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

Maximize
C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001 1973
n�
n�
E�
i=1 j¿i e=1
(log a e ij)
Subject to log a e ij = { (log a e ij)+(s e ij;R − s e ij;L) log a e ij;2
−(s e ij;R − s e ij;L)d e ij + s e ij;L log a e ij;1 }=se ij;R;
log ae ij + log ae ij;2 + de ij ¿ 0; vi;de ij;ae ij ¿ 0;
(i; j) ∈{(i; j) | 1 6 i¡j6 n};
(3.2)
where ae ij represents the eth decision-maker’s uncertainty assessment to which object i is
preferred more than object j; n depicts the number of objects, E represents the number of
decision-makers, fuzzy values log ae ij are conducted by (log aeij ), the term |log vi −log vj −log ae ij |
in (3.1) is the sum of deviations where decision-makers would like to minimize, and the term
(log ae ij ) in (3.2) is the sum of all membership functions’ grades where decision-makers want
to maximize.
To integrate (3.1) and (3.2), the following theorem is introduced.
Theorem 1. Problem 3 is equivalent to the following problem:
Problem 4.
Minimize
n�
n�
i=1 j¿i e=1
E�
| (log vi − log vj) − log ae ij |−
n�
n�
E�
i=1 j¿i e=1
(log a e ij)
Subject to log a e ij = { (log a e ij)+(s e ij;R − s e ij;L) log a e ij;2 − (se ij;R − s e ij;L)d e ij
+ s e ij;Llog a e ij;1 }=se ij;R;
log ae ij + log ae ij;2 + de ij ¿ 0;
vi;de ij;ae ij ¿ 0; (i; j) ∈{(i; j) | 1 6 i¡j6 n};
(3.3)
where is a trade-o weight speci ed by =(n×E)=|M 1 −M 0 | for n×E membership functions.
Proof. The lower ratio deviation within |log vi − log vj − log ae ij | in (3.1) implies that the higher
promising priority vector vij = vi=vj will be generated. The higher grade of membership functions
(log ae ij ) in (3.2) implies that the higher evaluators’ desirability will be derived. In solving
problems with multiple and non-commensurable goals a single solution capable of optimizing
all the goals generally does not exit. Consequently, in certain sense the optimal solution in
Problem 3 is a trade-o solution.
The optimization of simultaneously considering both an objective function and a group of
membership functions constitutes a trade-o problem. By assuming each (log aij) equals one
and after solving Problem 3, the obtained solution is denoted by M 1 . In contrast, when each
(log aij) equal zero, after Problem 3 is solved M 0 denotes the obtained solution. The technique
for order preference by similarity to ideal solution (TOPSIS) initially developed by Yoon et al.