3.1. Evaluation Index System
Based on the principles of comprehensiveness, hierarchy, and simplicity, the evaluation index system was built by using the frequency statistical approach and theoretical analysis method. Additionally, internal and external conditions, such as the spatiotemporal distribution of water resources and overall plan of socioeconomic development, were considered. The system consisted of the water resources subsystem, socioeconomic subsystem, and eco-environment subsystem. Twelve indices were finally selected in consonance with the nature and connotation of the problems (
Table 2).
Principal component analysis [
35,
36] is a multivariate statistical method that converts multiple unrelated indices into a few independent comprehensive indices. It possesses traits that can simplify complicated problems, render the problem analysis easier and more convenient, get more scientific results, and so forth. Entropy judges the level of valid information contained in the data by reacting to the disorder of the system [
37]. The entropy weight method is characterized by a positive relationship between the degree of disorder in the performance of an index and the entropy value, and an inverse relationship with the amount of information it can respond to [
38]. Hence, PCA and the entropy weight method were mixed to determine the weight.
Figure 3 is a chord diagram that shows the relationship between the selected 12 indices. If these indices belong to one subsystem, they will be converted into one point. The arc length of each index represents its corresponding weight. The specific weight value of each index is listed in
Table 3. Once the index system is established, each index needs to be analyzed to establish its reasonable value range and grading standard. The water carrying capacity of the study area was classified into five states, and an individual index was divided into five levels correspondingly (
Table 3). The grade classification and reflective meaning are illustrated in
Table 4.
3.2. Set Pair Analysis
A set pair analysis put forward by Zhao [
39] was primarily adopted to grapple with uncertainty problems [
40]. It is well-known for its ease of implementation, qualitative and quantitative analyses, and uncertainty consideration. Owing to the variability and complexity of the water resources, there is a complicated and uncertain relationship between each evaluation factor and the carrying capacity level [
41].
The main idea is to construct the research into two sets with a certain connection, and then systematically analyze the characteristics of the two sets in identity, discrepancy, and contradistinction using the degree of connection for quantitative description. Assuming set
and set
construct a set pair
, the expression describing the connection degree can be stated as
where
refers to the connection degree of sets
and
;
refers to the total number of elements;
refers to the number of identical elements, in which identity signifies that sets
and
are identical;
refers to the number of discrepant elements, in which discrepancy signifies that there are some subtle differences between sets
and
;
refers to the number of contradictory elements, in which contradistinction signifies that remarkable differences exist in sets
and
; accordingly,
,
, and
refer to the identical degree, discrepancy degree, and contrary degree, respectively;
refers to the uncertain coefficients of the discrepancy degree valuing in the range of −1 and 1; and
refers to the coefficient of the contrary degree and generally takes the value of −1, playing the role of contrary mark sometimes. Setting
,
, and
, then Equation (1) can be written as
where the values of the coefficients satisfy
. Equations (1) and (2) are the connection degrees that are commonly used (i.e., the three-element connection degree).
in Equation (2) can be expanded to
, so a multi-element connection degree can be attained.
A set pair
can be formed when the values of an index
(
l=1, 2, 3, …,
m;
m denotes the number of evaluation indices) in the evaluation are viewed as one set called
and the evaluation criteria for the corresponding index are taken as another set
(
k=1, 2, 3, …,
K;
K denotes the number of evaluation levels). Based on the principle of SPA, the
K-element connection degree of
can be defined as
where
refers to the weight of the
lth index, which can be assigned due to its contribution to
.
Let
,
,
,
,
be the thresholds for each index from Grades I to V, respectively, and the connection degree
of the sample
with its evaluation criteria at Grade I can be expressed as the line in
Figure 4. Then, the confidence criterion is utilized to judge the ranks of sample.
in which
where
λ refers to the confidence level. The greater the value of
λ is, the more conservative and safer the evaluation result is.
refers to the sum of the first
K-elements in the connection degree, and
,
,
,
,
refer to the identical degree, partial identical discrepancy degree, uncertainty discrepancy degree, partial contrary discrepancy degree, and contrary degree, respectively.
The set pair potential function of the connection degree is its adjoint function, which means the relative deterministic state and development trend of the study object at the macro level. Subtraction set pair potential (SSPP) is used for situation analysis [
42]. Based on the identity, discrepancy, and contradistinction of SPA, SSPP can be defined as
where the scope of the value of
is
.
a,
b, and
c refer to the same meaning in Equations (1) and (2).
is divided into five classes as inverse potential (
), partial inverse potential (
), symmetrical potential (
), partial identical potential (
), and identical potential (
). The major element degenerating the carrying status is what belongs to the inverse potential or partial inverse [
43]. Diagnosed as the vulnerability index, this index serves as the chief object for the regulation of the carrying capacity.
3.3. Distributed Hydrological Model
The SWAT model is a geographic information system (GIS)-based distributed hydrological model with a clear physical mechanism [
44]. The model can take advantage of GIS to extract a digital elevation model to form flow networks in sub-basins for simulating the hydrological cycle process. The model typically divides watersheds into several sub-basins with different soil types and land use attributes [
45]. Water balance is the premise and root of the hydrological cycle simulation in SWAT, the equation of which can be expressed as
where
refers to the final soil water content (mm),
refers to the initial soil water content (mm),
refers to the time with units of days,
refers to the precipitation on the
ith day (mm),
refers to the surface runoff on the
ith day (mm),
refers to the evapotranspiration on the
ith day (mm),
refers to seepage from the soil profile on the
ith day (mm), and
refers to the underground runoff on the
ith day (mm). Predominantly, the coefficient of the Nash–Sutcliffe efficiency
(NSE) and the relative error (
RE) are exploited to evaluate the results of the SWAT simulation.
where
represents the observed discharge at time
t,
represents the simulated discharge at time
t,
represents the mean of observed values, and
n represents the number of observed data. On the condition that the results satisfy
NSE > 0.5 and
RE < 15% during both calibration and validation periods, the SWAT model is applicable for this basin, and the simulation results are acceptable.
In this paper, the observed discharge data from four hydrological stations, including Ankang, Baihe, Danjiangkou, and Huangzhuang, were exploited for calibrating and validating the SWAT model. The Ankang and Baihe hydrological stations are located in the upper reaches of the basin, while the Danjiangkou hydrological station is in the middle reaches, and the Huangzhuang station lies in the lower reaches. The calibration began on 01 January 1980 and lasted until 31 December 1993, and validation was performed during the period 01 January 1994–31 December 2000.
3.4. Water Resources Development and Utilization Model
To capture the dynamic properties of the water resources subsystem, socioeconomic subsystem, and eco-environment subsystem, a compiled model was used to predict the values of evaluation indices scientifically in the planning year.
Water resources
Modulus of water resources production refers to annual water resources amount per unit area [
46]. The total water resources amount of the study area can be obtained according to the simulation result of the SWAT model. The corresponding formula for computing the entire amount of water resources is as follows:
where
denotes the entire amount of water resources (m
3),
denotes the surface runoff (i.e., the difference between streamflow and baseflow, m
3),
denotes the precipitation infiltration quantity (m
3),
denotes the streamflow (i.e., the surface water resources amount, m
3), and
denotes the baseflow (m
3).
Water supply and consumption
Water consumption refers to the sum of the water used by all types of off-stream water users, including losses from water transmission [
47]. The quota method was applied to assess the water consumption, the explicit steps of which are as follows: Step 1: explore the trends of the main factors affecting water consumption and determine water consumption indices and quotas. Step 2: compute the amount of water consumption in the planning year in terms of the indices and quotas derived from Step 1, such as population and industrial production. In the light of the classification of water users, the water consumption of each sector can be estimated based on the influence factors and quotas. The gross amount of water consumption refers to the sum of water consumption of each sector. Meanwhile, the estimated amount of water supply is equal to water consumption. The formula for estimating water consumption is as follows:
where
refers to the water consumption of one sector,
refers to the water quota per unit of one sector, and
and
refer to water use per activity level and the water transferring loss ratio of the sector, correspondingly. For example, when we estimate the domestic water consumption,
refers to the amount of domestic water consumption (m
3),
refers to the water consumption per person (m
3/person),
refers to the total population (number of persons), and
refers to the water transferring loss ratio of domesticity. As for the water consumption of an industry, agriculture, and ecological environment, the parameters represent the corresponding water consumption indices and quotas.
Water pollution and water environment
The rate of ecological water consumption is calculated by the outputs from the water consumption in the water resources development and utilization model. The wastewater discharge is calculated based on the water consumption of a domesticity and industry with the corresponding pollution discharging coefficients.
where
refers to the water consumption of an ecology, and
refers to the sum of water consumption of all sectors.
and
refer to the water consumption of a domesticity and industry, respectively.
and
refer to the pollution discharging coefficient of a domesticity and industry, respectively. Historical data from the Statistical Bulletin of the National Economic and Social Development and Environmental State Bulletin of the cities in the Han River basin are allowed for this study. Besides, the industrial structures, development trends, and environmental protection requirements are considered to predict the value comprehensively.