Environmental Engineering Reference
In-Depth Information
2.1 Linear Normalization Method
Simple linear normalization method is used. For both positive and negative attrib-
utes separate formulations have been used. Equations
1
and
2
have been used for
positive and negative attributes, respectively (Shakouri et al.
2014
):
a
ij
max
(
a
ij
)
Z
ij
=
(1)
1
a
ij
max
(
a
ij
)
Z
ij
=
(2)
2.2 Fuzzy Normalization Method
The fuzzy normalization formula for positive attributes and negative attributes are
describes in Eqs.
3
and
4
, respectively:
a
ij
−
min
a
ij
max
a
ij
−
min
a
ij
z
ij
=
(3)
max
a
ij
−
a
ij
max
a
ij
−
min
a
ij
z
ij
=
(4)
Usually the relative importance of attributes is different, so a weighting method
for attributes must be applied. In this study, two relevant methods has been used;
FAHP and Shannon's entropy.
2.3 Fuzzy-AHP
The procedure of standard FAHP has been fully described in Zangeneh et al. (
2014
).
2.4 Shannon's Entropy
The original Shannon's entropy comes with the concept of information theory,
where it is used to measure uncertainty expressed by distribution of
p
i
, and find
how much storage is needed to capture all uncertain information. The calcula-
tions can be expressed in a series of formulas that are expressed below (Shakouri
et al.
2014
):
P
ij
n
E
j
=−
k
P
ij
ln
∀
j
(5)
i
=
1
Search WWH ::
Custom Search