Algebras of Fuzzy Sets - Fuzzy Logic: An Introductory Course for Engineering Students - page 78

n

1
n

M ( x 1 ,..., x n ) = x i ,

i = 1

1
n

and with f ( x ) =− log x , and p 1 = ( 1 i n ) is the geometric mean

n √ p 1

M ( x 1 ,..., x n ) = · p 2 ··· p n .

x 1 x ʱ (ʱ > ,is f − 1

1

With f ( x ) = 0 ) ( x ) =

ʱ , and with p 1

=

n , it is obtained the

family of quasi-linear means,

x 1

1
ʱ

x n + ··· +

M

ʱ (

x 1 ,..., x n ) = .

n

In particular, with ʱ = 1, is M 1 the arithmetic mean, and with ʱ =− 1, it follows

n

M ( x 1 ,..., x n ) = ( provided x 1 ,..., x n = 0 ),

− 1

1 1

x n x 1 + ··· +

called Harmonic Mean . As it is easy to prove,

n √ x 1 ···

lim
ʱ ₒ

M

ʱ (

x 1 ,..., x n ) = x n

0

lim
ʱ ₒ∞

M

ʱ (

x 1 ,..., x n ) = max ( x 1 ,..., x n )

lim
ʱ ₒ−∞

M ʱ ( x 1 ,..., x n ) = min ( x 1 ,..., x n ).

2.3.2 Ordered Weighted Means

n

It is said that M :[ 0 , 1 ] ₒ[ 0 , 1 ] is a mean , when M is continuous, monotonic,

and verifies:

min ( x 1 ,..., x n ) M ( x 1 ,..., x n ) max ( x 1 ,..., x n ).

Since, min ( 0 ,..., 0 ) M ( 0 ,..., 0 ) max ( 0 ,..., 0 ) = 0, it results M ( 0 ,...,

0 ) = 0. Since, min ( 1 ,..., 1 ) M ( 1 ,..., 1 ) max ( 1 ,..., 1 ) = 1, it results

1. Hence, of course, quasi-linear means are means, but there are
more of such means. An important and useful example are the Ordered Weighted
Means (OWA). Its definition is the following:

M ( 1 ,..., 1 ) =