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

C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001 1979
Fig. 4. Ageneral separable linear membership function.
Proof. This proposition can be inspected as follows:
(i) If log aij 6 log aij;1 then
(log aij)=sij;1(log aij − log aij;0)+ sij;2 − sij;1
(|log aij − log aij;1| + log aij − log aij;1)
2
= sij;1(log aij − log aij;0):
(ii) If log aij;1 6 log aij 6 log aij;2 then
(log aij)=sij;1(log aij − log aij;0)+ sij;2 − sij;1
(|log aij − log aij;1| + log aij − log aij;1)
2
+ sij;3 − sij;2
(|log aij − log aij;2| + log aij − log aij;2)
2
= sij;1(log aij − log aij;0)+ sij;2 − sij;1
(|log aij − log aij;1| + log aij − log aij;1):
2
(iii) If log aij;k ′ −1 6 log aij 6 log aij;k ′ then �m−1 k¿k ′(|log aij −log aij;k ′ −1 | +log aij −log aij;k ′ −1)=0
and (log aij) becomes sij;1(log aij −log aij;0)+ � k ′ −1
k=2 (sij;k −sij;k−1)=2(|log aij −log aij;k−1|+
log aij − log aij;k−1).
This proposition is then proved. Consider the following example.
Example 2.
Maximize (log a 1 24)
Subject to a 1 24 ¿ 0;
where log a 1 24
is a fuzzy value depicted in Fig. 5.