Chap 2: Aggregation Operations on Fuzzy Sets u Goal: In this chapter we provide a number of different methodologies for aggregating fuzzy subsets. u t-norm and t-conorm generalize the intersection and union operations. u Ordered weighted aggregation (OWA)

2.1: Why Fuzzy Models? u Fuzzy set representation: X: universe of discourse (all the possible values that a variable can assume) A: a subset of X (a) (b) (c) u Fuzzy rules: If condition, then action.

Conventional Control vs Fuzzy Control Conventional: Fuzzy: measured data math-based model calculated result plant measured data fuzzy rule base Defuzzified output plant

Fuzzy Rule vs Conventional Control u Fuzzy expression: If brake temperature is WARM and speed is NOT VERY FAST, then brake pressure is SLIGHTLY DECREASED. u Conventional control: If brake temperature is greater than 280 and speed is less than 45, then brake pressure is 190.

Fuzzy Rules u If condition 1, then action 1. If condition 2, then action 2. … If condition N, then action N. u Ex: rule 1: If temperature is comfortable, then make fan speed low. rule 2: If temperature is warm, then make fan speed medium.

Membership functions T fuzzification process defuzzification process 1 cdctwm hot 0.4 low md rpm

Fuzzy Rules u Consequent parts: A singleton: If X is A1 and Y is B1, then Z is K. A fuzzy number: If X is A1 and Y is B1, then Z is C. A simple function: If X is A1 and Y is B1, then Z=f(X,Y).

2.2: Intersection and Union of Fuzzy Sets u Def: Consider A and B two fuzzy subsets of X. We can define E=A  B such that E(x)=A(x)B(x), the ordinary product, and F=A  B such that F(x)=A(x)+B(x)- A(x)B(x). u Ex: Assume A={1/a,.4/b,.8/c,.2/d,0/e,.5/f} and B={1/a,1/b,.8/c,.5/d,.3/e,0/f}. E1= A  B={1/a,.4/b,.8/c,.2/d,0/e,0/f} by Min operation. F1=A  B= {1/a,1/b,.8/c,.5/d,.3/e,.5/f} by Max operation. E2= A  B={1/a,.4/b,.64/c,.1/d,0/e,0/f} by Product operation. F2=A  B= {1/a,1/b,.96/c,.6/d,.3/e,.5/f} by Product operation.

2.2: t-norm and t-conorm u Def: An operator T T: [0,1]  [0,1]  [0,1] is called a t-norm operator if (1) T(a,b)=T(b,a) Commutativity (2) T(a, T(b,c))=T(T(a,b),c) Associativity (3) T(a,b)  T(c,d) if a  c and b  d Monotonicity (4) T(a,1)=a One identity u Def: An operator S S: [0,1]  [0,1]  [0,1] is called a t-conorm operator if (1) S(a,b)=S(b,a) Commutativity (2) S(a, S(b,c))=S(S(a,b),c) Associativity (3) S(a,b)  S(c,d) if a  c and b  d Monotonicity (4) S(a,0)=a Zero identity

2.2: t-norm and t-conorm u Assume T is any arbitrary t-norm operator; then an operator S defined by S(a,e)=1-T(ā,ē) can be shown to be a t-conorm. u If S is a t-conorm, then T(a,e)=1-S(ā,ē) is a t-norm. u The following is a short table of some commonly encountered t-norm and t-conorm duals. t-normt-conorm Min(a,b)Max(a,b)Min/Max aba+b-abProduct/ Probabilistic sum Max(0, a+b-1)Min(1, a+b)Bold union/ Bounded sum

2.3: Ordered Weighted Averaging (OWA) Operators u Def: An OWA operator of dimension n is a mapping that has an associated n vector W such that where with the jth largest of the u Ex: Assume f(.7,1,.3,.6)=(.4)(1)+(.3)(.7)+(.2)(.6)+(.1)(.3)=.76.

2.3: Ordered Weighted Averaging (OWA) Operators u Three important special cases of OWA aggregations can be pointed out. (1) in this case (2) in this case (3) in this case u It can easily be seen that

2.3: Ordered Weighted Averaging (OWA) Operators u For any OWA operator F, u The OWA operator is commutative. Let be a bag of aggregates and let be any permutation of the ; then for any OWA operator F, u The OWA operator allows us, by appropriate choice of the weighting vector, to move continuously from the and (Min) to or (Max) type aggregation. u The orness measure is defined as u

2.3: Ordered Weighted Averaging (OWA) Operators u A measure of andness can be defined as andness(W)=1-orness(W). u Theorem: Assume W and W’ are two n-dimensional OWA vectors such that where  >0 and j<k; then orness(W)  orness(W’). u Note that different vectors W can attain a given measure of orness. For example, the followings attain a measure of orness of 0.5.