October 2007 Volume 10 Number 4 - Educational Technology ...

Example 1: Let à and B ~ be two vague sets of the universe of discourse U,
U = {u1, u2, u3, u4, u5},
à = [0.2, 0.4]/u1 + [0.3, 0.5]/u2 + [0.5, 0.7]/u3 + [0.7, 0.9]/u4 + [0.8, 1]/u5
B ~ = [0.3, 0.5]/u1 + [0.4, 0.6]/u2 + [0.6, 0.8]/u3 + [0.7, 0.9]/u4 + [0.8, 1]/u5,
where
V A
~ ( u1
V ~ ( u
A 2
V ~ ( u
A 3
V ~ ( u
A 4
)
)
)
)
V ~ ( u
B 1
V ~ ( u
B 2
V ~ ( u
B 3
V ~ ( u
B 4
V ~ ( u
B 5
= [0.2, 0.4], ) = [0.3, 0.5],
= [0.3, 0.5], ) = [0.4, 0.6],
= [0.5, 0.7], ) = [0.6, 0.8],
= [0.7, 0.9], ) = [0.7, 0.9],
V ~ ( u ) = [0.8, 1], ) = [0.8, 1].
A 5
By applying Eq. (5), we can get
= 0.2 – 0.6 = –0.4,
S(
V ~ ( u ))
A 1
S V ( ))
( ~ u
A 2
S(
V ~ ( u
A 3
S(
V ~ ( u
A 4
S(
V ~ ( u
A 5
S( V ~ ( u
B 1
S( V ~ ( u
B 2
S( V ~ ( u
B 3
S( V ~ ( u
B 4
S( V ~ ( u
B 5
))
))
))
))
))
))
))
))
= 0.3 – 0.5 = –0.2,
= 0.5 – 0.3 = 0.2,
= 0.7 – 0.1 = 0.6,
= 0.8 – 0 = 0.8,
= 0.3 – 0.5 = –0.2,
= 0.4 – 0.4 = 0,
= 0.6 – 0.2 = 0.4,
= 0.7 – 0.1 = 0.6,
= 0.8 – 0 = 0.8.
By applying Eq. (7), the degree of similarity H(Ã, B ~ ) between the vague sets à and B ~ can be evaluated, shown as
follows:
H(Ã, B ~ ) ∑
= ⎟ ⎟
5 1 ⎛ S(
V ~ ( u )) − S(
V ~ ( u )) ⎞
= ⎜ A i B i
1−
5 i 1 ⎜
2
⎝
⎠
1 ⎡⎛
− 0.
4 − ( −0.
2)
⎞ ⎛ − 0.
2 − 0 ⎞ ⎛ 0.
2 − 0.
4 ⎞
= ⎢⎜( 1−
⎟ + ⎜1−
⎟ + ⎜1−
⎟ +
5 ⎣⎝
2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠
⎛ 0.
6 − 0.
6 ⎞ ⎛ 0.
8 − 0.
8 ⎞⎤
⎜1
− ⎟ + ⎜1−
⎟⎥
⎝ 2 ⎠ ⎝ 2 ⎠⎦
1
= ( 0.
9 + 0.
9 + 0.
9 + 1+
1)
5
= 0.94.
It indicates that the degree of similarity between the vague sets à and B ~ is equal to 0.94.
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