RecommendationMeetsall ofthedesires.(Max-min strategy)MinimumMinimum+!Secondminimum.Meets the desires asmanyas possible.AllW1W2W3AverageAs many asAtleasthalfAtleastone010.70.81.9/3=0.630.4020.80.50.51.8/3=0.60030.50.61.8/3=0.600.7040.60.30.91.8/3=0.60050.50.90.41.8/3=0.60Meetsalleasthalfofthedesires. MaximumMaximum+Secondmaximum.Meets at least one desire.(Max-max strategy)16
Recommendation 16 Average All As many as At least half At least one 0.7 0.8 0.4 1.9/3=0.63 0.8 0.5 0.5 1.8/3=0.60 0.5 0.6 0.7 1.8/3=0.60 0.6 0.3 0.9 1.8/3=0.60 0.5 0.9 0.4 1.8/3=0.60 o1 2 o o3 4 o o5 w1 w2 w3 Maximum 𝟐𝟐 𝟑𝟑 Maximum + 𝟏𝟏 𝟑𝟑 Second maximum 𝟐𝟐 𝟑𝟑 Minimum + 𝟏𝟏 𝟑𝟑 Second minimum Minimum • Meets all of the desires. (Max-min strategy) • Meets the desires as many as possible. • Meets al least half of the desires. • Meets at least one desire. (Max-max strategy)
Ordered Weighted Averaging Operators:Linguistic quantifiers (LQ) (Zadeh, 1983).LO=allZadehLA(1983)Acomputational approachtofuzzyquantifiers innatural.LQ=asmanyaspossiblelanguages,ComputersandMathematicswithApplications9(1),149-184.YagerRR,KacprzykJ(1997)TheOrderedWeightedAveragingOperators:.LO=at leasthalfTheoryandApplications,Kluwer,Norwell.LQ=at least one: Ordered weighted averaging operators (Yager and Kacprzyk, 1997)MembershipFunctionIncasethenumberofattributes(J)is3:LinguisticQuantifier[0, x±1LQ= allf(x)=μ=0, μ=0, ,=1,x=11.0,x≤0.5μi=0, μ2 =1/3, 3 = 2/3LQ=asmanyaspossiblef(x)=2x-1, x≥0.52x,x≤0.5LQ= at least halff(x)=μ =2/3, μ,=1/3, μ, =01.x≥0.50.x<1/JLQ= at least onef(x):μ=1, μ2=0,=0x≥1/J117
Ordered Weighted Averaging Operators • Linguistic quantifiers (LQ) (Zadeh, 1983) • LQ = all • LQ = as many as possible • LQ = at least half • LQ = at least one • Ordered weighted averaging operators (Yager and Kacprzyk, 1997) 17 Zadeh LA (1983) A computational approach to fuzzy quantifiers in natural languages, Computers and Mathematics with Applications 9(1), 149-184. Yager RR, Kacprzyk J (1997) The Ordered Weighted Averaging Operators: Theory and Applications, Kluwer, Norwell. ≥ < = = ≥ ≤ = = − ≥ ≤ = = = ≠ = = x J x J f x x x x f x x x x f x x x f x 1, 1 0, 1 LQ at least one ( ) 1, 0.5 2 , 0.5 LQ at least half ( ) 2 1, 0.5 0, 0.5 LQ as many as possible ( ) 1, 1 0, 1 LQ all ( ) Linguistic Quantifier Membership Function µ1 = 0, µ 2 = 1 3, µ3 = 2 3 In case the number of attributes (J) is 3: µ1 = 2 3, µ2 =1 3, µ3 = 0 µ1 = 0, µ2 = 0, µ3 =1 µ1 =1, µ2 = 0, µ3 = 0
Calculation of weights for“"as many as possiblef(x)Themembershipfunctionisgivenbyif0≤x≤1/20f(x) =[2x-1 if 1/2≤x≤113The weights are calculated by1/3j =1,2,3u,=fμ2x01/3 1/2 2/31μ = 0, μ2 =1/3, 3 = 2/3AsmanyaspossibleWhen thenumberof attributes (evaluation items)is 4,this3willbechanged to4.18
Calculation of weights for “as many as possible” 18 1 0 1 x f (x) 1 3 1 2 2 3 µ 2 µ3 1 3 As many as possible − ≤ ≤ ≤ ≤ = 2 1 if 1 2 1 0 if 0 1 2 ( ) x x x f x µ1 = 0, µ 2 = 1 3, µ3 = 2 3 The membership function is given by The weights are calculated by , 1,2,3 3 1 3 = − − = j j f j f µ j When the number of attributes (evaluation items) is 4, this 3 will be changed to 4
Calculation of weights for“at least half"f(x)Themembershipfunctionisgivenby[2x if 0≤x≤1/212f(x)=if 1/2≤x≤12/3The weights are calculated by川μ,=j = 1,2,3x01/3 1/2±2/31μ, =2/3,μz=1/3,μ= 0AtleasthalfWhenthenumberofattributes(evaluationitems)is4,this3willbechangedto4.19
Calculation of weights for “at least half” 19 x f (x) µ 2 At least half µ1 ≤ ≤ ≤ ≤ = 1 if 1 2 1 2 if 0 1 2 ( ) x x x f x The membership function is given by , 1,2,3 3 1 3 = − − = j j f j f µ j µ1 = 2 3, µ2 =1 3, µ3 = 0 The weights are calculated by When the number of attributes (evaluation items) is 4, this 3 will be changed to 4. 1 0 1 3 1 2 2 3 1 2 3
Question.Inthe case of J=4,calculate the weights in the following cases:(J:thenumberofevaluationitemsorattributes: LQ = all.LQ=asmanyaspossible.LQ= at least half.LQ=at least oneeight calculatio formula: u, = (4) - (1一),i =1,2,3,420
Question • In the case of J = 4, calculate the weights in the following cases: (J: the number of evaluation items or attributes • LQ = all • LQ = as many as possible • LQ = at least half • LQ = at least one 20 , 1,2,3,4 4 1 4 = − − = j j f j f Weight calculation formula: µ j