Surface Water Quality Evaluation Based on a Game Theory ... - MDPI

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Surface Water Quality Evaluation Based on a Game Theory-Based Cloud Model Bing Yang 1 1 2 3 4

*

ID

, Chengguang Lai 2 , Xiaohong Chen 1,3, *

ID

, Xiaoqing Wu 4 and Yanhu He 1,3

Center for Water Resource and Environment, Sun Yat-Sen University, Guangzhou 510275, China; [email protected] (B.Y.); [email protected] (Y.H.) School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510641, China; [email protected] Guangdong Engineering Technology Research Center of Water Security Regulation and Control for Southern China, Guangzhou 510640, China South China Institute of Environment Sciences, Ministry of Environment Protection of PRC, Guangzhou 510535, China; [email protected] Correspondence: [email protected]





Received: 27 March 2018; Accepted: 18 April 2018; Published: 20 April 2018

Abstract: Water quality evaluation is an essential measure to analyze water quality. However, excessive randomness and fuzziness affect the process of evaluation, thus reducing the accuracy of evaluation. Therefore, this study proposed a cloud model for evaluating the water quality to alleviate this problem. Analytic hierarchy process and entropy theory were used to calculate the subjective weight and objective weight, respectively, and then they were coupled as a combination weight (CW) via game theory. The proposed game theory-based cloud model (GCM) was then applied to the Qixinggang section of the Beijiang River. The results show that the CW ranks fecal coliform as the most important factor, followed by total nitrogen and total phosphorus, while biochemical oxygen demand and fluoride were considered least important. There were 19 months (31.67%) at grade I, 39 months (65.00%) at grade II, and one month at grade IV and grade V during 2010–2014. A total of 52 months (86.6%) of GCM were identical to the comprehensive evaluation result (CER). The obtained water quality grades of GCM are close to the grades of the analytic hierarchy process weight (AHPW) due to the weight coefficient of AHPW set to 0.7487. Generally, one or two grade gaps exist among the results of the three groups of weights, suggesting that the index weight is not particularly sensitive to the cloud model. The evaluated accuracy of water quality can be improved by modifying the quantitative boundaries. This study could provide a reference for water quality evaluation, prevention, and improvement of water quality assessment and other applications. Keywords: cloud model; game theory; water quality evaluation; randomness; fuzziness

1. Introduction With the rapid development of the economy and the continued growth of the global population, humans are currently exploring and utilizing limited water resources at an unprecedented rate and scale [1–3]. However, this accelerated human activity has had a negative effect on the water environment in recent decades, contributing to pollution of rivers, estuaries and oceans, especially in developing countries [4,5]. Undoubtedly, water quality degradation has been acknowledged as one of the most severe environmental issues worldwide since it disrupts the ecological balance of water bodies and threatens regional environmental security [6–8]. Therefore, this issue attracts significant attention across the world. Water quality evaluation is one of the basic methods to analyze water quality condition and has been considered as an essential measure and has since diversified. Nevertheless, two types of Water 2018, 10, 510; doi:10.3390/w10040510

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uncertainty should still be considered in water quality evaluation: the first is randomness, which is often exhibited in the monitoring and analysis of data related to water quality; the second is fuzziness, which often reflects the classification standard, evaluation class, and degree of pollution [9]. Most contemporary water quality evaluation methods or models, such as the single factor evaluation method, the Nemerow pollution index, the fuzzy comprehensive evaluation, and the artificial neural network method, can be divided into three types according to these two forms of uncertainties: the first type of model is mainly based on various statistical and stochastic techniques for randomness [10–12]; the second type of model is mainly based on fuzzy membership function, fuzzy logic, and fuzzy set theory for fuzziness [13,14]; and the last type of model is based on machine learning and artificial intelligence for unknown patterns that hardly capture key information during the assessment process [15–17]. The three types of methods or models provide superior tools to evaluate the water quality and greatly contribute to regional water quality protection. As mentioned before, since randomness and fuzziness are widely considered in water quality evaluation, it would be better if a method or model existed that could reduce both randomness and fuzziness. Fortunately, a new model, i.e., the cloud model was proposed with the purpose of reducing randomness and fuzziness in system evaluation [18]. This model can quantify both randomness and fuzziness via three fixed parameters and presents more advantages than a single randomness or fuzziness model. Due to these advantages, Wang et al. [19] first applied this model to the field of water quality evaluation and achieved satisfactory results. Nevertheless, water quality evaluation is a process of multi-criteria decision-making and therefore a key issue remains in the cloud model, i.e., how to determine appropriate index weights. Reasonable evaluation results mainly depend on index weight so that the determination of an appropriate index weight becomes a key step in the evaluation process [20,21]. Currently, subjective weight and objective weight are two common weights while both have their limitations. The former is strongly affected by expert knowledge as well as many biases, resulting in high subjectivity [22], while the later does not consider differences among indices, and it ignores practical situations [23]. Therefore, a combination weight, with the advantages of both subjective weight and objective weight, should be used to solve the aforementioned problems. Game theory is the mathematical modeling of strategic interaction among rational and irrational agents that specializes in solving conflicts among two or more participants [24,25]. In game theory, each participant’s objective is to maximize the expected value of his own payoff, and the decision made by all participants is rational for each individual participant [26,27]. Therefore, all participants reach an independent but collective decision that maximizes all of the participants’ expected utility payoffs, suggesting that the decision includes either a consensus or a compromise. This factor is known as the Nash Equilibrium [28]. Subjective weight and objective weight, analogously, can be regarded as two participants of the game, and the combination weight is the result of the ‘weight’ game. The most satisfied combination weight is considered that have reached the Nash Equilibrium, according to game theory. However, little attention has been paid to the concept of game theory for determining a comprehensive weight in water quality evaluation, let alone applying the combination weight to the cloud model of water quality evaluation. Therefore, the main objectives of this study are (1) to calculate a combination weight based on game theory integrating subjective weight and objective weight; (2) to construct an evaluation system of the cloud model; and (3) to analyze water quality grade in the study areas. This study aims to provide scientific and practical measures for water quality evaluation, prevention, and improvement of water quality assessment and other applications in the study areas. 2. Study Site and Data The study focuses on the Beijiang River, which is the second largest tributary of the Pearl River [29]. The Beijiang River originates from Xinfeng County in Jiangxi province, and flows through Shaoguan City, Qingyuan City, and Foshan City of Guangdong province. The total length of the Beijiang River

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Beijiang River is 573 km, with a catchment area of 52,068 km2, accounting for 10.3% of the total area is 573 km, withbasin. a catchment area of 52,068 km2 , accounting for 10.3% of the total area of the Pearl of the Pearl River River basin. quality data used in this study were collected from the Qixinggang section (QXG; The water The water quality data used in this study were collected from the Qixinggang section (QXG; 111°04′03″ E, 23°41′00″ N), i.e., the national control section of surface water in the Beijiang River 111◦ 040 0300 E, 23◦ 410 0000 N), i.e., the national control section of surface water in the Beijiang River (Figure 1). The QXG is located in the center of Qingyuan city and it is a key monitoring point of the (Figure 1). The QXG is located in the center of Qingyuan city and it is a key monitoring point of the entire Beijiang River. The employed values in this study consisted of nine evaluation indices, entire Beijiang River. The employed values in this study consisted of nine evaluation indices, including including dissolved oxygen (DO), permanganate index (PI), chemical oxygen (COD), dissolved oxygen (DO), permanganate index (PI), chemical oxygen demand (COD), demand biochemical + + -N), total (NH biochemical oxygen(BOD), demand (BOD), nitrogen ammonium 4 -N), total oxygen demand ammonium (NH4 nitrogen phosphorus (TP),phosphorus total nitrogen(TP), (TN), total nitrogen (TN), and fecal coliform (Fc),were and monitored these datamonthly were monitored monthly fluoride (F), fluoride and fecal (F), coliform (Fc), and these data from January 2010 to from January 2010 2014. to December The quality of the these datastandards satisfiedand theprofessional national standards December The quality2014. of these data satisfied national standards. and Additionally, we used the z-score method to the standardize indices, avoiding inconsistency professional standards. Additionally, we [30] used z-score these method [30]thus to standardize these indices, induced by measurement and calculation. thus avoiding inconsistency induced by measurement and calculation.

Figure Locationmap mapof of the the monitoring in in thethe study area. Figure 1. 1. Location monitoringsection section study area.

3. Methodology 3. Methodology framework waterquality quality cloud cloud model of the game theory-based cloud cloud model is TheThe framework ofof water modelevaluation evaluation of the game theory-based model illustrated in Figure 2. In this study, we first selected suitable water quality indices and transferred these is illustrated in Figure 2. In this study, we first selected suitable water quality indices and transferred to standardized dimensionless data; we then determined water quality criteria and calculated each these to standardized dimensionless data; we then determined water quality criteria and calculated index’s combination weight according to the game theory concept. The subjective weight and objective each index’s combination weight according to the game theory concept. The subjective weight and weight were combined via the analytic hierarchy process (AHP) and entropy theory, respectively. objective weight were combined via the analyticand hierarchy (AHP) entropy theory, Here, we determined the cloud model parameters input theprocess parameters into and the normal cloud respectively. we determined thethe cloud model parameters input parameters generator Here, to generate the cloud, and comprehensive certainty and degree was the computed using into the the normal cloud generator generate the cloud, and the comprehensive was of computed combination weight. to Lastly, the water quality grade was determined certainty accordingdegree to the rank the using the combination weight. Lastly, the water quality grade was determined according to the maximum certainty degree of each evaluation index calculated from the cloud model. This research rank

of the maximum certainty degree of each evaluation index calculated from the cloud model. This research divides the water quality into six grades: grades I, II, III, IV, V, and VI. The main implemented software includes R studio and Excel.

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divides the water quality into six grades: grades I, II, III, IV, V, and VI. The main implemented software Water 2018,R 10,studio x FOR PEER REVIEW includes and Excel.

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Part 1: Data collection & preprocessing Observed water quality data & Selection evaluation index

Data standardization

Part 2: Weights calculation Subjective Weight Entropy

Objective Weight AHP

Evaluation criteria determination

Part 3: Cloud Model Evaluation Model parameters determination Normal cloud generator

Combination Weight Game Theory

Certainty degree calculation

Final Water Quality Status (Grade)

Figure 2. Framework of the cloud model-based water quality evaluation approach.

3.1. 3.1. Cloud Cloud Model Model 3.1.1. Overview of the Model

The cloud model, first proposed by Li et al. [18,31], is a type of transformation transformation model that synthetically describes the randomness and fuzziness of concepts and can implement the uncertain transformation between a qualitative concept and its quantitative quantitative instantiations. instantiations. Given a qualitative concept T defined over a universe of discourse U == {u}, {u}, let let xx ∈ ∈ U is a random ∈ [0,1] is the the certainty certainty degree degree of x belonging to T, which [0, 1] is instantiation of the concept T and µ T ((x )) ∈ corresponds to a random number with a steady tendency. Then, the distribution of x in the universe U can be defined as a cloud cloud and and xx can can be be called called aa cloud cloud drop. drop. The point here is that the random instantiation in in the the sense sense of of probability probability theory. theory. Thus, Thus, ∀x ∈ U,, the the mapping mapping µ T(( x)) instantiation is the instantiation one-to-many mapping mapping in in nature, nature, i.e., i.e.,the thecertainty certaintydegree degreeofofx xbelonging belongingtotothe theconcept conceptT Tis is is a one-to-many a a probabilitydistribution distributionrather ratherthan thana afixed fixednumber. number. probability 3.1.2. Model Model Parameters Parameters 3.1.2. The cloud effectively integrate bothboth randomness and fuzziness of concepts and describe The cloudmodel modelcan can effectively integrate randomness and fuzziness of concepts and the overall quantitative property ofproperty a conceptofbyathe four numerical characteristics follows: describe the overall quantitative concept by the four numerical as characteristics as E (Expectation) represents the mathematical expectation that the cloud drops belong to a concept in x follows: the universe. It can be regarded as the representative and typical of the drops qualitative concept. Ex (Expectation) represents the most mathematical expectation thatsample the cloud belong to a En (Entropy) represents the uncertainty a qualitative concept. It is determined by concept in the universe. It can be regardedmeasurement as the most of representative and typical sample of the both randomness and the fuzziness of the concept. In one aspect, as the measurement of randomness, qualitative concept. En reflects the dispersing extent the cloud drops and in the aspect, itconcept. is also the measurement En (Entropy) represents theofuncertainty measurement ofother a qualitative It is determined of fuzziness, representing the scope of the universe that can be accepted by the concept. by both randomness and the fuzziness of the concept. In one aspect, as the measurement of He (Hyper represents theextent uncertain degree entropy n . the other aspect, it is also the randomness, En entropy) reflects the dispersing of the cloudofdrops andEin N represents the number of the repeat that that in Figure 3,accepted the x-axisby represents the measurement of fuzziness, representing thesimulations. scope of theNote universe can be the concept. values of the water quality index, and the y-axis represents the certainty degree of a water quality grade. He (Hyper entropy) represents the uncertain degree of entropy En.

N represents the number of the repeat simulations. Note that in Figure 3, the x-axis represents the values of the water quality index, and the y-axis represents the certainty degree of a water quality grade.

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Figure 3. Schematic diagram of the cloud (N = 50,000). Figure 3. Schematic diagram of the cloud (N = 50,000).

For the determination of (Ex, En, He), Du et al. [32] suggested Equation (1) for a bilateral boundary For the (B determination of (Ex , En , He ), Du et al. [32] suggested Equation (1) for a bilateral boundary of the form min, Bmax): of the form (Bmin , Bmax ):  )⁄2   Ex==(( Bmin + Bmax )/2 (1) (1) )⁄ 6 En==(( Bmax − −Bmin )/6   He = =k where Bmax areare the respective minimum and maximum values that can bethat accepted and Bmax the respective minimum and maximum values can qualitatively; be accepted whereBmin Bminand i.e., in water quality evaluation, the minimum and maximum values corresponding to a certain water qualitatively; i.e., in water quality evaluation, the minimum and maximum values corresponding to quality grade in quality each criterion the value Ex . the value of Ex. a certain water grade to in obtain each criterion toofobtain Note that k is a constant and can be adjusted according to the practical situation. In the present study, the algorithm as follows: entropy Healliscriteria the uncertainty degree of Tableof1. H Quantitative boundaries of waterhyper quality grades of [33]. e was modified entropy En ; however, in Equation (1) He is assumed as a constant irrelevant to En , and it is assumed Evaluation Factor Grade I Grade II Grade III Grade IV Grade V Grade VI that He can be adjusted by a linear relationship with En : DO (mg/L) ≥7.5 ≥6 ≥5 ≥3 ≥2 15 H = k× E (2) COD (mg/L) ≤15 ≤15 e ≤20 n ≤30 ≤40 >40 BOD (mg/L) ≤3 ≤3 ≤4 ≤6 ≤10 >10 In this way, k intuitively controls the≤0.5 ‘atomization’ of the normal (Figure 3) NH4+-N (mg/L) ≤0.15 ≤1 degree ≤1.5 ≤2 cloud model >2 and essentiallyTP reflects the variation of cognition of different evaluations. Here, k is assumed to be 0.1 (mg/L) ≤0.02 ≤0.1 ≤0.2 ≤0.3 ≤0.4 >0.4 to balance theTN variation and robustness of the assessment [33]. (mg/L) ≤0.2 ≤0.5 ≤1 ≤1.5 ≤2 >2 In this case, combined with≤1 standard ≤1 water quality (Table 1), the (Ex , En , F (mg/L) ≤1 criteria ≤1.5 ≤1.5three perimeters >1.5 Fc (Number) ≤200were calculated ≤2000 ≤10,000 ≤20,000 ≤40,000 >40,000 and He ) of each evaluation index according to Equations (1) and (2). The modified

equations are applicable for fixed intervals, noting that for grade VI of criterion PI, COD, BOD, k is constant and can adjusted according to the practical situation. In the present NH4 +Note -N, P,that N, F, Fc,a and for grade I of be criterion DO, Bmax is missing. To attain the pseudo boundary, the algorithm He was modified entropy He is the uncertainty degree of astudy, hypothetical processofindicates that Bmax as is follows: twice as hyper large as Bmin . For example, for grade II of PI, entropy E n ; however, in Equation (1) H e is assumed as a constant irrelevant to E n , and it is assumed Bmin = 2, and Bmax = 4 (see Table 1), the corresponding parameters can be obtained as Ex = 3 and e can begrade adjusted a linear with : Bmin = 80, the corresponding parameters can Ethat 1/3. For VI ofbyCOD, Bminrelationship = 40, and Bmax = 2En× n=H be obtained as Ex = 60 and En = 6.67. After attaining all = ×all quantitative boundaries of the grades of(2) criteria, all parameters are presented in Table 2. N was set to 2000 in this study to balance accuracy, In this and way,computational k intuitively controls the ‘atomization’ degree of the normal cloud model (Figure 3) robustness, expense. and essentially reflects the variation of cognition of different evaluations. Here, k is assumed to be 0.1 to balance the variation and robustness of the assessment [34]. In this case, combined with standard water quality criteria (Table 1), the three perimeters (Ex, En, and He) of each evaluation index were calculated according to Equations (1) and (2). The modified

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Table 1. Quantitative boundaries of water quality grades of all criteria [34]. Evaluation Factor

Grade I

Grade II

Grade III

Grade IV

Grade V

Grade VI

DO (mg/L) PI (mg/L) COD (mg/L) BOD (mg/L) NH4 + -N (mg/L) TP (mg/L) TN (mg/L) F (mg/L) Fc (Number)

≥7.5 ≤2 ≤15 ≤3 ≤0.15 ≤0.02 ≤0.2 ≤1 ≤200

≥6 ≤4 ≤15 ≤3 ≤0.5 ≤0.1 ≤0.5 ≤1 ≤2000

≥5 ≤6 ≤20 ≤4 ≤1 ≤0.2 ≤1 ≤1 ≤10,000

≥3 ≤10 ≤30 ≤6 ≤1.5 ≤0.3 ≤1.5 ≤1.5 ≤20,000

≥2 ≤15 ≤40 ≤10 ≤2 ≤0.4 ≤2 ≤1.5 ≤40,000

15 >40 >10 >2 >0.4 >2 >1.5 >40,000

Table 2. Cloud model parameters of water quality grades of all criteria. DO Grade I II III IV V VI

PI

Ex

En

He

Ex

En

He

Ex

En

He

11.25 6.75 5.5 4 2.5 1

1.25 0.25 0.17 0.33 0.17 0.33

0.13 0.025 0.017 0.033 0.017 0.033

1 3 5 8 12.5 22.5

0.33 0.33 0.33 0.67 0.83 2.5

0.03 0.03 0.03 0.067 0.083 0.25

7.5 15 17.5 25 35 60

2.5 2.5 0.83 1.67 1.67 6.67

0.3 0.3 0.1 0.17 0.17 0.7

NH4 + -N

BOD Grade I II III IV V VI

I II III IV V VI

TP

Ex

En

He

Ex

En

He

Ex

En

He

1.5 3 3.5 5 8 15

0.5 0.5 0.17 0.33 0.67 1.67

0.05 0.05 0.017 0.033 0.067 0.167

0.075 0.33 0.75 1.25 1.75 3

0.03 0.058 0.08 0.08 0.08 0.33

0.003 0.006 0.008 0.008 0.008 0.033

0.01 0.06 0.15 0.25 0.35 0.6

0.003 0.013 0.017 0.017 0.017 0.067

0.0003 0.001 0.002 0.002 0.002 0.007

TN Grade

COD

F

Fc

Ex

En

He

Ex

En

He

Ex

En

He

0.1 0.35 0.75 1.25 1.75 3

0.03 0.05 0.08 0.08 0.08 0.33

0.003 0.005 0.008 0.008 0.008 0.033

0.5 1 1 1.25 1.5 2.25

0.17 0.17 0.17 0.08 0.08 0.25

0.017 0.017 0.017 0.008 0.008 0.025

100 1100 6000 15,000 30,000 60,000

33 300 1333 1667 3333 6667

3.3 30 133.3 166.7 333.3 670

3.1.3. Formation of the Cloud Cloud models are executed by cloud generators. Generally, two types of cloud generators exist: forward and the backward cloud generators. A forward cloud generator is the transformation between the qualitative knowledge and the quantitative representation and is used to generate the cloud drops through these given cloud numerical descriptors and it can be denoted with CG. Given these numerical descriptors of cloud and the specified x = x0 , the combination to generator the cloud drops drop (x, µ( x )) is called the X-condition cloud, which can be denoted by XCG. The backward generator is a transferring process to derive the qualitative concept, represented by three descriptors from cloud drops and it can be denoted by CG−1 . The three cloud generators are shown in Figure 4. The combination of the two types of generators can be used interchangeably to derive various types of clouds to bridge the gap between qualitative concept and quantitative knowledge.

these numerical descriptors of cloud and the specified x = x0, the combination to generator the cloud drops drop (x, ( )) is called the X-condition cloud, which can be denoted by XCG. The backward generator is a transferring process to derive the qualitative concept, represented by three descriptors from cloud drops and it can be denoted by CG−1. The three cloud generators are shown in Figure 4. The combination of the two types of generators can be used interchangeably to derive various types Water 2018, 10, 510 7 of 15 of clouds to bridge the gap between qualitative concept and quantitative knowledge. Ex En

Ex Ex

CG

drops (x, u) Drops (x, u)

-1

CG

He

XCG

En En

drops (x0, u)

He He

Figure 4. Process Processofofforward forward cloud generators, backward cloud generators, and X-condition cloud Figure 4. cloud generators, backward cloud generators, and X-condition cloud generators. generators.

The normal cloud model has been shown to be universally applicable and has been applied in the field of water science [19]. It is also an important type of cloud model based on normal distribution and Gauss membership function. This type of model is used in this study, and the algorithm steps of the forward normal cloud generator are as follows: Input: Three parameters Ex , En , He , and the number of cloud drops N. Output: N cloud drops and their certainty degree. Steps: (1)

Generate a normally distributed random number En0 i with expectation En and variance He 2 ;

(2)

Generate a normally distributed random number xi with expectation Ex and variance En0 i ;

(3) (4)

−( x − Ex )2 2 2( En0 ) i

Calculate yi = e , where xi represents a cloud drop in the universe and yi is the certainty e and degree of xi belongs to the concept A; Repeat steps 1–3 until N cloud drops are generated.

3.2. Combination Weight Based on Game Theory Combination weight, integrating subjective weight and objective weight through a certain algorithm is more reasonable for the evaluation process [35]. The game theory calculation steps of the combination weight with two or more participants are as follows: Step 1. Because the weight results obtained by different weighting methods can vary widely and even contradict each other, it is necessary to verify the consistency of each weight before combining them. If the weights do not pass the consistency check, then adjustments have to be made to meet the requirements. Since this paper only uses two weight calculation methods, the following distance equation can be used for testing: 

"



d w (1) w (2) =

1 m (1) (2) 2 ( wi − wi ) ∑ 2 i =1

#1

2

(3)

where m represents the number of index vectors that are included in each set of weights and     ( 1 ) ( 2 ) ( 1 ) ( d w w represents the degree of consistency between the two groups. The smaller the d w w 2) ,   the higher the consistency. When d w(1) w(2) is below 0.2, the consistency test is passed, and the next coupling can be carried out after a consistency check. Step 2. Obtain n weights according to n types of weighting methods, and then construct a basic weight vector set U = {u1 , u2 , · · · , un }. A possible weight set is combined by n vectors with the form of arbitrary linear combination as: n

U=

∑ αk ukT (ak > 0)

(4)

k =1

where u represents a possible weight vector in set U and αk represents the weight coefficient. Step 3. Determine the most satisfied weight vector u∗ of the possible weight vector sets according to the concept of game theory, suggesting that a compromise is reached among n weights. Such a compromise can be regarded as optimization of the weight coefficient αk , which is a linear combination. The optimization aim is to minimize the deviation between u and uk using the following equation:

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n

min ∑ α j × u Tj − uiT (i = 1, 2, · · · , n)

j =1

(5)

2

According to the differentiation property of the matrix, the condition of the optimal first order derivative in Equation (5) is: n

∑ α j × ui × uTj = ui × · · · uiT (i = 1, 2, · · · , n)

(6)

j =1

The corresponding system of linear equations is:      

u1 ·u1T u2 ·u1T .. . un ·u1T

u1 ·u2T u2 ·u2T .. . un ·u2T

· · · u1 ·unT · · · u2 ·unT .. .. . . · · · un ·unT

     

α1 α2 .. . αn





    =    

u1 ·u1T u2 ·u2T .. . un ·unT

     

(7)

Step 4. Calculate the weight coefficient (α1 , α2 , · · · , αn ) according to Equation (7), and then normalize it with the following equation: α∗k =

αk n ∑ k =1

αk

(8)

Lastly, the combination weight will be obtained as: u∗ =

n

∑ α∗k ·ukT

(9)

k =1

In this case, the subjective weight was calculated via AHP [36–39]. Five experts were invited to participate in the judgment and then, the average of their judgment value was used. The judgment matrix was established with the consistency ratio (CR = 0.0268) which meets the condition CR < 0.1. The consistency ratio must be computed to check for discordances between the pairwise comparisons and the reliability of the obtained weights [40]. Then, the subjective weight based on AHP (AHPW) was finally determined. The objective weight was calculated via information entropy theory (EW) [41–44] using a program written in R language. The subjective  weight(AHPW) and objective weight (EW) passed the consistency test of Equation (3), with the d w(1) w(2) = 0.167 (