*2.1. SBM-DEA Model*

The DEA model can be used to evaluate the relative efficiency of different DMUs (decision-making units) through means of a specific mathematical programming model. The basic principle is to determine the production frontier surface with the help of a linear model and to determine the relative efficiency value of each decision unit by comparing the deviation of each decision unit relative to the production frontier surface. DEA models are widely used in efficiency assessment studies because they do not require assumptions about the functional relationships between variables and avoid too much subjectivity. However, the traditional radial DEA model is also partially flawed in that it does not consider slack variables in the efficiency measure of inefficient DMUs. In traditional radial DEA models (e.g., CCR-DEA, BCC-DEA, etc.), the assessment of the degree of DMU inefficiency only includes the proportional change of outputs and inputs. For the ineffective DMU, the gap between its current state and the state of the effective DMU on the production frontier should contain both equal proportional improvement and slack improvement, and the slack improvement part is not reflected in the traditional radial DEA model. Especially when the number of input-output indicators is relatively large, there will be more invalid DMUs, and the traditional radial DEA model loses the function of measuring the slack variables of invalid DMUs, and in such cases, the measurement results are not accurate to a certain extent. The slack variable-based efficiency measurement model (SBM) proposed by Tone [46] incorporated slack variables of input-output indicators into the calculation of decision unit efficiency, which effectively improved the problem of lack of consideration of slack variables in traditional DEA models.

Assuming there are n DMUs, each DMU contains m input X and q output Y. The a-th input of the k-th DMU is expressed as xak (a = 1, 2, ..., m; k = 1, 2, . . . , n), the b-th output is expressed as ybk (b = 1, 2, . . . , q; k = 1, 2, . . . , n), then the input-oriented SBM-DEA model with constant returns to scale can be expressed as [46]:

$$\begin{array}{ll}\min \theta = 1 - \frac{1}{m} \sum\_{i=1}^{m} s\_i^{-} / \mathfrak{x}\_{ik} \\ \text{s.t.} \quad \text{X} \lambda + \text{s}^{-} = \mathfrak{x}\_{k} \\ \text{Y} \lambda \geqslant y\_{k} \\ \lambda, \text{s}^{-} \geqslant 0 \end{array} \tag{1}$$

In Equation (1), *X<sup>k</sup>* = (*x*1*<sup>k</sup>* , *x*2*<sup>k</sup>* , . . . , *xmk*), *Y<sup>k</sup>* = *y*1*<sup>k</sup>* , *y*2*<sup>k</sup>* , . . . , *<sup>y</sup>qk* ; *s* − is the slack variable, i.e., the slack improvement value of the input index; *θ* and *λ* are the relative efficiency value and weight of the decision unit respectively, if *θ* ≥ 1, it stands that the DMU is in production frontier and belongs to the effective DMU.

#### *2.2. Window-DEA Model*

In most cases, efficiency values need to be dynamically assessed for multiple regions in different years. However, the SBM-DEA model cannot measure the whole panel data directly because the production frontier is different in different years, and the model can only decompose the panel data into cross-sectional data for static measurement separately. Therefore, the efficiency results measured by SBM-DEA model alone are not comparable between years. However, the Window-DEA model, also known as the DEA window analysis method, is a good solution to these problems [47].

The basic principle of DEA window analysis method is that the same decision unit in different years is regarded as multiple independent DMUs to participate in the calculation. Then, a number of reference sets are constructed based on the moving average method to realize the dynamic evaluation of efficiency values under the conditions of multiple decision units and long time series. Through the window analysis, it can meet the need of dynamic comparative analysis of each decision unit in time series. Moreover, the same cross-sectional data are repeatedly involved in the calculation, which can more fully explore the data value and reflect the real level. Thus, the DEA window analysis method is fully applied in the dynamic assessment of efficiency or performance of long time series in many fields. The difference between the Window-DEA model and the SBM-DEA model is only the change of reference set thus the DEA window analysis method can be well combined with the SBM-DEA model.

#### *2.3. Malmquist Index Model*

In order to analyze the evolution of CWUI in more depth and to explore the deepseated causes of efficiency changes, this study uses the Malmquist index model to explore the changes of TFP of water. Malmquist index is a non-parametric indicator used to dynamically measure the change in TFP of each DMU [48]. Moreover, Färe et al. [49] firstly used the Malmquist index for efficiency research in conjunction with DEA theory and proposed a decomposition method for total factor productivity change factors. t + 1 period compared to t period productivity change can be decomposed into integrated technical efficiency change (EC) and technical change (TC). TC refers to technological progress, which can also be understood as the forward movement of the production frontier. EC and TC are the two main causes of changes in TFP. Moreover, EC can be decomposed into deeper pure technical efficiency change (PEC) and scale efficiency change (SEC). The decomposition method can be expressed as [49]:

$$\begin{split} MI\_{t,t+1} &= \left[ \frac{TFP\_t(\mathbf{x}\_{t+1}, \mathbf{y}\_{t+1})}{TFP\_t(\mathbf{x}\_t, \mathbf{y}\_t)} \frac{TFP\_{t+1}(\mathbf{x}\_{t+1}, \mathbf{y}\_{t+1})}{TFP\_{t+1}(\mathbf{x}\_t, \mathbf{y}\_t)} \right]^{\frac{1}{2}} \\ &= \frac{TFP\_{t+1}(\mathbf{x}\_{t+1}, \mathbf{y}\_{t+1})}{TFP\_t(\mathbf{x}\_t, \mathbf{y}\_t)} \left[ \frac{TFP\_t(\mathbf{x}\_t, \mathbf{y}\_t)}{TFP\_{t+1}(\mathbf{x}\_t, \mathbf{y}\_t)} \frac{TFP\_t(\mathbf{x}\_{t+1}, \mathbf{y}\_{t+1})}{TFP\_{t+1}(\mathbf{x}\_{t+1}, \mathbf{y}\_{t+1})} \right]^{\frac{1}{2}} \\ &= E\mathbf{C}\_{t,t+1} \times T\mathbf{C}\_{t,t+1} = P\mathbf{C}\_{t,t+1} \times S\mathbf{C}\_{t,t+1} \times T\mathbf{C}\_{t,t+1} \end{split} \tag{2}$$

In Equation (2), MI is the Malmquist index, MI*t*,*t*+1 > 1 means that the TFP of water in period *t* + 1 has increased compared to period *t*; EC*t*,*t*+1 > 1 means that the DMU is closer to the production frontier in period *t* + 1 compared to in period *t*, representing an increase in overall technical efficiency; and TC*t*,*t*+1 > 1 means that the production frontier in period *t* + 1 has moved forward compared to period *t*, representing technological progress.

In this study, there is a strong link between the SBM-DEA model and the Malmquist index model. The SBM-DEA model measures integrated technical efficiency, and the Malmquist index model measures the rate of change in total factor productivity. The rate of change in TFP is not TFP itself. The Malmquist index model is a nonparametric model that cannot calculate TFP itself but can calculate the rate of change in TFP. The rate of change in TFP is also known as the Malmquist index (MI), and it can be further decomposed into the rate of change of integrated technical efficiency EC and technical progress TC. In efficiency studies involving long time series, the production frontier is constantly moving forward and can be understood as technical progress TC. The analysis of technical efficiency using only the SBM-DEA model is incomplete. Therefore, this study uses the Malmquist index model as an effective complement to the SBM-DEA model, and the two models together constitute the quantitative assessment model of water use level in this study.

#### *2.4. Spatial Matching Degree Calculation Method Based on Series Distance*

The matching degree calculation method based on the distance of the series proposed by Zuo et al. [50] can quantitatively describe the matching degree between different variables. The basic principle of this method is to create a new data series by using the proportion of a value in the series to the total value of the series and then characterize the match according to the ratio of the difference between the values of different variables and the maximum distance in the new data series. In this paper, the methodology is used to quantitatively assess the matching relationship between CWUI and economic and social development level (E-SDL) in nine provinces of the Yellow River Basin. Assuming that the study object contains a total of N subunits, a quantitative study of the match between two variables, A<sup>1</sup> and A<sup>2</sup> is carried out using a spatial matching degree calculation method based on the series distance, which is calculated as follows:

$$MD\_r = 1 - \frac{|\mathbf{x}\_1(r) - \mathbf{x}\_2(r)|}{\max\_{\substack{r=1\\r=1}}^N (\mathbf{x}\_1(r), \mathbf{x}\_2(r)) - \min\_{r=1}^N (\mathbf{x}\_1(r), \mathbf{x}\_2(r))} \tag{3}$$

In Equation (3), *<sup>x</sup>*1(*r*) = *<sup>A</sup>*<sup>1</sup> (*r*) *N* ∑ *r*=1 *A*1 (*r*) , *<sup>x</sup>*2(*r*) = *<sup>A</sup>*2(*r*) *N* ∑ *r*=1 *A*2(*r*) , (*r* = 1, 2, · · · , *N*); MD<sup>r</sup> is the degree

of the match between the two variables *A*<sup>1</sup> and *A*<sup>2</sup> in the rth subunits.

It is important to note that Equation (3) only applies to the quantification of the degree of the match when the two variables are positively correlated (i.e., the larger the value of *A***<sup>1</sup>** and the larger the value of *A***2**, the better the match between the two variables). The formula for calculating the degree of the match when the two variables are negatively correlated can be given similarly (i.e., the larger the value of A<sup>1</sup> and the smaller the value of A2, the better the match between the two variables):

$$MD\_r = 1 - \frac{\left| \mathbf{x}\_1(r) + \mathbf{x}\_2(r) - \max\_{r=1}^N (\mathbf{x}\_1(r), \mathbf{x}\_2(r)) - \min\_{r=1}^N (\mathbf{x}\_1(r), \mathbf{x}\_2(r)) \right|}{\max\_{r=1}^N (\mathbf{x}\_1(r), \mathbf{x}\_2(r)) - \min\_{r=1}^N (\mathbf{x}\_1(r), \mathbf{x}\_2(r))} \tag{4}$$

In Equation (4), the meaning of each symbol is the same as before.

#### **3. Case Study**

*3.1. Study Area*

The Yellow River is the longest river in north China. The Yellow River basin is an extremely important economic and cultural corridor and natural ecological barrier in China. It contains several basic energy bases and key ecological function areas, and it is also one of the basic water supply sources for northern China, playing an important role in the construction of China's ecological civilization and economic and social development. From the source to the mouth, the Yellow River passes through Qinghai, Sichuan, Gansu, Ningxia, Inner Mongolia, Shaanxi, Shanxi, Henan, and Shandong in turn, called the "nine provinces of the Yellow River basin," hereinafter referred to as the "nine provinces." Most of the nine provinces are arid or semi-arid areas, with a large geographical span and significant differences in altitude. There are large differences in resource and environmental endowments and economic development levels in different provinces and regions, and the difference in per capita GDP between the source and the mouth of the Yellow River is more than 10 times [51]. The elevation distribution of nine provinces of the Yellow River basin is shown in Figure 1.

**Figure 1.** Schematic view of the study area and elevation distribution.

The current situation of economic development of the nine provinces is lower than the overall level of the whole China. According to the "2019 Monitoring Report on China's Cities Completely Built a Well-off Society" issued by the Chinese government, only one of the top 20 prefecture-level cities in the Overall Well-off Index (which can represent the level of economic and social development to some extent) is located in the nine provinces. The nine provinces of the Yellow River Basin account for nearly one-third of the population and 34.8% of the arable land in China. However, the annual runoff of this river accounts for only 2% of China's total, with serious water supply and demand conflicts constraining regional development [52]. Especially in the past 20 years, under the general trend of highly rapid economic and social development in China, the water demand of the nine provinces has also increased dramatically. The Yellow River Basin is facing a severe situation of contradiction between water supply and demand. Shrinking natural incoming water and low levels of water utilization are also the causes of water shortages in the basin. The comprehensive improvement of CWUI has significance to the sustainable development of the nine provinces, especially to these provinces in the upper reach.

#### *3.2. Indicator Description and Data Sources*

When applying the DEA model, the reasonableness of the input-output indicators will directly affect the accuracy of the final efficiency measurement results. Thus, it is very important to construct a comprehensive and effective set of CWUI input-output indicators. The basic idea of efficient utilization of water is to satisfy both minimizations of resource consumption and maximization of production value. Guided by this idea, this study selects four representative indicators from four dimensions of resources, capital, labor, and ecological environment as input variables. The most direct input to reflect the level of water use is water consumption, and the total regional water consumption is used to represent water consumption [27,30,31,34]. The level of capital investment is also an important aspect in quantifying the level of resource utilization, using fixed-asset investment to represent social capital investment [27,29–31,34]. This study discusses the overall level of regional water utilization. Considering that all industries are inseparable from water utilization, the number of employees is taken as a variable reflecting human resource input [27,29–31]. The water use level without considering environmental pollution cannot reflect the real level of regional water utilization [30,34]. In order to evaluate the CWUI of nine provinces in the Yellow River basin more scientifically and comprehensively, this study adds ammonia nitrogen emission in wastewater as an input indicator reflecting environmental pollution. The most intuitive manifestation of a high-level area of water utilization is the ability to obtain higher economic benefits under certain conditions of resource inputs. Therefore, the gross regional product was selected as the output variable reflecting the value of water

resources production [27,29–31,34]. The final input-output indicator selection results are listed in Table 1.


**Table 1.** Water use level input-output indicator system.

Notes: <sup>a</sup> Investment in fixed assets data is processed through the fixed asset investment price index to eliminate price effects. <sup>b</sup> GDP data are processed through the GDP index to eliminate price effects.

High-quality economic and social development involves economic, social, and ecological aspects. Thus, the level of regional economic and social development needs to be characterized by a combination of indicators. In order to investigate the matching relationship between CWUI and E-SDL in the nine provinces, this study refers to relevant research results [14,50,51], as well as the "Statistical Monitoring Program for Building a Well-off Society" and the "Statistical Monitoring Indicator System for Building a Well-off Society in All Respects" issued by the Chinese government, and follows the principles of representativeness and dynamism to select a total of 12 representative indicators to characterize the relative level of economic and social development in terms of economic development, social harmony, and ecological friendliness. The indicator system is shown in Table 2.

**Table 2.** Quantitative indicator system of the relative level of economic and social development in nine provinces of the Yellow River Basin.


Notes: <sup>a</sup> The "+" in the indicator characteristic represents the "positive indicator," and the larger the value, the stronger the positive effect on E-SDL, and the "-" represents the "negative indicator," and the larger the value, the stronger the negative effect on E-SDL. <sup>b</sup> The weight of each indicator is combined with the corresponding indicator weights in the "Statistical Monitoring Program for Building a Well-off Society" to be converted to determine.

In order to ensure the reliability of data and the integrity of the time series, a total of seven research years from 2012 to 2018 were finally selected in this study to carry out the case study of nine provinces. The raw data used in this study were obtained by collating data from the China Statistical Yearbook, the China Science and Technology Statistical Yearbook, the Water Resources Bulletin of nine provinces, and the Statistical Yearbook of nine provinces.

#### **4. Results**

#### *4.1. Spatial–Temporal Evolution Characteristics of Water Use Level*

In this study, the values of CWUI of nine provinces during the period 2012–2018 needed to be dynamically evaluated. When applying the DEA window analysis method, there was no technical progress within a single window because all decision units within each window were involved in the calculation [53]. To increase the credibility of the results, DEA window analysis should be applied with as narrow a window width as possible [54]. Therefore, the optimal window width for the Window-DEA model was determined to be 3 years. The total length of the time series in this study was 7 years, thus for each decision unit, five windows needed to be established, namely Window1 (2012–2014), Window2 (2013–2015), Window3 (2014–2016), and Window4 (2015–2017), and Window5 (2016–2018). Then the SBM-DEA model was used to measure the CWUI for a total of 27 DMUs in 9 provinces under each window. Using the DEA-SOLVER PRO13.1 software (SAITECH, Tokyo, Japan) [55] based on the EXCEL macro program, the results of the window analysis of the SBM-DEA model for nine provinces during the period 2012–2018 were obtained (Figure 2).

**Figure 2.** *Cont.*

**Figure 2.** Results of window analysis of SBM-DEA model in nine provinces of the Yellow River Basin. ((**a**–**e**) represent Windows 1, Windows 2, Windows 3, Windows 4 and Windows 5, respectively. (**f**) represents the final result of CWUI of the nine provinces from 2012 to 2018).

As shown in Figure 2, there is an overall decreasing trend in the values for different windows within a given year during the study period. Based on the data distribution characteristics of CWUI, CWUI was classified into five levels of (0–0.5), (0.5–0.7), (0.7–0.8), (0.8–0.9), and (0.9–1.0), representing poor, medium, good, outstanding, and excellent water use levels, respectively. Taking Qinghai province as an example, in 2015, the value under Window2 was 0.506, CWUI was at a medium level. However, the value under Window3 dropped to 0.425, and under Window4 its value dropped to 0.394, under both windows, CWUI was at a poor level. Compared to Window3 and Window4, the CWUI under Window2 was at a relatively good level, but under Window4, which had a better production frontier, there was a significant decrease in its CWUI value. The CWUI values measured by the data envelopment analysis model were not absolute but were derived by

comparing the decision-making unit with the production frontier. The CWUI values of the same decision-making unit in the same year can vary depending on the production frontier, thus, the CWUI value has relativity. On the one hand, it shows that the production frontier surface of the nine provinces is moving forward, and the level of water use in the advanced provinces is improving rapidly; on the other hand, it also confirms the relative nature of CWUI values. The results of the window analysis of the SBM-DEA model in Table 3 were compiled to obtain the final results of the CWUI values of the nine provinces from 2012 to 2018, as shown in Figure 2f. The multi-year average values and rankings of CWUI for the nine provinces are listed in Table 3.

**Table 3.** Multi-year average values and ranking of water use level in nine provinces.


From an overall perspective: the overall CWUI values of the nine provinces were above 0.55 from 2012 to 2018, showing a trend of first decreasing and then increasing. The inflection point occurred in 2015, indicating that the overall CWUI level of the nine provinces deviated to the greatest extent from the production frontier surface in 2015, after which the degree of deviation gradually decreased. Subsequently, the CWUI value began to gradually rebound, rising significantly to 0.704 in the three years of 2016–2018, reaching the highest value in the study period.

From the perspective of each province: among the nine provinces, Shandong, Inner Mongolia, and Shanxi had a relatively high level of water utilization, with a multi-year average CWUI above 0.7, at a good level and above. Shandong was the only province that water use was at an excellent level and at a significant advantage in water use level compared to other provinces. Followed by Henan and Shaanxi provinces, with multi-year average CWUI above 0.6 and small fluctuations in CWUI, both provinces were at the medium level. The multi-year average CWUI of Sichuan, Gansu, Qinghai, and Ningxia were all below 0.5, which deviated from the production frontier surface to a large extent, and the water use level was at a poor level among the nine provinces. During the period 2012–2017, only Shandong province reached the production front surface among the nine provinces, and its CWUI was 1 in 2013, 2014, and 2015, indicating it was the only effective DMU among the nine provinces for three consecutive years. In 2018, the CWUI of Inner Mongolia and Shanxi also exceeded 1 and were jointly located on the production frontier surface with Shandong, while the remaining six provinces failed to reach the production frontier surface during 2012–2018, and the CWUI values of some provinces showed a decreasing trend in multiple years, which to some extent indicates while that the spatial variability of CWUI in the nine provinces of the Yellow River Basin was obvious, the synergy of water use level improvement was poor, and the gap between high and low CWUI provinces is gradually expanding, which is not conducive to the balanced and coordinated development of the nine provinces.

From the perspective of different regions: according to the dividing points of the upper, middle, and lower reaches of the Yellow River, the provinces of Inner Mongolia and above were divided into upstream areas, Shaanxi and Shanxi were divided into midstream areas. In order to facilitate the analysis of regional differences in water use levels in the Yellow River basin, Henan and Shandong were divided into downstream areas in this paper. Finally, the temporal evolution trends of CWUI in different regions are shown in Figure 3. The CWUI evolution trends of upstream and downstream provinces were basically the same, while the CWUI evolution trends of midstream provinces had more obvious fluctuations, and their water utilization levels were significantly improved in multiple years. The spatial distribution characteristics of CWUI in upstream, middle, and downstream provinces were basically consistent with the distribution of their economic and social

development levels. As can be seen from Figure 3, the CWUI of middle and downstream provinces was higher than that of upstream provinces in all years. Moreover, in 2014, the difference between the CWUI of upstream and downstream provinces reached 0.32, indicating that the industrial structure of upstream provinces needs to be optimized, there is a large redundancy of resource inputs. There is still a gap between the level of intensive use of water in the upstream provinces and the middle and downstream provinces.

**Figure 3.** Temporal evolution of CWUI in the upper, middle, and lower provinces, 2012–2018.
