Prof' Dr' Jaroslav Ramk

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Fuzzy sets II

• Prof. Dr. Jaroslav Ramík

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Content

• Extension principle
• Extended binary operations with fuzzy numbers
• Extended operations with L-R fuzzy numbers
• Extended operations with t-norms
• Probability, possibility and fuzzy measure
• Probability and possibility of fuzzy event
• Fuzzy sets of the 2nd type
• Fuzzy relations

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Extension principle (EP)by L. Zadeh, 1965

• EP makes possible to extend algebraical
  operations with NUMBERS to FUZZY SETS
• Even more EP makes possible to extend REAL
  FUNCTIONS of real variables to FUZZY FUNCTIONS
  with fuzzy variables
• Even more EP makes possible to extend CRISP
  CONCEPTS to FUZZY CONCEPTS
• (e.g. relations, convergence, derivative,
  integral, etc.)

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Example 1. Addition of fuzzy numbers
EP
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Theorem 1.

• Let
• the operation ? denotes or (add or multiply)
• - fuzzy numbers, ??0,1
• -
  ?-cuts
• Then is defined by its ?-cuts
  as follows

???0,1
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Extension principle for functions

• X1, X2,,Xn, Y - sets
• n - fuzzy sets on Xi , i 1,2,,n
• g X1?X2 ??Xn ? Y - function of n variables
• i.e. (x1,x2 ,,xn ) ? y g (x1,x2 ,,xn )
• Then the extended function
• is defined by

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Remarks

• g-1(y) (x1,x2 ,,xn ) y g (x1,x2 ,,xn )
  - co-image of y
• Special form of EP g (x1,x2) x1x2 or g
  (x1,x2) x1x2
• Instead of Min any t-norm T can be used - more
  general for of EP

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Example 2. Fuzzy Min and Max
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Extended operations with L-R fuzzy numbers

• L, R 0,?) ? 0,1 - decreasing functions -
  shape functions
• L(0) R(0) 1, m - main value, ? gt 0, ? gt 0
• (m, ?, ?)LR - fuzzy number of L-R-type if

Left spread Right spread
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Example 3. L-R fuzzy number About eight
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Example 4.

• L(u) Max(0,1 - u) R(u)

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Theorem 2.
Addition

• Let
• (m,?,?)LR , (n,?,?)LR
• where L, R are shape functions
• Then is defined as
• Example (2,3,4)LR (1,2,3)LR (3,5,7)LR

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Opposite FN

• (m,?,?)LR - FN of L-R-type
• (m,?, ?)LR - opposite FN of L-R-type to

Fuzzy minus
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Theorem 3.
Subtraction

• Let
• (m,?,?)LR , (n,?,?)LR
• where L, R are shape functions
• Then is defined as
• Example (2,3,4)LR (1,2,3)LR (1,6,6)LR

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Example 5. Subtraction
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Theorem 4.
Multiplication

• Let
• (m,?,?)LR , (n,?,?)LR
• where L, R are shape functions
• Then is defined by
  approximate formulae
• Example by 1. (2,3,4)LR (1,2,3)LR ? (2,7,10)LR
  

1.
2.
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Example 6. Multiplication

(2,1,2)LR , (4,2,2)LR
? (8,8,12)LR
? formula 1. - - - - formula 2. . exact
function
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Inverse FN

• (m,?,?)LR gt 0 - FN of L-R-type
• -
  approximate formula 1
• - approximate
  formula 2

We define inverse FN only for positive (or
negative) FN !
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Example 7. Inverse FN

(2,1,2)LR
f.2
f.1
? formula 1. - - - - formula 2. . exact
function
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Division

• (m,?,?)LR , (n,?,?)LR gt 0
• where L, R are shape functions
• Define
• Combinations of approximate formulae, e.g.

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Probability, possibility and fuzzy measure

• Sigma Algebra (?-Algebra) on ?
• F - collection of classical subsets of the set ?
• satisfying
• (A1) ? ? F
• (A2) if A ? F then CA ? F
• (A3) if Ai ? F, i 1, 2, ... then ?i Ai ? F

? - elementary space (space of outcomes -
elementary events) F - ?-Algebra of events of ?
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Probability measure

• F - ?-Algebra of events of ?
• p F ? 0,1 - probability measure on F
• satisfying
• (W1) if A ? F then p(A) ? 0
• (W2) p(?) 1
• (W3) if Ai ? F , i 1, 2, ..., Ai ?Aj ?, i?j
• then p(?i Ai ) ?i p(Ai ) - ?-additivity
• (W3) if A,B ? F , A?B ?,
• then p(A?B ) p(A ) p(B) - additivity

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Fuzzy measure

• F - ?-Algebra of events of ?
• g F ? 0,1 - fuzzy measure on F
• satisfying
• (FM1) p(?) 0
• (FM2) p(?) 1
• (FM3) if A,B ? F , A?B then p(A) ? p(B) -
  monotonicity
• (FM4) if A1, A2,... ? F , A1? A2 ? ...
• then g(Ai ) g( Ai )
  - continuity

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Properties

• Additivity condition (W3) is stronger than
  monotonicity (MP3) continuity (MP4) i.e.
• (W3) ? (MP3) (MP4)
• Consequence Any probability measure is a fuzzy
  measure but not contrary

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Possibility measure

• P(?) - Power set of ? (st of all subsets of ?)
• ? P(?) ? 0,1 - possibility measure on ?
• satisfying
• (P1) ?(?) 0
• (P2) ?(?) 1
• (P3) if Ai ? P(?) , i 1, 2, ...
• then ?(?i Ai ) Supi p(Ai )
• (P3) if A,B ? P(?) ,
• then ?(A?B ) Max?(A ), ?(B)

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Properties

• Condition (P3) is stronger than monotonicity
  (MP3) continuity (MP4) i.e.
• (P3) ? (MP3) (MP4)
• Consequence Any possibility measure is a fuzzy
  measure but not contrary

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Example 8.
? A?B?C F ?, A, B, C, A?B, B?C, A?C, A?B?C
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Possibility distribution

• ? - possibility measure on P(?)
• Function ? ? ? 0,1 defined by
• ?(x) ?(x) for ? x??
• is called a possibility distribution on ?
• Interpretation ? is a membership function of a
  fuzzy set , i.e. ?(x) ?A(x) ?x ??,
• ?A(x) is the possibility that x belongs to ?

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Probability and possibility of fuzzy event

• Example 1 What is the possibility (probability)
  that tomorrow will be a nice weather ?
• Example 2 What is the possibility (probability)
  that the profit of the firm A in 2003 will be
  high ?
• nice weather, high profit - fuzzy events

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Probability of fuzzy event Finite universe

• ? x1, ,xn - finite set of elementary outcomes
• F - ?-Algebra on ?
• P - probability measure on F
• - fuzzy set of ?, with the membership
• function ?A(x) - fuzzy event,
• A?? F for ?? ?0,1
• P( ) - probability of fuzzy
  event

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Probability of fuzzy event Real universe

• ? R - real numbers - set of elementary outcomes
• F - ?-Algebra on R
• P - probability measure on F given by density
  fction g
• - fuzzy set of R, with the membership
• function ?A(x) - fuzzy event
• A?? F for ?? ?0,1
• P( ) - probability of fuzzy
  event

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Example 9.

• (4, 1, 2)LR L(u) R(u) e-u

- around 4
- density function of random value
0,036
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Possibility of fuzzy event

• ? - set of elementary outcomes
• ? ? ? 0,1 - possibility distribution
• - fuzzy set of ?, with the membership
• function ?A(x) - fuzzy event
• A?? F for ?? ?0,1
• P( ) -
  possibility of fuzzy event

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Fuzzy sets of the 2nd type

• The function value of the membership function is
  again a fuzzy set (FN) of 0,1

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Example 10.
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Example 11.
Linguistic variable Stature- Height of the body
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Fuzzy relations

• X - universe
• - (binary) fuzzy (valued) relation on X
  fuzzy set on X?X
• is given by the membership function ?R X?X ?
  0,1
• FR is
• Reflexive ?R (x,x) 1 ?x?X
• Symmetric ?R (x,y) ?R (y,x) ?x,y?X
• Transitive SupzMin?R (x,z), ?R (z,y) ? ?R
  (x,y)
• Equivalence reflexive symmetric transitive

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Example 12.
Binary fuzzy relation x is much greater
than y
e.g. ?R(8,1) 7/9 0,77 - is antisymmetric
If ?R (x,y) gt 0 then ?R (y,x) 0 ?x,y?X
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Example 13.
Binary fuzzy relation x is similar to y
X 1,2,3,4,5
is equivalence !
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Summary

• Extension principle
• Extended binary operations with fuzzy numbers
• Extended operations with L-R fuzzy numbers
• Extended operations with t-norms
• Probability, possibility and fuzzy measure
• Probability and possibility of fuzzy event
• Fuzzy sets of the 2nd type
• Fuzzy relations

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References

• 1 J. Ramík, M. Vlach Generalized concavity in
  fuzzy optimization and decision analysis. Kluwer
  Academic Publ. Boston, Dordrecht, London, 2001.
• 2 H.-J. Zimmermann Fuzzy set theory and its
  applications. Kluwer Academic Publ. Boston,
  Dordrecht, London, 1996.
• 3 H. Rommelfanger Fuzzy Decision Support -
  Systeme. Springer - Verlag, Berlin Heidelberg,
  New York, 1994.
• 4 H. Rommelfanger, S. Eickemeier
  Entscheidungstheorie - Klassische Konzepte und
  Fuzzy - Erweiterungen, Springer - Verlag, Berlin
  Heidelberg, New York, 2002.