Arithmetic Operations on Intervals

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Arithmetic Operations on Intervals

• a, b ? d, e
• f ? g a f b, d g e
• ? any of the four arithmetic operations on closed
  intervals

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• The result of an arithmetic operation on closed
  intervals is again a closed interval.

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Addition

• a, b d, e a d, b e
• 2, 5 1, 3 3, 8
• 0, 1 -6, 5 -6, 6

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Subtraction -

• a, b - d, e a - e, b - d
• 2, 5 - -1, 4 -2, 6
• 3, 7 - 3, 7 -4, 4

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Multiplication

• a, b d, e min(ad, ae, bd, be), max(ad,
  ae, bd, be)
• -1, -1 -2, -0.5 -2, 2
• 3, 4 2, 2 6, 8

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Division /

• a, b / d, e min(a/d, a/e, b/d, b/e),
  max(a/d, a/e, b/d, b/e)
• -1, -1 / -2, -0.5 -2, 2
• 4, 10 / 1, 2 2, 10

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Properties

• For
• Aa1, a2, Bb1, b2, Cc1, c2,
• 0 0,0, 11,1

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• 1. A B B A (commutativity)
• A B B A
• 2. (A B) C A (B C) (associativity)
• (A B) C A (B C)
• 3. A 0 A A 0 (identity)
• A 1 A A 1

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• 4. A (B C) ? A B A C
  (subdistributivity)
• 5. If
• b c 0 for every b ? B and c ? C,
• then
• A (B C) A B A C
  (distributivity)
• If A a, a,
• then
• a (B C) a B a C

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• 6. 0 ? A - A and 1 ? A/A

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• 7. If
• A ? E and B ? F,
• then
• A B ? E F
• A - B ? E - F
• A B ? E F
• A / B ? E / F (inclusion monotonicity)

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Hamming distance
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Hamming distance

• d (A, B) ? (A(x) - B(x) for ? x ? X

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Hamming distance

• FS1 RandomFuzzySet0, 15, Type -gt Complete
• FS2 RandomFuzzySet0, 15, Type -gt Complete

In2HammingDistanceFS1,FS2 Out2
4.94825
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Fuzzy ranking methods example.

• Let A and B be fuzzy numbers whose
  triangular-type membership functions are given
  below. Illustrate three ranking methods.

A B
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Hamming distance ranking method.

• Determine MAX(A, B),
• Calculate d(MAX(A, B), A),
• Calculate d(MAX(A, B), B),
• IF d(MAX(A, B), A) ? d(MAX(A, B), B)
• Then A ? B.

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Hamming distance ranking.
HammingDistanceMAXA, B, A 4/3
HammingDistanceMAXA, B, B 1/3 A B
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MIN and MAX Operations
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Fuzzy number

• - a fuzzy set must be a normal
• - ?-cut must be a closed interval for every
• ? ? (0,1
• - the support of a fuzzy set must be bounded.

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Fuzzy numbers
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Triangular Membership Function
C 11
S 8
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Trapezoidal fuzzy numbers and its degenerated
cases
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Comparable fuzzy numbers.
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Lattice of real numbers R

• R, min, max

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Lattice of real numbers R
min(x,y) x if x y y if y
x max(x,y) y if x y x if y
x, for every pair x,y ? R
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Lattice of real numbers R

• The linear ordering of real numbers does not
  extend to fuzzy numbers.

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Lattice of fuzzy numbers ?

• ?, MIN, MAX

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Lattice of fuzzy numbers ?

• MIN(A,B) MIN (A, B)(z) max min A (x), B
  (y),
• for z min (x, y) and for all z ? R
• MAX(A,B) MAX (A, B)(z) max min A (x), B
  (y),
• for z max (x, y) and for all z ? R

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ShowGraphicsArrayAB, MIN
The two fuzzy numbers, A and B are not
comparable.
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ShowGraphicsArrayAB, MAX
The two fuzzy numbers, A and B are not
comparable.
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ShowGraphicsArrayAB, Min1
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ShowGraphicsArrayAB, Max1
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Properties of MIN and MAX

• Let MIN and MAX be binary operations on all fuzzy
  numbers ?,
• ?x??? and for every A,B,C ? ?
• the following properties hold

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1.Properties of MIN and MAX

• Commutativity
• MIN(A,B) MIN(B,A)
• MAX(A,B) MAX(B,A)

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2.Properties of MIN and MAX

• Associativity
• MINMIN(A,B),C MINA,MIN(B,C)
• MAXMAX(A,B),C MAXA,MAX(B,C)

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3.Properties of MIN and MAX

• Idempotence
• MIN(A,A) A
• MAX(A,A) A

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4.Properties of MIN and MAX

• Absorption
• MINA,MAX(A,B) A
• MAXA,MIN(A,B) A

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5.Properties of MIN and MAX

• Distributivity
• MINA,MAX(B,C) MAXMIN(A,B), MIN(A,C)
• MAXA,MIN(B,C) MINMAX(A,B), MAX(A,C)

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Lattice of fuzzy numbers ?

• MIN(A, B) MIN (A, B)(z) max min A (x), B
  (y),
• for z min (x, y) and for all z ? R
• MAX(A,B) MAX (A, B)(z) max min A (x), B
  (y),
• for z max (x, y) and for all z ? R

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Gaussian fuzzy numbers
FuzzyPlot NEG, ZERO, POS
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Gaussian fuzzy numbers
FuzzyPlot NEG, POS
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MINA,B
A B
MINA,B
The two fuzzy numbers, A and B are not
comparable.
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Fuzzy Arithmetic
A1, A2, A1A2
A1, A2, A1-A2