A First Course in FUZZY and NEURAL CONTROL

3.3. COMBINING FUZZY SETS 103

Definition 3.7The ìcompensatory andîoperatoris

≥Yn

i=1

xi

¥ 1 −γ≥

1 −

Yn

i=1

(1−xi)

¥γ

for someγ∈[0,1].

This operator is a combination of the algebraic product and the algebraic
sum. The parameter indicates where the actual operator is located between the
these two, giving the product whenγ=0and the algebraic sum whenγ=1.
When it is desirable to accommodate variations in the importance of in-
dividual items, we can useweighted generalized meansfor the average of
x 1 ,x 2 ,...,xnweighted by the vectorw 1 ,w 2 ,...,wnwith

Pn
i=1wi=1,asdefined
by the formula
√n
X

i=1

wixai

!^1 a

An aggregation technique, due to Yager [83], uses ordered weighted aver-
aging (OWA) operators, which gives the greatest weight to objects of greatest
magnitude.

Definition 3.8AnOWA operator of dimensionn, with associated vector
W=(w 1 ,...,wn)satisfyingwi≥ 0 and

Pn

j=1wi=1,isthemapping

FW:Rn→R:(x 1 ,...,xn) 7 →

Xn

j=1

xσ(i)wi

where

°

aσ(1),...,aσ(n)

¢

is a rearrangement of the coordinates of(x 1 ,...,xn)so
thatxσ(1)≥∑∑∑≥xσ(n).

The rearrangement of the coordinates into an ordered sequence is a crucial
part of this definition.

Example 3.3AssumeW=(0. 3 , 0. 4 , 0 .3).Then,

FW(1, 5 ,3) = (0. 3 , 0. 4 , 0 .3)∑(5, 3 ,1) = 3. 0

Yager pointed out the following special cases:

ïIfW=(1, 0 ,...,0),thenFW(a 1 ,...,an)=max{a 1 ,...,an}.

ïIfW=(0, 0 ,...,1),thenFW(a 1 ,...,an)=min{a 1 ,...,an}.

ïIfW=

° 1

n,

1

n,...,

1

n

¢

,thenFW(a 1 ,...,an)=n^1

Pn

j=1wi.

ïIfW=

≥

0 ,n−^12 ,...,n−^12 , 0

¥

,thenFW(a 1 ,...,an)=the ìOlympic average.î