Page 369 - Biofuels for a More Sustainable Future

Fuzzy MCDM for ranking the biofuels production pathways 325

n
j 1, if the ith alternativehasbeen rankedat thetth place
it (11.12)
φ ¼ 0, if the ith alternativehasnot been rankedat thetth place
j
j
φ ¼ φ (11.13)
it m m
j
j
where φ is the ranking matrix corresponding to the ( j)th criterion and φ it is
the element in the ranking matrix corresponding to the ( j)th criterion.
Step 4: Weighted ranking matrix.
The weighted ranking matrix R can be obtained by Eq. (11.14) and the
elements in the weighted ranking matrix can be calculated by Eqs. (11.15),
(11.16).
1 2 ⋯ m

A 1 r 11 r 12 ⋯ r 1m
R ¼ A 2 r 21 r 22 ⋯ r 2m (11.14)
⋮ ⋮ ⋮ ⋮ ⋮
A m r m1 r m2 ⋯ r mm
n
X
j
φ ω j (11.15)
r it ¼ it
j¼1
!
ω +2ω + ω U X ω +2ω + ω U
M
n
M
L
L
j j j j j j
= (11.16)
ω j ¼
4 4
j¼1
Step 5: Ranking the sequence of the alternatives.
Use the following linear 0–1 programming to rank the alternatives, and
the solutions of this programming are the elements in the final ranking
matrix Z, as shown in Eq. (11.21).If z it ¼1, it means that the (i)th alternative
has been ranked at the (t)th place.
m m
XX
(11.17)
r it z it
Max S ¼
j¼1 t¼1
n
1, if the ith alternativehasbeen rankedat thetth place
0, if the ith alternativehasnot been rankedat thetth place
z it ¼ (11.18)
m
X
z it ¼ 1, i ¼ 1,2,…,m (11.19)
t¼1
m
X
z it ¼ 1, t ¼ 1,2,…,m (11.20)
i¼1
(11.21)
Z ¼ z it m m
fg