1 / 4

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

uri
Télécharger la présentation

Standard fuzzy arithmetic (SFA)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related