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366 Biofuels for a More Sustainable Future

Table 13.3 Linguistic variables for rating the role of the decision-makers
(Zhang and Liu, 2011)

Linguistic terms Abbreviation IFN
Very important VI (0.90,0.05,0.05)
Important I (0.75,0.20,0.05)
Medium M (0.50,0.40,0.10)
Unimportant U (0.25,0.60,0.15)
Very unimportant VU (0.10,0.80,0.10)

R R R
Assume that R k ¼(μ k ,υ k ,π k ) is the role importance of the kth decision-
maker described by the intuitionistic fuzzy numbers, and the role impor-
tance of each decision-maker in crisp numbers can be determined by
Eq. (13.20) (Boran et al., 2009).
μ R

μ + π R k
R
R
k k μ + υ R
k k
K R
λ k ¼ (13.20)
X μ
R
μ + π R k
k k μ + υ R
R
k¼1 k k
where λ k represents the role weight of the kth decision-maker, and
λ 1 +λ 2 +⋯+λ K ¼1.
The decision-making matrices presented in Eq. (13.18) can be aggre-
gated into a unique decision-making matrix by using the intuitionistic fuzzy
¸
weighted averaging (IFWA) operator (B€uy€uk€ozkan and G€ocer, 2017), see
Eqs. (13.21), (13.22).

y
y,2
y,1
y
y,1
y,2
μ , υ , π y ij ¼ IFWA λ μ , υ , π y,1 , μ , υ , π y,2 , …, μ y,K , υ y,K , π y,K
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
!
K K K K
Y y,k λ k Y y,k λ k Y y,k λ k Y y,k λ k
1 μ , υ , 1 μ υ
¼ 1 ij ij ij ij
k¼1 k¼1 k¼1 k¼1
(13.21)
C 1 C 2 C n
y y y y y y y y y
1n
12
1n
1n
11
12
11
12
11
A 1 ð μ , υ , π Þ ð μ , υ , π Þ ⋯ ð μ , υ , π Þ
y y y y y y y y y
21 21 21 22 22 22 2n 2n 2n (13.22)
Y ¼ A 2 ð μ , υ , π Þ ð μ , υ , π Þ ⋯ ð μ , υ , π Þ
⋮ ⋮ ⋮ ⋱ ⋮
y
y
y
y
y
μ , υ , π Þ μ , υ , π Þ ⋯ μ , υ , π y
y
y
y
A m ð m1 m1 m1 ð m2 m2 m2 mn mn mn
y
y
y
where Y represents the aggregated decision-making matrix, and (μ ij ,υ ij ,π ij )
represents the aggregated score of the ith biofuel production pathway with
respect to the jth criterion.