Soft Computing and Its Applications, Volume One

Fuzzy Sets, Fuzzy Operators, and Fuzzy Relations 109

( , …, ) = .

The ordered weighted averaging operators as stated in Denition 2.39 can be fur-

ther explained by the following examples.

Example 2.39 Let a = [5 6 4]

T

and π is a permutation on {1, ……., n} such that

elements of a are changed in decreasing order. Therefore, = [6 5 4]

T

, and w = [0.2

0.7 0.1]

T

. Then, F(a) = 0.2 × 6 + 0.7 × 5 + 0.1 × 4 = 5.1.

Example 2.40 Let X = {1, 2, 3, 4} with fuzzy sets

[0.0 0.5 0.1 1.0]

T

and [0.1 0.4 1.0 0.0]

T

Hence,

[1.0 0.5 0.1 0.0]T and [1.0 0.4 0.1 0.0].

Moreover, let the associating vector for the OWA operator F be w

=[0.1 0.9]

T

. Then, for ...