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Standard fuzzy arithmetic (SFA)

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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  1. 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

  2. SFA - questionable examples A B A - A  0 A - A + A - A  0 A / A  1 B . B

  3. 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

  4. 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

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