Study Area

The Tao Jiang River is the primary tributary and main source of the Gan River, in Jiangxi Province, China. Its location is plotted in Fig. 1. It originates from the northeast slope of the Jiulian mountain range at the junction of Guangdong and Jiangxi and flows through Guangdong and Jiangxi provinces. Its main tributaries include the Thai Binh River, the Ottawa River, and the Huangtian River, which eventually joined the Gongjiang River in Gan County. The river basin is located in the southwest of Ganzhou, Jiangxi province (24°28′-25°53′N, 114°11′-115°19′ E). It covers the whole of Nan County, Longnan City, most of Xinfeng County, Gan County District, Anyuan County, and parts of Dingnan County. The basin area is approximately 8,440 km². The annual average precipitation is 2300mm, and the difference in rainfall in a year is noticeable; it mainly focuses between April and August. The interannual change is also apparent; the high and low-water years appear alternately.

Fig. 1.

The location of the study area.

The Determination of the Assessment Index

In this paper, six indicators are selected as the evaluation indexes of water quality based on the relevant norms and the research results of predecessors; they are, respectively, Dissolved oxygen (X₁), Permanganate index (X₂), BOD₅ (X₃, Ammonia nitrogen (X₄), Total phosphorus (X₅) and Fluoride (X₆). The evaluation indexes were divided into six quality evaluation grades: level I (excellent), level II (good), level III (common), level IV (inferior), level V (bad), and level VI (worse). Twelve groups of monitoring data and assessment results in the Tao Jiang River basin are selected as learning samples; the specific data is shown in Table 1.

The Game-Gray Target Model

The gray target decision-making approach is an essential method for solving multi-attribute decision-making problems from an objective angle, and it can effectively reduce the loss of original data information in the decision-making process [20]. Its basic idea is that the optimal data sequence is found to construct a standard model from the existing set of sequences. Then, the standard model is applied as the target, the gray target model is built by comparing other models with the standard one, the approaching degree between the models is evaluated, and the target is judged; finally, the target distance is calculated, and the evaluation grade is determined. Considering the accuracy defect of the single-weight gray target model, the game combination weighting method is adopted in the paper, the combination weight of the critic and entropy methods is optimized, and the optimal weight is obtained.

(1) The establishment of a decisive matrix

It is assumed that there are m samples to be evaluated A_i(i = 1, 2, ..., m), n evaluation indexes A_j(j = 1, 2, ..., m), and then the sample matrix is A = {a_ij}(i = 1, 2, ..., n).

Suppose that c_j is the mean value of the evaluation index, then:

Where, i = 1, 2, ..., m; j = 1, 2, ..., n.

For x_ij standardized processing results, the economic indicators can be expressed as:

For a cost-type indicator, the formula is

According to Equations (1)-(3), the decisive matrix can be expressed as:

(2) The calculation of the target center distance

For the decisive matrix X, if x_j⁰⁺ = max{x_ij|1<i<m}, then x⁰⁺ = {x₁⁰⁺, x₂⁰⁺, ..., x_m⁰⁺} is called the positive target center of the gray target; x^0– = {x₁^0–, x₂^0–, ..., x_m^0–} is defined as the negative target center. The distance between x⁰⁺ and x^0– is regarded as the interval d⁰ between the positive and negative target centers, and

Where, ω_i is the optimal weight of the jth index obtained based on the game theory.

The positive target center distance d_i⁺ is the distance between x_i and x⁰⁺, its formula is:

The negative target center distance d_i^– is the distance between x_i and x^0–, its formula is:

The distance from any sample point x_i to the positive target center is d_i⁺<d₀, then x_i is located in the sphere with x⁰⁺ as the center and d⁰ as the radius; the distance from the sample point x_i to the negative target center is d_i^–<d₀, then x_i is located in the sphere with x^0– as the center and d⁰ as the radius. Any sample point x_i, the positive target center d_i⁺, and the negative target center d_i^– are three points in space. The three points are collinear or triangular. Therefore, the dangerous degree of the samples can be measured using the projection size of the positive target center distance on the line between the positive and negative target centers. Assuming the angle between the positive target center distance and the line between the positive and negative target centers is θ, then according to the law of cosines, it can be obtained:

(3) The classification of quality grade

From the definition of target center distance, it can be found that the comprehensive target distance can quantitatively reflect the quality grade of samples. Assuming that samples to be evaluated have t quality grades, let D = (d₁, d₂, ..., d_m) be comprehensive target center distance set S of samples to be evaluated, the ordered set of t quality grades is B = (B₁, B₂, ..., B_t), let f is a positive integer, and 1≤f<t, then the threshold of the fth set g_f = max{B}, c_f = max{B_f}, then h_f = αc_f + (1 – α)g_f, and α∈(0,1), h₀ = 0, h_q = +∞, d = (d₁, d₂, ..., d_i) are the critical comprehensive target center distance of different quality grades, the interval distribution set of comprehensive target distance for t group quality grades can be obtained as follows:

The Determination of Index Weights

The critic method is a kind of weight-assignment method [21], which makes use of the variability and conflict of different evaluation indexes to determine weight, and it can comprehensively measure the evaluation index. Its procedure is listed as follows:

① The standardization and processing of data

Each evaluation index is quantified dimensionlessly to eliminate the influence of different variables. If the evaluation index is a benefit type, the calculation formula is:

If the evaluation index is a cost type, the calculation formula is

Where y is a normalized processing value. max(x_ij) and min(x_ij) are the maximum and minimum of a set of evaluation indicators.

② The variability of evaluation indexes is usually expressed using the standard deviation σ_j of the evaluation index, and the calculation formula is

Where x_j is the mean of jth evaluation index; n is the total number of the jth evaluation index.

③ The correlation coefficient of the evaluation index was calculated as:

④ The conflict coefficient of the evaluation index was calculated as:

⑤ Calculating the weight coefficient ω_j of the evaluation index:

The entropy weight method is a method to determine the entropy value of an index according to the variance size of the evaluation index; its specific calculation method is shown in the literature [22].

In order to avoid the information loss caused by a single weighting method and improve the accuracy of the weights, the combination weighting method of game [23, 24] is applied to optimize the weights obtained by several weighting methods, and the consistency is found out, and the optimal weight is obtained.

① The weight sets ω₁ and ω₂ are obtained by the entropy weight and CRITIC methods. It is assumed that α₁ and α₂ are the linear combination coefficients. Then, the weight sets ω₁ and ω₂ can be linearized as:

② According to game theory, the linear combination coefficients α₁ and α₂ in Formula (11) are optimized and expressed as:

③ According to the differential properties of the matrix, the linear differential equation group for optimizing the first derivative condition of formula (15) is:

④ The optimal combination coefficients α₁ and α₂ are obtained via Formula (16). The normalization process is obtained as a1*=a1/(a1+a2) and a2*=a2/(a1+a2). Then, based on the game theory, the comprehensive weight ω can be obtained as:

The Construction of the Model Frame

To determine the quality level of water in the Tao Jiang River, their calculative flowchart is plotted in Fig. 2, and their specific procedure is listed as follows:

(1) The sample matrix of the origin matrix is constructed

(2) The decisive matrix of original data is established based on Equations (2) and (3).

(3) The optimal combination weight of different samples is obtained according to Equations (9)-(17)

(4) The target centers of different samples can be solved based on the decisive matrix and Equation (4)

(5) The positive and negative target center distances of different samples can be determined according to Equations (5) and (6).

(6) Determining different samples’ synthetic target center distance based on Equation (7).

Fig. 2.

Flowchart of the assessment framework.

(7) The partitioning of quality level is determined according to the target center distance range in combination with Equation (8).

(8) The final grade of water quality is determined.

The Assessment Procession

Four indicators in Table 1 are all benefit types; based on Equations (1)-(3), and combined with Table 1, the decisive matrix is established, as shown in Table 2.

the entropy method

Based on Table 2, the corresponding weight coefficient can be obtained as:

② Calculating weight coefficient ω₂ based on the CRITIC method

According to Equations (9)-(11) and in conjunction with Table 1, the coefficients of correlation can be calculated as:

Based on Equation (12), the standard deviation of different columns is obtained as:

Similarly, Equation (13) calculates the weights of each evaluation index as:

③ Determination of combination weights

Based on Equations (14)-(17), the combination weight ω is obtained as:

The view of weights is plotted in Fig. 3.

(3) The determination of the target center distance

The positive and negative target centers are, respectively, x⁰⁺ = (1 0.88 0.67 0.3 0.85 0.82) and x^0– = [–0.7 – 1 – 1 – 1 – 1 – 1]. According to Equation (4), the interval distance between the positive and negative target centers is d₀ = 1.6637. According to Equations (5) and (6), the positive target center distance d⁺ and negative target center distance d^– of different samples can be calculated, and their results are shown in Table 3.

According to Equation (7), the magnitude of the synthetic target center distance is shown in Table 4.

(4) The determination of the quality grade for water quality

The comprehensive target distance of each sample is arranged according to the quality grade; the critical value of the target distance for each quality grade was obtained and is shown in Table 5. According to Equation (8), the target distance figure for each quality grade can be constructed; it is plotted in Fig. 4.

(5) The quality grade prediction of asphalt pavement

The Tao Jiang River is selected as an example in this paper to test the rationality and accuracy of the gray target evaluation model; the monitoring data for 13-18 samples are shown in Table 6.

The different parameters are substituted into the game gray target evaluation model, and the corresponding comprehensive target distance d_i^* is obtained. According to the different d_i^* ranges shown in Fig. 3, the quality grade of water is evaluated, and the results are shown in Table 7. The results obtained from the suggested model are compared with the other two methods, as shown in Fig. 5.

The game-gray target model is applied to evaluate the level of water quality. The assessment results are depicted in Table 7. It can be found in Table 7 that the grade of water quality in the Tao Jiang River watershed from Nos. 13–17# samples differ; the quality level at the No. 13# sample is II; one at the No. 14# sample is III; the rest are respectively IV, V, and VI. It means that the quality of the No. 13# sample is good, the quality of No. 14# is common, and the rest are inferior, bad, and worse, so its qualified rate is 40%. The corresponding remedial measures should be performed for the Nos. 15-17# samples section; no measurement needs to be adopted for other samples.

Fig. 3.

The view of weight coefficients.

Fig. 4.

The grade target distance for water quality.

Fig. 5.

The comparison of results from three methods.

The comparative results of the assessment model in Fig. 5 indicate that the proposed method is consistent with the investigations for five different samples. Its accuracy reaches 100%, which is more significant than the results from the BP Neural Network method (80%) [25]. Therefore, estimating water quality using the game-gray target model is feasible. The proposed approach provides additional details for assessing the level of water quality. For example, the quality level of No.7 is more likely to be level III than that of the No. 8 sample, as the synthetic target distance (0.9039) for level III is greater than the No.8 sample (0.8467). The results from the proposed model accurately demonstrate the water quality grade and further determine the risk grade rankings for different samples at the same level.

Discussion

The proposed method can accurately assess the water quality grade and has higher reliability and efficiency. However, its calculation process is complicated, and the randomness of evaluation indicators is not considered. So the proposed method can still be improved in the future. In future work, a three-dimensional gray target model will be applied, and it will be my future direction to assess the water quality level.