1. Introduction
2. Methods
2.1. RSWRS System
2.1.1. Index Determination
2.1.2. Index Ideal Set Partition
2.2. Rough Set
2.2.1. Method Introduction
2.2.2. Calculation Procedure
 (1)
 Calculate the division of the conditional attribute set C on the domain U, that is $U/C=\left\{{X}_{1},{X}_{2},\dots ,{X}_{n}\right\}$.
 (2)
 Calculate $\leftC\right={\sum}_{i=1}^{n}{\left{X}_{i}\right}^{2}$, among which $\leftX\right$ represents the cardinality of a knowledge particle, that is, the number of elements contained in a collection.
 (3)
 Calculate the division of the single attribute x on the domain U, that is $U/\left\{x\right\}=\left\{{Z}_{1},{Z}_{2},\dots ,{Z}_{n}\right\}$, thus calculate $Sig\left(x\right)$.
 (4)
 In turn, after each condition attribute x is removed, require the classification of attribute set X on the domain U, and calculate $\leftC\left\{c\right\}\right$.
 (5)
 Calculate the importance of attribute c, that is $SIG\left(x\right)$.
 (6)
 Calculate the weights ${\omega}_{j}$.
2.3. Fuzzy Theory
 (1)
 Set an index in the system as $P=\left\{{P}_{1},{P}_{2},\dots ,{P}_{n}\right\}$, and compare the elements ${P}_{k}$ and ${P}_{l}$ of the index set. If ${P}_{k}$ is more important than ${P}_{l}$, then ${e}_{kl}=1,{e}_{lk}=0$. If ${P}_{k}$ is as important as ${P}_{l}$, then ${e}_{kl}={e}_{lk}=0.5$. If ${P}_{l}$ is more important than ${P}_{k}$, then ${e}_{kl}=0,{e}_{lk}=1$. The binary comparison matrix E is finally obtained, and specific conversion relationships are shown in Table 4.
 (2)
 Add all rows of the comparison matrix E and arrange the results in descending order to obtain the order of importance of each index.
 (3)
 According to the importance of sorting, combining with relevant empirical knowledge, build the binary comparison judgment matrix $\beta $.$$\beta =\left\{\begin{array}{cccc}{\beta}_{11}& {\beta}_{12}& \cdots & {\beta}_{1m}\\ {\beta}_{21}& {\beta}_{22}& \cdots & {\beta}_{2m}\\ \vdots & \vdots & \ddots & \vdots \\ {\beta}_{m1}& {\beta}_{m2}& \cdots & \beta mm\end{array}\right\}=({\beta}_{st})\mathrm{In}\mathrm{the}\mathrm{formula}:\{\begin{array}{l}0\le {\beta}_{st}\le 1\\ {\beta}_{st}+{\beta}_{ts}=1\\ {\beta}_{st}=0.5,s=t\end{array}$$
 (4)
 Sum each row of the matrix $\beta $ (without the fuzzy scale value of selfcomparison which is 0.5), the result vector is the relative importance of target set P to quantify the feature vector:$${w}^{\prime}=\left({w}_{1}{}^{\prime},{w}_{2}{}^{\prime},\dots ,{w}_{m}{}^{\prime}\right)=\left({\displaystyle \sum}_{t=1}^{m}{\beta}_{1t},{\displaystyle \sum}_{t=1}^{m}{\beta}_{2t},\cdots ,{\displaystyle \sum}_{t=1}^{m}{\beta}_{mt}\right)$$
 (5)
 Normalize the vector to get the weight of the index set.
2.4. Other Methods
3. Analysis and Evaluation of Sustainable Use of Water Resources in Beijing
3.1. Data
3.2. Determination of the Index Weight
3.3. Determination of the Rule Layer Weight
3.4. Evaluation Results and Analysis
3.4.1. Evaluation Results
3.4.2. Interpretation of Results
4. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A
Step  Formula 

 $\left[\begin{array}{ccc}1& 1.618& 0.618\\ 0.618& 1& 0.382\\ 1.618& 2.618& 1\end{array}\right]$ 
 $\left[\begin{array}{cccc}1& 1& 1.618& 1.618\\ 1& 1& 1.618& 1.618\\ 0.618& 0.618& 1& 1\\ 0.618& 0.618& 1& 1\end{array}\right]$ 
 $\left[\begin{array}{cccc}1& 1& 1.618& 0.618\\ 1& 1& 1.618& 0.618\\ 0.618& 0.618& 1& 0.382\\ 1.618& 1.618& 2.618& 1\end{array}\right]$ 
 $\left[\begin{array}{ccccc}1& 1& 0.382& 0.382& 1\\ 1& 1& 0.382& 0.382& 1\\ 2.618& 2.618& 1& 1& 2.618\\ 2.618& 2.618& 1& 1& 2.618\\ 1& 1& 0.382& 0.382& 1\end{array}\right]$ 

Step  Formula 

 Benefit index: ${b}_{jm}=\left({x}_{jm}{x}_{jmin}\right)/\left({x}_{jmax}{x}_{jmin}\right)$ Cost index: ${b}_{jm}=\left({x}_{jmax}{x}_{jm}\right)/\left({x}_{jmax}{x}_{jmin}\right)$ 
 ${H}_{j}=({{\displaystyle \sum}}_{m=1}^{M}{f}_{jm}ln{f}_{jm})/lnM$ In the formula: ${f}_{jm}=\left(1+{b}_{jm}\right)/{\sum}_{m=1}^{M}\left(1+{b}_{jm}\right)$ 
 ${\omega}_{j}=(1{H}_{j})/(J{{\displaystyle \sum}}_{j=1}^{J}{H}_{j})$ 
Step  Formula 

 ${\mu}_{j1}=\{\begin{array}{c}1x\in \left[0,{S}_{j\left(1\right)}\right];\\ 1+\frac{2\left(x{S}_{j\left(1\right)}\right)}{{S}_{j\left(1\right)}{S}_{j\left(2\right)}}x\in \left({S}_{j\left(1\right)},{S}_{j\left(2\right)}\right];\\ 1x\in \left({S}_{j\left(2\right)},+\infty \right].\end{array}$ 
 ${\mu}_{j2}=\{\begin{array}{c}1+\frac{2\left(x{S}_{j\left(1\right)}\right)}{{S}_{j\left(1\right)}0}x\in \left[0,{S}_{j\left(1\right)}\right];\\ 1x\in \left({S}_{j\left(1\right)},{S}_{j\left(2\right)}\right];\\ 1+\frac{2\left(x{S}_{j\left(2\right)}\right)}{{S}_{j\left(2\right)}{S}_{j\left(3\right)}}x\in \left({S}_{j\left(2\right)},{S}_{j\left(3\right)}\right];\\ 1x\in \left({S}_{j\left(3\right)},+\infty \right].\end{array}$ 
 ${\mu}_{j3}=\{\begin{array}{c}1x\in \left[0,{S}_{j\left(1\right)}\right];\\ 1+\frac{2\left(x{S}_{j\left(2\right)}\right)}{{S}_{j\left(2\right)}{S}_{j\left(1\right)}}x\in \left({S}_{j\left(1\right)},{S}_{j\left(2\right)}\right];\\ 1x\in \left({S}_{j\left(2\right)},{S}_{j\left(3\right)}\right];\\ 1+\frac{2\left(x{S}_{j\left(3\right)}\right)}{{S}_{j\left(3\right)}{S}_{j\left(4\right)}}x\in \left({S}_{j\left(3\right)},{S}_{j\left(4\right)}\right];\\ 1x\in \left({S}_{j\left(4\right)},+\infty \right].\end{array}$ 
 ${\mu}_{j4}=\{\begin{array}{c}1x\in \left[0,{S}_{j\left(2\right)}\right];\\ 1+\frac{2\left(x{S}_{j\left(3\right)}\right)}{{S}_{j\left(3\right)}{S}_{j\left(2\right)}}x\in \left({S}_{j\left(2\right)},{S}_{j\left(3\right)}\right];\\ 1x\in \left({S}_{j\left(3\right)},{S}_{j\left(4\right)}\right];\\ 1+\frac{2\left(x{S}_{j\left(4\right)}\right)}{{S}_{j\left(4\right)}{S}_{j\left(5\right)}}x\in \left({S}_{j\left(4\right)},{S}_{j\left(5\right)}\right];\\ 1x\in \left({S}_{j\left(5\right)},+\infty \right].\end{array}$ 
 ${\mu}_{j5}=\{\begin{array}{c}1x\in \left[0,{S}_{j\left(3\right)}\right];\\ 1+\frac{2\left(x{S}_{j\left(4\right)}\right)}{{S}_{j\left(4\right)}{S}_{j\left(3\right)}}x\in \left({S}_{j\left(3\right)},{S}_{j\left(4\right)}\right];\\ 1x\in \left({S}_{j\left(4\right)},{S}_{j\left(5\right)}\right].\end{array}$ 
 ${\mu}_{k}={{\displaystyle \sum}}_{j=1}^{J}{\omega}_{j}\times {\mu}_{jk}$ 
 ${\delta}_{i}={\mu}_{k}\left(max\right)$ 
Step  Formula 

 ${\mu}_{jk}={e}^{\left(\frac{{x}_{j}{b}_{jk}}{{c}_{jk}}\right){}^{2}}$ In the formula: ${b}_{jk}=\left({S}_{j\left(k\right)}+{S}_{j\left(k+1\right)}\right)/2$ ${c}_{jk}=\left{S}_{j\left(k\right)}{S}_{j\left(k+1\right)}\right/\left(2\sqrt{ln2}\right)$ 
 ${\mu}_{k}={{\displaystyle \sum}}_{j=1}^{J}{\omega}_{j}\times {\mu}_{jk}$ 
 ${\delta}_{i}={\mu}_{k}\left(max\right)$ 
Step  Formula 

 Benefit index: ${d}_{jk}=1({S}_{j\left(k\right)}{S}_{j\left(5\right)})/{S}_{j\left(1\right)}$ Cost index: ${d}_{jk}={S}_{j\left(k\right)}/{S}_{j\left(5\right)}$ 
 Benefit index: ${v}_{jm}=1\left({x}_{jm}{S}_{j\left(5\right)}\right)/{S}_{j\left(1\right)}$ Cost index: ${v}_{jm}={x}_{jm}/{S}_{j\left(5\right)}$ 
 $\rho \left({v}_{jm},{D}_{jk}\right)=\left{v}_{jm}\frac{1}{2}\left({d}_{jk}+{d}_{jk+1}\right)\right\frac{1}{2}\left({d}_{jk+1}{d}_{jk}\right)$ 
 $\rho \left({v}_{jm},{D}_{jp}\right)=\left{v}_{jm}\frac{1}{2}\right\frac{1}{2}$ 
 ${\mu}_{jk}=\{\begin{array}{c}\frac{\rho \left({v}_{jm},{D}_{jk}\right)}{\left{d}_{jk+1}{d}_{jk}\right},{v}_{jm}\in {D}_{jk}\\ \frac{\rho \left({v}_{jm},{D}_{jk}\right)}{\rho \left({v}_{jm},{D}_{jp}\right)\rho \left({v}_{jm},{D}_{jk}\right)},{v}_{jm}\notin {D}_{jk}\end{array}$ 
 ${\omega}_{jk}=\raisebox{1ex}{${d}_{jk}$}\!\left/ \!\raisebox{1ex}{${\sum}_{j=1}^{J}{d}_{jk}$}\right.$ 
 ${\mu}_{k}={{\displaystyle \sum}}_{j=1}^{J}{\omega}_{jk}\times {\mu}_{jk}$ 
 ${\delta}_{i}={\mu}_{k}\left(max\right)$ 
Step  Formula 

Calculate the correlation coefficient.  $\mathrm{r}=1\frac{6{{\displaystyle \sum}}_{i=1}^{n}{t}_{i}{}^{2}}{n\left({n}^{2}1\right)}$ 
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Rule Layer  Criteria  Index 

Economic subsystem  Agriculture  The grain yield per unit water [24] 
Water efficiency of irrigation [25]  
Industry  The value of industrial output per unit water [24]  
Repeated use rate of industrial water [25]  
Industrial water quota [25]  
Tertiary industry  Third industry accounted for the proportion of total GDP [26]  
Comprehensive index  Ten thousand Yuan GDP water consumption [24]  
GDP growth rate [26]  
Environmental subsystem  Water quality  Water quality compliance rate in water function area [25] 
Water pollution comprehensive index [24]  
Reuse of water resources  Sewage disposal rate [27]  
The use of recycled water [28]  
Plant  Forest cover rate [29]  
Green ratio [30]  
The loss of soil and water  Soil erosion modulus of water loss and soil erosion in sloping land [25]  
The ratio of water and soil loss control [27]  
Groundwater  Funnel area/whole area [24]  
Groundwater level change rate [25]  
Social subsystem  Human  Density of population [27] 
Natural growth rate of population [27]  
Per capita water resources [27]  
Daily water consumption per capita [25]  
Consciousness of saving water [24]  
City  Urbanization level [31]  
The popular rate of tap water [25] 
Rule Layer  Index  Unit 

Economic subsystem  The grain yield per unit water u11  kg/m^{3} 
The value of industrial output per unit water u12  Yuan/m^{3}  
Ten thousand Yuan GDP water consumption u13  m^{3}/10^{4} Yuan  
Third industry accounted for the proportion of total GDP u14  %  
Environmental subsystem  Water quality compliance rate in water function area u21  % 
Forest cover rate u22  %  
Sewage disposal rate u23  %  
Soil erosion modulus of water loss and soil erosion in sloping land u24  t/km^{2}  
Funnel area/whole area u25  %  
Social subsystem  Density of population u31  person/km^{2} 
Urbanization level u32  %  
Per capita water resources u33  m^{3}/(person and year)  
Daily water consumption per capita u34  L/person 
Index  I  II  III  IV  V 

u11  >1.8  1.5–1.8  1.2–1.5  0.8–1.2  <0.8 
u12  >800  600–800  400–600  200–400  <200 
u13  <50  50–150  150–250  250–350  >350 
u14  >75  75–60  60–50  50–40  <40 
u21  >80  70–80  60–70  45–60  <45 
u22  >60  50–60  30–50  10–30  <10 
u23  >95  85–95  75–85  65–75  <65 
u24  <100  100–200  200–500  500–1000  >1000 
u25  <1  1–5  5–10  10–15  >15 
u31  <200  200–500  500–800  800–1200  >1200 
u32  >85  75–85  65–75  55–65  <55 
u33  >4000  3000–4000  2000–3000  1000–2000  <1000 
u34  <160  160–180  180–200  200–220  >220 
Fuzzy Mood  Equal  Slightly  Somewhat  Rather  Obvious  Remarkably 
Fuzzy Scale  0.5  0.55  0.6  0.65  0.7  0.75 
Fuzzy Mood  Very  Extra  Exceeding  Extreme  Incomparable  
Fuzzy Scale  0.8  0.85  0.9  0.95  1 
Methods  Introduction 

EW  The Entropy Weight (EW) method of objective weighting, according to the basic principle of information theory, can be interpreted that information is a measure of degree of order system. Entropy is a measure of the degree of disorder of the system. When the information entropy is very small, the amount of information provided by the index is larger, which plays the greater role in the comprehensive evaluation, and the weight is higher [36]. EW is implemented to determine weights in this study. 
AHP  Analytic hierarchy process (AHP) is a subjective method that allows the user to visually assess the relative weight of more than one standard or multiple options according to the given standard. Even if the quantitative rating is not available, the decisionmakers can still recognize that a standard is more important than another [17]. AHP is used to calculate weights in this study. 
SPA  The set pair analysis (SPA) method is a set of system theory to deal with the uncertainty of analysis, it is not only a technical means, but also a decisionmaking system that reflects dialectical thinking. SPA is used widely since it turns the problem of uncertainty into mathematical problems by introducing the concept of the connection degree [37]. The confirmed weights are analyzed with SPA for evaluation in this study. 
AR  The attribute recognition (AR) method is a qualitative measure of the relationship between the discussion questions and different methods of qualitative description and the relationship between the corresponding measure, which has the advantages of simple calculation and good operation. The results are more reliable. It has been applied quite extensive in the evaluation of soil and water resources and environment in the field of the gas system. The confirmed weights are analyzed with AR for evaluation in this study. 
MEM  The matterelement model (MEM) is a method of solving the multiple indicators evaluation problem. It can be abstracted as a complex problem visualization model. It indicates the results that may completely reflect the comprehensive quality of things by establishing a quality evaluation model of multipleindex performance parameters and using quantitative results. Additionally, it is easy to use a computer to solve [38]. The confirmed weights are analyzed with MEM for evaluation in this study. 
SRCC  The Spearman rank correlation coefficient (SRCC) method is used to describe the degree of correlation and direction between the two variables. Original variable distribution is not required in this method, which belongs to the nonparametric statistical method. In this study, this method was used to obtain the close degree of each evaluation results and other results of the evaluation, and the highest correlation coefficient is selected as the final evaluation result [20]. 
Index  2004  2005  2006  2007  2008  2009  2010  2011  2012  2013  2014  2015 

u11  0.54  0.75  0.91  0.87  1.11  1.1  1.07  1.19  1.22  1.06  0.78  0.98 
u12  203.2  251.0  293.8  362.2  409.9  442.9  546.2  608.5  673.7  696.6  736.1  976.6 
u13  57.27  49.5  42.25  35.35  31.56  29.21  24.94  22.12  20.07  18.92  17.08  16.6 
u14  68  70  72  73  75  76  75  76  77  78  76  80 
u21  57.9  54  58.7  49  47.5  46  48  51  48.5  46  47  53.4 
u22  35.4  35.5  35.9  36.5  36.5  36.7  37  37.6  38.6  40.1  41  41.6 
u23  53.9  62.4  73.2  76.2  78.9  80.3  81  82  83  84.6  86.1  87.9 
u24  117.2  151.3  145.3  139.4  219.7  65.8  94.9  103.4  747.9  294.4  194.4  238.5 
u25  5.95  5.95  6.23  6.26  6.27  6.38  6.44  6.45  6.39  6.44  6.45  6.43 
u31  910  937  976  1021  1079  1133  1196  1230  1261  1289  1311  1323 
u32  79  83.6  84.3  84.5  84.9  85  85.9  86.2  86.2  86.3  86.3  86.5 
u33  143.0  151.2  141.5  148.2  205.5  126.6  124.2  134.7  193.2  118.6  95.2  124.0 
u34  226.8  152.9  154.7  166.8  187.2  192.1  174.9  172.6  171.8  196.9  187.5  183.8 
Rule Layer  Index  $Si{\mathbf{g}}_{x}\left(X\right)$  $Sig\left(x\right)$  $SIG\left(x\right)$  Weight 

Economic subsystem  u11  0.37  0.64  1.02  0.30 
u12  0.45  0.75  1.20  0.36  
u13  0.07  0.48  0.55  0.16  
u14  0.00  0.62  0.62  0.18  
Environmental subsystem  u21  0.14  0.40  0.54  0.14 
u22  0.00  0.40  0.40  0.10  
u23  0.55  0.70  1.24  0.32  
u24  0.60  0.73  1.33  0.34  
u25  0.00  0.40  0.40  0.10  
Social subsystem  u31  0.27  0.64  0.91  0.27 
u32  0.13  0.64  0.77  0.23  
u33  0.00  0.40  0.40  0.12  
u34  0.54  0.74  1.28  0.38 
Index  I  II  III  IV  V 

u11  −1.00  −1.00  0.10  1.00  −0.10 
u12  1.00  0.64  −1.00  −1.00  −1.00 
u13  1.00  −0.34  −1.00  −1.00  −1.00 
u14  1.00  0.88  −1.00  −1.00  −1.00 
u21  −1.00  −1.00  0.26  1.00  −0.26 
u22  −1.00  0.39  1.00  −0.39  −1.00 
u23  −0.37  1.00  0.37  −1.00  −1.00 
u24  −1.00  0.74  1.00  −0.74  −1.00 
u25  −1.00  0.43  1.00  −0.43  −1.00 
u31  −1.00  −1.00  −1.00  −0.23  1.00 
u32  1.00  0.97  −1.00  −1.00  −1.00 
u33  −1.00  −1.00  −1.00  −0.40  1.00 
u34  −1.00  0.62  1.00  −0.62  −1.00 
Year  Weighted Connection Degrees  Evaluation Level  

I  II  III  IV  V  
2004  −0.60  −0.35  −0.47  −0.09  0.07  V 
2005  −0.24  −0.05  −0.46  −0.07  −0.30  II 
2006  −0.20  −0.06  −0.24  −0.03  −0.56  IV 
2007  −0.29  −0.04  −0.16  −0.05  −0.56  II 
2008  −0.58  −0.11  0.24  0.03  −0.66  III 
2009  −0.41  −0.17  0.02  0.00  −0.62  III 
2010  −0.33  0.20  −0.09  −0.33  −0.58  II 
2011  −0.29  0.26  −0.08  −0.42  −0.64  II 
2012  −0.35  0.12  −0.11  −0.29  −0.54  II 
2013  −0.45  −0.03  0.01  −0.22  −0.56  III 
2014  −0.38  0.18  −0.20  −0.48  −0.42  II 
2015  −0.29  0.17  −0.14  −0.50  −0.56  II 
Year  SPAro  ARro  SPAen  ARen  SPAan  ARan  MEM 

2004  V  V  V  V  V  V  V 
2005  II  IV  II  II  II  IV  IV 
2006  IV  IV  IV  IV  II  IV  IV 
2007  II  IV  II  IV  II  II  IV 
2008  III  III  III  III  III  III  III 
2009  III  III  III  III  III  III  III 
2010  II  III  II  III  II  II  III 
2011  II  II  II  III  II  II  II 
2012  II  II  II  III  II  II  II 
2013  III  III  II  III  III  III  III 
2014  II  II  II  II  II  III  III 
2015  II  I  II  I  II  I  I 
Methods  SPAro  ARro  SPAen  ARen  SPAan  ARan  MEM  Average 

SPAro  1.000  0.965  0.965  0.972  0.986  0.979  0.962  0.976 
Arro  0.965  1.000  0.930  0.979  0.951  0.979  0.997  0.972 
SPAen  0.965  0.930  1.000  0.937  0.951  0.944  0.927  0.951 
Aren  0.972  0.979  0.937  1.000  0.958  0.958  0.976  0.969 
SPAan  0.986  0.951  0.951  0.958  1.000  0.965  0.948  0.966 
Aran  0.979  0.979  0.944  0.958  0.965  1.000  0.983  0.973 
MEM  0.962  0.997  0.927  0.976  0.948  0.983  1.000  0.970 
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