A GP-AHP method for solving group decision-making fuzzy AHP ...

1984 C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001
Fig. 6. Afuzzy value log a 2 24.
Take (4.6) as an example. After switching (log a24) and log a24,
log a 1 24 =[ (log a 1 24)+14:60733d 1 24;1 +13:08556d 1 24;2
+10:81401d1 24;3 − 2:90424)]=(−11:94979)
where log a1 24 + d124;1 + d124;2 + d124;3 ¿ 0:17609.
In reality, fuzzy ratings obtained from the decision-maker(s) may be concave, convex, or
concave–convex mixed membership functions. Since Corollary 2 cannot treat convex membership
functions, the proposition of treating a convex membership function is introduced below.
Consider the following example.
Example 3.
Maximize (log a 2 24)
Subject to a 2 24 ¿ 0;
where log a 2 24
is a convex–concave mixed fuzzy value depicted in Fig. 6.
) can be expressed as follows:
(log a2 �
24)=1:99316 log a2 � ��
1
24 − log +
3
0:27839
��
���
log a
2
2 � ��
2 ���
24 − log
3
� ��
2
Referring to Proposition 3, the (log a 2 24
+ log a 2 24 − log
3
+ −3:40732
(|log a
2
2 24 − log 1|