# Fuzzy Comprehensive Assessment Method Based on the Entropy Weight Method and Its Application in the Water Environmental Safety Evaluation of the Heshangshan Drinking Water Source Area, Three Gorges Reservoir Area, China

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methodology

#### 2.1. Study Area

^{2}. According to the related division standard, the water quality of the HDWSA should meet the requirements of level III in environmental quality standards for surface water (GB3838-2002) [26].

#### 2.2. Methodology

#### 2.2.1. Fuzzy Comprehensive Assessment Method

- (1)
- Defining the factor set, i.e., U = {u
_{1}, u_{2}, …, u_{i}, …, u_{m}}, which is a set consisting of m kinds of evaluation factors, and u_{i}is the ith evaluation factor. - (2)
- Establishing the evaluation set V = {v
_{1}, v_{2}, …, v_{j}, …, v_{n}}, which is a discrete set made up of n levels of evaluation results, and v_{j}is the jth evaluation result. - (3)
- Building the original matrix X,$$X=\left[\begin{array}{ccccc}{{\displaystyle x}}_{11}& {{\displaystyle x}}_{12}& \dots & \dots & {{\displaystyle x}}_{1\text{}yr}\\ {{\displaystyle x}}_{21}& {{\displaystyle x}}_{22}& \dots & \dots & {{\displaystyle x}}_{2\text{}yr}\\ \vdots & \vdots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \vdots \\ {{\displaystyle x}}_{m1}& {{\displaystyle x}}_{m2}& \cdots & \cdots & {{\displaystyle x}}_{m\text{}yr}\end{array}\right],$$
_{ik}(1 ≤ I ≤ m, 1 ≤ k ≤ yr) is the value of the ith index in the kth year. - (4)
- Determining the weight matrix A = {a
_{1}, a_{2}, …, a_{m}}, which is a set composed of m kinds of index weights which indicate the importance of various evaluation indexes. - (5)
- Constructing the single factor evaluation matrix Q by membership function, where Q is a fuzzy relationship matrix that consists of the membership degrees of u
_{i}to v_{j}. The matrix is:$$Q=\left[\begin{array}{ccccc}{{\displaystyle q}}_{11}& {{\displaystyle q}}_{12}& \dots & \dots & {{\displaystyle q}}_{1\text{}yr}\\ {{\displaystyle q}}_{21}& {{\displaystyle q}}_{22}& \dots & \dots & {{\displaystyle q}}_{2\text{}yr}\\ \vdots & \vdots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \vdots \\ {{\displaystyle q}}_{m1}& {{\displaystyle q}}_{m2}& \cdots & \cdots & {{\displaystyle q}}_{m\text{}yr}\end{array}\right],$$_{ij}is the membership degree of factor u_{i}to v_{j}. - (6)
- Obtaining the comprehensive evaluation set B, which is a set made up of n kinds of evaluation results by fuzzy operating of the single factor evaluation matrix and weight matrix.$$B=({b}_{1},\text{}{b}_{2},\text{}......\text{},{b}_{n})=A\circ Q,$$

#### 2.2.2. Improved Fuzzy Comprehensive Assessment Method

- (1)
- Standardizing the indexes and building the standardization matrix Y = (y
_{ij})_{m × yr}(m is the number of indexes, yr is the year for evaluation), which will be explained specifically in Section 2.2.3. Then, the proportion of each index (p_{ik}) is determined as follows:$${p}_{ik}=\frac{{y}_{ik}}{{\displaystyle {\sum}_{j=1}^{yr}{y}_{ik}}}\text{}(1\le I\le m,\text{}1\le k\le yr),$$_{ik}is the proportion of the ith index in the kth year. - (2)
- Calculating the information entropy (e
_{i}) by$${e}_{i}=-c{\displaystyle {\sum}_{j=1}^{yr}{p}_{ij}}\mathrm{ln}{p}_{ij},$$_{ik}= 0, then p_{ik}lnp_{ik}= 0 (1 ≤ i ≤ m, 1 ≤ k ≤ yr). - (3)
- Obtaining the weights of indexes by$${a}_{i}=\frac{1-{e}_{i}}{m-{\displaystyle {\sum}_{i=1}^{m}{e}_{i}}},$$
_{i}is the final weight of the ith index, and the weight matrix is formed as A = {a_{1}, a_{2}, …, a_{m}}.

#### 2.2.3. Standardization

- (1)
- Building the original matrix X$$X=\left[\begin{array}{ccccc}{{\displaystyle x}}_{11}& {{\displaystyle x}}_{12}& \dots & \dots & {{\displaystyle x}}_{1\text{}yr}\\ {{\displaystyle x}}_{21}& {{\displaystyle x}}_{22}& \dots & \dots & {{\displaystyle x}}_{2\text{}yr}\\ \vdots & \vdots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \vdots \\ {{\displaystyle x}}_{m1}& {{\displaystyle x}}_{m2}& \cdots & \cdots & {{\displaystyle x}}_{m\text{}yr}\end{array}\right],$$
- (2)
- For a benefit index, the standardized x
_{ik}, i.e., y_{ik}is calculated as$${y}_{ik}=\frac{{x}_{ik}-\underset{1\le k\le yr}{\mathrm{min}}{x}_{ik}}{\underset{1\le k\le yr}{\mathrm{max}}{x}_{ik}-\underset{1\le k\le yr}{\mathrm{min}}{x}_{ik}},$$_{ik}(1 ≤ i ≤ m, 1 ≤ k ≤ yr) is the value of the ith index in the kth year, y_{ik}is the standardization value of a benefit index x_{ik}. - (3)
- As for a cost index, the standardization equation is$${y}_{ik}=\frac{\underset{1\le k\le yr}{\mathrm{max}}{x}_{ik}-{x}_{ik}}{\underset{1\le k\le yr}{\mathrm{max}}{x}_{ik}-\underset{1\le k\le yr}{\mathrm{min}}{x}_{ik}},$$

## 3. Results and Discussion

#### 3.1. Water Environmental Safety Evaluation Index System

#### 3.2. Water Environmental Safety Evaluation of the Heshangshan Drinking Water Source Area

#### 3.2.1. Construction of Factor Set U and Evaluation Set V

_{1}, u

_{2}, …, u

_{21}} = {per capita water resources, per capita domestic water consumption, …, proportion of tertiary industry}. According to the national standards and the status quo of the study area, the evaluation set V of water environmental safety in the HDWSA was divided into five levels (I, II, III, IV, and V) (i.e., V = {v

_{1}, v

_{2}, v

_{3}, v

_{4}, v

_{5}}). For the destination layer, the five levels meant the water environment of the HDWSA was very safe, relatively safe, substantially safe, relatively unsafe, and unsafe, respectively. Specifically, level I stood for the safest situation, which showed that the drinking water source area had a great capacity to contain pollutants and a small risk of losing drinking water supply ability. On the contrary, level V meant the worst situation, which indicated that water resources were over-explored, that the aqueous environment was deteriorative, and that water environmental safety was under great threat. Level III was a medium state, which demonstrated that the situation of the water environment was substantially safe and there were still potential risks to water resource utilization. Furthermore, level II (between level I and level III), as well as level IV (between level III and level V), expressed relatively safe and relatively unsafe grades, respectively. As for the other layers, five levels of evaluation set V represented a criterion, index, or subindex that were excellent, good, substantially good, poor, and very poor for ensuring water environmental safety. As shown in Table 3, V for each subindex had values in five levels according to previous studies, national conditions, regional regulations, and the status quo of the study area [36].

#### 3.2.2. Weight Determination

_{ik})

_{21×5}was standardized based on Formulas (8) and (9), and then the standardization matrix Y = (y

_{ik})

_{21×5}was obtained. According to Formulas (4)–(6), weights of the evaluation index system could be calculated as Table 4 shows. Furthermore, the index weight was the sum of those about its subindexes (e.g., the weight of P1 was the sum of weights of D1, D2, and D3), and the same was true of the criterion weight.

#### 3.2.3. Comprehensive Evaluation

_{j}(x) (j = 1, 2, 3, 4, 5) was the membership degree function, x

_{1}, x

_{2}, x

_{3},and x

_{4}were the boundary values of evaluation set V for the evaluation subindexes.

_{ij}was calculated on the basis of Equations (10) and (11). Moreover, each subindex had five membership degrees corresponding to five levels of evaluation set V, being excellent, good, substantially good, poor, and very poor for the water environmental safety of the HDWSA. Then, the single factor evaluation matrix Q was built, which was formed by the membership degrees of all subindexes to evaluation set V. In other words, Q reflected the evaluation results of the subindexes (Table 5). Based on Formula (3) and index weights showed in Table 4, comprehensive evaluation matrix B was obtained. Evaluation matrixes for index B

_{1}, B

_{2}, …, and B

_{7}were calculated by Q and the weights of the subindex layer. Similarly, those for criteria B

_{p}, B

_{s}, and B

_{r}were determined. Comprehensive evaluation results of the index and criterion layers were shown in Table 6. The final comprehensive evaluation matrix B (the evaluation matrix of the destination layer) for the HDWSA was achieved based on the weights and evaluation results of the criterion layer. The calculation process was as follows:

_{3}0.304 was the maximum value, which indicated that the water environmental safety level of the HDWSA belonged to level III. In other words, the water environment of the HDWSA was substantially safe in 2014. As for that of v

_{2}, it equaled 0.303 and meant that some of the indexes were considered relatively safe. As far as those of v

_{1}, v

_{4}, and v

_{5}were concerned, the value of v

_{1}meant several indexes were good for water environmental safety, whereas values of v

_{4}, and v

_{5}indicated that individual ones were undesired.

_{2}, v

_{4}, and v

_{5}were 0.453 (the maximum value), 0.072, and 0 respectively (Table 6), which indicated that the P criterion was good for the water environment of the HDWSA. According to the membership degrees of P1 and P2 to evaluation set V, P1 as well as P2 of the HDWSA were all in level II, and P2 was more beneficial for the water environmental safety. For membership degrees of the subindexes for pressure (D1–D6) to evaluation set V, they were in level II, except that of per capita water resources (D1). D1 was substantially good for the objective, and the reason was that the location area of the HDWSA has abundant water resources but a large population, meanwhile water resources per capita was not very high. In addition, comprehensive evaluation results concerning the state of various layers of the evaluation index system were discussed as follows: S was in level II and its membership degrees to v

_{4}and v

_{5}were small (0.129 and 0.152, respectively), which indicated that S was good for water environmental safety and there were still some aspects needed to be improved. By analyzing the membership degrees of individual subindexes for state (D7–D16) to evaluation set V, it could be found that the membership degrees of some subindexes in v

_{4}and v

_{5}were relatively large. Therefore, for promoting the water environmental safety of the HDWSA, more attention should be paid to those subindexes, including the natural population growth rate D7, water consumption per 10,000 yuan of value added by industry D12, and the eutrophication section percentage of influents in the Three Gorges reservoir area D16. Furthermore, the effects of R, R1, R2, as well as D17–D21 on the evaluation result were also revealed as below. As far as the response criterion was concerned, R was in level III for its membership degree to v

_{3}, in which 0.361 was the maximum. However, the membership degrees of R to v

_{4}and v

_{5}were 0.272 and 0.091, which meant efforts were also expected in some fields. In detail, the vegetation cover rate (D18) to v

_{5}was 0.390, and the investment rate of environmental protection (D20), as well as proportion of tertiary industry (D21) to v

_{4}, reached 0.464 and 0.258 respectively. Hence, more work should focus on those subindexes, especially the vegetation cover rate (D18). According to the actual situation of HDWSA, a water pollution accident has not occurred in recent years. Therefore, the evaluation result, that the water environment of HDWSA was substantially safe, is reasonable and reliable.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Frame of the Pressure–State–Response (PSR) model for water environmental safety in drinking water source areas.

**Table 1.**Water environmental safety evaluation index system of the Heshangshan drinking water source area.

Destination Layer | Criterion Layer | Index Layer | Subindex Layer |
---|---|---|---|

Water environmental safety of the Heshangshan drinking water sources area | Pressure (P) | Water resources P1 | Per capita water resources D1; |

Per capita domestic water consumption D2; | |||

Annual irrigation water consumption per hectare D3 | |||

Pollution source and its discharge P2 | COD discharge amount D4; | ||

Ammonia nitrogen discharge amount D5; | |||

The number of industrial enterprises beyond designed scale D6 | |||

State (S) | Social economy S1 | Natural population growth rate D7; | |

Population density D8; | |||

Per capita GDP D9 | |||

Water quantity S2 | Daily water supply amount of a project D10; | ||

Annual rainfall D11; | |||

Water consumption per 10,000 yuan of value-added by industry D12 | |||

Water quality S3 | Standard-meeting rate of drinking water for drinking water source area D13; | ||

Wastewater discharge of per unit GDP D14; | |||

Ratio of wastewater and runoff D15; | |||

Eutrophication section percentage of influents in Three Gorges reservoir area D16 | |||

Response (R) | Environmental protection R1 | Wastewater treatment rate of sewage plant D17; | |

Vegetation cover rate D18; | |||

Governance rate of soil and water loss D19 | |||

Industrial structure R2 | Investment rate of environmental protection D20; | ||

Proportion of tertiary industry D21 |

**Table 2.**Data for water environmental safety evaluation of the Heshangshan drinking water source area.

Subindex | Unit | 2010 | 2011 | 2012 | 2013 | 2014 | Type |
---|---|---|---|---|---|---|---|

D1 | M^{3}/person | 1405.50 | 1545.37 | 1425.07 | 1412.39 | 1903.82 | ↑ |

D2 | M^{3}/person | 47.51 | 49.45 | 50.72 | 52.63 | 48.18 | ↓ |

D3 | M^{3} | 251.00 | 291.00 | 287.00 | 270.00 | 307.00 | ↓ |

D4 | mg/L | 183.04 | 435.53 | 304.23 | 274.95 | 265.02 | ↓ |

D5 | mg/L | 19.51 | 57.47 | 40.33 | 36.63 | 35.19 | ↓ |

D6 | _ | 896.00 | 527.00 | 575.00 | 650.00 | 734.00 | ↓ |

D7 | % | 7.25 | 6.54 | 3.88 | 4.64 | 5.10 | ↓ |

D8 | person/km^{2} | 400.89 | 404.09 | 406.11 | 407.57 | 409.60 | ↓ |

D9 | 10^{4} yuan | 2.76 | 3.45 | 3.89 | 4.32 | 4.79 | ↑ |

D10 | 10^{4} m^{3} | 652.66 | 815.42 | 833.24 | 813.34 | 833.34 | ↑ |

D11 | 10^{8} m^{3} | 872.07 | 899.67 | 890.45 | 876.43 | 1046.52 | ↑ |

D12 | M^{3} | 128.00 | 92.00 | 76.00 | 77.00 | 71.00 | ↓ |

D13 | % | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | ↑ |

D14 | M^{3}/10^{4} yuan | 16.16 | 9.56 | 11.60 | 11.15 | 10.22 | ↓ |

D15 | % | 2.76 | 1.86 | 2.78 | 3.00 | 2.27 | ↓ |

D16 | % | 42.20 | 38.90 | 25.00 | 36.10 | 44.40 | ↓ |

D17 | % | 88.86 | 92.21 | 89.77 | 93.20 | 92.25 | ↑ |

D18 | % | 6.05 | 8.45 | 5.34 | 4.87 | 5.21 | ↑ |

D19 | % | 37.00 | 39.00 | 42.10 | 42.10 | 43.10 | ↑ |

D20 | % | 2.93 | 2.72 | 2.10 | 2.02 | 2.06 | ↑ |

D21 | % | 36.40 | 36.20 | 39.40 | 46.70 | 46.80 | ↑ |

Subindexes | Units | Excellent | Good | Substantially Good | Poor | Very Poor |
---|---|---|---|---|---|---|

D1 | M^{3}/person | [3000,∞) | [2000,3000) | [1000,2000) | [500,1000) | (0,500) |

D2 | m^{3}/person | [0,30] | (30,45] | (45,55] | (55,80] | (80,∞) |

D3 | m^{3} | [0,200] | (200,300] | (300,360] | (380,500] | (500,∞) |

D4 | mg/L | [0,100] | (100,300] | (300,700] | (700,1000] | (1000,∞) |

D5 | mg/L | [0,15] | (15,30] | (30,60] | (60,100] | (100,∞) |

D6 | _ | [0,400] | (400,800] | (800,1400] | (1400,2000] | (2000,∞) |

D7 | % | [0,0.7] | (0.7,1.2] | (1.2,3.5] | (3.5,5] | (5,∞) |

D8 | person/km^{2} | [0,300] | (300,400] | (400,500] | (500,2000] | (2000,∞) |

D9 | 10^{4} yuan | [5,15) | [3,5) | [1.5,3) | [1,1.5) | (0.5,1) |

D10 | 10^{4} m^{3} | [1200,∞) | [900,1200) | [600,900) | [300,600) | (0,300) |

D11 | 10^{8} m^{3} | [1100,∞) | [800,1100) | [650,800) | [400,650) | (0,400) |

D12 | m^{3} | [0,20] | (20,40] | (40,65] | (65,130] | (130,∞) |

D13 | % | [98,100) | [96,98) | [90,96) | [70,90) | (0,70) |

D14 | m^{3}/10^{4} yuan | [0,20] | (20,50] | (50.100] | (100,150] | (150,∞) |

D15 | % | [0,3] | (3,5.5] | (5.5,7.7] | (7.7,10] | (10,100) |

D16 | % | [0,5] | (5,10] | (10,15] | (15,45] | (45,∞) |

D17 | % | [98,100) | [90,98) | [80,90) | [70,80) | (0,70) |

D18 | % | [90,100) | [50,90) | [10,50) | [4,10) | (4,0) |

D19 | % | (50,100) | [35,50) | [20,35) | [10,20) | [0,10) |

D20 | % | [2.2,100) | [1.7,2.2) | [1.2,1.7) | [0.7,1.2) | (0,0.7) |

D21 | % | [70,100) | [50,70) | [30,50) | [20,30) | (0,20) |

**Table 4.**Weights of the water environmental safety evaluation index system in the Heshangshan drinking water source area.

Destination Layer | Criterion Layer | Index Layer | Subindex Layer | Weight |
---|---|---|---|---|

A 1.000 | P 0.278 | P1 0.182 | D1 | 0.111 |

D2 | 0.029 | |||

D3 | 0.041 | |||

P2 0.096 | D4 | 0.031 | ||

D5 | 0.033 | |||

D6 | 0.032 | |||

S 0.420 | S1 0.119 | D7 | 0.041 | |

D8 | 0.043 | |||

D9 | 0.036 | |||

S2 0.162 | D10 | 0.027 | ||

D11 | 0.106 | |||

D12 | 0.029 | |||

S3 0.139 | D13 | 0.000 | ||

D14 | 0.028 | |||

D15 | 0.053 | |||

D16 | 0.058 | |||

R 0.302 | R1 0.150 | D17 | 0.040 | |

D18 | 0.075 | |||

D19 | 0.035 | |||

R2 0.152 | D20 | 0.082 | ||

D21 | 0.070 |

Subindexes | Evaluation Set | ||||
---|---|---|---|---|---|

v_{1} | v_{2} | v_{3} | v_{4} | v_{5} | |

Levels of Water Environmental Safety | |||||

I | II | III | IV | V | |

D1 | 0.000 | 0.372 | 0.412 | 0.216 | 0.000 |

D2 | 0.047 | 0.518 | 0.434 | 0.000 | 0.000 |

D3 | 0.136 | 0.475 | 0.389 | 0.000 | 0.000 |

D4 | 0.221 | 0.490 | 0.289 | 0.000 | 0.000 |

D5 | 0.275 | 0.464 | 0.261 | 0.000 | 0.000 |

D6 | 0.090 | 0.546 | 0.364 | 0.000 | 0.000 |

D7 | 0.000 | 0.000 | 0.000 | 0.063 | 0.937 |

D8 | 0.000 | 0.032 | 0.532 | 0.436 | 0.000 |

D9 | 0.378 | 0.423 | 0.199 | 0.000 | 0.000 |

D10 | 0.190 | 0.429 | 0.381 | 0.000 | 0.000 |

D11 | 0.327 | 0.398 | 0.276 | 0.000 | 0.000 |

D12 | 0.000 | 0.000 | 0.370 | 0.407 | 0.222 |

D13 | 0.505 | 0.495 | 0.000 | 0.000 | 0.000 |

D14 | 0.662 | 0.338 | 0.000 | 0.000 | 0.000 |

D15 | 0.569 | 0.431 | 0.000 | 0.000 | 0.000 |

D16 | 0.000 | 0.000 | 0.015 | 0.496 | 0.489 |

D17 | 0.126 | 0.448 | 0.426 | 0.000 | 0.000 |

D18 | 0.000 | 0.000 | 0.103 | 0.508 | 0.390 |

D19 | 0.248 | 0.459 | 0.293 | 0.000 | 0.000 |

D20 | 0.000 | 0.057 | 0.478 | 0.464 | 0.000 |

D21 | 0.000 | 0.339 | 0.403 | 0.258 | 0.000 |

Layer | Evaluation set | |||||
---|---|---|---|---|---|---|

v_{1} | v_{2} | v_{3} | v_{4} | v_{5} | ||

Levels of Water Environmental Safety | ||||||

I | II | III | IV | V | ||

Index layer | P1 | 0.036 | 0.429 | 0.425 | 0.110 | 0.000 |

P2 | 0.201 | 0.498 | 0.302 | 0.000 | 0.000 | |

S1 | 0.170 | 0.204 | 0.318 | 0.089 | 0.218 | |

S2 | 0.248 | 0.333 | 0.309 | 0.072 | 0.039 | |

S3 | 0.321 | 0.215 | 0.007 | 0.230 | 0.227 | |

R1 | 0.097 | 0.240 | 0.241 | 0.239 | 0.183 | |

R2 | 0.000 | 0.216 | 0.478 | 0.305 | 0.000 | |

Criterion layer | P | 0.093 | 0.453 | 0.382 | 0.072 | 0.000 |

S | 0.250 | 0.257 | 0.212 | 0.129 | 0.152 | |

R | 0.048 | 0.228 | 0.361 | 0.272 | 0.091 |

**Table 7.**Evaluation result of the water environmental safety level of the Heshangshan drinking water source area.

Indicator | Level | ||||
---|---|---|---|---|---|

Water environmental safety level | I | II | III | IV | V |

Evaluation set V | v_{1} | v_{2} | v_{3} | v_{4} | v_{5} |

Membership degree of destination layer | 0.145 | 0.303 | 0.304 | 0.156 | 0.092 |

Evaluation result of water environmental safety level | — | — | √ | — | — |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ding, X.; Chong, X.; Bao, Z.; Xue, Y.; Zhang, S.
Fuzzy Comprehensive Assessment Method Based on the Entropy Weight Method and Its Application in the Water Environmental Safety Evaluation of the Heshangshan Drinking Water Source Area, Three Gorges Reservoir Area, China. *Water* **2017**, *9*, 329.
https://doi.org/10.3390/w9050329

**AMA Style**

Ding X, Chong X, Bao Z, Xue Y, Zhang S.
Fuzzy Comprehensive Assessment Method Based on the Entropy Weight Method and Its Application in the Water Environmental Safety Evaluation of the Heshangshan Drinking Water Source Area, Three Gorges Reservoir Area, China. *Water*. 2017; 9(5):329.
https://doi.org/10.3390/w9050329

**Chicago/Turabian Style**

Ding, Xiaowen, Xiao Chong, Zhengfeng Bao, Ying Xue, and Shanghong Zhang.
2017. "Fuzzy Comprehensive Assessment Method Based on the Entropy Weight Method and Its Application in the Water Environmental Safety Evaluation of the Heshangshan Drinking Water Source Area, Three Gorges Reservoir Area, China" *Water* 9, no. 5: 329.
https://doi.org/10.3390/w9050329