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322 Biofuels for a More Sustainable Future

m ~ ()x
a

1

0 a L a M a U X
U
M
L
Fig. 11.2 Triangular fuzzy number (a , a , a ).
The arithmetic operations between the triangular fuzzy numbers are pre-
sented in Table 11.1.
Definition 4 Comparisons of two triangular fuzzy numbers (Xu, 2002;
Wei, 2010)

L
M
L
M
x
The probability of e 1 ¼ x , x , x U being greater than e 2 ¼ x , x , x U
x
1 1 1 2 2 2
can be calculated by Eq. (11.10).
M L
x x 1
2
P e 1 ex 2 Þ ¼ λmax 1 max M L M L ,0 ,0 +
x ð
x x + x x
1 1 2 2
U M (11.10)
x x 1
2
1 λÞmax 1 max ,0 ,0
ð U M U M
x x + x x
1 1 2 2

L
M
where P e 1 ex 2 Þ represents the probability of ex 1 ¼ x , x , x U be
x ð
L M U 1 1 1
x
greater than e 2 ¼ x , x , x , and λ represents the attitudes of the
2 2 2
decision-makers on the risk, and it usually takes the value of 0.50 which
means that decision-makers are neutral to the risk. While a value which
is greater than 0.50 should be assigned to λ if the decision-makers tend to
pursue the risks, and a value which is smaller than 0.50 should be assigned
to λ when the decision-makers intend to reject the risks.
2.2 Fuzzy multicriteria decision making
A fuzzy Multicriteria Decision Making (FMCDM) method which allows
the stakeholders and decision-makers using linguistic terms/variables to par-
ticipate in the decision-making process has been developed for sustainability
assessment of biofuel production pathways. The procedures of FMCDM are
illustrated as follows based on the work of Li (2003). This method has been
used for sustainability ranking of biomass-based technologies for hydrogen