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320 Biofuels for a More Sustainable Future

methods (fuzzy ELECTRE and fuzzy PROMETHEE), and fuzzy
weighting methods. MODM techniques combined with fuzzy set theory
are also a major part of FMCDM, such as fuzzy multiobjective linear pro-
gramming, quasiconcave and nonconcave fuzzy multiobjective program-
ming, interactive fuzzy stochastic linear programming, fuzzy
multiobjective integer goal programming, gray fuzzy multiobjective opti-
mization, and fuzzy multiobjective geometric programming (Kahraman,
2008). The method applied in this research is a kind of fuzzy multiobjec-
tive programming approach. Framework for the FMCDM method is
shown in Fig. 11.1.

2.1 Fuzzy concept
Fuzzy set theory had been introduced by Zadeh (1965), and many improved
fuzzy methods have been developed to be used in many fields such as opti-
mization of multiobjective problem and multicriteria decision making
(Wang and Chen, 2011; Chen et al., 2011; Bajpai et al., 2010).
Definition 1 Fuzzy sets (Khrais et al., 2011)
Assume that X is a collection of objects presented by x, a fuzz set α in X is a
set of ordered pairs defined as shown in Eq. (11.1), and the bigger the value
of the membership function, it will be more certain that x belongs to α.

α
α ¼ ð f x, μ xðÞÞj x 2 Xg (11.1)
where μ α (x) is the membership function of x in α.
Definition 2 Triangular fuzzy numbers (Tsai and Hsiao, 2004)
The triangular Fuzzy number is usually used in fuzzy study, and e can be
a
L M U
defined by a triplet (a , a , a ). Its mathematical and graphic concepts
are shown in Eq. (11.2) and Fig. 11.2, respectively.
8 L
0 x a
>
> L
> x a
> L M
>
< a < x a
M
a a L
e a x a
μ xðÞ ¼ U (11.2)
> M U
> a < x a
>
M
> a a U
>
: U
0 x > a
Definition 3 Arithmetic operations (Chang, 1996; Yuen and Lau, 2011;
Chen, 2000)