???????(crisp set) ?

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????(Defuzzification)

• ???????,???????????????,????????????????,?????????
  ???,???????????,??????????????
• ???(Center of Gravity Method)
• ????????(Mean of Maximum Method)
• ???