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

1990 C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001
In real world AHP problems, a fuzzy rating should not be limited to concave shape or
convex shape only. Using Corollary 3 to treat concave, convex, or concave–convex mixed
fuzzy evaluations, this work introduces the following model.
5.4. The proposed GP-AHP model (4)
Minimize
⎧
⎨
⎩
m�
m�
q=1 q ′ ¿q e=1
+
m�
⎧⎡
⎨ m�
− ⎣
⎩
n�
E�
[(log vq − log vq ′) − log aeqq ′ +2 e qq ′]
n�
q=1 i=1 j¿i e=1
m�
E�
E�
[(log vqi − log vqj) − log ae qij +2 e ⎫
⎬
qij]
⎭
(log a e qq ′)+
q=1 q ′ ¿q e=1
q=1 i=1 j¿i e=1
Subject to log vq − log vq ′ − log aeqq ′ + e qq ′ ¿ 0; log vqi − log vqj − log ae qij + e qij ¿ 0;
log ae ⎧
⎨
qq ′ =
⎩ (log aeqq ′) −
−
log ae m−2 �
qq ′ +
k=1
�
m�
for k where s e
qq ′ ;k ¿se
qq ′ ;k−1
−ye qq ′ ;k−1 )+
m−1 �
n�
�
n�
E�
for k where s e
qq ′ ;k ¡se qq;;k−1
k=1
d e qq ′ ;k ¿ log ae qq ′ ;m−2
0 6 d e qq ′ ;k 6 log ae qq ′ ;k − log ae qq ′ ;k−1
ye qq ′ ;k−1 ¿ log aeqq ′ +(ve qq ′ ;k−1
ye qq ′ ;k−1 ¿ 0
log ae ⎧
⎨
qij =
⎩ (log aeqij) −
−
�
(log ae ⎤⎫
⎬
qij) ⎦
⎭
�
(se qq ′ ;k − se qq ′ ;k−1 )
k−1
‘=1
d e qq ′ ;‘
(s e qq ′ ;k − se qq ′ ;k−1 )(ve qq ′ ;k−1 log ae qq ′ ;k−1
(s e qq ′ ;k − se qq ′ ;k−1 ) log ae qq ′ ;k−1
− 1)M
for k where s e qij; k ¿se qij; k−1
�
for k where s e qij; k ¡se qij; k−1
for s e qq ′ ;k ¡se qq ′ ;k−1 ;
for s e qq ′ ;k ¿se qq ′ ;k−1 ;
(s e qij;k − se qij;k−1 )
⎫�
⎬
(s
⎭
e qq ′ ;m−1 );
(se qij;k − se qij;k−1 )
�k−1
‘=1
d e qij;‘