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Standard fuzzy arithmetic (SFA)

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- introduced by Zadeh, 1975
- basic aim: to extend the operations +, - , *, / to the domain of fuzzy quantities
- How to evaluate A+B?
1) a-cut representation a(A+B) = aA + aB

2) extension principle

(A+B)(z) = sup { min(A(x), B(y)) : x, y R, x + y = z }

A

B

A+B

A

B

A - A 0

A - A + A - A 0

A / A 1

B . B

- introduced by G.J.Klir, 1997
- motivation:
- to reduce the increase of vagueness
- to satisfy more of the classical laws of arithmetic

- equality constraint:
when one variable occurs more than once in the same expression

- different evaluations of AA
- CFA:
- SFA:

- undecomposability:
(A+A+B)CFA = (A+A)CFA+B A+(A+B)CFA

( A - A + A - A )CFA = 0

( A / A )CFA = 1

( B*B )CFA

( (A-1-4*B)/(1+A+B)+B )CFA

- On each a-level, we have to find both extremes inside the multidimensional interval formed by the cartesian product of a-levels of all distinct variables.

aB

aA

- The extreme may be anywhere inside the multidimensional interval non-algorithmizable task.
- Blind search: O(v.uv), where v is the number of distinct variables and u is resolution on the support
How to reduce the complexity?

- decompose, apply the SFA wherever possible, use the monotonicity
- Ex.: ((A-B)*(A-B) + C * (D+E) * E)CFA = ((A-B)* (A-B))CFA + C * ((D+E) * E)CFA
(all variables are fuzzy numbers, D and E are positive)

- (A-B)*(A-B) - “vertex” expression