Page 389 - Biofuels for a More Sustainable Future

Prioritization of biofuels production pathways under uncertainties 345

The geometric mean method (Ren, 2018) can be used to determine the
weights according to the matrices presented in Eqs. (12.10), (12.11), and the
weight vectors determined by these two matrices are presented in
Eqs. (12.12), (12.13), respectively.

W L ¼ ω L ω L ⋯ ω L (12.12)
1 2 n
W U ¼ ω U ω U ⋯ ω U (12.13)

1 2 n
where W L and W U represent the weight vectors determined by the matrices
L U
presented in Eqs. (12.10), (12.11), respectively. ω j and ω j are the weights of
the jth metric in W L and W U , respectively.
Step 3: Determining the interval weights. The interval weights of each
metric can be determined by Eqs. (12.14)–(12.16).
v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
u
X 1
u
(12.14)
n
k ¼ u
u X
j¼1 +
t q ij
i¼1
v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n
u
X 1
u
(12.15)
n
m ¼ u
u
X
j¼1
q ij
t
i¼1
It is worth pointing out that if k and m satisfy 0<k 1 m, then the users
can use Eq. (12.16) to determine the interval weight of the jth metric, or the
users should modify the interval pair-wise comparison matrix to make k and
m satisfy this condition.
h i
ω ¼ ω ω ¼ kω L mω U (12.16)

j j,L j,U j j
where ω j represents the interval weight of the jth criterion and ω j, L and

ω j, U are the lower and upper bounds of ω j , respectively.

2.3 Interval gray relational analysis
The interval gray relational analysis (GRA) method was presented in the fol-
lowing six steps based on the work of Zhang (2005), Wang et al. (2017), and
Manzardo et al. (2012).
Step 1: Establishing the interval decision-making matrix. This step is to
determine the weights of the decision criteria by using the IAHP method
and to collect the data of the alternatives with respect to the decision criteria.