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344 Biofuels for a More Sustainable Future

depict the relative importance/preference of one criterion over another
accurately. For instance, it is difficult or even impossible to depict the rel-
ative weight/priority of a criterion over another when the stakeholders/
decision-makers think that the relative importance of a criterion over
another is between “moderate importance” (corresponding to number 3)
and “essential importance” (corresponding to number 5). Accordingly,
the interval number [3 5] should be used to depict this judgment. In a similar
way, the interval comparison matrix for determining the relative importance
(weights) of the n metrics can be established:
⋯
C 1 C 2 C n
L U L U
C 1 1 m , m ⋯ m , m
12 12 1n
L U L 1n
U
m , m 1 ⋯ m , m (12.8)
M ¼ C 2 21 21 2n 2n
⋮ ⋮ ⋮ ⋱ ⋮
L U L U
m , m m , m ⋯ 1
C n
n1 n1 n2 n2
where M represents the interval pair-wise matrix for determining the rel-

L
U
ative weights of the n criteria, [m ij ,m ij ] which is an interval number repre-
L
sents the relative preference of the ith criterion over the jth criterion, and m ij
U
U
L
and m ij are the lower and upper boundary of the interval number [q ij ,q ij ].
The relative preference of the jth criterion over ith metric can be deter-
mined by Eq. (12.9).
" #
h i 1 1 1
L U
m , m , , i, j ¼ 1,2,…,n (12.9)
ji ji ¼ h L U i ¼ m U m L
m , m ij ij
ij ij
Step 2: Decomposing the interval pair-wise comparison matrix into two
crisp nonnegative matrices.
The interval pair-wise comparison matrix in Eq. (12.8) can be decom-
posed into two crisp nonnegative matrices, as presented in Eqs. (12.10),
(12.11), respectively.
L
1 m ⋯ m
L
12
U 1n
1=m 1 ⋯ m
21 L
⋮ ⋮ ⋱ ⋮
2n (12.10)
M L ¼

U U
1=m 1=m ⋯ 1
n1 n2
U
1 m ⋯ m
12 U
L 1n
1=m 1 ⋯ m
21 U
⋮ ⋮ ⋱ ⋮
2n (12.11)
M U ¼

L L
1=m 1=m ⋯ 1
n1 n2