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)
= 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
A-A 0
A-A+A-A 0
A/A 1
B.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.
a
B
a
•
•
A
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
•
+
C
+
E
D E
-
*
*
-
A B A B
(A-B)*(A-B) - “vertex” expression
Core(B)
Core (A)
C
+
D E
E