Page 300 - Materials Chemistry, Second Edition

14.3 Methods 299
The development of a Bayesian hierarchical model can refer to (Mohammadi and Rezaei,
2019). To compute the posterior distribution of a Bayesian hierarchical model, the Markov-
chain Monte Carlo (MCMC) techniques need to be used, and one of the best available prob-
abilistic languages “just another Gibbs sampler” (JAGS) is employed.
For BBWM, the optimization problem of the original BWM is substituted with a probabi-
listic model, and the above-mentioned model will replace Step 5 of the original BWM. The
inputs between the BWM and BBWM are identical, but the output of BBWM can provide
more information including the confidence of the relation between each pair of evaluation
criteria. Meanwhile, the credal ordering and ranking are also introduced. Interested readers
can refer to (Mohammadi and Rezaei, 2019).

14.3.3 Fuzzy TOPSIS method for sustainability ranking of different
electrochemical energy storage technologies from the perspective of life cycle

The original TOPSIS method combined with fuzzy set theory, namely fuzzy TOPSIS, is
employed to deal with the ambiguity and uncertainty of sustainability ranking of different
electrochemical energy storage technologies. The fuzzy TOPSIS conducts the combination
of fuzzy set theory and traditional TOPSIS method, which uses triangular fuzzy numbers
to represent the value of criteria (Guo and Zhao, 2015).
Fuzzy set theory, proposed by Zadeh, is an extension of the classical set theory (Zadeh,

1965). A fuzzy set a is a pair (U,m) where U is a set and m:U![0,1] is the membership func-
tion, denoted by μ xðÞ. A triangular fuzzy number (TFN) is represented as a triplet
a
L M R
a¼ a , a , a , and its membership function μ xðÞ is expressed as:
a
8 L
0 x < a
>
x a
> L
>
> L M
> a x < a
< M L
a a
μ xðÞ ¼ R (14.13)
a a x
> M R
> a x a
> R M
a a
>
>
: R
0 x > a
L
R
R
L
M
R
M
L
where a , a , and a are crisp numbers ( ∞ <a a a <∞); a and a are the lower and
upper bounds of available area for evaluation data, respectively.
To transform the linguistic variables of decision-makers into TFN, the transformation rules
between the linguistic terms and fuzzy ratings need to be set first, as listed in Table 14.2 (Liao
et al., 2013; Zhao and Guo, 2014).
For fuzzy TOPSIS method, the entry in the decision matrix is represented by TFN, which
can characterize the fuzzy and uncertainty issues (Hwang et al., 1993; Wang, 2015). The de-
tailed steps of fuzzy TOPSIS method are introduced as bellow.
Step 1: Calculate the values of the quantitative criteria and the qualitative criteria respec-
tive to alternatives. Suppose that there are m alternatives A¼{A 1 ,A 2 ,⋯A m } to be ranked. The
quantitative criteria of m alternatives can be valued based on the practical calculations and
survey. The qualitative criteria can be calculated based on the aggregate fuzzy linguistic rat-
ings for criteria performance of alternatives.

L M R L M R
Let a ikj ¼ a , a , a ,0 a ikj a ikj a ikj 1, i¼1, 2, ⋯, m, k¼1, 2, ⋯, n, j¼1, 2, ⋯, r be
ikj ikj ikj
the superiority linguistic rating on criteria performance assigned to alternative A i by