# Application of a New Improved Weighting Method, ESO Method Combined with Fuzzy Synthetic Method, in Water Quality Evaluation of Chagan Lake

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

_{Mn}), biochemical oxygen demand (BOD

_{5}), ammo-nia nitrogen (NH

_{3}-N), chemical oxygen demand (COD

_{Cr}), and the total phosphorus (TP), total nitrogen (TN), and fluoride (F) contents. The location of Chagan lake is as shown in Figure 1.

_{Cr}, COD

_{Mn}, BOD

_{5}, TP, and TN contents are the main indicators affecting the aquatic environment of the lake. After selecting the evaluation indicators, it was necessary to select their standard values for all levels. As Chagan Lake is located in China, the Environmental Quality Standard of Surface Water (GB3838-2002) issued by the Ministry of Ecology and Environment of the People’s Republic of China in 2002 was selected as the evaluation criterion.

#### 2.2. Method and Improvement

#### 2.2.1. Fuzzy Synthetic Evaluation Method

- Determination of the evaluation indicator, A = {DO, F, CODCR, CODEMN, TP, TN}
- Determination of the standard value of each grade of the evaluation indicator, V = {v1, v2, …, vm}.
- Determination of the distribution function and calculation of the membership degree to construct the fuzzy matrix R [22].

_{j}and v

_{j+1}are the standard values of indicator i, x is the monitoring value of indicator i, and r

_{ij}represents the membership degree of a certain evaluation indicator A

_{i}in a sample to the water quality v

_{j}.

- 4.
- Calculation of the weight vector of all indicators, W = {w
_{1}, w_{2}, …, w_{n}}. - 5.
- In the fuzzy operation, matrix multiplication is used to calculate the weight W and membership R matrices, and the fuzzy synthetic operation result B is obtained.$$B=W\times R=\left({w}_{1},{w}_{2},\dots ,{w}_{n}\right)\left[\begin{array}{cccc}{r}_{11}& {r}_{12}& \dots & {r}_{1m}\\ {r}_{21}& {r}_{22}& \dots & {r}_{2m}\\ \dots & \dots & \dots & \dots \\ {r}_{n1}& {r}_{n2}& \dots & {r}_{nm}\end{array}\right]=\left({b}_{1},{b}_{2},\dots ,{b}_{m}\right)$$
- 6.
- Comprehensive evaluation. The results of the fuzzy operation are calculated and evaluated according to the principle of weighted average comprehensive evaluation in this study, which regards the water quality grade as a continuous relative position, and values {1, 2, 3, 4, 5} represent water quality grades {I, II, III, IV, V}, and these values are the Z-rank of each water quality grade. The relative position B
_{T}, which expresses the grade of water quality by rational numbers rather than integers, of the water quality grade of the indicator can then be obtained by weighting the Z-rank of each water quality grade for each indicator [23]. Additionally, compared to the maximum membership principle, this principle can preserve as much information in the assessment coefficients as possible [24].$${B}_{T}=\frac{{{\displaystyle \sum}}_{j=1}^{5}{b}_{j}^{\beta}\times j}{{{\displaystyle \sum}}_{j=1}^{5}{b}_{j}^{\beta}}$$_{T}is relative water quality grade for a sample, b_{j}is the weighted membership degree of the sample to the j-grade water quality, and β is the weighted coefficient with a value of 2.

#### 2.2.2. ESO Method

- Step 1—Entropy weight calculation

- 1
- Calculation of the original matrix, Q = (qij)
_{(m×n)}, I = 1, 2, …, m; j = 1, 2, …, n.For the “bigger the better” indicator:$${q}_{ij}=\frac{{x}_{ij}-{x}_{min,j}}{{x}_{max,j}-{x}_{min,j}}$$For the “smaller the better” indicator:$${q}_{ij}=\frac{{x}_{max,j}-{x}_{ij}}{{x}_{max,j}-{x}_{min,j}}$$ - 2
- The original matrix is normalized to obtain the matrix Cn = (cn
_{ij})_{(m×n)}$$c{n}_{ij}=\frac{{q}_{ij}}{{{\displaystyle \sum}}_{t=1}^{m}{q}_{tj}}$$ - 3
- As the logarithm will appear in the entropy calculation, matrix Cn must be modified to prevent its zero element from being unable to be calculated. The revision is as follows:$${{c}^{\prime}}_{ij}=\frac{1+c{n}_{ij}}{{{\displaystyle \sum}}_{t=1}^{m}\left(1+c{n}_{tj}\right)}$$
- 4
- Entropy calculation:$${E}_{j}=\frac{-{{\displaystyle \sum}}_{i=1}^{m}{{c}^{\prime}}_{ij}\times ln{{c}^{\prime}}_{ij}}{lnm}$$
- 5
- Determination of entropy weight:$${W}_{j}=\frac{1-{E}_{j}}{n-{{\displaystyle \sum}}_{j=1}^{n}{E}_{j}}$$
_{ij}is the scale value of indicator j in sample i that expresses the degree of discreteness of an index, x_{i}_{j}is the measured value of indicator j in sample i, x_{min,j}is the minimum value in the measured value of evaluation indicator j, x_{max,j}is the maximum value in the measured value of evaluation indicator j, cn_{ij}is the normalized scale value of indicator j in sample i, c’_{ij}is the normalized scale value of indicator j in revised sample i, E_{j}is the information entropy of evaluation indicator j, and W_{j}is the weight value of evaluation indicator j.

- Step 2—Weight correction based on the single-factor evaluation results

- 1
- Determination of the amendment data. The amendment data refers to the basic data used to modify the entropy weight, which can reflect the user’s attitude towards the influencing factors of the parameters. This study thinks that the pollution degree of each parameter is an important factor affecting its weight, so the single-factor evaluation results are used as amendment data. The choice of the amendment data can be determined according to the actual demand and, therefore, it is subjective. However, the ESO method is objective after determining the amendment data. If users agree with the amendment rule of this study, they also could take the single factor assessment results as amendment data, then the subjectivity disappears.
- 2
- Calculation of the amendment entropy weight according to the single-factor evaluation results using the amendment rule. The amendment data can be selected to the requirement. As grade III is used to indicate whether the water quality is qualified or not, the weights of sample indices better than grade III will be reduced, while those of sample indices inferior to grade III will increase, taking grade III water as the reference point in the algorithm proposed in this study. This work is to avoid ignorance to indicators with high pollution to cause an unreasonable result. The formula is as follows.$$W{n}_{j}={W}_{j}\times \frac{{{\displaystyle \sum}}_{i=1}^{3}m\times n}{N}$$
_{j}is the amendment weight of indicator j, W_{j}is the entropy weight of indicator j, m is the amendment value according to the single-factor evaluation results, n is the number of samples corresponding to different amendment values, and N is the number of samples.

- 3
- The amendment entropy weight calculated by formula (11) was normalized, then, amendment weight Wn
^{’}_{j}based on the result of single-factor evaluation was obtained.$${{W}^{\prime}}_{j}=\frac{W{n}_{j}}{{{\displaystyle \sum}}_{j=1}^{n}W{n}_{j}}$$_{j}is the amendment weight of indicator j, W’j is the normalized amendment weight of indicator j (amendment entropy weight).

- Step 3—OSE weight calculation

- 1
- Calculation of the degree by which each indicator exceeds the standard in each sample.For the “bigger the better” indicator:$${k}_{ij}=\frac{\frac{1}{5}\times {{\displaystyle \sum}}_{t=1}^{5}{v}_{jt}}{{x}_{ij}}$$For the “smaller the better” indicator:$${k}_{ij}=\frac{{x}_{ij}}{\frac{1}{5}\times {{\displaystyle \sum}}_{t=1}^{5}{v}_{jt}}$$
- 2
- Determination of the weight value W″
_{ij}:$${{W}^{\u2033}}_{ij}=\frac{{k}_{ij}}{{{\displaystyle \sum}}_{t=1}^{n}{k}_{tj}}$$_{ij}is the degree by which sample i exceeds the standard of indicator j, v_{jt}represents the standard value of the t-grade water quality in indicator j, x_{ij}is the measured value of indicator j in sample i, and W″_{ij}represents the weight value of indicator j in sample i.

- Step 4—The arithmetic average method was used to calculate the combined weights. The weights obtained here are called ESO weights, as we used the ESO method in this study [17].$$W{c}_{ij}=\frac{\left({{W}^{\prime}}_{j}+{{W}^{\u2033}}_{ij}\right)}{2}$$
_{ij}is the combination weight (ESO weight) of the amendment entropy and over-standard weights, W^{′}_{j}is the normalized amendment weight of indicator j, and W^{″}_{ij}is the over-standard weight of indicator j in sample i.

## 3. Results and Discussion

#### 3.1. Entropy Weight and Amendment Entropy Weight

_{Cr}results showed that there were eight class V and two class IV samples; thus, F and COD

_{Cr}were the two factors that exceeded the standard most severely. Although the COD

_{Mn}and TN results indicated that all samples were inferior to class III, there were few class V samples, while there were three class V samples for TP and five class III samples. Therefore, the over-standard degrees of COD

_{Mn}, TN, and TP were slightly better than those of F and COD

_{Cr}. Only two BOD

_{5}samples failed to meet the standard, and the other eight samples were class III, which was a slightly over-standard factor. All the DO and NH

_{3}-N samples reached the standard of class III, which was the qualification factor. Therefore, the pollution degree of each factor was decreased in the following order: F and COD

_{Cr}> COD

_{Mn}, TN, and TP > BOD

_{5}> NH

_{3}-N > DO.

_{5}, and COD

_{Cr}were the main indicators influencing the entropy weight, while the weight values of other indicators were relatively evenly distributed, which is consistent with the data dispersion of samples (expressed by variance) for each indicator in Table 3 (The indicators with underline are the main pollution indicators determined by each method). This result demonstrates that the entropy weight values can be determined from the dispersion of the sample data [29].

_{Cr}and F, and the weight values of the indicators which exceeded the standard seriously in the single-factor evaluation, such as F, COD

_{Cr}, COD

_{Mn}, TP, and TN, were increased to a certain extent compared with the entropy weight values, while those of DO, NH

_{3}-N, and BOD

_{5}, which did not exceed the standard or slightly exceeded the standard, were reduced to a certain extent, and the DO even became 0. This result shows that the amendment entropy weight is affected by the results of single-factor evaluation to a certain extent; that is, the higher the degree of exceeding the standard of the indicator, the greater the increase of the amendment weight value, and vice versa. The weight value of DO became 0 was because all its samples were better than class III, and it did not need to be concerned according to the amendment rules in step 2, so it was given a weight of 0. However, the result of amendment entropy weight was not completely consistent with single-factor evaluation; for example, TP was not the most serious over-standard factor in single-factor evaluation, but it was given the largest amendment weight value because it was affected by dispersion degree according to Table 3.

_{Cr}, COD

_{Mn}, TP, and TN, were higher than the entropy weight values, while those of DO, NH

_{3}-N, and BOD

_{5}, which did not exceed the standard or slightly exceeded the standard, were reduced to a certain extent, and the DO even reached 0. This result indicates that the amendment entropy weight is affected by the results of single-factor evaluation, to a certain extent. That is, the higher the degree by which the indicator exceeds the standard, the greater the increase in the amendment weight value, and vice versa. The weight value of DO was 0 because the quality of all DO samples was better than class III and, therefore, it did not need to be considered according to the amendment rules in step 2 and was given a weight value of 0. However, the amendment entropy weight result was not fully consistent with the single-factor evaluation; for example, TP was not the most severe over-standard factor in single-factor evaluation, but it was given the largest amendment weight value as it was affected by the dispersion degree, as shown in Table 3.

#### 3.2. Over-Standard and ESO Weights

_{5}was 0.067 in 2011, but 0.12 in 2014. Although some differences in the ESO weight values of these indicators in different samples remained, they were more similar. For example, the weight values of TP were 0.215 and 0.143 in 2010 and 2014, while those of BOD

_{5}became 0.091 and 0.111 in 2011 and 2014, respectively. The data of other indicators were also aggregated, to a certain extent.

#### 3.3. Fuzzy Synthetic Evaluation Results

#### 3.4. Applicability of the ESO Method

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of the study area: (

**a**) Jilin Province, China; (

**b**) the position of the study area in Jilin Province; and (

**c**) Chagan Lake wetland ecological environment reserve.

Grade | I | II | III | IV | V | Inferior V |
---|---|---|---|---|---|---|

m | 0 | 0 | 1 | 2 | 3 | 4 |

Year | DO | F | NH_{3}-N | COD_{Cr} | COD_{MN} | BOD_{5} | TP | TN |
---|---|---|---|---|---|---|---|---|

2007 | I | IV | III | IV | IV | III | V | IV |

2008 | I | IV | III | IV | IV | III | V | IV |

2009 | I | Inferior V | III | V | IV | III | IV | IV |

2010 | I | IV | III | V | IV | III | V | IV |

2011 | I | IV | II | V | IV | III | III | IV |

2012 | I | IV | II | V | IV | III | III | IV |

2013 | I | IV | II | V | IV | III | III | IV |

2014 | I | IV | II | V | IV | IV | III | IV |

2015 | II | Inferior V | II | V | IV | III | IV | IV |

2016 | II | Inferior V | II | V | IV | IV | III | V |

DO | F | NH_{3}-N | COD_{Cr} | COD_{Mn} | BOD_{5} | TP | TN | |
---|---|---|---|---|---|---|---|---|

Entropy weight | 0.114 | 0.105 | 0.096 | 0.122 | 0.110 | 0.140 | 0.207 | 0.105 |

Amendment entropy Weight | 0.000 | 0.168 | 0.024 | 0.208 | 0.135 | 0.103 | 0.228 | 0.135 |

Variance (10^{−4}) | 3.18 | 2.94 | 2.68 | 3.54 | 3.04 | 4.25 | 5.97 | 2.82 |

Year | DO | F | NH3-N | COD_{Cr} | COD_{Mn} | BOD_{5} | TP | TN |
---|---|---|---|---|---|---|---|---|

2007 | 0.069 | 0.122 | 0.092 | 0.154 | 0.133 | 0.079 | 0.195 | 0.155 |

2008 | 0.07 | 0.139 | 0.066 | 0.158 | 0.118 | 0.079 | 0.183 | 0.182 |

2009 | 0.061 | 0.162 | 0.063 | 0.193 | 0.113 | 0.085 | 0.165 | 0.158 |

2010 | 0.071 | 0.115 | 0.071 | 0.176 | 0.130 | 0.080 | 0.202 | 0.156 |

2011 | 0.077 | 0.168 | 0.069 | 0.201 | 0.158 | 0.067 | 0.095 | 0.164 |

2012 | 0.075 | 0.167 | 0.049 | 0.220 | 0.168 | 0.097 | 0.071 | 0.153 |

2013 | 0.082 | 0.155 | 0.065 | 0.203 | 0.158 | 0.104 | 0.076 | 0.155 |

2014 | 0.078 | 0.185 | 0.059 | 0.203 | 0.140 | 0.120 | 0.059 | 0.156 |

2015 | 0.085 | 0.194 | 0.054 | 0.179 | 0.116 | 0.084 | 0.139 | 0.150 |

2016 | 0.080 | 0.166 | 0.061 | 0.169 | 0.133 | 0.097 | 0.082 | 0.212 |

Year | DO | F | NH3-N | COD_{Cr} | COD_{Mn} | BOD_{5} | TP | TN |
---|---|---|---|---|---|---|---|---|

2007 | 0.035 | 0.145 | 0.058 | 0.181 | 0.134 | 0.091 | 0.212 | 0.145 |

2008 | 0.038 | 0.153 | 0.045 | 0.183 | 0.126 | 0.091 | 0.205 | 0.159 |

2009 | 0.030 | 0.165 | 0.043 | 0.201 | 0.124 | 0.094 | 0.196 | 0.146 |

2010 | 0.035 | 0.141 | 0.047 | 0.192 | 0.132 | 0.091 | 0.215 | 0.146 |

2011 | 0.039 | 0.168 | 0.047 | 0.204 | 0.146 | 0.085 | 0.161 | 0.150 |

2012 | 0.037 | 0.167 | 0.036 | 0.214 | 0.152 | 0.100 | 0.149 | 0.144 |

2013 | 0.041 | 0.161 | 0.045 | 0.206 | 0.147 | 0.104 | 0.152 | 0.145 |

2014 | 0.039 | 0.176 | 0.041 | 0.205 | 0.137 | 0.111 | 0.143 | 0.146 |

2015 | 0.042 | 0.181 | 0.039 | 0.194 | 0.125 | 0.094 | 0.183 | 0.142 |

2016 | 0.040 | 0.167 | 0.042 | 0.188 | 0.134 | 0.100 | 0.155 | 0.174 |

Year | Single-Factor Evaluation Method | Over-Standard Method | Entropy Method | Amendment Entropy Method | ESO Method |
---|---|---|---|---|---|

2007 | V | 3.344 | 2.988 | 3.512 | 3.438 |

2008 | V | 3.802 | 3.535 | 3.867 | 3.847 |

2009 | Inferior V | 4.028 | 3.542 | 4.100 | 4.079 |

2010 | V | 3.587 | 3.299 | 3.663 | 3.639 |

2011 | V | 3.413 | 2.731 | 3.392 | 3.411 |

2012 | V | 3.700 | 2.987 | 3.555 | 3.649 |

2013 | V | 3.302 | 2.905 | 3.275 | 3.303 |

2014 | V | 3.402 | 2.859 | 3.314 | 3.385 |

2015 | Inferior V | 4.026 | 3.491 | 4.043 | 4.050 |

2016 | Inferior V | 4.021 | 3.281 | 3.735 | 3.895 |

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**MDPI and ACS Style**

Zhao, W.; Xiao, C.; Chai, Y.; Feng, X.; Liang, X.; Fang, Z. Application of a New Improved Weighting Method, ESO Method Combined with Fuzzy Synthetic Method, in Water Quality Evaluation of Chagan Lake. *Water* **2021**, *13*, 1424.
https://doi.org/10.3390/w13101424

**AMA Style**

Zhao W, Xiao C, Chai Y, Feng X, Liang X, Fang Z. Application of a New Improved Weighting Method, ESO Method Combined with Fuzzy Synthetic Method, in Water Quality Evaluation of Chagan Lake. *Water*. 2021; 13(10):1424.
https://doi.org/10.3390/w13101424

**Chicago/Turabian Style**

Zhao, Wenbin, Changlai Xiao, Yunxu Chai, Xiaoya Feng, Xiujuan Liang, and Zhang Fang. 2021. "Application of a New Improved Weighting Method, ESO Method Combined with Fuzzy Synthetic Method, in Water Quality Evaluation of Chagan Lake" *Water* 13, no. 10: 1424.
https://doi.org/10.3390/w13101424