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

C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001 1983
By using LINDO [35] or EXCEL [36], the obtained solution set is ( (log a 1 24 )=1, log a1 24 =0,
a 1 24 =1, d1 24;1 =0, d1 24;2 = 0, and d1 24;3 =0:17609).
Similar to Corollary 1, Corollary 2 is described as follows.
Corollary 2. A general concave fuzzy value log aij can be interpreted as
�
m−2 �
m−1 �
log aij = (log aij) − (sij;m−1 − sij;k)dij;k +
k=1
where log aij + � m−2
‘=1 dij;‘ ¿ log aij;m−2; aij ¿ 0; and sij;0 =0.
k=1
(sij;k − sij;k−1)log aij;k−1
��
(sij;m−1);
Proof. Regarding Proposition 3, a piecewise linear membership function (log aij) can be represented
by
m−1 �
sij;1(log aij − log aij;0)+
k=2
sij;k − sij;k−1
(|log aij − log aij;k−1| + log aij − log aij;k−1):
2
After utilizing Proposition 4 to linearize the absolute terms with negative coe cients, (log aij)
can then be interpreted as
�
�
m−1 �
�k−1
(log aij)=sij;1(log aij − log aij;0)+ (sij;k − sij;k−1) log aij − log aij;k−1 + ;
where log aij + � m−2
‘=1 dij;‘ ¿ log aij;m−2 and aij ¿ 0.
Accordingly, after expanding (2:7), the following occurs:
k=2
(log aij)=sij;1 × log aij − sij;1 × log aij;0 +(sij;2 − sij;1)(log aij − log aij;1 + dij;1)
+(sij;3 − sij;2)(log aij − log aij;2 + dij;1 + dij;2)
‘=1
dij;‘
(4.7)
+(sij;4 − sij;3)(log aij − log aij;3 + dij;1 + dij;2 + dij;3)+···+(sij;m−1 − sij;m−2)
�
m−2 �
log aij − log aij;m−2 +
�
: (4.8)
‘=1
dij;‘
As a result, after reorganizing (4.8), we obtain
m−2 �
m−1 �
(log aij)= (sij;m−1 − sij;k)dij;k − (sij;k − sij;k−1)log aij;k−1 + sij;m−1 × log aij: (4.9)
k=1
k=1
After reversing (log aij) and log aij in (4.9),
log aij =
�
m−2 �
(log aij) −
k=1
�
m−1
(sij;m−1 − sij;k)dij;k +
where log aij + � m−2
‘=1 dij;‘ ¿ log aij;m−2;aij ¿ 0, and sij;0 =0.
Therefore, Corollary 2 is completed.
k=1
(sij;k − sij;k−1)log aij;k−1
��
(sij;m−1);