z.n. Fuzzy Numbers - JulTob/Mathematics GitHub Wiki

Fuzzy Numbers

Fuzzy numbers are a mathematical formulation of vague statements about real numbers.

For example, a statement expressing that a number is approximately 2 is modelled by a fuzzy number

fuzzy subset A is called a fuzzy (real) number when the universe on which A is defined is the set of all real numbers R and it satisfies the following conditions:

  • All the 𝛼-cuts of A are not empty for 0≤𝛼≤1;
  • All the 𝛼-cuts of A are closed intervals of R;
  • The support of A, that is, the set
    • [0+]A = {x|x ∈ ℝ and A(x) > 0}
    • is bounded

A fuzzy number A is called positive (negative), if A(x) = 0, for all x < 0 (x > 0).

Each x ∈ R can be considered as a fuzzy number x defined by x̃(t) { 1, if t = x, { 0, if t ≠ x.

Triangular Fuzzy Numbers _/r_

Trapezoidal Fuzzy Numbers _/‘’’r’’’_

Gaussian Fuzzy Number ( r )

Generalized Fuzzy Numbers

  • Ã is a continuous mapping from R to the closed interval [0,1]
  • Ã(x) = 0 when −∞<x≤a;
  • Ã is strictly increasing in [a,b]
  • Ã=𝑤 when b≤x≤c
  • Ã is strictly decreasing in [c,d]
  • Ã(x)=0 when d≤x<∞. Here 𝑤 is supposed to be the degree of confidence of some expert’s opinion. If 𝑤 = 1, then the generalized fuzzy number à is called a normal trapezoidal fuzzy number. If a=b and c=d, then à is called a crisp interval. If b=c, then à is called a generalized triangular fuzzy number. If a = b = c = d and 𝑤 = 1, then à is called a real number

Arithmetic of Fuzzy Numbers I = [a, b] and J = [c, d] ★ denotes any of the four arithmetic operations, then

I★J = {x★y|x ∈ I and y ∈ J}.

For example, if I = [1, 4] and J = [2, 3] and ★ denotes addition, then [1,4] + [2,3] = {x+y|x ∈ [1,4] and y ∈ [2,3]}, = [3,7]

[a,b]+[d,e] = [a+d,b+e], [a,b]−[d,e] = [a−e,b−d], [a, b] ⋅ [d, e] = [min(ad, ae, bd, be), max(ad, ae, bd, be)],

0 ∉ [d, e], [a, b]∕[d, e] = [a, b] ⋅ [1∕e, 1∕d]
= [min(a∕d, a∕e, b∕d, b∕e), max(a∕d, a∕e, b∕d, b∕e)].

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