72
Tecnología y Ciencias del Agua
, vol. VIII, núm. 2, marzo-abril de 2017, pp. 71-76
Zheng & Yu,
Improvement of vertical “scatter degree” method and its application in evaluating water environmental carrying capacity
•
ISSN 2007-2422
addition to analysis of social, environmental,
and ecological factors. These factors can be
multi-layered, dubious and complex. Because
of this complexity, this paper combines the
analytic hierarchy process (AHP) and the verti-
cal “scatter degree” method, which considers
objective and subjective information fully in its
evaluation. Through the comparison with the
vertical “scatter degree” method, this method is
determined to be more reasonable. In addition,
it can provide technical support for improving
regional WECC.
Improvement of “scatter degree” method
The theory of the “scatter degree” method
(Guo, 2012)
The synthetic function is the linear function of
greatest-type indicators (
x
1
,
x
2
,…,
x
m
):
y
= ω
1
x
1
+ ω
2
x
2
+ + ω
m
x
m
= ω
T
x
(1)
where
w
= (
w
1
,
w
2
...
w
m
)
T
is an
m
-dimensional
pending positive vector.
Inserting the standard observed values (
x
i
1
,
x
i
2
,…,
x
im
) of evaluation object
S
i
into Eq. (1), Eq.
(2) is acquired:
y
=
1
x
i
1
+
2
x
i
2
+ +
m
x
im
=
j
x
ij
j
=
1
m
(2)
If and, Eq. (2) is converted to Eq. (3):
y
=
A
w
(3)
The variance of evaluation objects
y
=
w
T
x
:
s
2
=
1
n
(
y
i
y
)
2
i
=
1
n
=
y
T
y
n y
2
(4)
The principle of determining the weight
coefficient of the “scatter degree” method is
getting the linear function
w
T
x
of index vector
x
and making the variance of the evaluated
object’s values as high as possible;
y
=
A
w
is
substituted into Eq. (4). When
y
= 0, the new
formula is as follows:
ns
2
=
T
A
T
A
=
T
H
(5)
where
H
=
A
T
A
is a real symmetric matrix.
When
w
is unlimited, formula (5) can be
an arbitrarily large value. Here,
w
T
w
= 1, and
the maximum value of formula (5) is acquired.
Namely, the value
w
is acquired to meet the
constraints:
max
T
H
s
.
t
.
T
=
1
>
0
(6)
For formula (5), when
w
is standard feature
vector of the largest eigenvalues of H,
w
T
H
w
achieves its maximum value. The weight co-
efficient vector (
w
1
,
w
2
, ...
w
m
)
T
is obtained by
normalizing
w
, and . The comprehensive values
y
i
of the evaluation object are obtained by using
Eq. (2).
The improvement of the “scatter degree”
method
The “scatter degree” method emphasizes that
the method should highlight the differences
among the evaluation objects as a whole, and it
runs under the premise of the same importance
of the evaluation target to every evaluation
indicator. In fact, the degree of the importance
is not same. The “function drive” theory pro-
vides the weighting coefficient
r
j
(
j
= 1,2, ...,
m
)
of each indicator
x
j
. On this basis, each indicator
is weighted by the following formula:
x
ij
*
=
r
j
x
ij
,
j
=
1,2, ,
m
;
i
=
1,2, ,
n
(7)
where
x
ij
are the standard observed values. The
mean value and mean square deviation of
x
ij
*
are
0 and
r
j
2
, respectively. The weighting coefficient
of weighted data {
x
ij
*
} is acquired by using the
“scatter degree” method. Then, the comprehen-
sive evaluation values can be obtained.
