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

C.-S. Yu / Computers & Operations Research 29 (2002) 1969–2001 1997
= −0:083683478 (log a 2 24) − 0:023296643y 2 24;1 − 0:004102164v 2 24;1 − 1:19009d 2 24;2
− 0:904953978d 2 24;3 +0:243037743; (6.25)
where y2 24;1 ¿ log a224 +(v2 24;1 − 1)M, log a224 + d224;2 + d224;3 6 0:17609, 0 6 d224;2 6 0:17609,
0 6 d2 24;3 ¿ 0:17609, a224 ;y2 24;1 ¿ 0, v2 24;1 is a 0–1 variable, M is a big value.
Step 2: In the form of the proposed model 4, Example 5 is formulated as follows:
Minimize
+
⎧
⎨
⎩
⎧
⎨
−
⎩
4�
4�
q=1 q ′ ¿q e=1
4�
3�
3�
q=1 i=1 j¿i e=1
4�
4�
q=1 q ′ ¿q e=1
3�
(log vq − log vq ′ − log aeqq ′ +2 e qq ′)
3�
(log vqi − log vqj − log ae qij +2 e ⎫
⎬
qij)
⎭
3�
(log a e qq ′)+
Subject to (6:1)–(6:7); (6:10)–(6:25);
4�
3�
3�
3�
q=1 i=1 j¿i e=1
(log ae ⎫
⎬
qij)
⎭
log vq − log vq ′ − log ae qq ′ + e qq ′ ¿ 0; log vqi − log vqj − log a e qij + e qij ¿ 0;
( 3
2 ) 6 log a3 5
24 6 ( 2 );vq;vqi;ae qq ′;ae e e
qij; qq ′; qij;de ij;de qij ¿ 0;
(q; q ′ ) ∈{(q; q ′ ) | 1 6 q¡q ′ 6 4}; (i; j) ∈{(i; j) | 1 6 i¡j6 3}:
Step 3: Based on Theorem 1, the obtained M 1 is 2.05645, M 0 is 0.5751544, and value is
31.72898104.
Step 4: By executing the formulated model on LINDO or EXCEL, the acquired solution set
is (log v1 =0:23571, log v2 =0:41353; log v3 =0, log v4 =0:21752, log v11 =0:17592, log v12 =0,
log v13 =0:27616, log v21 =0:47664, log v22 =0, log v23 =0, log v31 =0:47664, log v32 =0,
log v33 =0, log v41 =0:10024, log v42 =0:20049, log v43 =0, v1 =1:72072, v2 =2:59162, v3 =1,
v4 =1:65014, v11 =1:49939, v12 =1, v13 =1:88869, v21 =2:99671, v22 =1, v23 =1, v31 =2:99671,
v32 =1, v33 =1, v41 =1:25963, v42 =1:58667 and v43 = 1), where the average grade of total
membership functions is computed as 0.74416.
Step 5: After calculating normalized the vector V ,
⎡
⎢
[v1;v2;v3;v4] ⎢
⎣
v11 v12 v13
v21 v22 v23
v31 v32 v33
v41 v42 v43
⎤
⎥
⎦