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

C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001 1977
By applying Proposition 2, (log aij) can then be linearized as
where
(log aij)=sij;R log aij − (sij;R − sij;L) log aij;2 +(sij;R − sij;L)dij − sij;L log aij;1;
log aij − log aij;2 + dij ¿ 0 and aij;2;dij ¿ 0: (4.4)
Accordingly, after inverting (log aij) and log aij in (4.4),
log aij = { (log aij)+(sij;R − sij;L) log aij;2 − (sij;R − sij;L)dij + sij;L log aij;1}=sij;R;
where log aij + log aij;2 + dij ¿ 0; aij;2;dij ¿ 0.
By doing so, this corollary is completed.
View (4.3) as an instance. After reversing (log a12) and log a12,
log a12 = − 0:06694704 (log a12) − 1:380183542d12 +0:54406825351;
where log a12 − log 3+d12 ¿ 0; a12;d12 ¿ 0.
Herein LLSM was employed to reduce the fractional form wi=wj in Problem 1 to linear form
log wi − log wj. Axioms 1, 2 and Theorem 2 are thereby introduced to prove that all decision
contents retain their original meaning after transformation of a fuzzy number to a log value.
Axiom 1. For a fuzzy number x, let xL; xM; xU are the lower, modal, and upper values of
x, respectively. In a restricted domain [xL;xU], the possibility of (xL) is 0, the possibility of
(xM) is 1, and the possibility of (xU) is 0. The slope between (xL) and (xM) is increasing
while the slope between (xM) and (xU) is decreasing where xL ¡xM ¡xU.
Axiom 2. After transforming a fuzzy number x to a log value, let log xL; log xM; log xU represent
the lower, modal, and upper values of log x, respectively. In a restricted domain [log xL; log xU],
the possibilities of (log xL); (log xM), and (log xU) are separately 0; 1, and 0. The slope
between (log xL) and (log xM) increases while the slope between (log xM) and (log xU)
decreases where log xL ¡ log xM ¡ log xU.
Theorem 2. An optimal solution for the problem PP1 is the same as that for PP2; PP1 and
PP2 are presented below in which xL; xM; xU are the lower; modal; and upper values of fuzzy
number x; respectively;
PP1: Maximize x
Subject to x ¿ 0;
where x is a fuzzy number depicted in Fig:3a:
PP2: Maximize log x
Subject to x ¿ 0;
where log x is a fuzzy number depicted in Fig:3b: