73
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
•
Analytic Hierarchy Process (Deng, Li, Zeng,
Chen, & Zhao, 2012)
In this article, the Analytic Hierarchy Process
(AHP) is used to confirm the weighting coef-
ficients
r
j
of each indicator.
AHP was discovered by Satty while per-
forming operational research in the 1970’s. The
indicators are classified into several levels of
objective, criterion, and index. This method is
based on both quantitative and qualitative ap-
proaches.
The steps of AHP include building a hier-
archical model, building a matrix of each level,
performing a consistent check, and verifying
the total arrangement of each hierarchy with a
consistent check.
The process and results of the evaluation
The evaluating data and its standards
Reference data was adopted from (Geng, 2012).
All the nine indicators are classified into positive
and negative classes, and need to be standard-
ized to remove the influence of inverse indices
and different dimensions.
For the positive index:
x
ij
*
=
x
ij
m
j
M
j
m
j
=
i
=
1,2,
…
,
n
;
j
=
1,2,
…
,
m
(
)
(8)
For the negative index:
x
ij
*
=
M
j
x
ij
M
j
m
j
=
i
=
1,2,
…
,
n
;
j
=
1,2,
…
,
m
(
)
(9)
where
M
j
=
max
x
i
i
j
{ }
and
m
j
=
min
x
i
i
j
{ }
Matrix A is calculated based on the raw data
as well as Eqs. (8) and (9). The results shown as
follows:
A
=
1.000 0.000 0.000 0.000 0.785 0.000 0.832 0.436 0.604
0.000 0.137 0.256 0.293 1.000 0.247 1.000 0.000 1.000
0.448 0.360 0.587 0.698 0.981 0.258 0.000 0.876 0.328
0.069 0.801 0.877 0.788 0.238 0.852 0.857 1.000 0.000
0.552 1.000 1.000 1.000 0.000 1.000 0.814 0.907 0.029
The application of vertical “scatter degree”
Symmetric matrix H is calculated by using the
formula
H
=
A
T
A
.
H
=
1.5101 0.7685 0.8753 0.9191 1.2414 0.7261 1.3404 1.3987 0.7675
0.7685 1.7898 1.9485 1.9230 0.6820 1.8091 1.6378 2.0239 0.2852
0.8753 1.9485 2.1783 2.1757 1.0404 1.9615 1.8212 2.2984 0.4777
0.9191 1.9230 2.1757 2.1946 1.1658 1.9241 1.7826 2.3076 0.5514
1.2414 0.6820 1.0404 1.1658 2.6361 0.7035 1.8573 1.4407 1.7966
0.7261 1.8091 1.9615 1.9241 0.7035 1.8536 1.7918 1.9855 0.3614
1.3404 1.6378 1.8212 1.7826 1.8573 1.7918 3.0896 1.9588 1.5265
1.3987 2.0239 2.2984 2.3076 1.4407 1.9855 1.9588 2.7817 0.5780
0.7675 0.2852 0.4777 0.5514 1.7966 0.3614 1.5265 0.5780 1.4737
The maximal eigenvalue and eigenvector
of matrix H’ is calculated by using MATLAB:
l
max
(
H
) = 13.9332,
w
(
H
) = (0.0768, 0.1124, 0.1280,
0.1287, 0.0980, 0.1144, 0.1391, 0.1436, 0.0591)
T
.
The values of WECC in the years of 2005 and
2009 are calculated by plugging
x
j
(
t
k
) and
w
j
into
Eq. (2).The results are shown in table 1.
The application of AHP-Vertical “scatter
degree”
Calculation of the weight using AHP
1. A system of evaluation is established accord-
ing to the reference, which is as follows: 1)
Level of objective: WECC; 2) Level of indica-
tors: ratio of water resource to utilization and
exploitation
X
1
, gross regional output per
capita
X
2
, water consumption of GDP
X
3
,
water consumption of industrial output
X
4
,
discharged volume of wastewater of indus-
trial output
X
5
, discharged chemical oxygen
demand of industrial output
X
6
, repeated
utilization rate of industrial wastewater
X
7
,
attainment rate of the industrial wastewater
X
8
, daily water consumption per capita
X
9
.
2. The weight of the index: The relative impor-
tance of all indicators was determined with
