Effect of Weight of Water Resources Carrying Capacity Evaluation Index on Its Evaluation Results in Xinjiang, China

Abstract

To analyze the effect of the weight of water resources carrying capacity evaluation index on its evaluation results, taking Xinjiang, China, as an example, this paper selects nine evaluation indicators related to the water resources system, social and economic development system, and ecosystem. The analytic hierarchy process (AHP), the coefficient of variation method, the geometric average method, and Murphy’s averaging method are used to calculate the weight of evaluation indexes. Subsequently, the fuzzy comprehensive evaluation model is used to evaluate the water resources carrying capacity of Xinjiang from 2011 to 2015. The results show that Murphy’s averaging method improves the shortcomings of single subjective or objective weight values and rankings that are inconsistent with the actual situation. The water resources carrying capacity of the study area was in a weak overload state for all years except 2014, and the water resources carrying capacity showed a downward trend year by year. According to the actual situation of precipitation change, Murphy’s averaging method is more reasonable in terms of index assignment and evaluation results. This method is more suitable for the water resources carrying capacity evaluation in Xinjiang.

1. Introduction

Water resources carrying capacity is used to quantitatively judge the development and utilization of regional water resources using a mathematical model. Its evaluation results have significance in terms of practical guidance for optimizing regional water resources allocation and promoting sustainable development of the regional ecological environment. The most widely used evaluation methods include the fuzzy comprehensive evaluation method [1], the system dynamics method [2], and the principal component analysis method [3], among others. However, the system dynamics method has the shortcoming of low calculation accuracy, while the principal component analysis method lacks a reliable basis to determine the number of principal components. In contrast, the fuzzy comprehensive evaluation method can comprehensively analyze the regional water resources carrying capacity through quantitative evaluation indexes. Wu et al. (2021) applied the fuzzy comprehensive evaluation model to evaluate the water resources carrying capacity of Shandong Province, China [3]. Their results indicated that the fuzzy comprehensive evaluation method is scientific and applicable. Sun and Yang (2019) constructed a system dynamics model of regional water resources carrying capacity. The model passed the rationality test and historical test [4]. However, the system dynamics method also has its limitations, such as regarding qualitative evaluation. Wu et al. (2021) used the principal component analysis method to evaluate the water resources carrying capacity of Huai’an in Jiangsu Province from 2013 to 2019 [3]. However, principal component analysis can only determine whether the water resources development and utilization is in a situation of overload, but cannot quantify the degree or grade of overload.

Determining the weight of each evaluation index is extremely important during the process of water resources carrying capacity evaluation. According to the principle of the quantitative evaluation index, weight calculation methods can be divided into subjective methods and objective methods. The subjective method usually determines the relative importance of weights among indicators by experts. The most common subjective weighting methods include the analytic hierarchy process (AHP) [5,6,7], the Delphi method [8,9], and the binomial coefficient method [10], among others. The advantage of the subjective weighting method is that decision-makers can reasonably determine the weight ranking of each indicator according to the actual situation and their professional knowledge and experience. It can effectively avoid conflict between the theoretical weight of factors and the actual importance of factors. Its results depend on the comprehensive knowledge of decision-makers. Therefore, the factor weight determined by this method is subjective but not objective. Among these methods, AHP is widely used because of its simple operation [11,12]. Feng et al. (2020) used AHP to determine the weight of indicators of water resources carrying capacity [11]. The scientific rationality of AHP was verified by examples. However, this method lacks analysis of the interaction between factors. The expert investigation method can avoid being influenced by one’s opinion, but it also has a certain degree of subjectivity [13].

The objective method assigns weights based on the information provided by the sample data for each index. The widely used objective weighting methods are the coefficient of variation method [14,15,16] and the entropy method [17,18,19]. The CRITIC method [20,21], the multi-objective programming method [22], the geometric average method, and Murphy’s averaging method are used to calculate the weight in recent years. The coefficient of variation method has the advantages of objectivity and adaptability. To eliminate the influence of human factors, Wang et al. (2019) used the coefficient of variation method to evaluate water resource sustainability [23]. However, the index weight value obtained by this method will change with the number of samples. Ding et al. (2019) evaluated the water resources carrying capacity of Hefei city using the coupled Philo model, calculating the weight of evaluation indicators using the entropy weight method [24]. Although this method can be used to calculate the index weight, the evaluation results are greatly affected by the extremely large or extremely small data values. Standard deviation is used in the CRITIC method to reflect the degree of data variation, but the shortcomings of low precision and large error limit the application of this method [25]. Li (2015) and Gao (2020) assessed the domestic drinking water quality and water resources carrying capacity using the geometric average method and Murphy’s averaging method. When the two different weights obtained from two methods differ greatly, the geometric average method can weaken the differences [26,27]. Murphy’s averaging method is mainly to square the mean value of two different weights. The weight obtained from the Murphy’s averaging method is heavily influenced by extreme values [27].

Different weighting methods may lead to different weighting results, which may affect the evaluation results for water resources carrying capacity. At present, a single method is mostly used to determine index weight in the evaluation of water resources carrying capacity. In order to evaluate the influence of index weight on the evaluation results of water resources carrying capacity, this paper selects Xinjiang Uygur Autonomous Region (Xinjiang for short), located in northwest China, as an example. Based on the detailed analysis of the water resources conditions, ecological and environmental problems, and socio-economic development of the study area from 2011 to 2015, the evaluation model of water resources carrying capacity is built using the fuzzy comprehensive evaluation theory. AHP, the coefficient of variation, the geometric average, and Murphy’s averaging method are used to calculate the weight of the evaluation index, respectively. Subsequently, the carrying capacity of water resources is evaluated according to the index weight calculated by each method, and the effect of index weight on the evaluation result of water resources carrying capacity is compared and analyzed. Finally, by comparing the evaluation results with the actual situation of water resources development and utilization in the study area, an appropriate weight calculation method is selected to provide scientific references for water resources carrying capacity evaluation in other similar areas.

2. Methods

2.1. Evaluation Method of Water Resources Carrying Capacity

This paper aims to analyze the weight effect of the water resources carrying capacity evaluation index on its evaluation results. The evaluation method for water resources carrying capacity is not the focus of this study. Therefore, in order to enhance the reliability of evaluation results, this paper adopts the widely used fuzzy comprehensive evaluation method to evaluate the carrying capacity of water resources in Xinjiang. The basic principle of this method can be found in Li et al. (2012) [28]. The calculation process of the method is shown in Figure 1.

First, the set of indicators U = {u₁, u₂, …, u_m} and evaluated grades V = {v₁, v₂, …, v_n} of the subject are determined. Next, the weights of each indicators are calculated. According to the grade membership formula of each evaluation index $r_i^{\left(j\right)}$ (j = 1, 2, …, n), the membership vector r_ij and affiliation matrix R = (r_ij)_m×n are obtained. The membership matrix and the weight vector A = {a₁, a₂, …, a_m} of the indicators are then subjected to obtain the vector B = A·R = (b₁, b₂, …, b_n). Finally, the evaluation results are achieved from the following three methods.

2.1.1. Maximum Membership Degree Method

The maximum value of the judgment indexes b₁, b₂, …, b_n obtained according to the above steps is set as maxb_i and used as the representative value. Then, the grading limits k₁, k₂, …, k_j are calculated based on the evaluation level values v₁, v₂, …, v_n. The element k_L (1 ≤ L ≤ j) that is closest to the maxb_i value is then selected as the evaluation result, which is called the maximum affiliation method, expressed as:

$$b=k_L\left(k_L\mathrm{is}\mathrm{closest}\mathrm{to}max{b}_i\right)$$

(1)

The evaluation result obtained by this method only considers the index corresponding to the maximum weight and ignores the information provided by other indexes.

2.1.2. Weighted Average Method

The weighted average of v_j is carried out with b_j as the weight, and the calculated average value is used as the evaluation result. The calculation equation is expressed as:

$$b=\frac{{\sum }_{j=1}^n{b}_j{v}_j}{{\sum }_{j=1}^n{b}_j}$$

(2)

If the evaluation object is data, the evaluation result can be calculated according to Equation (2); if the evaluation object is not data, it should be quantified by assigning value first, and then calculated by Equation (2).

2.1.3. Fuzzy Distribution Method

In this method, the evaluation grade of the evaluation index is directly regarded as the evaluation result b. Each judging index specifically reflects the distribution of the evaluation object in the characteristics of the evaluation.

To comprehensively consider the influence of all indicators and retain all useful information, the weighted average method is used to calculate the results of fuzzy comprehensive evaluation.

2.2. Calculation Method of Index Weight

2.2.1. Analytic Hierarchy Process (AHP)

The principle of AHP for solving the index weights is to decompose the complex system into several subsystems and merge the problem into an ordered hierarchical structure. Each index in the same hierarchy is pairwise compared and assigned a value according to the degree of importance. Subsequently, a judgment matrix is established based on the assigned values. After passing the consistency test, the weights of each subsystem are calculated.

There are many studies on weight calculation using the AHP method, which will not be repeated in this paper (for details, see Panchal and Shrivastava 2022; Xiao et al., 2022) [29,30].

The weights calculated by the AHP method are denoted as $w_1\left(i\right)$.

2.2.2. The Coefficient of Variation Method

The coefficient of variation method is a method to determine the index weights according to the degree of variation of each evaluation index data. If the data of an evaluation index can clearly distinguish the differences among the evaluated objects, it indicates that the evaluation index has abundant distinguishing information, and the evaluation index is assigned a larger weight than other indexes. On the contrary, if the differences among the evaluated objects on a certain index are small, the evaluation index has a weak impact on the evaluation target, and the evaluation index should be assigned a smaller weight.

The calculation formula of the coefficient of variation method is as follows:

$$V_i=\frac{{\sigma }_i}{\overline{x_i}}$$

(3)

$$w_2\left(i\right)=\frac{V_i}{{\sum }_{i=1}^n{V}_i}$$

(4)

where ${\sigma }_i$ is the standard deviation of index i; n is the total number of indexes; $\overline{x_i}$ is the average of index i; $V_i$ is the coefficient of variation of index i; and $w_2\left(i\right)$ is the weight of index i.

2.2.3. Combined Weight of Geometric Average Method

The calculation formula of the combined weight of geometric average method [26] is as follows:

$$w_i=\frac{\sqrt{w_1\left(i\right)w_2\left(i\right)}}{{\sum }_{i=1}^n\sqrt{w_1\left(i\right)w_2\left(i\right)}}$$

(5)

where $w_i$ is the weight of index i using the geometric average method.

2.2.4. Combined Weight of Murphy’s Averaging Method

The calculation formula of the combined weight for Murphy’s averaging method [27] is as follows:

$$w_i^{\prime }=\frac{{\left(\frac{w_1\left(i\right)+w_2\left(i\right)}{2}\right)}^2}{{\sum }_{i=1}^n{\left(\frac{w_1\left(i\right)+w_2\left(i\right)}{2}\right)}^2}$$

(6)

where $w_i^{\prime }$ is the weight of index i obtained from Murphy’s averaging method.

3. Case Study

3.1. Overview of the Study Area

The study area of Xinjiang is in northwest China, with the Altai Mountains in the north, the Kunlun Mountains in the south, and the Tianshan Mountains running across the middle part of Xinjiang (Figure 2). There are more than 100 lakes and 570 rivers in the case study. Except for the Ili River and the Irtysh River, all other rivers are endorheic. Alpine glaciers and snow are widespread and abundant, which are the main sources of recharge for most rivers. As Xinjiang is far away from the ocean, it has a typical temperate continental arid climate (dry and little rain).

From 1956 to 2015, the average annual precipitation and evaporation was 159.69 mm and 2450 mm, respectively, in the study area. The average total water resources were 85.309 billion m³, of which 80.766 billion m³ were surface water resources and 51.767 billion m³ were groundwater resources. The average water production modulus is 51,200 m³/km², which was about one-sixth of the average value for China. The industrialization rate of the area is only 29.39%. In addition, the spatial and temporal distribution of water resources is unbalanced. The contradiction between water supply and water demand is prominent. The phenomenon of desertification is serious, and the ecological environment is extremely fragile.

In 2015, the total water supply in Xinjiang was 57.719 billion m³. The water consumption from agriculture, urban and rural life, industry, and ecological water outside the river accounted for 94.7%, 2.3%, 2.0%, and 1.0%, respectively. The surface water supply was 45.688 billion m³, accounting for 79.1%; the groundwater supply was 11.941 billion m³, accounting for 20.7%; and the recycled water utilization was 0.90 billion m³, accounting for 0.2% [27].

In 2015, the total population of Xinjiang was 23,597,000, including an 11,145,000 urban population. The urbanization rate was 47.23%. Xinjiang’s GDP totaled 932.5 billion RMB, and its industrial added value was 269 billion RMB [27].

3.2. Evaluation Index of Water Resources Carrying Capacity

According to the characteristics of uneven distribution of water resources in the study area, the local industrial structure dominated by agriculture, the differences in the natural stock of water resources in the region, and the different ways of development and utilization are taken into full consideration. By referring to the evaluation standards of water resources and the index system in the national water resources supply and demand analysis, the following evaluation indexes were selected from the three sub-systems of water resources, social economy and ecological environment [27], as shown in

Table 1. Evaluation indexes for water resources carrying capacity.

The Target Layer	The System Layer	The Index Layer	Index Types	Implication of Indexes
Water resources carrying capacity	Water-resources system	u1 Water resources per capita (m3)	+	Reflecting the regional water resources development and utilization situation
		u2 Water production modulus (10 thousand m3/km2)	+	Reflecting the quantity of water resources
		u3 Precipitation (mm)	+	Reflecting regional precipitation
		u4 Proportion of groundwater supply (%)	–	Reflecting groundwater resources development and utilization
	Socio-economic system	u5 Urbanization rate (%)	–	Reflecting the level of social development and population distribution
		u6 Domestic water consumption per capita (L/d)	–	Reflecting water consumption per capita
		u7 Water consumption per ten thousand RMB industrial added value (m3/10 thousand RMB)	–	Reflecting the level of industrial water use
		u8 Average irrigation water per 666.67 m2 (m3/666.67 m2)	–	Reflecting agricultural water consumption
	Eco-environment system	u9 Ecological water use rate (%)	+	Reflecting the water consumption of ecological environment

. According to the relationship between the value of the evaluation index and the strength of the carrying capacity of water resources, the evaluation index was divided into a positive index and a reverse index. The positive indicator indicated that the larger the index value, the stronger the carrying capacity of water resources, which is represented by “+”. The reverse indicator meant that the direction of change of the index value was opposite to that of the carrying capacity of water resources, i.e., the larger the index value, the weaker the carrying capacity of water resources. It is represented by “–”.

4. Results and Discussion

4.1. Grade of Evaluation Index

Referring to the existing evaluation results for water resources carrying capacity in arid areas similar to this case study [1,31], the grade of water resources carrying capacity evaluation index is divided into five levels. Level v₁ indicates that the regional water resources carrying capacity is extremely poor. The water demand is much larger than the water supply, and the carrying capacity of water resources is in a state of serious excess. Level v₂ indicates that the regional water resources carrying capacity is poor. The water demand is slightly larger than the water supply. Level v₃ indicates that the regional water resources carrying capacity is in a balanced state, and the water demand of water resources is close to the water supply. Level v₄ means the regional water resources carrying capacity is good, and the water demand of water resources is smaller than the water supply. Level v₅ shows that the regional water resources carrying capacity is excellent, and the water demand of water resources is much smaller than the water supply.

According to the China Water Resources Bulletin, Xinjiang Water Resources Bulletin, and Xinjiang Statistical Yearbook, the graded values of the five grades corresponding to the nine indicators were obtained, as shown in

Table 2. Graded values of the five grades corresponding to the nine indicators.

	u1	u2	u3	u4	u5	u6	u7	u8	u9
v1	<2500	<20	<500	>20	>80	>100	>70	>520	<1
v2	2500–3000	20–26.67	500–600	16.67–20	70–80	83.33–100	53.33–70	436.67–520	1–2
v3	3000–3500	26.67–33.3	600–700	13.33–16.67	60–70	66.67–83.33	36.67–53.33	353.33–436.67	2–3
v4	3500–4000	33.3–40	700–800	10–13.33	50–60	50–66.67	20–36.67	270–353.33	3–5
v5	>4000	>40	>800	<10	<50	<50	<20	<270	>5

.

4.2. Calculating the Weights of Evaluation Indexes

The weights of the evaluation indexes were calculated according to Equations (3)–(6), as shown in

Table 3. The weights of the evaluation indexes.

Index	AHP		Coefficient of Variation Method		Geometric Average Method		Murphy’s Averaging Method
	Weight	Order	Weight	Order	Weight	Order	Weight	Order
Per capita water resources (m3)	0.2000	1	0.1034	2	0.1601	2	0.1615	2
Water production modulus (10 thousand m3/km2)	0.2000	1	0.0978	3	0.1557	3	0.1556	3
Precipitation (mm)	0.1000	4	0.0924	5	0.1070	5	0.0649	5
The proportion of groundwater supply (%)	0.1665	3	0.0849	6	0.1323	4	0.1109	4
Urbanization rate (%)	0.0833	6	0.0334	8	0.0587	7	0.0239	7
Per capita domestic water consumption (L/d)	0.0418	8	0.0930	4	0.0694	6	0.0320	6
Water consumption per ten thousand yuan industrial added value (m3/10 thousand yuan)	0.0418	8	0.0628	7	0.0570	8	0.0192	8
Average irrigation water per mu (m3/mu)	0.0833	6	0.0312	9	0.0567	9	0.0230	9
Ecological water use rate (%)	0.0835	5	0.3993	1	0.2032	1	0.4089	1

.

It can be seen from Table 3 that the order of index weight obtained by the AHP method is quite different from that of the other three methods. The order of indexes weight obtained from geometric average and Murphy’s averaging method is the same, but the weight value is different. The indicators corresponding to the maximum weight obtained by AHP are water resources per capita and water production modulus, both of which are 0.2000. Further, the index corresponding to the maximum weight calculated by the coefficient of variation method, the geometric average method, and Murphy’s averaging method is ecological water consumption, but the weight values are different (0.3993, 0.2032, and 0.4089, respectively). The weight of ecological water consumption obtained by AHP is 0.0835, ranking fifth.

It is necessary to analyze the reasons leading to the differences in the calculation results for the above four methods. AHP is a subjective weighting method. Due to experts’ different familiarity with the conditions of the study area, the weight of some indexes may be too large or too small, which will affect the calculation results. The coefficient of variation method is an objective assignment method, which leads to a comparison according to numerical values without considering the physical significance of the indicators. The combined weights obtained by the geometric averaging and Murphy’s averaging method can complement the strengths and weaknesses of the subjective and objective methods, avoiding the shortcomings of a single method. However, the calculation principles of these two combination methods are different, resulting in different weights for some indicators. It can be seen from Equations (5) and (6) that Murphy’s averaging method considers the effect of extreme values when calculating index weight. When there is a large difference between subjective and objective weights, it will amplify the difference by the square calculation of the value, while the geometric average method will weaken the difference by the square root. For example, the weight of ecological water consumption obtained from the coefficient of variation method is 0.3993, which is quite different from the value for the other eight indicators. The weights achieved by Murphy’s averaging method and the geometric average method are 0.4089 and 0.2032, respectively. Murphy’s averaging method calculated the value of 0.4089, which is greater than 0.3993, while the geometric average method calculated the value of 0.2032, which is less than 0.3993. This is completely consistent with the theoretical analysis. Considering the fragile ecosystem in the case study, ecological water consumption should be guaranteed, and the weight of this index should not be too small. Therefore, the weight calculated by Murphy’s averaging method was selected in this study to evaluate the carrying capacity of water resources in Xinjiang.

4.3. Water Resources Capacity Evaluation

4.3.1. Water Resources Carrying Capacity Classification

In this study, the carrying capacity of water resources is divided into five grades: extremely poor; poor; medium; good; and excellent. To quantify the carrying ability, the five grades are assigned as 0.1, 0.3, 0.5, 0.7, and 0.9 [32,33]. The higher the value, the stronger the carrying capacity of water resources. According to Figure 1, the corresponding carrying capacity interval of water resources at each carrying level is calculated, as shown in

Table 4. Carrying capacity interval of water resources at each carrying level.

Carrying Capacity Interval b	[0, 0.2)	[0.2, 0.4)	[0.4, 0.6)	[0.6, 0.8)	[0.8, 1.0]
Carrying status	Serious overload	Weak overload	Balance	Loadable	Suitable load
Carrying grade	Extremely poor	Poor	Medium	Good	Excellent

.

When the value of b is [0, 0.2), it indicates that the water demand is much larger than the water supply in the case study, and the carrying capacity of water resources is defined as serious overload and graded as extremely poor. When the b value is [0.2, 0.4), it indicates that the water demand is slightly greater than the water supply, but that the development and utilization of water resources has not caused obvious ecological environment problems. The carrying capacity of water resources is weak overload and the grade is poor. When the value of b is [0.4, 0.6), it means the supply and demand of water resources are equal, and the carrying capacity of water resources is balanced and the grade is medium. When the b value is [0.6, 0.8), it means that the water demand is slightly smaller than the water supply, and the carrying capacity of water resources is not overloaded and the grade is good. When the b value is [0.8, 1], it indicates that there are abundant water resources and that water demand is much less than water supply; the carrying capacity of water resources is suitable and the grade is excellent.

4.3.2. Water Resources Carrying Capacity Analysis

According to the membership function $r_i^{\left(j\right)}$, the membership matrix R_i of each index in the study area from 2011 to 2015 was calculated (i = 1, 2, …, 5, respectively, 2011 to 2015). The affiliation function below can be found in An et al. (2019) [34]:

$$\begin{array}{c}{R}_1=\left[\begin{array}{lllll}0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 1.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.101\hfill & 0.899\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 1.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.096\hfill & 0.904\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.670\hfill & 0.330\hfill & 0.000\hfill & 0.000\hfill \end{array}\right]R_2=\left[\begin{array}{lllll}0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 1.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.137\hfill & 0.863\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 1.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.000\hfill & 0.646\hfill & 0.354\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \end{array}\right]\\ R_3=\left[\begin{array}{lllll}0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 1.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.131\hfill & 0.869\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 1.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.000\hfill & 0.844\hfill & 0.156\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \end{array}\right]R_4=\left[\begin{array}{lllll}0.000\hfill & 0.175\hfill & 0.825\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 1.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.000\hfill & 0.796\hfill & 0.204\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \end{array}\right]\\ R_5=\left[\begin{array}{lllll}0.000\hfill & 0.000\hfill & 0.000\hfill & 0.723\hfill & 0.277\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 1.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.000\hfill & 0.000\hfill & 0.886\hfill & 0.114\hfill & 0.000\hfill \\ 1.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \\ 0.500\hfill & 0.500\hfill & 0.000\hfill & 0.000\hfill & 0.000\hfill \end{array}\right]\end{array}$$

The evaluation result b for Xinjiang water resources carrying capacity can be calculated based on the fuzzy compound operation carried out in Figure 1 and Equation (2), as shown in

Table 5. Evaluation results of water resources carrying capacity in Xinjiang.

	AHP			Coefficient of Variation Method			Geometric Average Method			Murphy’s Averaging Method
	Carrying Capacity	Carrying Status	Carrying Grade	Carrying Capacity	Carrying Status	Carrying Grade	Carrying Capacity	Carrying Status	Carrying Grade	Carrying Capacity	Carrying Status	Carrying Grade
2011	0.3946	Weak overload	Poor	0.3553	Weak overload	Poor	0.3746	Weak overload	Poor	0.3844	Weak overload	Poor
2012	0.3750	Weak overload	Poor	0.2539	Weak overload	Poor	0.3247	Weak overload	Poor	0.2765	Weak overload	Poor
2013	0.3735	Weak overload	Poor	0.2515	Weak overload	Poor	0.3226	Weak overload	Poor	0.2759	Weak overload	Poor
2014	0.2580	Weak overload	Poor	0.1923	Serious overload	Extremely poor	0.2305	Weak overload	Poor	0.1335	Serious overload	Extremely poor
2015	0.3237	Weak overload	Poor	0.2613	Weak overload	Poor	0.2963	Weak overload	Poor	0.2740	Weak overload	Poor

.

As can be seen from Table 5, the carrying capacity values for water resources calculated by the four methods gradually decreased from 2011 to 2014, and the carrying capacity values for 2015 were all larger than those for 2014. This shows that, with the development of the social economy, the carrying capacity of water resources in Xinjiang shows a downward trend from 2011 to 2014. After China implemented the “strictest water resources management system” in 2015, water use efficiency was improved and the water resources carrying capacity began to increase.

The carrying capacity of water resources obtained by the subjective weighting method was greater than that obtained by the other three methods. The values for carrying capacity obtained by the coefficient of variation method and Murphy’s averaging method in 2014 were 0.1923 and 0.1335, respectively (less than 0.2). According to the classification in Table 4, it belonged to the overload state. The evaluation values in other years were between 0.2 and 0.4, indicating a weak overload state. The evaluation results obtained by the geometric average method were in the range of 0.2–0.4, indicating a weak overload state. In 2014, the precipitation in Xinjiang decreased, and the total water resources decreased by 15.23% compared with the average water resources. The actual condition was overloading in 2014. Therefore, the evaluation results of the variation coefficient method and Murphy’s averaging method are consistent with the actual situation. The values for the carrying capacity obtained by Murphy’s averaging method in 2013 and 2015 were 0.2759 and 0.2740, respectively, and the value in 2015 was smaller than that in 2013. However, the values for the carrying capacity obtained by the coefficient of variation method in 2013 and 2015 were 0.2515 and 0.2613, respectively, and the value in 2015 was greater than that in 2013. As the precipitation in 2015 decreased by 5.7 mm compared with that in 2013, the per capita water resources also decreased by 312.67 m³. The carrying capacity for 2015 should be less than that for 2013. Therefore, compared with the other three methods, Murphy’s averaging method is more suitable for the evaluation of water resources carrying capacity in Xinjiang.

4.3.3. Validation

From 2011 to 2015, the rainfall was 167 mm, 182.5 mm, 184.7 mm, 145.6 mm, and 179mm, and the per capita water resources were 4010.03 m³, 4045.18 m³, 4201.3 m³, 3162.54 m³, and 3888.63 m³, respectively. Although the rainfall and per capita water resources increased from 2011 to 2013, due to the rapid development of Xinjiang’s social economy, the carrying capacity of Xinjiang’s water resources was in a declining trend from 2011 to 2013. Due to the decrease in precipitation in 2014 compared with other years and the aggravation of groundwater overextraction, Xinjiang’s water resources carrying capacity in 2014 was seriously overloaded. Although AHP and the geometric mean method were also in line with the changing trend of water resources carrying capacity in Xinjiang, the accuracy of the calculation results of water resources carrying capacity was insufficient. Therefore, Murphy’s averaging method was closer to the actual situation of water resources carrying capacity in Xinjiang.

5. Conclusions

There are many methods to calculate the weight of evaluation indicators, and different methods may obtain different weights. To analyze the effect of the weight of evaluation indicators on the evaluation results, this study focused on Xinjiang, China, and selected nine evaluation indicators from the aspects of the water resources system, social and economic development system, and ecosystem. AHP, the coefficient of variation method, the geometric average method, and Murphy’s averaging method were used to calculate the weight of evaluation indexes, respectively. Subsequently, the fuzzy comprehensive evaluation model was used to evaluate the water resources carrying capacity of Xinjiang from 2011 to 2015.

The index weight calculation results revealed that the order of index weight obtained by the AHP method was quite different from the other three methods. The order of indexes weight obtained from the geometric average and Murphy’s averaging method was the same, but the weight value was different. In particular, the index of ecological water consumption should not be given too little weight, considering the fragile ecosystem in this case study.

The water resources carrying capacity evaluation results revealed that the carrying capacity of water resources decreased gradually from 2011 to 2014 in Xinjiang, and that it was in a state of overload. The water demand for water resources had exceeded the water supply, resulting in a serious contradiction between water supply and water demand. Xinjiang was facing a serious water resources shortage. However, China implemented the “strictest water resource management system” in 2015. The water use efficiency of industry and agriculture has been greatly improved, alleviating the contradiction between the supply and demand of water resources. The carrying capacity of water resources in Xinjiang is now presenting an increasing trend.

For the water resources carrying capacity evaluation in Xinjiang, the four weight calculation methods are all feasible, but there are certain differences in the impact on the evaluation results for water resources carrying capacity. The AHP method and coefficient of variation method were introduced to calculate Murphy’s averaging weight of the evaluation index. The calculation method of the evaluation model was simple and suitable for multi-index water resources carrying capacity evaluation. Murphy’s averaging method had a large difference in the weight of each indicator, which was more conducive to screening the main indicators, and the calculation results were consistent with the actual situation.

Based on the main framework of water resources carrying capacity, this paper has systematically discussed the effect of different index weight methods on the evaluation results from the construction of the index system and weight assignment methods on the water resources evaluation results. In the next step, our group will continue to study whether the evaluation method of water resources carrying capacity will have an impact on the selection of weight methods.