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

C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001 1975
Fig. 2. Afuzzy value log a12.
Proof. This proposition can be veri ed as follows:
(i) If log aij 6 log aij;2 then
(log aij)=sij;L(log aij − log aij;1)+ sij;R − sij;L
(|log aij − log aij;2| + log aij − log aij;2)
2
= sij;L(log aij − log aij;1):
(ii) If log aij;2 6 log aij 6 log aij;3 then
(log aij)=sij;L(log aij − log aij;1)+ sij;R − sij;L
(|log aij − log aij;2| + log aij − log aij;2)
2
= sij;L(log aij − log aij;1)+(sij;R − sij;L)(log aij − log aij;2)
= sij;L(log aij;2 − log aij;1)+sij;R(log aij − log aij;2):
This proposition is then proved. Consider the following example.
Example 1.
Maximize (log a12)
Subject to a12 ¿ 0;
where log a12 is a fuzzy value depicted in Fig. 2.
Through Proposition 2, (log a12) can be expressed as follows:
−14:93718 − 5:67887
(log a12)=5:67887(log a12 − log 2) +
2
(|log a12 − log 3| + log a12 − log 3)