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

1982 C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001
Substitute rij;k−1 by � k−1
m−1 �
PP6: Maximize z =
‘=1 dij;‘, PP5 then becomes
k=2
ck
�
�k−1
log aij − log aij;k−1 +
Subject to log aij + dij;1 ¿ log aij;1;
log aij + dij;1 + dij;2 ¿ log aij;2;
.
.
.
.
‘=1
dij;‘
log aij + dij;1 + dij;2 + ···+ dij;m−3 ¿ log aij;m−3;
log aij + dij;1 + dij;2 + ···+ dij;m−2 ¿ log aij;m−2;
log aij ∈ F(a feasible set); aij ¿ 0; ck 6 0:
Since the rst (m − 2) constraints display log aij ¿ log aij;m−2 − � m−2
‘=1 dij;‘ ¿ log aij;m−3 −
� m−3
‘=1 dij;‘ ¿ ···¿ log aij;2 − dij;1 − dij;2 ¿ log ij;1 − dij;1 ¿ 0, the rst (m − 3) constraints in PP6
are covered by the (m − 2)th constraint in PP6. As a result, Proposition 4 is proved.
Hence, consider (log a 1 24
where
) in Example 2, expression (4.5) can be linearized as
(log a1 �
24)=2:65754 log a1 � �� �
1
24 − log − 1:52177 log a
3
1 �
2
24 − log
3
−2:27155(log a 1 24 − log 1+d 1 24;1 + d 1 24;2)
−10:81401
�
log a 1 24 − log
�
3
2
�
�
+ d 1 24;1 + d 1 24;2 + d 1 24;3
= −11:94979 log a 1 24 − 14:60733d 1 24;1 − 13:08556d 1 24;2
−10:81401d 1 24;3 +2:90424;
�
�
+ d 1 24;1
log a 1 24 + d 1 24;1 + d 1 24;2 + d 1 24;3 ¿ 0:17609: (4.6)
Nevertheless, Example 2 can be reformulated as
Maximize (log a 1 24)
Subject to (log a 1 24)=− 11:94979 log a 1 24 − 14:60733d 1 24;1 − 13:08556d 1 24;2
−10:81401d1 24;3 +2:90424
� �
3
=0:17609; a
2
1 24 ¿ 0:
log a 1 24 + d 1 24;1 + d 1 24;2 + d 1 24;3 ¿ log
�