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Standard fuzzy arithmetic (SFA). 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) = a A + a B 2) extension principle

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Standard fuzzy arithmetic (SFA)

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### Standard fuzzy arithmetic (SFA)

• 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

### SFA - questionable examples

A

B

A - A  0

A - A + A - A  0

A / A  1

B . B

### Constrained fuzzy arithmetic (CFA)

• 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 AA

• 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

### CFA - computational problems

• 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