Processingoffuzzysets-from theory to numerical procedures
Michael Beer 1 / 30 Processing of fuzzy sets – from theory to numerical procedures
ProcessingofFuzzySetsEXTENSIONPRINCIPLEDefinitionGiven are: - n+l fundamental sets X,.., X,....X..z- n fuzzy sets A, onthefundamental sets X withthemembership functions μ(x)= μa.(x); i=1, ., n- the mapping X, ×... × X × ... × X, - Z with z= f(x, ..., x,) andZEZThe mapping f(A,....A,) leads to the fuzzy set B on Z (Fig. 2.12)B - (z,g(2)/z-f(xXi.... ,);zeZ;(x.....)eX,..xX.)(2.32)withthemembershipfunctionsup min [μ(x,)....μ(x,)], if=z=f(x,...x)z-f(xi..,x,)(2.33)μ(z) = Jlg(z) =0otherwiseMichaelBeer2/30
Michael Beer 2 / 30 EXTENSION PRINCIPLE Definition Processing of Fuzzy Sets
ProcessingofFuzzySetsEXTENSIONPRINCIPLEIllustrationCartesian product K = A, × A,(Fuzzy input set)FuzzyresultsetBHk(α)He(z) AA1.01.00.00.0z = f(X; ×2)zA,ExtensionprincipleXB = (z, μg(2)Iz -f(X... );zez;(X...)eX, ..X,)3/30MichaelBeer
Michael Beer 3 / 30 EXTENSION PRINCIPLE Illustration Processing of Fuzzy Sets
ProcessingofFuzzySetsEXTENSIONPRINCIPLEExample:discrete caseZ = f(x,x) = 3 X,-X, +5Inputμ(x)μ(x2)Fuzzysetx, =AFuzzy set &, = A,1.01.00.90.80.80.70.60.30.20.20.10.10.00.0141618/ 20 X2123869ALX113151719MichaelBeer4/30
Michael Beer 4 / 30 EXTENSION PRINCIPLE Example: discrete case Processing of Fuzzy Sets Input
Processing of Fuzzy SetsEXTENSIONPRINCIPLEExample:discretecase,analysisA,A2(8, 0.2)(3, 0.1)(4, 0.3)(5, 0.8)(6, 1.0)(7. 0.7)(9, 0.1)(12, 0.1)(2, 0.1)(5. 0.1)(8, 0.1)(11, 0.1)(14, 0.1)(17.0.1)(20, 0.1)(13, 0.1)(1, 0.1)(4, 0.1)(7, 0.1)(10. 0.1)(16, 0.1)(19.0.1)(13, 0.1)(14, 0.6)(0, 0.1)(3, 0.3)(6, 0.6)(9, 0.6)(12,0.6)(15, 0.2)(18.0.1)(15, 0.6)(8. 0.6)(17, 0.1)(-1, 0.1)(2, 0.3)(5, 0.6)(11,0.6)(14.0.2)(16, 1.0)(-2, 0.1)(1,0.3)(4.0.8)(7. 1.0)(10. 0.7)(13, 0.2)(16, 0.1)(17, 0.9)(3, 0.8)(6. 0.9)(9,0.7)(12, 0.2)(15. 0.1)(-3, 0.1)(0.0.3)(5.0.8)(8, 0.7)(18, 0.8)(-4, 0.1)(-1, 0.3)(2, 0.8)(11, 0.2)(14, 0.1)(1, 0.2)(4, 0.2)(19, 0.2)(-5, 0.1)(-2, 0.2)(7, 0.2)(10, 0.2)(13, 0.1)(20, 0.1)(-6, 0.1)(-3, 0.1)(0, 0.1)(3, 0.1)(6, 0.1)(9, 0.1)(12, 0.1)MichaelBeer5/30
Michael Beer 5 / 30 EXTENSION PRINCIPLE Processing of Fuzzy Sets Example: discrete case, analysis
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,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