ProcessingofFuzzySetsEXTENSIONPRINCIPLEExample:discrete case,resultHg(z)4Fuzzy result setZ=B1.00.90.80.70.60.3826-20410121416118120z-4-61571391113151719-1-5-3Michael Beer6/30
Michael Beer 6 / 30 EXTENSION PRINCIPLE Processing of Fuzzy Sets Example: discrete case, result
ProcessingofFuzzySetsEXTENSIONPRINCIPLEIllustration of a continuous case,resultμ(z) AExactsolution1.0Approximationbysmoothing0.80.60.40.20.00.2742.5705.859 zMichaelBeer7/30
Michael Beer 7 / 30 EXTENSION PRINCIPLE Illustration of a continuous case, result Processing of Fuzzy Sets
ProcessingofFuzzySetsFUZZYANALYSISASNESTEDINTERVALANALYSISX1X2区=(..X.)→2=(..2)fuzzy inputfuzzy inputvariablex=variableX, =XXi,arX2,αEXi,emapping1,0modelX2,aX1,αl-μ(x1)μ(×2)1Zj e Zj,o0.00.01.01.0ααα-leveloptimizationExact ifμ(z,)1.0Dfuzzy input is convexα(α-level setsareconnected sets)fuzzy result variable Z, =Z(ii)mapping is continuous0.0Zi,d1Ziiar1.08/30MichaelBeer
Michael Beer 8 / 30 FUZZY ANALYSIS AS NESTED INTERVAL ANALYSIS ~ fuzzy result variable zj = Zj ~ 0.0 1.0 α zj Zj,α zj,α l zj,α r µ(zj ) zj ∈ Zj,α mapping model x1 ∈ X1,α x2 ∈ X2,α x = (., xi , .) z = (., zj , .) ~ ~ → ~ ~ 1.0 α 0.0 µ(x1) µ(x2) fuzzy input variable x2 = X2 ~ fuzzy input variable x1 = X1 ~ X1,α x1,α l x1,α r x1 X2,α x2,α l x2,α r x2 ~ ~ 0.0 α 1.0 α-level optimization Processing of Fuzzy Sets Exact if (i) fuzzy input is convex (α-level sets are connected sets) (ii) mapping is continuous
ProcessingofFuzzySetsFUZZYANALYSISASNESTEDINTERVALANALYSISExample:non-continuousmappingP= 10kNM=2kNmX0Xk文2.502.50 m1+5X[kNm] I X,<XkM(x.24-5.x.[KNm] |X,>Xk9/30Michael Beer
Michael Beer 9 / 30 Example: non-continuous mapping Processing of Fuzzy Sets FUZZY ANALYSIS AS NESTED INTERVAL ANALYSIS
Processing of Fuzzy SetsFUZZYANALYSISASNESTEDINTERVALANALYSISExample:non-continuousmappinginputresult fromresultfromextensionprincipleintervalapproachμ(M,)4 μ(x)μ(M,)1.01.01.00.50.50.5Aakakak0.00.00.0000112.32.42.52.6X[m]1112121313M,[kNm]M,[kNMichael Beer10/30
Michael Beer 10 / 30 Processing of Fuzzy Sets Example: non-continuous mapping FUZZY ANALYSIS AS NESTED INTERVAL ANALYSIS input result from extension principle result from interval approach